PrototypeFilterDesignBasedonChannelEstimationfor FBMC ...FBMC/OQAM system transmits the real-valued...
Transcript of PrototypeFilterDesignBasedonChannelEstimationfor FBMC ...FBMC/OQAM system transmits the real-valued...
Research ArticlePrototype Filter Design Based on Channel Estimation forFBMCOQAM Systems
Yongjin Liu Xihong Chen and Yu Zhao
Air and Missile Defense College Air Force Engineering University Xirsquoan 710051 China
Correspondence should be addressed to Yongjin Liu liuyongjindw163com
Received 25 September 2019 Revised 31 October 2019 Accepted 8 November 2019 Published 27 November 2019
Academic Editor Francesco Cannizzaro
Copyright copy 2019 Yongjin Liu et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
A prototype filter design for FBMCOQAM systems is proposed in this study e influence of both the channel estimation andthe stop-band energy is taken into account in this method An efficient preamble structure is proposed to improve the per-formance of channel estimation and save the frequency spectral efficiency e reciprocal of the signal-to-interference plus noiseratio (RSINR) is derived to measure the influence of the prototype filter on channel estimation After that the process of prototypefilter design is formulated as an optimization problem with constraint on the RSINR To accelerate the convergence and obtainglobal optimal solution an improved genetic algorithm is proposed Especially the History Network and pruning operator areadopted in this improved genetic algorithm Simulation results demonstrate the validity and efficiency of the prototype filterdesigned in this study
1 Introduction
As an alternative modulation scheme for the fifth-generation(5G) communication offset quadrature amplitude modu-lation-based orthogonal frequency division multiplexing(FBMCOQAM) outperforms orthogonal frequency di-vision multiplexing (OFDM) with respect to spectral effi-ciency side lobes and sensitivity to asynchronoustransmission [1 2] As one of the important components ofFBMCOQAM systems the prototype filter determines theperformance of the whole system
In recent years many effective methods about prototypefilter design have been brought up All these methods can bedivided into two classes [3] direct approaches and indirectapproaches Indirect approaches mainly include the win-dowing-based method [4 5] and the frequency-samplingmethod [6ndash8] Direct approaches optimize all coefficients ofthe prototype filter Due to many more degrees of designfreedom direct approaches have the potential to achievebetter performance than indirect approaches [9]
However the complexity grows rapidly along with thenumber of coefficients in the direct approach In order toreduce the computational complexity the α-based branch
and bound (αBB) algorithm is used to design the prototypefilter in [10] In [11] the box-based branch and bound (Box-BB) method based on αBB algorithm is proposed to decreasethe number of iterations Spectral factorization is employedto design the final prototype filter in [12] but this method isunstable because the factor is chosen randomly e exis-tence of unknowns in the aforementioned methods makes itdifficult to obtain the global optimum
e genetic algorithm (GA) is a powerful evolutionaryalgorithm which is able to solve this difficulty [13 14] GA isadopted to handle the thinning optimization problem in[15] e Baldwin effect is added into GA via a HistoryNetwork to avoid the inappropriate answer in [16] but thereare so many random operators in this method In [17] therandom assignments of value during initialization andmutation operator are deleted by the pruning operator toreach the answer more quickly In this study an improvedGA based on the History Network and pruning operator isproposed to deal with these random operators
Furthermore accurate channel estimation (CE) at thereceiver is indispensable for reliable signal detection erehave already been some studies on the CE for the FBMCOQAM system
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3730759 11 pageshttpsdoiorg10115520193730759
In [18] two modified CE methods were proposed byplacing guard symbols at different sides of reference pre-amble symbol the existence of guard symbols reducesspectrum efficiency In order to increase spectrum efficiencycoded auxiliary pilot symbols that also carry information aredesigned in [19] However at least two to four pilot symbolsare needed to achieve acceptable performance in practice In[20] an efficient preamble structure is proposed to achieve abetter CE with less preamble consumption Because there isno pilot in the compressed sensing method CE methodsbased on compressed sensing are proposed further to in-crease the spectral efficiency In [21] when prior sparseknowledge is unknown an adaptive regularized com-pressive sampling matching pursuit chooses the support setto achieve better CE accuracy Inspired by the same theory asin [21] two improved CE methods are derived respectivelybased on the interference approximation method (IAM) andpair of real pilots in [22] e key to these methods is thathow to prove that the FBMCOQAM system is sparseHowever whether the system like the scattering channel issparse is debatable
Few studies have taken the CE into account in prototypefilter design In [23] the prototype filter is designed byutilizing the mean square error (MSE) of the CE Howeverzero symbols inserted in the preambles will reduce thesystemrsquos spectral efficiency Inspired by the preamblestructure described in [20] preamble without guard symbolsis considered in these studies to acquire higher transmissionefficiency e proposed structure will improve the accuracyof CE by increasing pseudo pilot power
e influence of noise may be also ignored in prototypefilter design For example in [24] the weight signal-to-in-terference ratio is treated as the utility function to design theprototype filter under highly doubly dispersive channelwithout noise However the noise power is generally equalto or even larger than the interference power in practicalsystems erefore the noise influence should be givensufficient attention in prototype filter design Usually thesignal-to-interference plus noise ratio (SINR) is adopted tomeasure the noise influence A bigger SINR means bettersystem performance erefore it will be difficult to set thethreshold when the SINR is added to the constraints Underthis condition in this study the reciprocal of the signal-to-interference plus noise ratio (RSINR) is adopted to measurethe influence of noise
Simulation results indicate that the proposed preamblestructure achieves outperformance compared to the IAMthe improved GA shows less convergence time and averagegenerations than the proposed GAs in [16 17] the designedprototype filter in this study is better than in [23 24] when itcomes to the performance of the stop-band energy and themagnitude response
e rest of the study is organized as follows In Section 2the FBMCOQAM model is described the preamble isproposed and the CE is obtained In Section 3 the process ofprototype filter design is formulated by minimizing the stop-band energy with constraints on the RSINR and the GAwith the History Network and pruning operator is utilized tosolve the optimization problem e results of the
simulations are discussed in Section 4 and finally a con-clusion is given in Section 5
2 System Model
As one of the contributions of this paper the preamblestructure with larger pseudo pilot power and spectrum ef-ficiency is introduced in this section
At first the FBMCOQAM systemmodel is built Insteadof transmitting complex symbols cmn cRmn + jcImn theFBMCOQAM system transmits the real-valued symbolsamn which are from the constellation mapping of thereal components cRmn and imaginary components cImnAccording to [25] the continuous-time baseband equivalentof the transmitting signal in FBMCOQAM can be written as
s(t) 1113944+infin
nminus infin1113944
Mminus 1
m0amn e
jϕmn ej2πm]0t
g t minus nτ0( 11138571113980radicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradic1113981
gmn(t)
(1)
where M denotes the number of subcarriers ]0 1T0
12τ0 denotes the subcarrier spacing T0 denotes the con-stellation interval and τ0 denotes the offset between the realpart and imaginary part of the OQAM symbol g(t) denotesthe prototype filter function ϕmn is an additional phaseterm
ϕmn ϕ0 +π2
(m + n)(mod π) (2)
where ϕ0 can be chosen arbitrarily For simplification in thisstudy ϕ0 0
Considering the multipath channel with an additivewhite Gaussian noise which is denoted as η(t) when thetransmitted FBMCOQAM signal is passed through thebaseband version of the received signal is described as
y(t) 1113946Δ
0h(τ)s(t minus τ)dτ + η(t)
1113944+infin
nminus infin1113944
Mminus 1
m0amn 1113946
Δ
0h(τ)e
minus j2πm]0τejπ(m+n)2
ej2πm]0t
times g t minus τ minus nτ0( 1113857dτ + η(t)
(3)
where h(t) denotes the channel impulse response (CIR) andΔ presents the maximum delay spread of the channel Es-pecially we assume that g(t minus τ minus nτ0) asymp g(t minus nτ0) in thetime interval τ isin [0Δ] for the flat fading channel In thatcase rewrite (3) as
y(t) 1113944+infin
nminus infin1113944
Mminus 1
m0amne
j(π(m+n)2)e
j2πm]0tg t minus nτ0( 1113857
times 1113946Δ
0h(τ)e
minus j2πm]0τdτ + η(t)
1113944+infin
nminus infin1113944
Mminus 1
m0amngmn(t)Hm + η(t)
(4)
where Hm 1113938Δ0 h(t)eminus j2πm]0tdt represents the channel im-
pulse frequency responsee demodulation signal at the (m0 n0) position can be
written as
2 Mathematical Problems in Engineering
ym0 n0langy(t) gm0 n0
rang
1113944+infin
nminus infin1113944
Mminus 1
m0amnHmlanggmn(t) gm0 n0
rang + η1m0 n0
(5)
where langx yrang represents the inner product of x and y andη1m0 n0
langη(t) gm0 n0rang
e orthogonal condition in FBMCOQAM is
R langgmn gm0 n0rang1113966 1113967 δmm0
δnn0 (6)
where R middot denotes real part of the argument and δ(t)
denotes the Diracrsquos delta functionus the FBMCOQAM systemmodel has been built After
that in order to design the new preamble structure the re-lationship between preamble and CE is necessary to be analyzed
21 Relationship between Preamble and CE We definem m0 + p n n0 + q where p q isin Z and then the innerproduct of the pulse shaping filter in equation (5) can berewritten as
langgmn(t) gm0 n0rang 1113946
+infin
minus infingm0+pn0+q(t)g
lowastm0+n0
(t)dt
1113946+infin
minus infine
j(π2) m0+n0p+q( )ej2π m0+p( )vot
middot g t minus n0τ0 minus qτ0( 1113857 times eminus j(π2) m0+n0( )
middot eminus j2πm0v0t
glowast
t minus n0τ0( 1113857dt
j(p+q)
1113946+infin
minus infing t minus n0τ0 minus qτ0( 1113857
middot glowast
t minus n0τ0( 1113857ej2πpv0tdt
(7)
where lowast denotes conjugate operation Since ]oτ0 12t minus n0τ0 minus qτ0 l minus (q2)τ0 and t minus n0τ0 l + (q2)τ0equation (7) can be rewritten as
langgmn(t) gm0 n0rang j
(p+q)1113946
+infin
minus infing l +(minus q2)τ0( 1113857
middot glowast
l minusminus q
2τ01113874 1113875 times e
minus j2πp]0 l+n0τ0+qτ02( )dl
jp+q+pq+2pn0( )Ag minus qτ0 p]0( 1113857
(8)
where Ag(τ ]) denotes the ambiguity function of the filterwhich can be expressed as
Ag(τ ]) 1113946 g t +τ2
1113874 1113875glowast
t minusτ2
1113874 1113875ej2π]tdt (9)
When g(t) is an even function its instantaneous au-tocorrelation function cg(τ t) g(t + (τ2))glowast(t minus (τ2))
is also an even conjugate Ag(τ ]) is the inverse Fouriertransformation of cg(τ t) and therefore Ag(τ ]) is a real-valued function [26]
We can get equation (10) from equation (5)
ym0 n0 1113944
(pq)isinZj
p+q+pq+2pn0( )Ag minus qτ0 p]0( 1113857
times am0+pn0+qHm0+p + η1m0 n0
am0 n0Hm0
+ 1113944(pq)ne(00)
jp+q+pq+2pn0( )
times Ag minus qτ0 p]0( 1113857am0+pn0+qHm0+p + η1m0 n0
(10)
According to the conventional OFDM CE method thechannel frequency response 1113954Hm0
can be estimated by1113954Hm0 n0
(ym0 n0am0 n0
) when the pilot symbol am0 n0is
transmitted at time-frequency position (m0 n0)We define Cpq j(p+q+pq+2pn0)Ag(minus qτ0 p]0) and Δm0
and Δn0 are respectively the neighborhoods of m0 and n0ΩΔm0 Δn0 (p q) |p|leΔm0 |q|leΔn0 | Hm0+pn0+q asymp Hm0 n0
1113966 1113967
and ΩlowastΔm0 Δn0 ΩΔm0 Δn0 minus (0 0) where ΩΔm0 Δn0 denotes theneighborhood of position (m0 n0) With the increase in |p|
and |q| Cpq becomes very close to zero For example for anisotropic orthogonal transform algorithm (IOTA) functionwhen (p q) notin Ω11 we have [27]
1113936(pq)notinΩ11Cpq
11138681113868111386811138681113868
111386811138681113868111386811138682
1113936(pq)notinΩ11Cpq
11138681113868111386811138681113868
111386811138681113868111386811138682 asymp 002 (11)
From equation (11) we can conclude that one column ofzero symbols is enough to prevent the preamble from beinginterfered by adjacent unknown data symbols However thezero symbols affect the CE performance e structure of theIAM is shown in Figure 1 It can be seen from Figure 1 thatthere are two columns of zero symbols for IAM According to[27] when only one column of zero symbols is inserted in thepreamble its CE performance is worse than the IAM structure
Since am0minus 1n0and am0+1n0
are of inverse signs equation(10) can be rewritten as
ym0 n0 am0 n0
Hm0+ Cminus 10am0minus 1n0
Hm0minus 1 + C10am0+1n0Hm0+1
+ 1113944(pq)ne(00)(pq)ne(plusmn10)qneplusmn1
Cpqam0+pn0+qHm0+p
+ η1m0 n0
(12)
We assume Hm0 Hm0+1 Hm0minus 1 e CE based on the
IAM can be obtained by
1113954Hm0
ym0 n0
bm0 n0
Hm0+ 1113944
(pq)ne(00)(pq)ne(plusmn10)qneplusmn1
Cpqam0+pn0+qHm0+p
bm0 n0
+η1m0 n0
bm0 n0
(13)
Mathematical Problems in Engineering 3
where bm0 n0 am0 n0
+ C10am0+1n0+ Cminus 10am0minus 1n0
bm0 n0can
be treated as the pseudo pilot Equation (13) implies that alarger pseudo pilot power leads to a better CE performancee power of pseudo pilot bm0 n0
for the IAM is shown in thefollowing equation
E1 bm0 n0
11138681113868111386811138681113868
111386811138681113868111386811138682
1113882 1113883 a2m0 n0
+ Cminus 10am0minus 1n0+ C10am0+1n0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2a 1 + 4Ag + 4A2g1113872 1113873
(14)
where σ2a denotes the variance of the transmitted FBMCOQAM symbols amn
Obviously CE based on the IAM has two drawbacks (1)the preamble duration is 3τ0 which leads to the lowspectrum efficiency and (2) the second item in equation (13)cannot be ignored when both the signal-to-noise ratio andconstellation mapping order are high
e fact is that the power of pseudo pilot bm0 n0is mainly
affected by the pilot symbols that surround am0 n0for the
IAM erefore it can be increased by changing thestructure Additionally the zero symbols in the IAM will beredundant if the intersymbol interference is alleviated by thepreamble
In order to overcome two drawbacks of the IAM a newpreamble structure is proposed in this paper It is shown inFigure 2 For sufficient accuracy and concise description onlythree-tap interference of neighborhood symbols is considered
e pseudo pilot power of the proposed structure is
E2 bm0 n0
11138681113868111386811138681113868
111386811138681113868111386811138682
1113882 1113883 a2m0 n0
+ 1113944(pq)isinΩlowast33
Cpqam0+pn0+q
11138681113868111386811138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868111386811138681113868
2
gtE1
(15)
From (15) we can conclude that the proposed preamblestructure is able to improve CE performance for its largerpseudo pilot power Obviously it also increases the fre-quency spectrum efficiency
22 RSINR In order to take the influence of prototype filteron CE into account while designing the prototype filter theRSINR is chosen to measure these influencese expressionof RSINR based on the CIR is derived in this section
First we rewrite equation (10) as
ym0 n0 am0 n0
Hm0+ a
(i1)m0n0
Hm0+ a
(i2)m0 n0
Hm0+ η1m0 n0
(16)
where
a(i1)m0 n0
1113944(pq)isinΩlowast33 q0
Cpqam0+pn0+q (17)
a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpqam0+pn0+q (18)
and then the channel frequency impulse response 1113954Hm0is
estimated by
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + a
(i2)m0 n0
(19)
Because am0+pn0+q in equation (18) belongs to a randomsignal its value cannot be determined during transmissionAs a result the term a(i2)
m0 n0is unknown Equation (19) will be
invalid if a(i2)m0 n0
is unknown To solve this problem thepredecision method is adopted e process of estimationbased on the predecision method is shown as follows
m0 minus 3
m0 minus 2
m0 minus 1
m0
m0 + 1
m0 + 2
m0 + 3
+1
ndash1
j
Data
ndashj
Figure 2 Proposed structure of preamble
m0 minus 3
m0 minus 2
m0 minus 1
m0
m0 + 1
m0 + 2
m0 + 3
+1
ndash1
j
Data
Zero
ndashj
Figure 1 Structure of the IAM
4 Mathematical Problems in Engineering
First the initial channel frequency response Hm0can be
obtained from (16)
Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0
Hm0+
a(i2)m0 n0
Hm0+ η1m0 n0
am0 n0+ a
(i1)m0 n0
(20)
Next am0+pn0+q in the term a(i2)m0 n0
can be rebuilt based onzero-forcing equalization Estimate am0+pn0+q based on thepredecision method which is shown as follows
1113954am0+pn0+q Dym0+pn0+q
Hm0
⎡⎣ ⎤⎦ (p q) isin Ωlowast33 qne 0 (21)
Here D[middot] denotes the predecision operator
en a(i2)m0 n0
can be estimated by
1113954a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpq1113954am0+pn0+q (22)
Finally since a(i2)m0 n0
has been obtained the CE is obtainedby substituting a(i2)
m0 n0into equation (19) Equation (23)
shows the final channel frequency response expression
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + 1113954a(i2)
m0 n0
(23)
By inverse Fourier transform the CIR of the FBMCOQAM system is obtained as follows
1113954h(t) 12π
1113946Δ
01113954Hme
minus j2πm]0tdt (24)
We then discretize the CIR 1113954h(t) into form 1113954hl as follows
1113954h(t) 1113944Pminus 1
i0
1113954hiδ τ minus τi( 1113857
1113954hl
1113954hi l lceilτi
Tsrceil
0 otherwise
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(25)
where x denotes the smallest integer larger than or equal to x
and Ts (T02M) denotes the sampling periode desired power of the system is closely related to the
transmitted symbol e transmitted symbol at the position(m0 n0) can be estimated as follows
1113954am0 n0 R
ym0 n0
1113954Hm0
⎛⎝ ⎞⎠ (26)
e QAM symbols are reconstructed by combining1113954cm0 n0
1113954am0 2n0+ j1113954am0 2n0+1 (27)
We then define
Hpqm0
1113944
Lhminus 1
l0
1113954hleminus jπ 2m0+p( )lM
Ag minus l minus qM
2 minus p1113876 1113877 (28)
where Lh is the number of channel taps at the sampling rateand Ag[k l] Ag(kTs (lMTs))
e desired power of the system is
Pd m0( 1113857 σ2c RHpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(29)
where σ2c denotes the variance of the complex symbol cAccording to the definition of the variance σ2c is the realnumber
Similarly the interference power and the noise power areexpressed as [28]
PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)
Rej(π2)(p+q+pq)Hpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(30)
Pnoise m0( 1113857 var η1m0 n01113960 1113961
σ2η1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682 (31)
where σ2η denotes the variance of the noise ηus the expression of RSINR is obtained by equations
(29)ndash(31)
RSINR PICI+ISI + Pnoise
Pd m0( 1113857
2σ2a1113936(pq)ne(00) R ej(π2)(p+q+pq)Hpq
m0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
+ σ2η 1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2c R Hpqm0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
(32)
3 Prototype Filter Design
31 ProblemFormulation e prototype filter in the FBMCOQAM system is assumed to be real-valued symmetric andof unit energy According to the previous discussion whenthere is intrinsic interference and noise in the system theprototype filter will influence the CE erefore the RSINRwill be chosen as an important constraint in problem for-mulation At the same time the prototype filter designshould minimize the stop-band energy us the prototypefilter design problem can be formulated as
P1 ming(k)
1113946π
2πMG e
jω1113872 1113873
11138681113868111386811138681113868
111386811138681113868111386811138682dω
st
g[k] g Lg minus 1 minus k1113960 1113961 k 0 1 Lg minus 1
1113944
Lgminus 1
k0g2[k] 1
RSINRleTH
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(33)
where |G(ejω)| |1113936Lgminus 1k0 g(k)eminus jωk| is the magnitude re-
sponse of the prototype filter TH represents a constantthreshold and Lg denotes the length of the prototype filterAccording to [29] g[k]
Ts
1113968g((k minus (Lg minus 12))Ts)
e prototype filter used to be designed without con-sidering the interaction between prototype filter and CE
Mathematical Problems in Engineering 5
erefore utilizing RSINR to measure the influence ofprototype filter on CE and regarding RSINR as the con-straint of prototype filter design problem are also one of thecontributions of this paper
ere are many variables in P1 e objective functionand constraint functions all have quadratic formsereforeit is difficult to solve the optimization problem P1 directlyAs everyone knows orthogonal transformation does notchange the nature of the original function However it canbe employed to change the form of (33)
First of all the first constraint function in (33) impliesthat only half of the filter coefficients are independent Inorder to reduce the number of variables we substitute g[k]
with a new variable x
x x1 x2 xL1113858 1113859T
g[0] g[1] gLgminus 121113876 11138771113876 1113877
T
(34)
e objective function can be rewritten with a vectorform as f0(x) xTCx where C is a real symmetric positive-definite matrix λ1 λ2 λL respectively denote positiveeigenvalues in ascending order and v1 v2 vL re-spectively denote corresponding unit orthogonaleigenvectors
en the orthogonal transformation x vz is applied tothe variable x where z z1 z2 zL1113858 1113859
T denotes the neworthogonal variable
Finally the optimization problem after the variablesubstitution and orthogonal transformation is expressed as
P2 min1113944L
j1λjz
2j
stc1(z) minus 1113944
Lg minus 1
k0g2[k] minus 1⎛⎝ ⎞⎠le 0
c2(z) RSINRleTH
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(35)
32 Problem Solution In this part an improved GA withHistory Network and pruning operator is proposed to solvethe optimization problem expressed in (35)
All the potential global answers of GA exist in a searchspace Gorges in GA will expand this search space Furtherthe inappropriate space may be searched many times duringiterations ese increase the convergence time of GA
In [16] the knowledge of search space learned by theprevious generation is passed on to the next generation viaan History Network to narrow the search space and theBaldwin effect is applied to avoid searching the same space indifferent iterations Figure 3 shows the flowchart of theproposed GA in [16]
However the influence of random operators ignored in[16] will slow down the convergence speed of GA erandom assignment operators may result in invalid iterationsearly termination and jumping out of the loop with localoptimum Inspired by [17] the pruning operator is applied inthis paper to control these random operators Pruning
operator is used to take away the inappropriate amountsresulting in increment convergence time from population andreach the global optimal answer more quickly Flowchart ofimproved GA in this study is shown in Figure 4
It can be seen from Figure 4 that the improvement of theimproved GA proposed in this paper mainly exists in fourparts initialization DRS History Network and mutation
e pruning operator of the initialization part directlydeletes the invalid subset of solution to narrow the domain atthe beginning
e time order of the pruned initialization is shown as
initialization O(NplowastNglowastNc) (36)
where Ng presents the number of genomes in each chro-mosome Nc presents the number of chromosomes of eachgeneration and Np is determined by the pruning function Itis shown as
Npruning η ηleNglowastNc
NglowastNc ηgtNglowastNc1113896 (37)
where η is the reduced ratio of search spaceIn DRS part the range of direction change is limited by
pruning operator As a result those individuals are discardedwho are so different to avoid being too far from the lastreasonable direction
ere are many exemplars stored in the History Net-work It is required to limit the variation of the exemplars toa certain range Pruning operator is applied to avoid eitherduplication with existing exemplars or drastic changesduring the updating process
Mutation is critical for reaching optimal solution In theprocess of mutation the value of genome is replaced ran-domly to create new generation for the next iterationHowever the random substitution likely results in in-appropriate value It requires a lot of iterations to eliminatethe impact of inappropriate values on the search As a resultiterations in the whole algorithm are increased Pruningoperator is able to help appoint appropriate value as much aspossible In all circumstances the better the appointment ofvalue during mutation the faster it is to reach the best answer
e time order of the pruned search loop is shown as
search loop ONsηlowastNclowastNclowastNp1113888 1113889 (38)
where Ns presents the solution spaceSince the redundant random operators have been
pruned the computational complexity of the whole algo-rithm is certainly reduced
Start Initialization
Fitness
Probability calculation
Increase the ageDRS
Update History NetworkComparison
AboveBelow
Breed individuals
Best individual
Figure 3 Flowchart of the proposed GA in [16]
6 Mathematical Problems in Engineering
4 Simulation Result
41ProblemFormulation eCE performance based on theproposed preamble is simulated in this simulation etruncation of an IOTA filter with 4T0 length [30] is chosen asthe prototype filtere fundamental parameters are listed inTable 1
Figure 5 illustrates the CE performances of the proposedstructure (A) and the IAM (B) for two kinds of constellationmapping For 16-QAM modulation when the signal-to-noise ratio (SNR) is low (SNRlt 15 dB) the CE performancesof A are slightly better than those of B Along with the SNRincreasing the CE performance of A becomes better andbetter and the advantages become more and more obviousWhen SNRgt 15 dB the IAM performance curve tends toflatten due to the performance platform Similarly for 4-QAM modulation the IAM performance curve tends toflatten when SNRgt 20 dB
e CE performances of two methods with differentnumber of subcarriers for 4-QAMmodulation and 16-QAMmodulation are also respectively compared in Figures 6 and7 It is obvious that the curve of A with 512 subcarriers islower than the one with 256 subcarriers in Figure 6 so is B Itimplies that a larger number of subcarriers mean a better CEperformance ese two curves of A are both lower thanthose of B for the same number of subcarriers Curves inFigure 6 show that A is better than B about 04 dB for 256subcarriers and 1 dB for 512 subcarriers at BER 10minus 3Modulation of curve 16-QAM in Figure 7 also illustrates thesame conclusion erefore the proposed preamble struc-ture has advantages with respect to CE
In order to lower the impact of predecision errors CEwith different number of iterations is simulated in Figure 8Perfect bound represents the upper bound of performancee curves with iterations show better performance than thecurve without iterations which indicates iteration can re-duce the predecision errors and improve the CE perfor-mancee difference between curves with different numberof iterations is obvious when the number is less than fourBut the curve with four iterations is almost overlapped withthe curve with five iterations e reason why curves areoverlapped when the number of iterations are more thanfour is that the value of am0+pn0+q is close to its true valueafter four iterations erefore there is no need to carry outmore than four iterations in order to save time and reducethe complexity
42 Improved GA performance e performance of theimproved GA is analyzed in this section According to [16]values of parameters in History Network are assigned asfollows the Good reshold the Bad reshold and theUpdate reshold are set to 10 25 and 10 of themutation radius respectivelyemutation radius is definedas the maximum Euclidean distance from the old searchspace position before mutation to the new position aftermutatione rest of the parameters of the improved GA areexhibited in Table 2
e average generations and the convergence time ofdifferent algorithms are respectively shown in Figure 9 andTable 3 Figure 9 implies that the generations of other threealgorithms are much more than the improved GA proposedin this paper It illustrates that the pruning operation in theHistory Network deletes the invalid random value assign-ment and avoids redundant individuals As can been seenfrom Table 3 the improved GA proposed in this paperreduces the convergence time varying from 8 to 221when compared to the standard GA the accelerated GA in[16] and the pruned GA in [17] It reveals that the improvedGA proposed in this paper can reduce the time to reach thebest answer and accelerate the convergence speed In con-clusion the combination of pruning operation and HistoryNetwork is more effective than just utilizing any one of them
Table 1 Fundamental parameters of CE
Parameter ValueSubcarrier spacing ]0 1094 kHzFilter length Lg 4M + 1Channel coding Convolution codingChannel mode ITU-A channel [31]Channel path delay (us) [0 16 35 50]Frame length 20 OQAM symbols
A16QAMB16QAM
A4QAMB4QAM
5 10 15 20 25 30 350SNR (dB)
ndash50
ndash45
ndash40
ndash35
ndash30
ndash25
ndash20
ndash15
ndash10
ndash5
0
NM
SE
Figure 5 Normalized mean square error (NMSE) performance ofthe proposed structure (A) and the IAM structure (B) for 4-QAMand 16-QAM in CE
Start Initialization
Fitness Crossover
Mutation
Pruning Pruning
DRS
Update History NetworkTerminate
Yes No
Increasethe age
Best individual
Pruning
Pruning
Probability calculation
Figure 4 Flowchart of the improved GA in this paper
Mathematical Problems in Engineering 7
43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations
e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller
than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11
Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one
In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper
Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study
Without iteration2 iterations3 iterations
4 iterations5 iterationsPerfect bound
ndash20
ndash18
ndash16
ndash14
ndash12
ndash10
ndash8
ndash6
ndash4
ndash2
NM
SE
2 4 6 8 10 12 14 16 18 200SNR (dB)
Figure 8 NMSE performance with different number of iterations
A256B256
A512B512
10ndash4
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 250SNR (dB)
Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM
A256B256
B512A512
ndash15
ndash10
ndash5
NM
SE
2 4 6 8 100SNR (dB)
Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM
Table 2 Parameters of the improved GA
Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9
8 Mathematical Problems in Engineering
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
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In [18] two modified CE methods were proposed byplacing guard symbols at different sides of reference pre-amble symbol the existence of guard symbols reducesspectrum efficiency In order to increase spectrum efficiencycoded auxiliary pilot symbols that also carry information aredesigned in [19] However at least two to four pilot symbolsare needed to achieve acceptable performance in practice In[20] an efficient preamble structure is proposed to achieve abetter CE with less preamble consumption Because there isno pilot in the compressed sensing method CE methodsbased on compressed sensing are proposed further to in-crease the spectral efficiency In [21] when prior sparseknowledge is unknown an adaptive regularized com-pressive sampling matching pursuit chooses the support setto achieve better CE accuracy Inspired by the same theory asin [21] two improved CE methods are derived respectivelybased on the interference approximation method (IAM) andpair of real pilots in [22] e key to these methods is thathow to prove that the FBMCOQAM system is sparseHowever whether the system like the scattering channel issparse is debatable
Few studies have taken the CE into account in prototypefilter design In [23] the prototype filter is designed byutilizing the mean square error (MSE) of the CE Howeverzero symbols inserted in the preambles will reduce thesystemrsquos spectral efficiency Inspired by the preamblestructure described in [20] preamble without guard symbolsis considered in these studies to acquire higher transmissionefficiency e proposed structure will improve the accuracyof CE by increasing pseudo pilot power
e influence of noise may be also ignored in prototypefilter design For example in [24] the weight signal-to-in-terference ratio is treated as the utility function to design theprototype filter under highly doubly dispersive channelwithout noise However the noise power is generally equalto or even larger than the interference power in practicalsystems erefore the noise influence should be givensufficient attention in prototype filter design Usually thesignal-to-interference plus noise ratio (SINR) is adopted tomeasure the noise influence A bigger SINR means bettersystem performance erefore it will be difficult to set thethreshold when the SINR is added to the constraints Underthis condition in this study the reciprocal of the signal-to-interference plus noise ratio (RSINR) is adopted to measurethe influence of noise
Simulation results indicate that the proposed preamblestructure achieves outperformance compared to the IAMthe improved GA shows less convergence time and averagegenerations than the proposed GAs in [16 17] the designedprototype filter in this study is better than in [23 24] when itcomes to the performance of the stop-band energy and themagnitude response
e rest of the study is organized as follows In Section 2the FBMCOQAM model is described the preamble isproposed and the CE is obtained In Section 3 the process ofprototype filter design is formulated by minimizing the stop-band energy with constraints on the RSINR and the GAwith the History Network and pruning operator is utilized tosolve the optimization problem e results of the
simulations are discussed in Section 4 and finally a con-clusion is given in Section 5
2 System Model
As one of the contributions of this paper the preamblestructure with larger pseudo pilot power and spectrum ef-ficiency is introduced in this section
At first the FBMCOQAM systemmodel is built Insteadof transmitting complex symbols cmn cRmn + jcImn theFBMCOQAM system transmits the real-valued symbolsamn which are from the constellation mapping of thereal components cRmn and imaginary components cImnAccording to [25] the continuous-time baseband equivalentof the transmitting signal in FBMCOQAM can be written as
s(t) 1113944+infin
nminus infin1113944
Mminus 1
m0amn e
jϕmn ej2πm]0t
g t minus nτ0( 11138571113980radicradicradicradicradicradicradicradicradic11139791113978radicradicradicradicradicradicradicradicradic1113981
gmn(t)
(1)
where M denotes the number of subcarriers ]0 1T0
12τ0 denotes the subcarrier spacing T0 denotes the con-stellation interval and τ0 denotes the offset between the realpart and imaginary part of the OQAM symbol g(t) denotesthe prototype filter function ϕmn is an additional phaseterm
ϕmn ϕ0 +π2
(m + n)(mod π) (2)
where ϕ0 can be chosen arbitrarily For simplification in thisstudy ϕ0 0
Considering the multipath channel with an additivewhite Gaussian noise which is denoted as η(t) when thetransmitted FBMCOQAM signal is passed through thebaseband version of the received signal is described as
y(t) 1113946Δ
0h(τ)s(t minus τ)dτ + η(t)
1113944+infin
nminus infin1113944
Mminus 1
m0amn 1113946
Δ
0h(τ)e
minus j2πm]0τejπ(m+n)2
ej2πm]0t
times g t minus τ minus nτ0( 1113857dτ + η(t)
(3)
where h(t) denotes the channel impulse response (CIR) andΔ presents the maximum delay spread of the channel Es-pecially we assume that g(t minus τ minus nτ0) asymp g(t minus nτ0) in thetime interval τ isin [0Δ] for the flat fading channel In thatcase rewrite (3) as
y(t) 1113944+infin
nminus infin1113944
Mminus 1
m0amne
j(π(m+n)2)e
j2πm]0tg t minus nτ0( 1113857
times 1113946Δ
0h(τ)e
minus j2πm]0τdτ + η(t)
1113944+infin
nminus infin1113944
Mminus 1
m0amngmn(t)Hm + η(t)
(4)
where Hm 1113938Δ0 h(t)eminus j2πm]0tdt represents the channel im-
pulse frequency responsee demodulation signal at the (m0 n0) position can be
written as
2 Mathematical Problems in Engineering
ym0 n0langy(t) gm0 n0
rang
1113944+infin
nminus infin1113944
Mminus 1
m0amnHmlanggmn(t) gm0 n0
rang + η1m0 n0
(5)
where langx yrang represents the inner product of x and y andη1m0 n0
langη(t) gm0 n0rang
e orthogonal condition in FBMCOQAM is
R langgmn gm0 n0rang1113966 1113967 δmm0
δnn0 (6)
where R middot denotes real part of the argument and δ(t)
denotes the Diracrsquos delta functionus the FBMCOQAM systemmodel has been built After
that in order to design the new preamble structure the re-lationship between preamble and CE is necessary to be analyzed
21 Relationship between Preamble and CE We definem m0 + p n n0 + q where p q isin Z and then the innerproduct of the pulse shaping filter in equation (5) can berewritten as
langgmn(t) gm0 n0rang 1113946
+infin
minus infingm0+pn0+q(t)g
lowastm0+n0
(t)dt
1113946+infin
minus infine
j(π2) m0+n0p+q( )ej2π m0+p( )vot
middot g t minus n0τ0 minus qτ0( 1113857 times eminus j(π2) m0+n0( )
middot eminus j2πm0v0t
glowast
t minus n0τ0( 1113857dt
j(p+q)
1113946+infin
minus infing t minus n0τ0 minus qτ0( 1113857
middot glowast
t minus n0τ0( 1113857ej2πpv0tdt
(7)
where lowast denotes conjugate operation Since ]oτ0 12t minus n0τ0 minus qτ0 l minus (q2)τ0 and t minus n0τ0 l + (q2)τ0equation (7) can be rewritten as
langgmn(t) gm0 n0rang j
(p+q)1113946
+infin
minus infing l +(minus q2)τ0( 1113857
middot glowast
l minusminus q
2τ01113874 1113875 times e
minus j2πp]0 l+n0τ0+qτ02( )dl
jp+q+pq+2pn0( )Ag minus qτ0 p]0( 1113857
(8)
where Ag(τ ]) denotes the ambiguity function of the filterwhich can be expressed as
Ag(τ ]) 1113946 g t +τ2
1113874 1113875glowast
t minusτ2
1113874 1113875ej2π]tdt (9)
When g(t) is an even function its instantaneous au-tocorrelation function cg(τ t) g(t + (τ2))glowast(t minus (τ2))
is also an even conjugate Ag(τ ]) is the inverse Fouriertransformation of cg(τ t) and therefore Ag(τ ]) is a real-valued function [26]
We can get equation (10) from equation (5)
ym0 n0 1113944
(pq)isinZj
p+q+pq+2pn0( )Ag minus qτ0 p]0( 1113857
times am0+pn0+qHm0+p + η1m0 n0
am0 n0Hm0
+ 1113944(pq)ne(00)
jp+q+pq+2pn0( )
times Ag minus qτ0 p]0( 1113857am0+pn0+qHm0+p + η1m0 n0
(10)
According to the conventional OFDM CE method thechannel frequency response 1113954Hm0
can be estimated by1113954Hm0 n0
(ym0 n0am0 n0
) when the pilot symbol am0 n0is
transmitted at time-frequency position (m0 n0)We define Cpq j(p+q+pq+2pn0)Ag(minus qτ0 p]0) and Δm0
and Δn0 are respectively the neighborhoods of m0 and n0ΩΔm0 Δn0 (p q) |p|leΔm0 |q|leΔn0 | Hm0+pn0+q asymp Hm0 n0
1113966 1113967
and ΩlowastΔm0 Δn0 ΩΔm0 Δn0 minus (0 0) where ΩΔm0 Δn0 denotes theneighborhood of position (m0 n0) With the increase in |p|
and |q| Cpq becomes very close to zero For example for anisotropic orthogonal transform algorithm (IOTA) functionwhen (p q) notin Ω11 we have [27]
1113936(pq)notinΩ11Cpq
11138681113868111386811138681113868
111386811138681113868111386811138682
1113936(pq)notinΩ11Cpq
11138681113868111386811138681113868
111386811138681113868111386811138682 asymp 002 (11)
From equation (11) we can conclude that one column ofzero symbols is enough to prevent the preamble from beinginterfered by adjacent unknown data symbols However thezero symbols affect the CE performance e structure of theIAM is shown in Figure 1 It can be seen from Figure 1 thatthere are two columns of zero symbols for IAM According to[27] when only one column of zero symbols is inserted in thepreamble its CE performance is worse than the IAM structure
Since am0minus 1n0and am0+1n0
are of inverse signs equation(10) can be rewritten as
ym0 n0 am0 n0
Hm0+ Cminus 10am0minus 1n0
Hm0minus 1 + C10am0+1n0Hm0+1
+ 1113944(pq)ne(00)(pq)ne(plusmn10)qneplusmn1
Cpqam0+pn0+qHm0+p
+ η1m0 n0
(12)
We assume Hm0 Hm0+1 Hm0minus 1 e CE based on the
IAM can be obtained by
1113954Hm0
ym0 n0
bm0 n0
Hm0+ 1113944
(pq)ne(00)(pq)ne(plusmn10)qneplusmn1
Cpqam0+pn0+qHm0+p
bm0 n0
+η1m0 n0
bm0 n0
(13)
Mathematical Problems in Engineering 3
where bm0 n0 am0 n0
+ C10am0+1n0+ Cminus 10am0minus 1n0
bm0 n0can
be treated as the pseudo pilot Equation (13) implies that alarger pseudo pilot power leads to a better CE performancee power of pseudo pilot bm0 n0
for the IAM is shown in thefollowing equation
E1 bm0 n0
11138681113868111386811138681113868
111386811138681113868111386811138682
1113882 1113883 a2m0 n0
+ Cminus 10am0minus 1n0+ C10am0+1n0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2a 1 + 4Ag + 4A2g1113872 1113873
(14)
where σ2a denotes the variance of the transmitted FBMCOQAM symbols amn
Obviously CE based on the IAM has two drawbacks (1)the preamble duration is 3τ0 which leads to the lowspectrum efficiency and (2) the second item in equation (13)cannot be ignored when both the signal-to-noise ratio andconstellation mapping order are high
e fact is that the power of pseudo pilot bm0 n0is mainly
affected by the pilot symbols that surround am0 n0for the
IAM erefore it can be increased by changing thestructure Additionally the zero symbols in the IAM will beredundant if the intersymbol interference is alleviated by thepreamble
In order to overcome two drawbacks of the IAM a newpreamble structure is proposed in this paper It is shown inFigure 2 For sufficient accuracy and concise description onlythree-tap interference of neighborhood symbols is considered
e pseudo pilot power of the proposed structure is
E2 bm0 n0
11138681113868111386811138681113868
111386811138681113868111386811138682
1113882 1113883 a2m0 n0
+ 1113944(pq)isinΩlowast33
Cpqam0+pn0+q
11138681113868111386811138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868111386811138681113868
2
gtE1
(15)
From (15) we can conclude that the proposed preamblestructure is able to improve CE performance for its largerpseudo pilot power Obviously it also increases the fre-quency spectrum efficiency
22 RSINR In order to take the influence of prototype filteron CE into account while designing the prototype filter theRSINR is chosen to measure these influencese expressionof RSINR based on the CIR is derived in this section
First we rewrite equation (10) as
ym0 n0 am0 n0
Hm0+ a
(i1)m0n0
Hm0+ a
(i2)m0 n0
Hm0+ η1m0 n0
(16)
where
a(i1)m0 n0
1113944(pq)isinΩlowast33 q0
Cpqam0+pn0+q (17)
a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpqam0+pn0+q (18)
and then the channel frequency impulse response 1113954Hm0is
estimated by
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + a
(i2)m0 n0
(19)
Because am0+pn0+q in equation (18) belongs to a randomsignal its value cannot be determined during transmissionAs a result the term a(i2)
m0 n0is unknown Equation (19) will be
invalid if a(i2)m0 n0
is unknown To solve this problem thepredecision method is adopted e process of estimationbased on the predecision method is shown as follows
m0 minus 3
m0 minus 2
m0 minus 1
m0
m0 + 1
m0 + 2
m0 + 3
+1
ndash1
j
Data
ndashj
Figure 2 Proposed structure of preamble
m0 minus 3
m0 minus 2
m0 minus 1
m0
m0 + 1
m0 + 2
m0 + 3
+1
ndash1
j
Data
Zero
ndashj
Figure 1 Structure of the IAM
4 Mathematical Problems in Engineering
First the initial channel frequency response Hm0can be
obtained from (16)
Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0
Hm0+
a(i2)m0 n0
Hm0+ η1m0 n0
am0 n0+ a
(i1)m0 n0
(20)
Next am0+pn0+q in the term a(i2)m0 n0
can be rebuilt based onzero-forcing equalization Estimate am0+pn0+q based on thepredecision method which is shown as follows
1113954am0+pn0+q Dym0+pn0+q
Hm0
⎡⎣ ⎤⎦ (p q) isin Ωlowast33 qne 0 (21)
Here D[middot] denotes the predecision operator
en a(i2)m0 n0
can be estimated by
1113954a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpq1113954am0+pn0+q (22)
Finally since a(i2)m0 n0
has been obtained the CE is obtainedby substituting a(i2)
m0 n0into equation (19) Equation (23)
shows the final channel frequency response expression
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + 1113954a(i2)
m0 n0
(23)
By inverse Fourier transform the CIR of the FBMCOQAM system is obtained as follows
1113954h(t) 12π
1113946Δ
01113954Hme
minus j2πm]0tdt (24)
We then discretize the CIR 1113954h(t) into form 1113954hl as follows
1113954h(t) 1113944Pminus 1
i0
1113954hiδ τ minus τi( 1113857
1113954hl
1113954hi l lceilτi
Tsrceil
0 otherwise
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(25)
where x denotes the smallest integer larger than or equal to x
and Ts (T02M) denotes the sampling periode desired power of the system is closely related to the
transmitted symbol e transmitted symbol at the position(m0 n0) can be estimated as follows
1113954am0 n0 R
ym0 n0
1113954Hm0
⎛⎝ ⎞⎠ (26)
e QAM symbols are reconstructed by combining1113954cm0 n0
1113954am0 2n0+ j1113954am0 2n0+1 (27)
We then define
Hpqm0
1113944
Lhminus 1
l0
1113954hleminus jπ 2m0+p( )lM
Ag minus l minus qM
2 minus p1113876 1113877 (28)
where Lh is the number of channel taps at the sampling rateand Ag[k l] Ag(kTs (lMTs))
e desired power of the system is
Pd m0( 1113857 σ2c RHpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(29)
where σ2c denotes the variance of the complex symbol cAccording to the definition of the variance σ2c is the realnumber
Similarly the interference power and the noise power areexpressed as [28]
PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)
Rej(π2)(p+q+pq)Hpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(30)
Pnoise m0( 1113857 var η1m0 n01113960 1113961
σ2η1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682 (31)
where σ2η denotes the variance of the noise ηus the expression of RSINR is obtained by equations
(29)ndash(31)
RSINR PICI+ISI + Pnoise
Pd m0( 1113857
2σ2a1113936(pq)ne(00) R ej(π2)(p+q+pq)Hpq
m0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
+ σ2η 1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2c R Hpqm0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
(32)
3 Prototype Filter Design
31 ProblemFormulation e prototype filter in the FBMCOQAM system is assumed to be real-valued symmetric andof unit energy According to the previous discussion whenthere is intrinsic interference and noise in the system theprototype filter will influence the CE erefore the RSINRwill be chosen as an important constraint in problem for-mulation At the same time the prototype filter designshould minimize the stop-band energy us the prototypefilter design problem can be formulated as
P1 ming(k)
1113946π
2πMG e
jω1113872 1113873
11138681113868111386811138681113868
111386811138681113868111386811138682dω
st
g[k] g Lg minus 1 minus k1113960 1113961 k 0 1 Lg minus 1
1113944
Lgminus 1
k0g2[k] 1
RSINRleTH
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(33)
where |G(ejω)| |1113936Lgminus 1k0 g(k)eminus jωk| is the magnitude re-
sponse of the prototype filter TH represents a constantthreshold and Lg denotes the length of the prototype filterAccording to [29] g[k]
Ts
1113968g((k minus (Lg minus 12))Ts)
e prototype filter used to be designed without con-sidering the interaction between prototype filter and CE
Mathematical Problems in Engineering 5
erefore utilizing RSINR to measure the influence ofprototype filter on CE and regarding RSINR as the con-straint of prototype filter design problem are also one of thecontributions of this paper
ere are many variables in P1 e objective functionand constraint functions all have quadratic formsereforeit is difficult to solve the optimization problem P1 directlyAs everyone knows orthogonal transformation does notchange the nature of the original function However it canbe employed to change the form of (33)
First of all the first constraint function in (33) impliesthat only half of the filter coefficients are independent Inorder to reduce the number of variables we substitute g[k]
with a new variable x
x x1 x2 xL1113858 1113859T
g[0] g[1] gLgminus 121113876 11138771113876 1113877
T
(34)
e objective function can be rewritten with a vectorform as f0(x) xTCx where C is a real symmetric positive-definite matrix λ1 λ2 λL respectively denote positiveeigenvalues in ascending order and v1 v2 vL re-spectively denote corresponding unit orthogonaleigenvectors
en the orthogonal transformation x vz is applied tothe variable x where z z1 z2 zL1113858 1113859
T denotes the neworthogonal variable
Finally the optimization problem after the variablesubstitution and orthogonal transformation is expressed as
P2 min1113944L
j1λjz
2j
stc1(z) minus 1113944
Lg minus 1
k0g2[k] minus 1⎛⎝ ⎞⎠le 0
c2(z) RSINRleTH
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(35)
32 Problem Solution In this part an improved GA withHistory Network and pruning operator is proposed to solvethe optimization problem expressed in (35)
All the potential global answers of GA exist in a searchspace Gorges in GA will expand this search space Furtherthe inappropriate space may be searched many times duringiterations ese increase the convergence time of GA
In [16] the knowledge of search space learned by theprevious generation is passed on to the next generation viaan History Network to narrow the search space and theBaldwin effect is applied to avoid searching the same space indifferent iterations Figure 3 shows the flowchart of theproposed GA in [16]
However the influence of random operators ignored in[16] will slow down the convergence speed of GA erandom assignment operators may result in invalid iterationsearly termination and jumping out of the loop with localoptimum Inspired by [17] the pruning operator is applied inthis paper to control these random operators Pruning
operator is used to take away the inappropriate amountsresulting in increment convergence time from population andreach the global optimal answer more quickly Flowchart ofimproved GA in this study is shown in Figure 4
It can be seen from Figure 4 that the improvement of theimproved GA proposed in this paper mainly exists in fourparts initialization DRS History Network and mutation
e pruning operator of the initialization part directlydeletes the invalid subset of solution to narrow the domain atthe beginning
e time order of the pruned initialization is shown as
initialization O(NplowastNglowastNc) (36)
where Ng presents the number of genomes in each chro-mosome Nc presents the number of chromosomes of eachgeneration and Np is determined by the pruning function Itis shown as
Npruning η ηleNglowastNc
NglowastNc ηgtNglowastNc1113896 (37)
where η is the reduced ratio of search spaceIn DRS part the range of direction change is limited by
pruning operator As a result those individuals are discardedwho are so different to avoid being too far from the lastreasonable direction
ere are many exemplars stored in the History Net-work It is required to limit the variation of the exemplars toa certain range Pruning operator is applied to avoid eitherduplication with existing exemplars or drastic changesduring the updating process
Mutation is critical for reaching optimal solution In theprocess of mutation the value of genome is replaced ran-domly to create new generation for the next iterationHowever the random substitution likely results in in-appropriate value It requires a lot of iterations to eliminatethe impact of inappropriate values on the search As a resultiterations in the whole algorithm are increased Pruningoperator is able to help appoint appropriate value as much aspossible In all circumstances the better the appointment ofvalue during mutation the faster it is to reach the best answer
e time order of the pruned search loop is shown as
search loop ONsηlowastNclowastNclowastNp1113888 1113889 (38)
where Ns presents the solution spaceSince the redundant random operators have been
pruned the computational complexity of the whole algo-rithm is certainly reduced
Start Initialization
Fitness
Probability calculation
Increase the ageDRS
Update History NetworkComparison
AboveBelow
Breed individuals
Best individual
Figure 3 Flowchart of the proposed GA in [16]
6 Mathematical Problems in Engineering
4 Simulation Result
41ProblemFormulation eCE performance based on theproposed preamble is simulated in this simulation etruncation of an IOTA filter with 4T0 length [30] is chosen asthe prototype filtere fundamental parameters are listed inTable 1
Figure 5 illustrates the CE performances of the proposedstructure (A) and the IAM (B) for two kinds of constellationmapping For 16-QAM modulation when the signal-to-noise ratio (SNR) is low (SNRlt 15 dB) the CE performancesof A are slightly better than those of B Along with the SNRincreasing the CE performance of A becomes better andbetter and the advantages become more and more obviousWhen SNRgt 15 dB the IAM performance curve tends toflatten due to the performance platform Similarly for 4-QAM modulation the IAM performance curve tends toflatten when SNRgt 20 dB
e CE performances of two methods with differentnumber of subcarriers for 4-QAMmodulation and 16-QAMmodulation are also respectively compared in Figures 6 and7 It is obvious that the curve of A with 512 subcarriers islower than the one with 256 subcarriers in Figure 6 so is B Itimplies that a larger number of subcarriers mean a better CEperformance ese two curves of A are both lower thanthose of B for the same number of subcarriers Curves inFigure 6 show that A is better than B about 04 dB for 256subcarriers and 1 dB for 512 subcarriers at BER 10minus 3Modulation of curve 16-QAM in Figure 7 also illustrates thesame conclusion erefore the proposed preamble struc-ture has advantages with respect to CE
In order to lower the impact of predecision errors CEwith different number of iterations is simulated in Figure 8Perfect bound represents the upper bound of performancee curves with iterations show better performance than thecurve without iterations which indicates iteration can re-duce the predecision errors and improve the CE perfor-mancee difference between curves with different numberof iterations is obvious when the number is less than fourBut the curve with four iterations is almost overlapped withthe curve with five iterations e reason why curves areoverlapped when the number of iterations are more thanfour is that the value of am0+pn0+q is close to its true valueafter four iterations erefore there is no need to carry outmore than four iterations in order to save time and reducethe complexity
42 Improved GA performance e performance of theimproved GA is analyzed in this section According to [16]values of parameters in History Network are assigned asfollows the Good reshold the Bad reshold and theUpdate reshold are set to 10 25 and 10 of themutation radius respectivelyemutation radius is definedas the maximum Euclidean distance from the old searchspace position before mutation to the new position aftermutatione rest of the parameters of the improved GA areexhibited in Table 2
e average generations and the convergence time ofdifferent algorithms are respectively shown in Figure 9 andTable 3 Figure 9 implies that the generations of other threealgorithms are much more than the improved GA proposedin this paper It illustrates that the pruning operation in theHistory Network deletes the invalid random value assign-ment and avoids redundant individuals As can been seenfrom Table 3 the improved GA proposed in this paperreduces the convergence time varying from 8 to 221when compared to the standard GA the accelerated GA in[16] and the pruned GA in [17] It reveals that the improvedGA proposed in this paper can reduce the time to reach thebest answer and accelerate the convergence speed In con-clusion the combination of pruning operation and HistoryNetwork is more effective than just utilizing any one of them
Table 1 Fundamental parameters of CE
Parameter ValueSubcarrier spacing ]0 1094 kHzFilter length Lg 4M + 1Channel coding Convolution codingChannel mode ITU-A channel [31]Channel path delay (us) [0 16 35 50]Frame length 20 OQAM symbols
A16QAMB16QAM
A4QAMB4QAM
5 10 15 20 25 30 350SNR (dB)
ndash50
ndash45
ndash40
ndash35
ndash30
ndash25
ndash20
ndash15
ndash10
ndash5
0
NM
SE
Figure 5 Normalized mean square error (NMSE) performance ofthe proposed structure (A) and the IAM structure (B) for 4-QAMand 16-QAM in CE
Start Initialization
Fitness Crossover
Mutation
Pruning Pruning
DRS
Update History NetworkTerminate
Yes No
Increasethe age
Best individual
Pruning
Pruning
Probability calculation
Figure 4 Flowchart of the improved GA in this paper
Mathematical Problems in Engineering 7
43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations
e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller
than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11
Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one
In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper
Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study
Without iteration2 iterations3 iterations
4 iterations5 iterationsPerfect bound
ndash20
ndash18
ndash16
ndash14
ndash12
ndash10
ndash8
ndash6
ndash4
ndash2
NM
SE
2 4 6 8 10 12 14 16 18 200SNR (dB)
Figure 8 NMSE performance with different number of iterations
A256B256
A512B512
10ndash4
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 250SNR (dB)
Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM
A256B256
B512A512
ndash15
ndash10
ndash5
NM
SE
2 4 6 8 100SNR (dB)
Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM
Table 2 Parameters of the improved GA
Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9
8 Mathematical Problems in Engineering
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
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ym0 n0langy(t) gm0 n0
rang
1113944+infin
nminus infin1113944
Mminus 1
m0amnHmlanggmn(t) gm0 n0
rang + η1m0 n0
(5)
where langx yrang represents the inner product of x and y andη1m0 n0
langη(t) gm0 n0rang
e orthogonal condition in FBMCOQAM is
R langgmn gm0 n0rang1113966 1113967 δmm0
δnn0 (6)
where R middot denotes real part of the argument and δ(t)
denotes the Diracrsquos delta functionus the FBMCOQAM systemmodel has been built After
that in order to design the new preamble structure the re-lationship between preamble and CE is necessary to be analyzed
21 Relationship between Preamble and CE We definem m0 + p n n0 + q where p q isin Z and then the innerproduct of the pulse shaping filter in equation (5) can berewritten as
langgmn(t) gm0 n0rang 1113946
+infin
minus infingm0+pn0+q(t)g
lowastm0+n0
(t)dt
1113946+infin
minus infine
j(π2) m0+n0p+q( )ej2π m0+p( )vot
middot g t minus n0τ0 minus qτ0( 1113857 times eminus j(π2) m0+n0( )
middot eminus j2πm0v0t
glowast
t minus n0τ0( 1113857dt
j(p+q)
1113946+infin
minus infing t minus n0τ0 minus qτ0( 1113857
middot glowast
t minus n0τ0( 1113857ej2πpv0tdt
(7)
where lowast denotes conjugate operation Since ]oτ0 12t minus n0τ0 minus qτ0 l minus (q2)τ0 and t minus n0τ0 l + (q2)τ0equation (7) can be rewritten as
langgmn(t) gm0 n0rang j
(p+q)1113946
+infin
minus infing l +(minus q2)τ0( 1113857
middot glowast
l minusminus q
2τ01113874 1113875 times e
minus j2πp]0 l+n0τ0+qτ02( )dl
jp+q+pq+2pn0( )Ag minus qτ0 p]0( 1113857
(8)
where Ag(τ ]) denotes the ambiguity function of the filterwhich can be expressed as
Ag(τ ]) 1113946 g t +τ2
1113874 1113875glowast
t minusτ2
1113874 1113875ej2π]tdt (9)
When g(t) is an even function its instantaneous au-tocorrelation function cg(τ t) g(t + (τ2))glowast(t minus (τ2))
is also an even conjugate Ag(τ ]) is the inverse Fouriertransformation of cg(τ t) and therefore Ag(τ ]) is a real-valued function [26]
We can get equation (10) from equation (5)
ym0 n0 1113944
(pq)isinZj
p+q+pq+2pn0( )Ag minus qτ0 p]0( 1113857
times am0+pn0+qHm0+p + η1m0 n0
am0 n0Hm0
+ 1113944(pq)ne(00)
jp+q+pq+2pn0( )
times Ag minus qτ0 p]0( 1113857am0+pn0+qHm0+p + η1m0 n0
(10)
According to the conventional OFDM CE method thechannel frequency response 1113954Hm0
can be estimated by1113954Hm0 n0
(ym0 n0am0 n0
) when the pilot symbol am0 n0is
transmitted at time-frequency position (m0 n0)We define Cpq j(p+q+pq+2pn0)Ag(minus qτ0 p]0) and Δm0
and Δn0 are respectively the neighborhoods of m0 and n0ΩΔm0 Δn0 (p q) |p|leΔm0 |q|leΔn0 | Hm0+pn0+q asymp Hm0 n0
1113966 1113967
and ΩlowastΔm0 Δn0 ΩΔm0 Δn0 minus (0 0) where ΩΔm0 Δn0 denotes theneighborhood of position (m0 n0) With the increase in |p|
and |q| Cpq becomes very close to zero For example for anisotropic orthogonal transform algorithm (IOTA) functionwhen (p q) notin Ω11 we have [27]
1113936(pq)notinΩ11Cpq
11138681113868111386811138681113868
111386811138681113868111386811138682
1113936(pq)notinΩ11Cpq
11138681113868111386811138681113868
111386811138681113868111386811138682 asymp 002 (11)
From equation (11) we can conclude that one column ofzero symbols is enough to prevent the preamble from beinginterfered by adjacent unknown data symbols However thezero symbols affect the CE performance e structure of theIAM is shown in Figure 1 It can be seen from Figure 1 thatthere are two columns of zero symbols for IAM According to[27] when only one column of zero symbols is inserted in thepreamble its CE performance is worse than the IAM structure
Since am0minus 1n0and am0+1n0
are of inverse signs equation(10) can be rewritten as
ym0 n0 am0 n0
Hm0+ Cminus 10am0minus 1n0
Hm0minus 1 + C10am0+1n0Hm0+1
+ 1113944(pq)ne(00)(pq)ne(plusmn10)qneplusmn1
Cpqam0+pn0+qHm0+p
+ η1m0 n0
(12)
We assume Hm0 Hm0+1 Hm0minus 1 e CE based on the
IAM can be obtained by
1113954Hm0
ym0 n0
bm0 n0
Hm0+ 1113944
(pq)ne(00)(pq)ne(plusmn10)qneplusmn1
Cpqam0+pn0+qHm0+p
bm0 n0
+η1m0 n0
bm0 n0
(13)
Mathematical Problems in Engineering 3
where bm0 n0 am0 n0
+ C10am0+1n0+ Cminus 10am0minus 1n0
bm0 n0can
be treated as the pseudo pilot Equation (13) implies that alarger pseudo pilot power leads to a better CE performancee power of pseudo pilot bm0 n0
for the IAM is shown in thefollowing equation
E1 bm0 n0
11138681113868111386811138681113868
111386811138681113868111386811138682
1113882 1113883 a2m0 n0
+ Cminus 10am0minus 1n0+ C10am0+1n0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2a 1 + 4Ag + 4A2g1113872 1113873
(14)
where σ2a denotes the variance of the transmitted FBMCOQAM symbols amn
Obviously CE based on the IAM has two drawbacks (1)the preamble duration is 3τ0 which leads to the lowspectrum efficiency and (2) the second item in equation (13)cannot be ignored when both the signal-to-noise ratio andconstellation mapping order are high
e fact is that the power of pseudo pilot bm0 n0is mainly
affected by the pilot symbols that surround am0 n0for the
IAM erefore it can be increased by changing thestructure Additionally the zero symbols in the IAM will beredundant if the intersymbol interference is alleviated by thepreamble
In order to overcome two drawbacks of the IAM a newpreamble structure is proposed in this paper It is shown inFigure 2 For sufficient accuracy and concise description onlythree-tap interference of neighborhood symbols is considered
e pseudo pilot power of the proposed structure is
E2 bm0 n0
11138681113868111386811138681113868
111386811138681113868111386811138682
1113882 1113883 a2m0 n0
+ 1113944(pq)isinΩlowast33
Cpqam0+pn0+q
11138681113868111386811138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868111386811138681113868
2
gtE1
(15)
From (15) we can conclude that the proposed preamblestructure is able to improve CE performance for its largerpseudo pilot power Obviously it also increases the fre-quency spectrum efficiency
22 RSINR In order to take the influence of prototype filteron CE into account while designing the prototype filter theRSINR is chosen to measure these influencese expressionof RSINR based on the CIR is derived in this section
First we rewrite equation (10) as
ym0 n0 am0 n0
Hm0+ a
(i1)m0n0
Hm0+ a
(i2)m0 n0
Hm0+ η1m0 n0
(16)
where
a(i1)m0 n0
1113944(pq)isinΩlowast33 q0
Cpqam0+pn0+q (17)
a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpqam0+pn0+q (18)
and then the channel frequency impulse response 1113954Hm0is
estimated by
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + a
(i2)m0 n0
(19)
Because am0+pn0+q in equation (18) belongs to a randomsignal its value cannot be determined during transmissionAs a result the term a(i2)
m0 n0is unknown Equation (19) will be
invalid if a(i2)m0 n0
is unknown To solve this problem thepredecision method is adopted e process of estimationbased on the predecision method is shown as follows
m0 minus 3
m0 minus 2
m0 minus 1
m0
m0 + 1
m0 + 2
m0 + 3
+1
ndash1
j
Data
ndashj
Figure 2 Proposed structure of preamble
m0 minus 3
m0 minus 2
m0 minus 1
m0
m0 + 1
m0 + 2
m0 + 3
+1
ndash1
j
Data
Zero
ndashj
Figure 1 Structure of the IAM
4 Mathematical Problems in Engineering
First the initial channel frequency response Hm0can be
obtained from (16)
Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0
Hm0+
a(i2)m0 n0
Hm0+ η1m0 n0
am0 n0+ a
(i1)m0 n0
(20)
Next am0+pn0+q in the term a(i2)m0 n0
can be rebuilt based onzero-forcing equalization Estimate am0+pn0+q based on thepredecision method which is shown as follows
1113954am0+pn0+q Dym0+pn0+q
Hm0
⎡⎣ ⎤⎦ (p q) isin Ωlowast33 qne 0 (21)
Here D[middot] denotes the predecision operator
en a(i2)m0 n0
can be estimated by
1113954a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpq1113954am0+pn0+q (22)
Finally since a(i2)m0 n0
has been obtained the CE is obtainedby substituting a(i2)
m0 n0into equation (19) Equation (23)
shows the final channel frequency response expression
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + 1113954a(i2)
m0 n0
(23)
By inverse Fourier transform the CIR of the FBMCOQAM system is obtained as follows
1113954h(t) 12π
1113946Δ
01113954Hme
minus j2πm]0tdt (24)
We then discretize the CIR 1113954h(t) into form 1113954hl as follows
1113954h(t) 1113944Pminus 1
i0
1113954hiδ τ minus τi( 1113857
1113954hl
1113954hi l lceilτi
Tsrceil
0 otherwise
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(25)
where x denotes the smallest integer larger than or equal to x
and Ts (T02M) denotes the sampling periode desired power of the system is closely related to the
transmitted symbol e transmitted symbol at the position(m0 n0) can be estimated as follows
1113954am0 n0 R
ym0 n0
1113954Hm0
⎛⎝ ⎞⎠ (26)
e QAM symbols are reconstructed by combining1113954cm0 n0
1113954am0 2n0+ j1113954am0 2n0+1 (27)
We then define
Hpqm0
1113944
Lhminus 1
l0
1113954hleminus jπ 2m0+p( )lM
Ag minus l minus qM
2 minus p1113876 1113877 (28)
where Lh is the number of channel taps at the sampling rateand Ag[k l] Ag(kTs (lMTs))
e desired power of the system is
Pd m0( 1113857 σ2c RHpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(29)
where σ2c denotes the variance of the complex symbol cAccording to the definition of the variance σ2c is the realnumber
Similarly the interference power and the noise power areexpressed as [28]
PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)
Rej(π2)(p+q+pq)Hpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(30)
Pnoise m0( 1113857 var η1m0 n01113960 1113961
σ2η1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682 (31)
where σ2η denotes the variance of the noise ηus the expression of RSINR is obtained by equations
(29)ndash(31)
RSINR PICI+ISI + Pnoise
Pd m0( 1113857
2σ2a1113936(pq)ne(00) R ej(π2)(p+q+pq)Hpq
m0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
+ σ2η 1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2c R Hpqm0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
(32)
3 Prototype Filter Design
31 ProblemFormulation e prototype filter in the FBMCOQAM system is assumed to be real-valued symmetric andof unit energy According to the previous discussion whenthere is intrinsic interference and noise in the system theprototype filter will influence the CE erefore the RSINRwill be chosen as an important constraint in problem for-mulation At the same time the prototype filter designshould minimize the stop-band energy us the prototypefilter design problem can be formulated as
P1 ming(k)
1113946π
2πMG e
jω1113872 1113873
11138681113868111386811138681113868
111386811138681113868111386811138682dω
st
g[k] g Lg minus 1 minus k1113960 1113961 k 0 1 Lg minus 1
1113944
Lgminus 1
k0g2[k] 1
RSINRleTH
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(33)
where |G(ejω)| |1113936Lgminus 1k0 g(k)eminus jωk| is the magnitude re-
sponse of the prototype filter TH represents a constantthreshold and Lg denotes the length of the prototype filterAccording to [29] g[k]
Ts
1113968g((k minus (Lg minus 12))Ts)
e prototype filter used to be designed without con-sidering the interaction between prototype filter and CE
Mathematical Problems in Engineering 5
erefore utilizing RSINR to measure the influence ofprototype filter on CE and regarding RSINR as the con-straint of prototype filter design problem are also one of thecontributions of this paper
ere are many variables in P1 e objective functionand constraint functions all have quadratic formsereforeit is difficult to solve the optimization problem P1 directlyAs everyone knows orthogonal transformation does notchange the nature of the original function However it canbe employed to change the form of (33)
First of all the first constraint function in (33) impliesthat only half of the filter coefficients are independent Inorder to reduce the number of variables we substitute g[k]
with a new variable x
x x1 x2 xL1113858 1113859T
g[0] g[1] gLgminus 121113876 11138771113876 1113877
T
(34)
e objective function can be rewritten with a vectorform as f0(x) xTCx where C is a real symmetric positive-definite matrix λ1 λ2 λL respectively denote positiveeigenvalues in ascending order and v1 v2 vL re-spectively denote corresponding unit orthogonaleigenvectors
en the orthogonal transformation x vz is applied tothe variable x where z z1 z2 zL1113858 1113859
T denotes the neworthogonal variable
Finally the optimization problem after the variablesubstitution and orthogonal transformation is expressed as
P2 min1113944L
j1λjz
2j
stc1(z) minus 1113944
Lg minus 1
k0g2[k] minus 1⎛⎝ ⎞⎠le 0
c2(z) RSINRleTH
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(35)
32 Problem Solution In this part an improved GA withHistory Network and pruning operator is proposed to solvethe optimization problem expressed in (35)
All the potential global answers of GA exist in a searchspace Gorges in GA will expand this search space Furtherthe inappropriate space may be searched many times duringiterations ese increase the convergence time of GA
In [16] the knowledge of search space learned by theprevious generation is passed on to the next generation viaan History Network to narrow the search space and theBaldwin effect is applied to avoid searching the same space indifferent iterations Figure 3 shows the flowchart of theproposed GA in [16]
However the influence of random operators ignored in[16] will slow down the convergence speed of GA erandom assignment operators may result in invalid iterationsearly termination and jumping out of the loop with localoptimum Inspired by [17] the pruning operator is applied inthis paper to control these random operators Pruning
operator is used to take away the inappropriate amountsresulting in increment convergence time from population andreach the global optimal answer more quickly Flowchart ofimproved GA in this study is shown in Figure 4
It can be seen from Figure 4 that the improvement of theimproved GA proposed in this paper mainly exists in fourparts initialization DRS History Network and mutation
e pruning operator of the initialization part directlydeletes the invalid subset of solution to narrow the domain atthe beginning
e time order of the pruned initialization is shown as
initialization O(NplowastNglowastNc) (36)
where Ng presents the number of genomes in each chro-mosome Nc presents the number of chromosomes of eachgeneration and Np is determined by the pruning function Itis shown as
Npruning η ηleNglowastNc
NglowastNc ηgtNglowastNc1113896 (37)
where η is the reduced ratio of search spaceIn DRS part the range of direction change is limited by
pruning operator As a result those individuals are discardedwho are so different to avoid being too far from the lastreasonable direction
ere are many exemplars stored in the History Net-work It is required to limit the variation of the exemplars toa certain range Pruning operator is applied to avoid eitherduplication with existing exemplars or drastic changesduring the updating process
Mutation is critical for reaching optimal solution In theprocess of mutation the value of genome is replaced ran-domly to create new generation for the next iterationHowever the random substitution likely results in in-appropriate value It requires a lot of iterations to eliminatethe impact of inappropriate values on the search As a resultiterations in the whole algorithm are increased Pruningoperator is able to help appoint appropriate value as much aspossible In all circumstances the better the appointment ofvalue during mutation the faster it is to reach the best answer
e time order of the pruned search loop is shown as
search loop ONsηlowastNclowastNclowastNp1113888 1113889 (38)
where Ns presents the solution spaceSince the redundant random operators have been
pruned the computational complexity of the whole algo-rithm is certainly reduced
Start Initialization
Fitness
Probability calculation
Increase the ageDRS
Update History NetworkComparison
AboveBelow
Breed individuals
Best individual
Figure 3 Flowchart of the proposed GA in [16]
6 Mathematical Problems in Engineering
4 Simulation Result
41ProblemFormulation eCE performance based on theproposed preamble is simulated in this simulation etruncation of an IOTA filter with 4T0 length [30] is chosen asthe prototype filtere fundamental parameters are listed inTable 1
Figure 5 illustrates the CE performances of the proposedstructure (A) and the IAM (B) for two kinds of constellationmapping For 16-QAM modulation when the signal-to-noise ratio (SNR) is low (SNRlt 15 dB) the CE performancesof A are slightly better than those of B Along with the SNRincreasing the CE performance of A becomes better andbetter and the advantages become more and more obviousWhen SNRgt 15 dB the IAM performance curve tends toflatten due to the performance platform Similarly for 4-QAM modulation the IAM performance curve tends toflatten when SNRgt 20 dB
e CE performances of two methods with differentnumber of subcarriers for 4-QAMmodulation and 16-QAMmodulation are also respectively compared in Figures 6 and7 It is obvious that the curve of A with 512 subcarriers islower than the one with 256 subcarriers in Figure 6 so is B Itimplies that a larger number of subcarriers mean a better CEperformance ese two curves of A are both lower thanthose of B for the same number of subcarriers Curves inFigure 6 show that A is better than B about 04 dB for 256subcarriers and 1 dB for 512 subcarriers at BER 10minus 3Modulation of curve 16-QAM in Figure 7 also illustrates thesame conclusion erefore the proposed preamble struc-ture has advantages with respect to CE
In order to lower the impact of predecision errors CEwith different number of iterations is simulated in Figure 8Perfect bound represents the upper bound of performancee curves with iterations show better performance than thecurve without iterations which indicates iteration can re-duce the predecision errors and improve the CE perfor-mancee difference between curves with different numberof iterations is obvious when the number is less than fourBut the curve with four iterations is almost overlapped withthe curve with five iterations e reason why curves areoverlapped when the number of iterations are more thanfour is that the value of am0+pn0+q is close to its true valueafter four iterations erefore there is no need to carry outmore than four iterations in order to save time and reducethe complexity
42 Improved GA performance e performance of theimproved GA is analyzed in this section According to [16]values of parameters in History Network are assigned asfollows the Good reshold the Bad reshold and theUpdate reshold are set to 10 25 and 10 of themutation radius respectivelyemutation radius is definedas the maximum Euclidean distance from the old searchspace position before mutation to the new position aftermutatione rest of the parameters of the improved GA areexhibited in Table 2
e average generations and the convergence time ofdifferent algorithms are respectively shown in Figure 9 andTable 3 Figure 9 implies that the generations of other threealgorithms are much more than the improved GA proposedin this paper It illustrates that the pruning operation in theHistory Network deletes the invalid random value assign-ment and avoids redundant individuals As can been seenfrom Table 3 the improved GA proposed in this paperreduces the convergence time varying from 8 to 221when compared to the standard GA the accelerated GA in[16] and the pruned GA in [17] It reveals that the improvedGA proposed in this paper can reduce the time to reach thebest answer and accelerate the convergence speed In con-clusion the combination of pruning operation and HistoryNetwork is more effective than just utilizing any one of them
Table 1 Fundamental parameters of CE
Parameter ValueSubcarrier spacing ]0 1094 kHzFilter length Lg 4M + 1Channel coding Convolution codingChannel mode ITU-A channel [31]Channel path delay (us) [0 16 35 50]Frame length 20 OQAM symbols
A16QAMB16QAM
A4QAMB4QAM
5 10 15 20 25 30 350SNR (dB)
ndash50
ndash45
ndash40
ndash35
ndash30
ndash25
ndash20
ndash15
ndash10
ndash5
0
NM
SE
Figure 5 Normalized mean square error (NMSE) performance ofthe proposed structure (A) and the IAM structure (B) for 4-QAMand 16-QAM in CE
Start Initialization
Fitness Crossover
Mutation
Pruning Pruning
DRS
Update History NetworkTerminate
Yes No
Increasethe age
Best individual
Pruning
Pruning
Probability calculation
Figure 4 Flowchart of the improved GA in this paper
Mathematical Problems in Engineering 7
43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations
e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller
than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11
Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one
In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper
Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study
Without iteration2 iterations3 iterations
4 iterations5 iterationsPerfect bound
ndash20
ndash18
ndash16
ndash14
ndash12
ndash10
ndash8
ndash6
ndash4
ndash2
NM
SE
2 4 6 8 10 12 14 16 18 200SNR (dB)
Figure 8 NMSE performance with different number of iterations
A256B256
A512B512
10ndash4
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 250SNR (dB)
Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM
A256B256
B512A512
ndash15
ndash10
ndash5
NM
SE
2 4 6 8 100SNR (dB)
Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM
Table 2 Parameters of the improved GA
Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9
8 Mathematical Problems in Engineering
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
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where bm0 n0 am0 n0
+ C10am0+1n0+ Cminus 10am0minus 1n0
bm0 n0can
be treated as the pseudo pilot Equation (13) implies that alarger pseudo pilot power leads to a better CE performancee power of pseudo pilot bm0 n0
for the IAM is shown in thefollowing equation
E1 bm0 n0
11138681113868111386811138681113868
111386811138681113868111386811138682
1113882 1113883 a2m0 n0
+ Cminus 10am0minus 1n0+ C10am0+1n0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2a 1 + 4Ag + 4A2g1113872 1113873
(14)
where σ2a denotes the variance of the transmitted FBMCOQAM symbols amn
Obviously CE based on the IAM has two drawbacks (1)the preamble duration is 3τ0 which leads to the lowspectrum efficiency and (2) the second item in equation (13)cannot be ignored when both the signal-to-noise ratio andconstellation mapping order are high
e fact is that the power of pseudo pilot bm0 n0is mainly
affected by the pilot symbols that surround am0 n0for the
IAM erefore it can be increased by changing thestructure Additionally the zero symbols in the IAM will beredundant if the intersymbol interference is alleviated by thepreamble
In order to overcome two drawbacks of the IAM a newpreamble structure is proposed in this paper It is shown inFigure 2 For sufficient accuracy and concise description onlythree-tap interference of neighborhood symbols is considered
e pseudo pilot power of the proposed structure is
E2 bm0 n0
11138681113868111386811138681113868
111386811138681113868111386811138682
1113882 1113883 a2m0 n0
+ 1113944(pq)isinΩlowast33
Cpqam0+pn0+q
11138681113868111386811138681113868111386811138681113868111386811138681113868
11138681113868111386811138681113868111386811138681113868111386811138681113868
2
gtE1
(15)
From (15) we can conclude that the proposed preamblestructure is able to improve CE performance for its largerpseudo pilot power Obviously it also increases the fre-quency spectrum efficiency
22 RSINR In order to take the influence of prototype filteron CE into account while designing the prototype filter theRSINR is chosen to measure these influencese expressionof RSINR based on the CIR is derived in this section
First we rewrite equation (10) as
ym0 n0 am0 n0
Hm0+ a
(i1)m0n0
Hm0+ a
(i2)m0 n0
Hm0+ η1m0 n0
(16)
where
a(i1)m0 n0
1113944(pq)isinΩlowast33 q0
Cpqam0+pn0+q (17)
a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpqam0+pn0+q (18)
and then the channel frequency impulse response 1113954Hm0is
estimated by
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + a
(i2)m0 n0
(19)
Because am0+pn0+q in equation (18) belongs to a randomsignal its value cannot be determined during transmissionAs a result the term a(i2)
m0 n0is unknown Equation (19) will be
invalid if a(i2)m0 n0
is unknown To solve this problem thepredecision method is adopted e process of estimationbased on the predecision method is shown as follows
m0 minus 3
m0 minus 2
m0 minus 1
m0
m0 + 1
m0 + 2
m0 + 3
+1
ndash1
j
Data
ndashj
Figure 2 Proposed structure of preamble
m0 minus 3
m0 minus 2
m0 minus 1
m0
m0 + 1
m0 + 2
m0 + 3
+1
ndash1
j
Data
Zero
ndashj
Figure 1 Structure of the IAM
4 Mathematical Problems in Engineering
First the initial channel frequency response Hm0can be
obtained from (16)
Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0
Hm0+
a(i2)m0 n0
Hm0+ η1m0 n0
am0 n0+ a
(i1)m0 n0
(20)
Next am0+pn0+q in the term a(i2)m0 n0
can be rebuilt based onzero-forcing equalization Estimate am0+pn0+q based on thepredecision method which is shown as follows
1113954am0+pn0+q Dym0+pn0+q
Hm0
⎡⎣ ⎤⎦ (p q) isin Ωlowast33 qne 0 (21)
Here D[middot] denotes the predecision operator
en a(i2)m0 n0
can be estimated by
1113954a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpq1113954am0+pn0+q (22)
Finally since a(i2)m0 n0
has been obtained the CE is obtainedby substituting a(i2)
m0 n0into equation (19) Equation (23)
shows the final channel frequency response expression
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + 1113954a(i2)
m0 n0
(23)
By inverse Fourier transform the CIR of the FBMCOQAM system is obtained as follows
1113954h(t) 12π
1113946Δ
01113954Hme
minus j2πm]0tdt (24)
We then discretize the CIR 1113954h(t) into form 1113954hl as follows
1113954h(t) 1113944Pminus 1
i0
1113954hiδ τ minus τi( 1113857
1113954hl
1113954hi l lceilτi
Tsrceil
0 otherwise
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(25)
where x denotes the smallest integer larger than or equal to x
and Ts (T02M) denotes the sampling periode desired power of the system is closely related to the
transmitted symbol e transmitted symbol at the position(m0 n0) can be estimated as follows
1113954am0 n0 R
ym0 n0
1113954Hm0
⎛⎝ ⎞⎠ (26)
e QAM symbols are reconstructed by combining1113954cm0 n0
1113954am0 2n0+ j1113954am0 2n0+1 (27)
We then define
Hpqm0
1113944
Lhminus 1
l0
1113954hleminus jπ 2m0+p( )lM
Ag minus l minus qM
2 minus p1113876 1113877 (28)
where Lh is the number of channel taps at the sampling rateand Ag[k l] Ag(kTs (lMTs))
e desired power of the system is
Pd m0( 1113857 σ2c RHpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(29)
where σ2c denotes the variance of the complex symbol cAccording to the definition of the variance σ2c is the realnumber
Similarly the interference power and the noise power areexpressed as [28]
PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)
Rej(π2)(p+q+pq)Hpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(30)
Pnoise m0( 1113857 var η1m0 n01113960 1113961
σ2η1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682 (31)
where σ2η denotes the variance of the noise ηus the expression of RSINR is obtained by equations
(29)ndash(31)
RSINR PICI+ISI + Pnoise
Pd m0( 1113857
2σ2a1113936(pq)ne(00) R ej(π2)(p+q+pq)Hpq
m0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
+ σ2η 1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2c R Hpqm0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
(32)
3 Prototype Filter Design
31 ProblemFormulation e prototype filter in the FBMCOQAM system is assumed to be real-valued symmetric andof unit energy According to the previous discussion whenthere is intrinsic interference and noise in the system theprototype filter will influence the CE erefore the RSINRwill be chosen as an important constraint in problem for-mulation At the same time the prototype filter designshould minimize the stop-band energy us the prototypefilter design problem can be formulated as
P1 ming(k)
1113946π
2πMG e
jω1113872 1113873
11138681113868111386811138681113868
111386811138681113868111386811138682dω
st
g[k] g Lg minus 1 minus k1113960 1113961 k 0 1 Lg minus 1
1113944
Lgminus 1
k0g2[k] 1
RSINRleTH
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(33)
where |G(ejω)| |1113936Lgminus 1k0 g(k)eminus jωk| is the magnitude re-
sponse of the prototype filter TH represents a constantthreshold and Lg denotes the length of the prototype filterAccording to [29] g[k]
Ts
1113968g((k minus (Lg minus 12))Ts)
e prototype filter used to be designed without con-sidering the interaction between prototype filter and CE
Mathematical Problems in Engineering 5
erefore utilizing RSINR to measure the influence ofprototype filter on CE and regarding RSINR as the con-straint of prototype filter design problem are also one of thecontributions of this paper
ere are many variables in P1 e objective functionand constraint functions all have quadratic formsereforeit is difficult to solve the optimization problem P1 directlyAs everyone knows orthogonal transformation does notchange the nature of the original function However it canbe employed to change the form of (33)
First of all the first constraint function in (33) impliesthat only half of the filter coefficients are independent Inorder to reduce the number of variables we substitute g[k]
with a new variable x
x x1 x2 xL1113858 1113859T
g[0] g[1] gLgminus 121113876 11138771113876 1113877
T
(34)
e objective function can be rewritten with a vectorform as f0(x) xTCx where C is a real symmetric positive-definite matrix λ1 λ2 λL respectively denote positiveeigenvalues in ascending order and v1 v2 vL re-spectively denote corresponding unit orthogonaleigenvectors
en the orthogonal transformation x vz is applied tothe variable x where z z1 z2 zL1113858 1113859
T denotes the neworthogonal variable
Finally the optimization problem after the variablesubstitution and orthogonal transformation is expressed as
P2 min1113944L
j1λjz
2j
stc1(z) minus 1113944
Lg minus 1
k0g2[k] minus 1⎛⎝ ⎞⎠le 0
c2(z) RSINRleTH
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(35)
32 Problem Solution In this part an improved GA withHistory Network and pruning operator is proposed to solvethe optimization problem expressed in (35)
All the potential global answers of GA exist in a searchspace Gorges in GA will expand this search space Furtherthe inappropriate space may be searched many times duringiterations ese increase the convergence time of GA
In [16] the knowledge of search space learned by theprevious generation is passed on to the next generation viaan History Network to narrow the search space and theBaldwin effect is applied to avoid searching the same space indifferent iterations Figure 3 shows the flowchart of theproposed GA in [16]
However the influence of random operators ignored in[16] will slow down the convergence speed of GA erandom assignment operators may result in invalid iterationsearly termination and jumping out of the loop with localoptimum Inspired by [17] the pruning operator is applied inthis paper to control these random operators Pruning
operator is used to take away the inappropriate amountsresulting in increment convergence time from population andreach the global optimal answer more quickly Flowchart ofimproved GA in this study is shown in Figure 4
It can be seen from Figure 4 that the improvement of theimproved GA proposed in this paper mainly exists in fourparts initialization DRS History Network and mutation
e pruning operator of the initialization part directlydeletes the invalid subset of solution to narrow the domain atthe beginning
e time order of the pruned initialization is shown as
initialization O(NplowastNglowastNc) (36)
where Ng presents the number of genomes in each chro-mosome Nc presents the number of chromosomes of eachgeneration and Np is determined by the pruning function Itis shown as
Npruning η ηleNglowastNc
NglowastNc ηgtNglowastNc1113896 (37)
where η is the reduced ratio of search spaceIn DRS part the range of direction change is limited by
pruning operator As a result those individuals are discardedwho are so different to avoid being too far from the lastreasonable direction
ere are many exemplars stored in the History Net-work It is required to limit the variation of the exemplars toa certain range Pruning operator is applied to avoid eitherduplication with existing exemplars or drastic changesduring the updating process
Mutation is critical for reaching optimal solution In theprocess of mutation the value of genome is replaced ran-domly to create new generation for the next iterationHowever the random substitution likely results in in-appropriate value It requires a lot of iterations to eliminatethe impact of inappropriate values on the search As a resultiterations in the whole algorithm are increased Pruningoperator is able to help appoint appropriate value as much aspossible In all circumstances the better the appointment ofvalue during mutation the faster it is to reach the best answer
e time order of the pruned search loop is shown as
search loop ONsηlowastNclowastNclowastNp1113888 1113889 (38)
where Ns presents the solution spaceSince the redundant random operators have been
pruned the computational complexity of the whole algo-rithm is certainly reduced
Start Initialization
Fitness
Probability calculation
Increase the ageDRS
Update History NetworkComparison
AboveBelow
Breed individuals
Best individual
Figure 3 Flowchart of the proposed GA in [16]
6 Mathematical Problems in Engineering
4 Simulation Result
41ProblemFormulation eCE performance based on theproposed preamble is simulated in this simulation etruncation of an IOTA filter with 4T0 length [30] is chosen asthe prototype filtere fundamental parameters are listed inTable 1
Figure 5 illustrates the CE performances of the proposedstructure (A) and the IAM (B) for two kinds of constellationmapping For 16-QAM modulation when the signal-to-noise ratio (SNR) is low (SNRlt 15 dB) the CE performancesof A are slightly better than those of B Along with the SNRincreasing the CE performance of A becomes better andbetter and the advantages become more and more obviousWhen SNRgt 15 dB the IAM performance curve tends toflatten due to the performance platform Similarly for 4-QAM modulation the IAM performance curve tends toflatten when SNRgt 20 dB
e CE performances of two methods with differentnumber of subcarriers for 4-QAMmodulation and 16-QAMmodulation are also respectively compared in Figures 6 and7 It is obvious that the curve of A with 512 subcarriers islower than the one with 256 subcarriers in Figure 6 so is B Itimplies that a larger number of subcarriers mean a better CEperformance ese two curves of A are both lower thanthose of B for the same number of subcarriers Curves inFigure 6 show that A is better than B about 04 dB for 256subcarriers and 1 dB for 512 subcarriers at BER 10minus 3Modulation of curve 16-QAM in Figure 7 also illustrates thesame conclusion erefore the proposed preamble struc-ture has advantages with respect to CE
In order to lower the impact of predecision errors CEwith different number of iterations is simulated in Figure 8Perfect bound represents the upper bound of performancee curves with iterations show better performance than thecurve without iterations which indicates iteration can re-duce the predecision errors and improve the CE perfor-mancee difference between curves with different numberof iterations is obvious when the number is less than fourBut the curve with four iterations is almost overlapped withthe curve with five iterations e reason why curves areoverlapped when the number of iterations are more thanfour is that the value of am0+pn0+q is close to its true valueafter four iterations erefore there is no need to carry outmore than four iterations in order to save time and reducethe complexity
42 Improved GA performance e performance of theimproved GA is analyzed in this section According to [16]values of parameters in History Network are assigned asfollows the Good reshold the Bad reshold and theUpdate reshold are set to 10 25 and 10 of themutation radius respectivelyemutation radius is definedas the maximum Euclidean distance from the old searchspace position before mutation to the new position aftermutatione rest of the parameters of the improved GA areexhibited in Table 2
e average generations and the convergence time ofdifferent algorithms are respectively shown in Figure 9 andTable 3 Figure 9 implies that the generations of other threealgorithms are much more than the improved GA proposedin this paper It illustrates that the pruning operation in theHistory Network deletes the invalid random value assign-ment and avoids redundant individuals As can been seenfrom Table 3 the improved GA proposed in this paperreduces the convergence time varying from 8 to 221when compared to the standard GA the accelerated GA in[16] and the pruned GA in [17] It reveals that the improvedGA proposed in this paper can reduce the time to reach thebest answer and accelerate the convergence speed In con-clusion the combination of pruning operation and HistoryNetwork is more effective than just utilizing any one of them
Table 1 Fundamental parameters of CE
Parameter ValueSubcarrier spacing ]0 1094 kHzFilter length Lg 4M + 1Channel coding Convolution codingChannel mode ITU-A channel [31]Channel path delay (us) [0 16 35 50]Frame length 20 OQAM symbols
A16QAMB16QAM
A4QAMB4QAM
5 10 15 20 25 30 350SNR (dB)
ndash50
ndash45
ndash40
ndash35
ndash30
ndash25
ndash20
ndash15
ndash10
ndash5
0
NM
SE
Figure 5 Normalized mean square error (NMSE) performance ofthe proposed structure (A) and the IAM structure (B) for 4-QAMand 16-QAM in CE
Start Initialization
Fitness Crossover
Mutation
Pruning Pruning
DRS
Update History NetworkTerminate
Yes No
Increasethe age
Best individual
Pruning
Pruning
Probability calculation
Figure 4 Flowchart of the improved GA in this paper
Mathematical Problems in Engineering 7
43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations
e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller
than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11
Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one
In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper
Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study
Without iteration2 iterations3 iterations
4 iterations5 iterationsPerfect bound
ndash20
ndash18
ndash16
ndash14
ndash12
ndash10
ndash8
ndash6
ndash4
ndash2
NM
SE
2 4 6 8 10 12 14 16 18 200SNR (dB)
Figure 8 NMSE performance with different number of iterations
A256B256
A512B512
10ndash4
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 250SNR (dB)
Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM
A256B256
B512A512
ndash15
ndash10
ndash5
NM
SE
2 4 6 8 100SNR (dB)
Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM
Table 2 Parameters of the improved GA
Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9
8 Mathematical Problems in Engineering
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
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Submit your manuscripts atwwwhindawicom
First the initial channel frequency response Hm0can be
obtained from (16)
Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0
Hm0+
a(i2)m0 n0
Hm0+ η1m0 n0
am0 n0+ a
(i1)m0 n0
(20)
Next am0+pn0+q in the term a(i2)m0 n0
can be rebuilt based onzero-forcing equalization Estimate am0+pn0+q based on thepredecision method which is shown as follows
1113954am0+pn0+q Dym0+pn0+q
Hm0
⎡⎣ ⎤⎦ (p q) isin Ωlowast33 qne 0 (21)
Here D[middot] denotes the predecision operator
en a(i2)m0 n0
can be estimated by
1113954a(i2)m0 n0
1113944(pq)isinΩlowast33 qne 0
Cpq1113954am0+pn0+q (22)
Finally since a(i2)m0 n0
has been obtained the CE is obtainedby substituting a(i2)
m0 n0into equation (19) Equation (23)
shows the final channel frequency response expression
1113954Hm0
ym0 n0
am0 n0+ a
(i1)m0 n0 + 1113954a(i2)
m0 n0
(23)
By inverse Fourier transform the CIR of the FBMCOQAM system is obtained as follows
1113954h(t) 12π
1113946Δ
01113954Hme
minus j2πm]0tdt (24)
We then discretize the CIR 1113954h(t) into form 1113954hl as follows
1113954h(t) 1113944Pminus 1
i0
1113954hiδ τ minus τi( 1113857
1113954hl
1113954hi l lceilτi
Tsrceil
0 otherwise
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(25)
where x denotes the smallest integer larger than or equal to x
and Ts (T02M) denotes the sampling periode desired power of the system is closely related to the
transmitted symbol e transmitted symbol at the position(m0 n0) can be estimated as follows
1113954am0 n0 R
ym0 n0
1113954Hm0
⎛⎝ ⎞⎠ (26)
e QAM symbols are reconstructed by combining1113954cm0 n0
1113954am0 2n0+ j1113954am0 2n0+1 (27)
We then define
Hpqm0
1113944
Lhminus 1
l0
1113954hleminus jπ 2m0+p( )lM
Ag minus l minus qM
2 minus p1113876 1113877 (28)
where Lh is the number of channel taps at the sampling rateand Ag[k l] Ag(kTs (lMTs))
e desired power of the system is
Pd m0( 1113857 σ2c RHpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(29)
where σ2c denotes the variance of the complex symbol cAccording to the definition of the variance σ2c is the realnumber
Similarly the interference power and the noise power areexpressed as [28]
PISI+ICI m0( 1113857 2σ2a 1113944(pq)ne(00)
Rej(π2)(p+q+pq)Hpq
m0
1113954Hm0
⎧⎨
⎩
⎫⎬
⎭
111386811138681113868111386811138681113868111386811138681113868
111386811138681113868111386811138681113868111386811138681113868
2
(30)
Pnoise m0( 1113857 var η1m0 n01113960 1113961
σ2η1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682 (31)
where σ2η denotes the variance of the noise ηus the expression of RSINR is obtained by equations
(29)ndash(31)
RSINR PICI+ISI + Pnoise
Pd m0( 1113857
2σ2a1113936(pq)ne(00) R ej(π2)(p+q+pq)Hpq
m0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
+ σ2η 1113954Hm0
11138681113868111386811138681113868
111386811138681113868111386811138682
σ2c R Hpqm0 1113954Hm0
1113966 111396711138681113868111386811138681113868
111386811138681113868111386811138682
(32)
3 Prototype Filter Design
31 ProblemFormulation e prototype filter in the FBMCOQAM system is assumed to be real-valued symmetric andof unit energy According to the previous discussion whenthere is intrinsic interference and noise in the system theprototype filter will influence the CE erefore the RSINRwill be chosen as an important constraint in problem for-mulation At the same time the prototype filter designshould minimize the stop-band energy us the prototypefilter design problem can be formulated as
P1 ming(k)
1113946π
2πMG e
jω1113872 1113873
11138681113868111386811138681113868
111386811138681113868111386811138682dω
st
g[k] g Lg minus 1 minus k1113960 1113961 k 0 1 Lg minus 1
1113944
Lgminus 1
k0g2[k] 1
RSINRleTH
⎧⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎩
(33)
where |G(ejω)| |1113936Lgminus 1k0 g(k)eminus jωk| is the magnitude re-
sponse of the prototype filter TH represents a constantthreshold and Lg denotes the length of the prototype filterAccording to [29] g[k]
Ts
1113968g((k minus (Lg minus 12))Ts)
e prototype filter used to be designed without con-sidering the interaction between prototype filter and CE
Mathematical Problems in Engineering 5
erefore utilizing RSINR to measure the influence ofprototype filter on CE and regarding RSINR as the con-straint of prototype filter design problem are also one of thecontributions of this paper
ere are many variables in P1 e objective functionand constraint functions all have quadratic formsereforeit is difficult to solve the optimization problem P1 directlyAs everyone knows orthogonal transformation does notchange the nature of the original function However it canbe employed to change the form of (33)
First of all the first constraint function in (33) impliesthat only half of the filter coefficients are independent Inorder to reduce the number of variables we substitute g[k]
with a new variable x
x x1 x2 xL1113858 1113859T
g[0] g[1] gLgminus 121113876 11138771113876 1113877
T
(34)
e objective function can be rewritten with a vectorform as f0(x) xTCx where C is a real symmetric positive-definite matrix λ1 λ2 λL respectively denote positiveeigenvalues in ascending order and v1 v2 vL re-spectively denote corresponding unit orthogonaleigenvectors
en the orthogonal transformation x vz is applied tothe variable x where z z1 z2 zL1113858 1113859
T denotes the neworthogonal variable
Finally the optimization problem after the variablesubstitution and orthogonal transformation is expressed as
P2 min1113944L
j1λjz
2j
stc1(z) minus 1113944
Lg minus 1
k0g2[k] minus 1⎛⎝ ⎞⎠le 0
c2(z) RSINRleTH
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(35)
32 Problem Solution In this part an improved GA withHistory Network and pruning operator is proposed to solvethe optimization problem expressed in (35)
All the potential global answers of GA exist in a searchspace Gorges in GA will expand this search space Furtherthe inappropriate space may be searched many times duringiterations ese increase the convergence time of GA
In [16] the knowledge of search space learned by theprevious generation is passed on to the next generation viaan History Network to narrow the search space and theBaldwin effect is applied to avoid searching the same space indifferent iterations Figure 3 shows the flowchart of theproposed GA in [16]
However the influence of random operators ignored in[16] will slow down the convergence speed of GA erandom assignment operators may result in invalid iterationsearly termination and jumping out of the loop with localoptimum Inspired by [17] the pruning operator is applied inthis paper to control these random operators Pruning
operator is used to take away the inappropriate amountsresulting in increment convergence time from population andreach the global optimal answer more quickly Flowchart ofimproved GA in this study is shown in Figure 4
It can be seen from Figure 4 that the improvement of theimproved GA proposed in this paper mainly exists in fourparts initialization DRS History Network and mutation
e pruning operator of the initialization part directlydeletes the invalid subset of solution to narrow the domain atthe beginning
e time order of the pruned initialization is shown as
initialization O(NplowastNglowastNc) (36)
where Ng presents the number of genomes in each chro-mosome Nc presents the number of chromosomes of eachgeneration and Np is determined by the pruning function Itis shown as
Npruning η ηleNglowastNc
NglowastNc ηgtNglowastNc1113896 (37)
where η is the reduced ratio of search spaceIn DRS part the range of direction change is limited by
pruning operator As a result those individuals are discardedwho are so different to avoid being too far from the lastreasonable direction
ere are many exemplars stored in the History Net-work It is required to limit the variation of the exemplars toa certain range Pruning operator is applied to avoid eitherduplication with existing exemplars or drastic changesduring the updating process
Mutation is critical for reaching optimal solution In theprocess of mutation the value of genome is replaced ran-domly to create new generation for the next iterationHowever the random substitution likely results in in-appropriate value It requires a lot of iterations to eliminatethe impact of inappropriate values on the search As a resultiterations in the whole algorithm are increased Pruningoperator is able to help appoint appropriate value as much aspossible In all circumstances the better the appointment ofvalue during mutation the faster it is to reach the best answer
e time order of the pruned search loop is shown as
search loop ONsηlowastNclowastNclowastNp1113888 1113889 (38)
where Ns presents the solution spaceSince the redundant random operators have been
pruned the computational complexity of the whole algo-rithm is certainly reduced
Start Initialization
Fitness
Probability calculation
Increase the ageDRS
Update History NetworkComparison
AboveBelow
Breed individuals
Best individual
Figure 3 Flowchart of the proposed GA in [16]
6 Mathematical Problems in Engineering
4 Simulation Result
41ProblemFormulation eCE performance based on theproposed preamble is simulated in this simulation etruncation of an IOTA filter with 4T0 length [30] is chosen asthe prototype filtere fundamental parameters are listed inTable 1
Figure 5 illustrates the CE performances of the proposedstructure (A) and the IAM (B) for two kinds of constellationmapping For 16-QAM modulation when the signal-to-noise ratio (SNR) is low (SNRlt 15 dB) the CE performancesof A are slightly better than those of B Along with the SNRincreasing the CE performance of A becomes better andbetter and the advantages become more and more obviousWhen SNRgt 15 dB the IAM performance curve tends toflatten due to the performance platform Similarly for 4-QAM modulation the IAM performance curve tends toflatten when SNRgt 20 dB
e CE performances of two methods with differentnumber of subcarriers for 4-QAMmodulation and 16-QAMmodulation are also respectively compared in Figures 6 and7 It is obvious that the curve of A with 512 subcarriers islower than the one with 256 subcarriers in Figure 6 so is B Itimplies that a larger number of subcarriers mean a better CEperformance ese two curves of A are both lower thanthose of B for the same number of subcarriers Curves inFigure 6 show that A is better than B about 04 dB for 256subcarriers and 1 dB for 512 subcarriers at BER 10minus 3Modulation of curve 16-QAM in Figure 7 also illustrates thesame conclusion erefore the proposed preamble struc-ture has advantages with respect to CE
In order to lower the impact of predecision errors CEwith different number of iterations is simulated in Figure 8Perfect bound represents the upper bound of performancee curves with iterations show better performance than thecurve without iterations which indicates iteration can re-duce the predecision errors and improve the CE perfor-mancee difference between curves with different numberof iterations is obvious when the number is less than fourBut the curve with four iterations is almost overlapped withthe curve with five iterations e reason why curves areoverlapped when the number of iterations are more thanfour is that the value of am0+pn0+q is close to its true valueafter four iterations erefore there is no need to carry outmore than four iterations in order to save time and reducethe complexity
42 Improved GA performance e performance of theimproved GA is analyzed in this section According to [16]values of parameters in History Network are assigned asfollows the Good reshold the Bad reshold and theUpdate reshold are set to 10 25 and 10 of themutation radius respectivelyemutation radius is definedas the maximum Euclidean distance from the old searchspace position before mutation to the new position aftermutatione rest of the parameters of the improved GA areexhibited in Table 2
e average generations and the convergence time ofdifferent algorithms are respectively shown in Figure 9 andTable 3 Figure 9 implies that the generations of other threealgorithms are much more than the improved GA proposedin this paper It illustrates that the pruning operation in theHistory Network deletes the invalid random value assign-ment and avoids redundant individuals As can been seenfrom Table 3 the improved GA proposed in this paperreduces the convergence time varying from 8 to 221when compared to the standard GA the accelerated GA in[16] and the pruned GA in [17] It reveals that the improvedGA proposed in this paper can reduce the time to reach thebest answer and accelerate the convergence speed In con-clusion the combination of pruning operation and HistoryNetwork is more effective than just utilizing any one of them
Table 1 Fundamental parameters of CE
Parameter ValueSubcarrier spacing ]0 1094 kHzFilter length Lg 4M + 1Channel coding Convolution codingChannel mode ITU-A channel [31]Channel path delay (us) [0 16 35 50]Frame length 20 OQAM symbols
A16QAMB16QAM
A4QAMB4QAM
5 10 15 20 25 30 350SNR (dB)
ndash50
ndash45
ndash40
ndash35
ndash30
ndash25
ndash20
ndash15
ndash10
ndash5
0
NM
SE
Figure 5 Normalized mean square error (NMSE) performance ofthe proposed structure (A) and the IAM structure (B) for 4-QAMand 16-QAM in CE
Start Initialization
Fitness Crossover
Mutation
Pruning Pruning
DRS
Update History NetworkTerminate
Yes No
Increasethe age
Best individual
Pruning
Pruning
Probability calculation
Figure 4 Flowchart of the improved GA in this paper
Mathematical Problems in Engineering 7
43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations
e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller
than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11
Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one
In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper
Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study
Without iteration2 iterations3 iterations
4 iterations5 iterationsPerfect bound
ndash20
ndash18
ndash16
ndash14
ndash12
ndash10
ndash8
ndash6
ndash4
ndash2
NM
SE
2 4 6 8 10 12 14 16 18 200SNR (dB)
Figure 8 NMSE performance with different number of iterations
A256B256
A512B512
10ndash4
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 250SNR (dB)
Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM
A256B256
B512A512
ndash15
ndash10
ndash5
NM
SE
2 4 6 8 100SNR (dB)
Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM
Table 2 Parameters of the improved GA
Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9
8 Mathematical Problems in Engineering
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
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erefore utilizing RSINR to measure the influence ofprototype filter on CE and regarding RSINR as the con-straint of prototype filter design problem are also one of thecontributions of this paper
ere are many variables in P1 e objective functionand constraint functions all have quadratic formsereforeit is difficult to solve the optimization problem P1 directlyAs everyone knows orthogonal transformation does notchange the nature of the original function However it canbe employed to change the form of (33)
First of all the first constraint function in (33) impliesthat only half of the filter coefficients are independent Inorder to reduce the number of variables we substitute g[k]
with a new variable x
x x1 x2 xL1113858 1113859T
g[0] g[1] gLgminus 121113876 11138771113876 1113877
T
(34)
e objective function can be rewritten with a vectorform as f0(x) xTCx where C is a real symmetric positive-definite matrix λ1 λ2 λL respectively denote positiveeigenvalues in ascending order and v1 v2 vL re-spectively denote corresponding unit orthogonaleigenvectors
en the orthogonal transformation x vz is applied tothe variable x where z z1 z2 zL1113858 1113859
T denotes the neworthogonal variable
Finally the optimization problem after the variablesubstitution and orthogonal transformation is expressed as
P2 min1113944L
j1λjz
2j
stc1(z) minus 1113944
Lg minus 1
k0g2[k] minus 1⎛⎝ ⎞⎠le 0
c2(z) RSINRleTH
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
(35)
32 Problem Solution In this part an improved GA withHistory Network and pruning operator is proposed to solvethe optimization problem expressed in (35)
All the potential global answers of GA exist in a searchspace Gorges in GA will expand this search space Furtherthe inappropriate space may be searched many times duringiterations ese increase the convergence time of GA
In [16] the knowledge of search space learned by theprevious generation is passed on to the next generation viaan History Network to narrow the search space and theBaldwin effect is applied to avoid searching the same space indifferent iterations Figure 3 shows the flowchart of theproposed GA in [16]
However the influence of random operators ignored in[16] will slow down the convergence speed of GA erandom assignment operators may result in invalid iterationsearly termination and jumping out of the loop with localoptimum Inspired by [17] the pruning operator is applied inthis paper to control these random operators Pruning
operator is used to take away the inappropriate amountsresulting in increment convergence time from population andreach the global optimal answer more quickly Flowchart ofimproved GA in this study is shown in Figure 4
It can be seen from Figure 4 that the improvement of theimproved GA proposed in this paper mainly exists in fourparts initialization DRS History Network and mutation
e pruning operator of the initialization part directlydeletes the invalid subset of solution to narrow the domain atthe beginning
e time order of the pruned initialization is shown as
initialization O(NplowastNglowastNc) (36)
where Ng presents the number of genomes in each chro-mosome Nc presents the number of chromosomes of eachgeneration and Np is determined by the pruning function Itis shown as
Npruning η ηleNglowastNc
NglowastNc ηgtNglowastNc1113896 (37)
where η is the reduced ratio of search spaceIn DRS part the range of direction change is limited by
pruning operator As a result those individuals are discardedwho are so different to avoid being too far from the lastreasonable direction
ere are many exemplars stored in the History Net-work It is required to limit the variation of the exemplars toa certain range Pruning operator is applied to avoid eitherduplication with existing exemplars or drastic changesduring the updating process
Mutation is critical for reaching optimal solution In theprocess of mutation the value of genome is replaced ran-domly to create new generation for the next iterationHowever the random substitution likely results in in-appropriate value It requires a lot of iterations to eliminatethe impact of inappropriate values on the search As a resultiterations in the whole algorithm are increased Pruningoperator is able to help appoint appropriate value as much aspossible In all circumstances the better the appointment ofvalue during mutation the faster it is to reach the best answer
e time order of the pruned search loop is shown as
search loop ONsηlowastNclowastNclowastNp1113888 1113889 (38)
where Ns presents the solution spaceSince the redundant random operators have been
pruned the computational complexity of the whole algo-rithm is certainly reduced
Start Initialization
Fitness
Probability calculation
Increase the ageDRS
Update History NetworkComparison
AboveBelow
Breed individuals
Best individual
Figure 3 Flowchart of the proposed GA in [16]
6 Mathematical Problems in Engineering
4 Simulation Result
41ProblemFormulation eCE performance based on theproposed preamble is simulated in this simulation etruncation of an IOTA filter with 4T0 length [30] is chosen asthe prototype filtere fundamental parameters are listed inTable 1
Figure 5 illustrates the CE performances of the proposedstructure (A) and the IAM (B) for two kinds of constellationmapping For 16-QAM modulation when the signal-to-noise ratio (SNR) is low (SNRlt 15 dB) the CE performancesof A are slightly better than those of B Along with the SNRincreasing the CE performance of A becomes better andbetter and the advantages become more and more obviousWhen SNRgt 15 dB the IAM performance curve tends toflatten due to the performance platform Similarly for 4-QAM modulation the IAM performance curve tends toflatten when SNRgt 20 dB
e CE performances of two methods with differentnumber of subcarriers for 4-QAMmodulation and 16-QAMmodulation are also respectively compared in Figures 6 and7 It is obvious that the curve of A with 512 subcarriers islower than the one with 256 subcarriers in Figure 6 so is B Itimplies that a larger number of subcarriers mean a better CEperformance ese two curves of A are both lower thanthose of B for the same number of subcarriers Curves inFigure 6 show that A is better than B about 04 dB for 256subcarriers and 1 dB for 512 subcarriers at BER 10minus 3Modulation of curve 16-QAM in Figure 7 also illustrates thesame conclusion erefore the proposed preamble struc-ture has advantages with respect to CE
In order to lower the impact of predecision errors CEwith different number of iterations is simulated in Figure 8Perfect bound represents the upper bound of performancee curves with iterations show better performance than thecurve without iterations which indicates iteration can re-duce the predecision errors and improve the CE perfor-mancee difference between curves with different numberof iterations is obvious when the number is less than fourBut the curve with four iterations is almost overlapped withthe curve with five iterations e reason why curves areoverlapped when the number of iterations are more thanfour is that the value of am0+pn0+q is close to its true valueafter four iterations erefore there is no need to carry outmore than four iterations in order to save time and reducethe complexity
42 Improved GA performance e performance of theimproved GA is analyzed in this section According to [16]values of parameters in History Network are assigned asfollows the Good reshold the Bad reshold and theUpdate reshold are set to 10 25 and 10 of themutation radius respectivelyemutation radius is definedas the maximum Euclidean distance from the old searchspace position before mutation to the new position aftermutatione rest of the parameters of the improved GA areexhibited in Table 2
e average generations and the convergence time ofdifferent algorithms are respectively shown in Figure 9 andTable 3 Figure 9 implies that the generations of other threealgorithms are much more than the improved GA proposedin this paper It illustrates that the pruning operation in theHistory Network deletes the invalid random value assign-ment and avoids redundant individuals As can been seenfrom Table 3 the improved GA proposed in this paperreduces the convergence time varying from 8 to 221when compared to the standard GA the accelerated GA in[16] and the pruned GA in [17] It reveals that the improvedGA proposed in this paper can reduce the time to reach thebest answer and accelerate the convergence speed In con-clusion the combination of pruning operation and HistoryNetwork is more effective than just utilizing any one of them
Table 1 Fundamental parameters of CE
Parameter ValueSubcarrier spacing ]0 1094 kHzFilter length Lg 4M + 1Channel coding Convolution codingChannel mode ITU-A channel [31]Channel path delay (us) [0 16 35 50]Frame length 20 OQAM symbols
A16QAMB16QAM
A4QAMB4QAM
5 10 15 20 25 30 350SNR (dB)
ndash50
ndash45
ndash40
ndash35
ndash30
ndash25
ndash20
ndash15
ndash10
ndash5
0
NM
SE
Figure 5 Normalized mean square error (NMSE) performance ofthe proposed structure (A) and the IAM structure (B) for 4-QAMand 16-QAM in CE
Start Initialization
Fitness Crossover
Mutation
Pruning Pruning
DRS
Update History NetworkTerminate
Yes No
Increasethe age
Best individual
Pruning
Pruning
Probability calculation
Figure 4 Flowchart of the improved GA in this paper
Mathematical Problems in Engineering 7
43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations
e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller
than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11
Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one
In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper
Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study
Without iteration2 iterations3 iterations
4 iterations5 iterationsPerfect bound
ndash20
ndash18
ndash16
ndash14
ndash12
ndash10
ndash8
ndash6
ndash4
ndash2
NM
SE
2 4 6 8 10 12 14 16 18 200SNR (dB)
Figure 8 NMSE performance with different number of iterations
A256B256
A512B512
10ndash4
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 250SNR (dB)
Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM
A256B256
B512A512
ndash15
ndash10
ndash5
NM
SE
2 4 6 8 100SNR (dB)
Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM
Table 2 Parameters of the improved GA
Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9
8 Mathematical Problems in Engineering
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
4 Simulation Result
41ProblemFormulation eCE performance based on theproposed preamble is simulated in this simulation etruncation of an IOTA filter with 4T0 length [30] is chosen asthe prototype filtere fundamental parameters are listed inTable 1
Figure 5 illustrates the CE performances of the proposedstructure (A) and the IAM (B) for two kinds of constellationmapping For 16-QAM modulation when the signal-to-noise ratio (SNR) is low (SNRlt 15 dB) the CE performancesof A are slightly better than those of B Along with the SNRincreasing the CE performance of A becomes better andbetter and the advantages become more and more obviousWhen SNRgt 15 dB the IAM performance curve tends toflatten due to the performance platform Similarly for 4-QAM modulation the IAM performance curve tends toflatten when SNRgt 20 dB
e CE performances of two methods with differentnumber of subcarriers for 4-QAMmodulation and 16-QAMmodulation are also respectively compared in Figures 6 and7 It is obvious that the curve of A with 512 subcarriers islower than the one with 256 subcarriers in Figure 6 so is B Itimplies that a larger number of subcarriers mean a better CEperformance ese two curves of A are both lower thanthose of B for the same number of subcarriers Curves inFigure 6 show that A is better than B about 04 dB for 256subcarriers and 1 dB for 512 subcarriers at BER 10minus 3Modulation of curve 16-QAM in Figure 7 also illustrates thesame conclusion erefore the proposed preamble struc-ture has advantages with respect to CE
In order to lower the impact of predecision errors CEwith different number of iterations is simulated in Figure 8Perfect bound represents the upper bound of performancee curves with iterations show better performance than thecurve without iterations which indicates iteration can re-duce the predecision errors and improve the CE perfor-mancee difference between curves with different numberof iterations is obvious when the number is less than fourBut the curve with four iterations is almost overlapped withthe curve with five iterations e reason why curves areoverlapped when the number of iterations are more thanfour is that the value of am0+pn0+q is close to its true valueafter four iterations erefore there is no need to carry outmore than four iterations in order to save time and reducethe complexity
42 Improved GA performance e performance of theimproved GA is analyzed in this section According to [16]values of parameters in History Network are assigned asfollows the Good reshold the Bad reshold and theUpdate reshold are set to 10 25 and 10 of themutation radius respectivelyemutation radius is definedas the maximum Euclidean distance from the old searchspace position before mutation to the new position aftermutatione rest of the parameters of the improved GA areexhibited in Table 2
e average generations and the convergence time ofdifferent algorithms are respectively shown in Figure 9 andTable 3 Figure 9 implies that the generations of other threealgorithms are much more than the improved GA proposedin this paper It illustrates that the pruning operation in theHistory Network deletes the invalid random value assign-ment and avoids redundant individuals As can been seenfrom Table 3 the improved GA proposed in this paperreduces the convergence time varying from 8 to 221when compared to the standard GA the accelerated GA in[16] and the pruned GA in [17] It reveals that the improvedGA proposed in this paper can reduce the time to reach thebest answer and accelerate the convergence speed In con-clusion the combination of pruning operation and HistoryNetwork is more effective than just utilizing any one of them
Table 1 Fundamental parameters of CE
Parameter ValueSubcarrier spacing ]0 1094 kHzFilter length Lg 4M + 1Channel coding Convolution codingChannel mode ITU-A channel [31]Channel path delay (us) [0 16 35 50]Frame length 20 OQAM symbols
A16QAMB16QAM
A4QAMB4QAM
5 10 15 20 25 30 350SNR (dB)
ndash50
ndash45
ndash40
ndash35
ndash30
ndash25
ndash20
ndash15
ndash10
ndash5
0
NM
SE
Figure 5 Normalized mean square error (NMSE) performance ofthe proposed structure (A) and the IAM structure (B) for 4-QAMand 16-QAM in CE
Start Initialization
Fitness Crossover
Mutation
Pruning Pruning
DRS
Update History NetworkTerminate
Yes No
Increasethe age
Best individual
Pruning
Pruning
Probability calculation
Figure 4 Flowchart of the improved GA in this paper
Mathematical Problems in Engineering 7
43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations
e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller
than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11
Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one
In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper
Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study
Without iteration2 iterations3 iterations
4 iterations5 iterationsPerfect bound
ndash20
ndash18
ndash16
ndash14
ndash12
ndash10
ndash8
ndash6
ndash4
ndash2
NM
SE
2 4 6 8 10 12 14 16 18 200SNR (dB)
Figure 8 NMSE performance with different number of iterations
A256B256
A512B512
10ndash4
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 250SNR (dB)
Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM
A256B256
B512A512
ndash15
ndash10
ndash5
NM
SE
2 4 6 8 100SNR (dB)
Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM
Table 2 Parameters of the improved GA
Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9
8 Mathematical Problems in Engineering
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
43 Performance of the Designed Prototype Filter In thissection the validity of the designed prototype filter is verifiedthrough a series of simulations
e average BER performances of FBMCOQAM systemwith different number of subcarriers for three prototypefilters are respectively shown in Figures 10 and 11 eprototype filters designed by method A and method B arerespectively described in [23 24] and method C is the onein this paper e simulation results in Figure 10 reveal thatthe average BER performance of the system with the pro-totype filter designed by method C is superior to the othertwo methods forM 256 It can be seen from Figure 10 thatmethod A and method B show similar BER performancewhen SNR is larger than 14 dB Meanwhile method B andmethod C show similar performance when SNR is smaller
than 8 dB Although there are some intersections betweenthese three lines in Figure 10 the outperformance of methodC is obvious when SNR is larger than 10 dB Similar con-clusion can be drawn from Figure 11
Since the proposed filter is designed with a constraint onthe RSINR not only the influence of noise is considered toguarantee the effectiveness of the method but also the de-sired power is increased as the interference power decreasesHowever the influence of noise which should not be ignoredin practice was not considered in [23] e existence of thenoise destroys the efficiency of the method proposed in [23]especially when SNR of the system is at the low level edesired symbol power was also not considered in [23] whichmay have obtained the solution for the optimizationproblem not being the minimum one
In [24] the objective of the prototype filter design is tomaximize the signal-to-weighted interference radio Largestop-band energy at high SNR level destroys the orthogonalcondition resulting in a much poorer BER performancethan the method proposed in this paper
Figure 12 shows the magnitude responses of IOTA thefilter designed in [23] and in [24] and that designed in thisstudy with M 256 e red curve in Figure 12 shows thatthe first side lobe of the prototype filter designed in this study
Without iteration2 iterations3 iterations
4 iterations5 iterationsPerfect bound
ndash20
ndash18
ndash16
ndash14
ndash12
ndash10
ndash8
ndash6
ndash4
ndash2
NM
SE
2 4 6 8 10 12 14 16 18 200SNR (dB)
Figure 8 NMSE performance with different number of iterations
A256B256
A512B512
10ndash4
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 250SNR (dB)
Figure 6 Bit error ratio (BER) performance of the proposedstructure (A) and the IAM structure (B) with different number ofsubcarriers in CE for 4-QAM
A256B256
B512A512
ndash15
ndash10
ndash5
NM
SE
2 4 6 8 100SNR (dB)
Figure 7 NMSE performance of the proposed structure (A) andthe IAM structure (B) with different number of subcarriers in CEfor 16-QAM
Table 2 Parameters of the improved GA
Parameters ValuePopulation Double vectorSelection TournamentCrossover Two pointMutation Creep mutationCrossover probability 08Reject probability 095Calculation accuracy 1 times 10minus 9
8 Mathematical Problems in Engineering
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
is about minus 40 dB It is better than the filter designed in [23]and that designed in [24] of about 8 dB and 9 dB It is alsosuperior to IOTA of about 15 dB A lower side lobe means abetter time-frequency localization property In order tofurther reveal the outperformance of the prototype filterproposed in this study Table 4 represents the stop-band
energy of different filters It shows that the stop-band en-ergies of the IOTA filter the filter designed in [23] and thatdesigned in [24] are minus 32 dB minus 43 dB and minus 41 dB re-spectively e proposed prototype filter achieves the loweststop-band energy (minus 51 dB) It means the proposed prototypefilter outperforms three other filters in filtering clutter
Table 3 Convergence time of different algorithms
Methods e standard GA e accelerated GA in [16] e pruned GA in [17] e improved GA in this paperConvergence time 163275 s 139682 s 143233 s 127354 s
Method AMethod BMethod C
5 10 15 20 25 300SNR (dB)
10ndash3
10ndash2
10ndash1
100
BER
Figure 10 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 256
10ndash3
10ndash2
10ndash1
100
BER
5 10 15 20 25 300SNR (dB)
Method AMethod BMethod C
Figure 11 Average BER performance of FBMCOQAM systemswith prototype filters designed by different methods for M 512
The s
tand
ard
GA
The a
ccel
erat
edG
A in
[16]
The p
rune
dG
A in
[17]
The i
mpr
oved
GA
in th
is stu
dy
0
20
40
60
80
100
120
Figure 9 Average generations of different GAs
Mathematical Problems in Engineering 9
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Because the proposed prototype filter is designed withconstraints on RSINR and the noise influence has been takeninto account the proposed prototype filter in this studyshows better performance in filtering clutter
5 Conclusion
In this study a new prototype filter was designed based onthe CE for the FBMCOQAM system in frequency-selectivechannels e proposed method is able to minimize thestop-band energy and increase the CE performanceRSINR which is utilized to measure the influence of theprototype filter on the CE is an important constraint forprototype filter design A new preamble structure wasproposed to save the frequency spectrum before the RSINRcalculation en an optimization problem of prototypefilter design was formulated to minimize the stop-bandenergy with constraint on the RSINR Finally an improvedGA is employed to solve this optimization problem byutilizing a History Network and a pruning operation inorder to obtain low convergence time and the global op-timal solution Simulation results verified that the preamblestructure proposed in this paper increased the CE accuracyand spectral efficiency the improved GA accelerated theconvergence speed and the designed prototype filteroutperforms other filters
ere are still many works that are worth doing in thefuture On the one hand if the preamble symbol is encodedthe spectrum efficiency can be further increased on theother hand the History Network is the key part of theproposed GA how to set up the History Network and chooseits exemplar more efficiently relate to the performance of thewhole algorithm
Data Availability
No data were used to support this study
Conflicts of Interest
e authors declare that they have no conflicts of interest
Acknowledgments
is work was supported by the National Natural ScienceFoundation of China under Grant no 61671468
References
[1] X P Liu X H Chen and Z D Xie ldquoApplication of OQAMOFDM modulation in troposcatter communicationrdquo inProceedings of the Fifth International Conference on NetworkCommunication and ComputingmdashICNCC rsquo16 Kyoto JapanDecember 2016
[2] Y Zhao X Chen L Xue J Liu and Z Xie ldquoIterative pre-amble-based time domain channel estimation for OFDMOQAM systemsrdquo IEICE Transactions on Communicationsvol E99B no 10 pp 2221ndash2227 2016
[3] A Viholainen T Ihalainen T Stitz M Renfors andM Bellanger ldquoPrototype filter design for filter bank basedmulticarrier transmissionrdquo in Proceedings of the 17th Euro-pran Signal Processing Conference pp 1359ndash1363 PiscatawayNJ USA August 2009
[4] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approachfor designing transmultiplexersrdquo IEEE Transactions on Cir-cuits and Systems I Regular Papers vol 55 no 9 pp 2696ndash2706 2008
[5] S Alphan G Ismail and A Huseyin ldquoA survey on multi-carrier communications prototype filter lattice structures
Table 4 Stop-band energy of different filters
Methods IOTA Prototype filter in [23] Prototype filter in [24] Proposed filterStop-band energy minus 32 dB minus 43 dB minus 41 dB minus 51 dB
0 0005 001 0015 002 0025 003 0035 004
ndash80
ndash60
ndash40
ndash20
0
Normalized frequency (timesπ radsample)
Mag
nitu
de (d
B)
IOTAFilter designed in [23]
Filter designed in [24]Filter designed in this study
Figure 12 Magnitude response of IOTA and the filter designed in [23] and in [24] and in this study
10 Mathematical Problems in Engineering
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
and implementation aspectsrdquo IEEE Communications Surveysamp Tutorials vol 16 no 3 pp 1312ndash1338 2014
[6] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Pro-ceedings of the 2001 IEEE International Conference onAcoustics Speech and Signal Processing Proceedings (CatNo01CH37221) pp 2417ndash2420 Salt Lake City UT USAMay2001
[7] S Mirablasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems IIAnalog and Digital Signal Processing vol 50 no 8pp 456ndash469 2003
[8] F Cruz-Roldan C Heneghan J B Saez-Landete M Blanco-Velasco and P Amo-Lopez ldquoMulti-objective optimizationtechnique to design digital filter for modulated multi-ratesystemsrdquo Electronics Letters vol 44 no 13 pp 827-828 2008
[9] J Z Jiang B W Ling and S Ouyang ldquoEfficient design ofprototype filter for large scale filter bank-based multicarriersystemsrdquo IET Signal Processing vol 11 no 5 pp 521ndash5262014
[10] I P Androulakis C D Maranas and C A Floudas ldquoαBB aglobal optimization method for general constrained non-convex problemsrdquo Journal of Global Optimization vol 7no 4 pp 337ndash363 1995
[11] Y T Wu D Chen and T Jiang ldquoEfficient branch and boundalgorithms for prototype filter optimization in OQAM-OFDM systemsrdquo International Journal of CommunicationSystems vol 30 no 5 2017
[12] J G Wen J Y Hua F Li D M Wang and J M Li ldquoDesignof FBMC waveform by exploiting a NPR prototype filterrdquo inProceedings of the 2017 Advances in Wireless and OpticalCommunications (RTUWO) pp 156ndash161 Riga Latvia No-vember 2017
[13] Q S Qian Q Liu S Xu W Sun and H Li ldquoAn evolutionarymethod to achieve the maximum efficiency tracking withmulti-objective optimization based on the genetic algorithmrdquoin Proceedings of the 2019 IEEE Applied Power ElectronicsConference and Exposition (APEC) Anaheim CA USA May2019
[14] A Tamimi ldquoIntelligent traffic light based on genetic algo-rithminrdquo in Proceedings of the 2019 IEEE Jordan InternationalJoint Conference on Electrical Engineering and InformationTechnology (JEEIT) Amman Jordan May 2019
[15] G M Kakandikar and V M Nandedkar ldquoPrediction andoptimization of thinning in automotive sealing cover usinggenetic algorithmrdquo Journal of Computational Design andEngineering vol 3 no 1 pp 63ndash70 2016
[16] J R Podlena and T Hendtlass ldquoAn accelerated genetic al-gorithmrdquo Applied Intelligence vol 8 no 2 pp 103ndash111 1998
[17] S M Hedjazi and S S Marjani ldquoPruned genetic algorithmrdquoLecture Notes in Computer Science vol 6320 no 1pp 193ndash200 2010
[18] S Hu ldquoEffectiveness of preamble based channel estimation forOFDMOQAM systemrdquo in Proceedings of the 2009 In-ternational Conference on Networks Security Wireless Com-munications and Trusted Computing Wuhan China April2009
[19] W Cui D Qu T Jiang and B Farhang-Boroujeny ldquoCodedauxiliary pilots for channel estimation in FBMC-OQAMsystemsrdquo IEEE Transactions on Vehicular Technology vol 65no 5 pp 2936ndash2946 2016
[20] G Cheng Y Xiao S Hu and S Li ldquoInterference cancellationaided channel estimation for OFDMOQAM systemrdquo
SCIENCE CHINA Information Sciences vol 56 no 12pp 1ndash8 2013
[21] H Wang W C Du and L W Xu ldquoA new sparse adaptivechannel estimation method based on compressive sensing forFBMCOQAM transmission networkrdquo Sensors vol 16 no 7p 966 2016
[22] X M Liu ldquoA novel channel estimation method based oncompressive sensing for OFDMOQAM Systemrdquo Journal ofComputational Information Systems vol 9 no 15pp 5955ndash5963 2013
[23] Y Tian D Chen K Luo and T Jiang ldquoPrototype filter designto minimize stopband energy with constraint on channelestimation performance for OQAMFBMC systemsrdquo IEEETransactions on Broadcasting vol 65 no 2 pp 260ndash2692018
[24] S M J Tabatabaee and H Z Jafarian ldquoPrototype filter designfor FBMC systems via evolutionary PSO algorithm in highlydoubly dispersive channelsrdquo Transactions on Emerging Tele-communications Technologies vol 28 no 4 Article ID e30482017
[25] P Achaichia M Le Bot and P Siohan ldquoOFDMOQAM asolution to efficiently increase the capacity of future plcnetworksrdquo IEEE Transactions on Power Delivery vol 26 no 4pp 2443ndash2455 2011
[26] X D Zhan Modern Signal Processing (in Chinese) pp 442ndash475 Tsinghua University Press Beijing China 1994
[27] C LeLe P Siohan R Legouable and J P Javaudin ldquoPre-amble-based channel estimation techniques for OFDMOQAM over the powerlinerdquo in Proceedings of the 2007 IEEEInternational Symposium on Power Line Communications andits Applications pp 59ndash64 Pisa Italy March 2007
[28] H Lin and P Siohan ldquoCapacity analysis for indoor PLC usingdifferent multi-carrier modulation schemesrdquo IEEE Trans-actions on Power Delivery vol 25 no 1 pp 113ndash124 2010
[29] P Siohan C Siclet and N Lacaille ldquoAnalysis and design ofOFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash1183 2002
[30] B L Floch M Alard and C Berrou ldquoCoded orthogonalfrequency divisionmultiplexrdquo Proceedings of the IEEE vol 83no 6 pp 982ndash996 1995
[31] International Telecommunication Union Guidelines forevaluation of radio transmission technologies for IMT-2000vol 1225 International Telecommunication Union GenevaSwitzerland 1997
Mathematical Problems in Engineering 11
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Hindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018Volume 2018
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom