Design of time-limited pulses
3
IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 8, AUGUST 2009 573 Design of Time-Limited Pulses Seung Joon Lee, Senior Member, IEEE Abstract—An approach for designing time-limited transmit pulses is proposed. In the approach, a time-limited pulse is represented by a weighted sum of cosine functions whose fre- quencies are integer multiples of one over the time duration of the transmit pulse. The weighting coefficients are derived to minimize the mean square error of received symbols for a given symbol timing offset, while restricting the out-of-band energy to a predefined level. The minimization is numerically performed by using the Zoutendijk’s feasible direction method. Performance comparison results are presented showing that the pulses derived by this approach not only have smaller mean square errors, but also smaller symbol error probabilities for BPSK signaling compared to the time-truncated versions of conventional square- root Nyquist pulses. Index Terms—Feasible direction method, mean square error, out-of-band energy, pulse shaping, time-limited pulse. I. I NTRODUCTION N YQUIST pulses are known to have no inter-symbol interference (ISI) at a perfect symbol timing epoch. The most popular example of a Nyquist pulse is the raise-cosine (RC) pulse [1, p. 560]. Recently, several other Nyquist pulses [2]–[4] have been proposed which perform better than the RC pulse in terms of symbol error probability when there is a symbol timing offset. A long time ago, a Nyquist pulse called the double-jump (DJ) pulse was analytically derived to have the minimum mean square error (MSE) of received symbols in a small timing offset [5]. It was also shown in [6] that the DJ pulse has smaller symbol error probability for BPSK signaling than any other pulse considered therein in timing offsets less than a certain value (e.g., 5% of the symbol duration). As indicated in [3], most practical systems employ a time- limited transmit pulse and a receive filter of an impulse response function equal to the transmit pulse. Therefore, performance comparison was also examined for time-truncated square-root Nyquist pulses 1 in [3], where it was observed that the truncated 2 versions of the recently proposed square-root Nyquist pulses are worse than the truncated square-root raised- cosine pulse in zero timing offset, while the former are still better than the latter in timing offsets larger than a certain value. In this letter, we investigate how to obtain improved time- limited pulses. The remainder of this letter is organized as Manuscript received May 20, 2009. The associate editor coordinating the review of this letter and approving it for publication was P. Cotae. This study was supported by 2008 Research Grant from Kangwon National University. S. J. Lee is with the Department of Electrical & Electronic Engineer- ing, Kangwon National University, Chuncheon 200-701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/LCOMM.2009.091095 1 ‘A square-root Nyquist pulse in the time-domain’ means a time-function of which the squared Fourier transform meets the Nyquist criterion [1, eq. (9.2-13)] for zero intersymbol interference. 2 The word ‘truncated’ implies throughout this letter that a pulse in the time-domain is truncated. 0 0.2 0.4 0.6 0.8 1 0 0.5 0.675 1 RC BTRC ASECH POLY DJ P (f)/T fT 2 Fig. 1. Spectra of conventional square-root Nyquist pulses truncated in [−5.5T, 5.5T ] for α =0.35. follows. In Section II, a system model is described and truncation of conventional square-root Nyquist pulses is re- viewed. In Section III, a new design method of time-limited pulses is proposed. In Section IV, performance comparison results are presented and some discussions are made. We draw conclusions in Section V. II. SYSTEM MODEL AND TRUNCATION OF CONVENTIONAL SQUARE-ROOT NYQUIST PULSES We assume that the symbol interval is T and the symmetric transmit pulse p(t) has the time-duration of [−LT/2,LT/2] and a unit energy, that is, LT/2 −LT/2 p 2 (t)dt =1. The receive filter has an impulse response p ∗ (−t)= p(t). Let q(t) denote the matched-filtered transmit pulse q(t)= ∞ −∞ p(s)p(t−s)ds. We first consider that time-limited transmit pulses are made by truncating conventional square-root Nyquist pulses p N (t) and normalizing their energy to one: p(t)= 1 √ K p N (t) p N (t), |t|≤ LT 2 0, elsewhere, (1a) K pN(t) = LT 2 − LT 2 p 2 N (t)dt. (1b) For the excess bandwidth α =0.35 and L = 11, 3 the squared Fourier transforms of p(t), P 2 (f ), are shown in Fig. 1 when P 2 N (f ) are the RC shaping [1, p. 560], the better than raised- cosine (BTRC) shaping [2], [6], the flipped-inverse hyperbolic secant (ASECH) shaping [3], the polynomial (POLY) shaping of the asymptotic decay rate t −2 [4, Table I], and the double- jump (DJ) shaping [5]. A dotted straight line is used to delimit the in-band frequency range [0, 1+ α 2T ]. The out-of-band energy defined as E o 1 − 1+α 2T − 1+α 2T P 2 (f )df is tabulated in Table I. 3 The value L = 11 is considered due to its use in [3]. 1089-7798/09$25.00 c 2009 IEEE
-
Upload
seung-joon-lee -
Category
Documents
-
view
223 -
download
3
Transcript of Design of time-limited pulses
IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 8, AUGUST 2009 573
Design of Time-Limited PulsesSeung Joon Lee, Senior Member, IEEE
Abstract—An approach for designing time-limited transmitpulses is proposed. In the approach, a time-limited pulse isrepresented by a weighted sum of cosine functions whose fre-quencies are integer multiples of one over the time durationof the transmit pulse. The weighting coefficients are derived tominimize the mean square error of received symbols for a givensymbol timing offset, while restricting the out-of-band energy toa predefined level. The minimization is numerically performedby using the Zoutendijk’s feasible direction method. Performancecomparison results are presented showing that the pulses derivedby this approach not only have smaller mean square errors,but also smaller symbol error probabilities for BPSK signalingcompared to the time-truncated versions of conventional square-root Nyquist pulses.
Index Terms—Feasible direction method, mean square error,out-of-band energy, pulse shaping, time-limited pulse.
I. INTRODUCTION
NYQUIST pulses are known to have no inter-symbolinterference (ISI) at a perfect symbol timing epoch. The
most popular example of a Nyquist pulse is the raise-cosine(RC) pulse [1, p. 560]. Recently, several other Nyquist pulses[2]–[4] have been proposed which perform better than the RCpulse in terms of symbol error probability when there is asymbol timing offset.
A long time ago, a Nyquist pulse called the double-jump(DJ) pulse was analytically derived to have the minimum meansquare error (MSE) of received symbols in a small timingoffset [5]. It was also shown in [6] that the DJ pulse hassmaller symbol error probability for BPSK signaling than anyother pulse considered therein in timing offsets less than acertain value (e.g., 5% of the symbol duration).
As indicated in [3], most practical systems employ a time-limited transmit pulse and a receive filter of an impulseresponse function equal to the transmit pulse. Therefore,performance comparison was also examined for time-truncatedsquare-root Nyquist pulses1 in [3], where it was observed thatthe truncated2 versions of the recently proposed square-rootNyquist pulses are worse than the truncated square-root raised-cosine pulse in zero timing offset, while the former are stillbetter than the latter in timing offsets larger than a certainvalue.
In this letter, we investigate how to obtain improved time-limited pulses. The remainder of this letter is organized as
Manuscript received May 20, 2009. The associate editor coordinating thereview of this letter and approving it for publication was P. Cotae.
This study was supported by 2008 Research Grant from Kangwon NationalUniversity.
S. J. Lee is with the Department of Electrical & Electronic Engineer-ing, Kangwon National University, Chuncheon 200-701, Korea (e-mail:[email protected]).
Digital Object Identifier 10.1109/LCOMM.2009.0910951‘A square-root Nyquist pulse in the time-domain’ means a time-function
of which the squared Fourier transform meets the Nyquist criterion [1, eq.(9.2-13)] for zero intersymbol interference.
2The word ‘truncated’ implies throughout this letter that a pulse in thetime-domain is truncated.
0
0.2
0.4
0.6
0.8
1
0 0.5 0.675 1
RCBTRC
ASECHPOLY
DJ
P (f)/T
fT
2
Fig. 1. Spectra of conventional square-root Nyquist pulses truncated in[−5.5T, 5.5T ] for α = 0.35.
follows. In Section II, a system model is described andtruncation of conventional square-root Nyquist pulses is re-viewed. In Section III, a new design method of time-limitedpulses is proposed. In Section IV, performance comparisonresults are presented and some discussions are made. We drawconclusions in Section V.
II. SYSTEM MODEL AND TRUNCATION OF
CONVENTIONAL SQUARE-ROOT NYQUIST PULSES
We assume that the symbol interval is T and the symmetrictransmit pulse p(t) has the time-duration of [−LT/2, LT/2]and a unit energy, that is,
∫ LT/2
−LT/2p2(t)dt = 1. The receive
filter has an impulse response p∗(−t) = p(t). Let q(t) denotethe matched-filtered transmit pulse q(t) =
∫ ∞−∞ p(s)p(t−s)ds.
We first consider that time-limited transmit pulses are madeby truncating conventional square-root Nyquist pulses pN(t)and normalizing their energy to one:
p(t) =
{1√
KpN (t)pN(t), |t| ≤ LT
2
0, elsewhere,(1a)
KpN (t) =∫ LT
2
−LT2
p2N (t)dt. (1b)
For the excess bandwidth α = 0.35 and L = 11,3 the squaredFourier transforms of p(t), P 2(f), are shown in Fig. 1 whenP 2
N (f) are the RC shaping [1, p. 560], the better than raised-cosine (BTRC) shaping [2], [6], the flipped-inverse hyperbolicsecant (ASECH) shaping [3], the polynomial (POLY) shapingof the asymptotic decay rate t−2 [4, Table I], and the double-jump (DJ) shaping [5]. A dotted straight line is used to delimitthe in-band frequency range [0, 1+α
2T ]. The out-of-band energy
defined as Eo � 1− ∫ 1+α2T
− 1+α2T
P 2(f)df is tabulated in Table I.
3The value L = 11 is considered due to its use in [3].
1089-7798/09$25.00 c© 2009 IEEE
574 IEEE COMMUNICATIONS LETTERS, VOL. 13, NO. 8, AUGUST 2009
TABLE IOUT-OF-BAND ENERGY OF SQUARE-ROOT NYQUIST PULSES
TRUNCATED IN [−5.5T, 5.5T ] FOR α = 0.35 AND 0.25
α Eo,RC Eo,BTRC Eo,ASECH Eo,POLY Eo,DJ0.35 4.6266e-5 5.1107e-4 1.0687e-3 1.7782e-3 3.6713e-30.25 1.1798e-4 8.4187e-4 1.1540e-3 2.1878e-3 3.2395e-3
III. NEW TIME-LIMITED PULSES
We derive new time-limited pulses by reducing the meansquare error of the received symbols, while restricting theout-of-band energy to a predefined level. Neglecting additivenoise, let the received data modulated signal be described as
r(t) =∞∑
m=−∞amq(t−mT ) (2)
where am is a complex data symbol with E[|am|2] = 1. Giventhe symbol timing offset ε, the MSE of received symbols is
MSE( ε
T
)�E
[|r(ε)−a0|2
]=
� εT + L
2 �∑m=� ε
T −L2 �
q2(ε−mT )+1−2q(ε)
(3)where �x� and �x� denote the smallest integer no smaller thanx and the greatest integer no greater than x, respectively.
It looks difficult to analytically derive p(t) by minimizingthe MSE. Alternatively, we consider the following numericalapproach. Let the time-limited transmit pulse be representedby a weighted sum of cosine functions of frequencies 0,1/(LT ), · · · , and K/(LT ), resulting in
p(t) =
{1√
Kx(t)x(t), |t| ≤ LT
2
0, elsewhere(4a)
x(t) =K∑
k=0
wk cos(
2πkt
LT
)(4b)
Kx(t) =∫ LT
2
−LT2
x2(t)dt. (4c)
The Fourier transform of x(t) is obtained as
X(f) =K∑
k=0
wk(−1)kfL2T 2 sin(πfLT )
π (f2L2T 2 − k2). (5)
Note that MSE(ε/T ) and Eo depend on w0, w1, · · · , wK .Let the constant Eo,Th denote the allowable maximum out-of-band energy. The weighting coefficients {wk} are numer-ically derived by the following iterative procedure using theZoutendijk’s feasible direction method [7, Section 7.5.1] forsolving a constrained (inequality constraint Eo ≤ Eo,Th)optimization problem:
1) Initialize wk = w(0)k for k = 0, 1, · · · , K and set i = 0.
Also, initialize the step size μ = μ0 > 0.2) Select a feasible direction D = [D0, D1, · · · , DK ]T as
follows:a) If Eo�Eo,Th (that is,−δ<Eo−Eo,Th≤0 for a very
small δ), then D = argminξ α subject to
ξT ·∇MSE(ε/T )/‖∇MSE(ε/T )‖2 + α < 0 (6a)
ξT ·∇Eo/‖∇Eo‖2 + α < 0 (6b)
TABLE IINUMERICALLY OPTIMIZED NEW TIME-LIMITED TRANSMIT PULSE
REPRESENTED BY∑8
k=0 wk cos (2πkt/(11T )) FOR α = 0.35
ε/T = 0 ε/T =0.05 ε/T =0.1 ε/T =0.2
(NewEo,RC0 ) (New
Eo,ASECH0.05 ) (New
Eo,POLY0.1 ) (New
Eo,DJ0.2 )
w0 9.1420e-2 9.0569e-2 9.0105e-2 8.8653e-2w1 1.8284e-1 1.8135e-1 1.8100e-1 1.7725e-1w2 1.8285e-1 1.8077e-1 1.7940e-1 1.7484e-1w3 1.8283e-1 1.8162e-1 1.8014e-1 1.7220e-1w4 1.8130e-1 1.6463e-1 1.6130e-1 1.5877e-1w5 1.5804e-1 1.2782e-1 1.3040e-1 1.4150e-1w6 9.1951e-2 1.3088e-1 1.3225e-1 1.3699e-1w7 2.3680e-2 7.4052e-2 8.3454e-2 1.0080e-1w8 −1.7124e-3 7.6662e-4 1.0117e-3 3.5996e-3
0
0.2
0.4
0.6
0.8
1
0.5 1
New
P (f)/T2
0
New0.05
New0.1
New0.2
0 0.675 fT
Eo,RC
o,ASECH
o,POLY
o,DJ
E
E
E
Fig. 2. Spectra of the new time-limited transmit pulses for α = 0.35.
where ∇MSE(ε/T) =[
∂MSE(ε/T)∂w0
, · · · , ∂MSE(ε/T)∂wK
]Tand ∇Eo = [∂Eo/∂w0, · · · , ∂Eo/∂wK ]T .
b) Otherwise,
Dk =
{∂Eo/∂wk, Eo > Eo,Th,
∂MSE(ε/T )/∂wk, Eo < Eo,Th.(7)
When Eo � Eo,Th, we also reinitialize μ = μ0.3) Update wk as wk = w
(i+1)k by
w(i+1)k =w
(i)k −μDk. (8)
Compute ΔMSE=MSE(i+1)(ε/T )−MSE(i)(ε/T ) andE
(i+1)o where MSE(i)(ε/T ) and E
(i)o are the values of
MSE(ε/T ) and Eo, respectively, when wk = w(i)k .
a) If ΔMSE< 0 and E(i+1)o ≤ 0, increase i by 1,
update μ← 1.1μ, and go to Step (2).b) Otherwise, discard w
(i+1)k and update μ← μ/2.
i) If μ is too small and there has been no valid4
update for {wk} since the last feasible directionD was found by using (6), then stop.
ii) Otherwise, repeat Step (3).
Note that
∂MSE(
εT
)∂wk
=2� ε
T + L2 �∑
m=�εT−L
2�q(ε−mT)
∂q(ε−mT )∂wk
+1−2∂q(ε)∂wk
(9a)
4The word ‘valid’ means ‘not discarded’.
LEE: DESIGN OF TIME-LIMITED PULSES 575
TABLE IIIMSES OF RECEIVED SYMBOLS WITHOUT NOISE FOR α = 0.35
ε/T=0 ε/T=0.05 ε/T=0.1 ε/T=0.2RC 2.1469e-5 4.5329e-3 1.8335e-2 7.7265e-2
BTRC 8.5371e-5 3.3559e-3 1.3584e-2 6.0313e-2ASECH 2.4495e-4 3.2614e-3 1.2758e-2 5.7005e-2POLY 5.6293e-4 3.2488e-3 1.1799e-2 5.2871e-2
DJ 1.3628e-3 3.6265e-3 1.0976e-2 4.8170e-2
NewEo,RC0 1.2831e-9 4.9812e-3 2.0140e-2 8.3752e-2
NewEo,ASECH0.05 9.8764e-5 2.8959e-3 1.1773e-2 5.4076e-2
NewEo,POLY0.1 7.9360e-4 3.1412e-3 1.0733e-2 4.8789e-2
NewEo,DJ0.2 1.1423e-2 1.2306e-2 1.5706e-2 3.9874e-2
∂q(t)∂wk
= 2∫ LT
2
t−LT2
∂p(u)∂wk
p(t− u)du (9b)
∂p(t)∂wk
=cos 2πkt
LT√Kx(t)
−x(t)√
Kx(t)
∫ LT2
−LT2
x(u)cos2πku
LTdu (9c)
∂Eo
∂wk= −4(−1)kL2T 2
πKx(t)
∫ 1+α2T
− 1+α2T
X3(f)f sin(πfLT )f2L2T 2 − k2
df. (10)
The convergence point of w(i)k may depend on the initial-
ization. One useful initialization method is to set the initialcoefficients w
(0)k such that a weighted sum of cosine functions
is a truncated Fourier series of the truncated square-root RCpulse. The step size μ is dynamically updated for stability(μ← μ/2) and convergence speed (μ← 1.1μ).
When α = 0.35, the new transmit pulses with K = 8 andL = 11 numerically optimized for ε/T = 0, 0.05, 0.1, and0.2 are presented in Table II5, where the out-of-band energyconstraints are considered to be Eo,Th = Eo,RC, Eo,ASECH,Eo,POLY, and Eo,DJ, respectively6. Their power spectra areshown in Fig 2.
IV. PERFORMANCE & DISCUSSION
When α = 0.35 and L = 11, the MSEs of received symbolswithout noise are compared in Table III for the truncatedsquare-root Nyquist pulses and the new pulses obtained bythe proposed method. The symbol error probabilities (SEPs)for BPSK signaling are also evaluated by using [8] to comparethose pulses in Table IV7 when SNR =15 dB. The pulsewith the smallest MSE and SEP for timing offset ε/T isobserved to be the new pulse optimized for the same timingoffset. Table V shows that, also for α = 0.25, the proposedmethod can be used to derive new time-limited transmitpulses8 outperforming truncated conventional ones.
Usually communication systems performance is dominatedby a worst case. If a specific value of ε/T is provided as arepresentative worst case, then we can derive the time-limited
5As the number of summands K increases, the performance improves ingeneral. However, even if K is larger than 8 for the transmit pulse of duration[−5.5T, 5.5T ], the improvement has been experimentally found not to besignificant.
6Such choice is motivated by the fact that RC, ASECH, POLY, and DJhave the smallest symbol error probability among the considered conventionaltruncated pulses with α = 0.35 for ε/T = 0, 0.05, 0.1, and 0.2, respectively.
7The SEPs of truncated conventional pulses given here are slightly differentfrom those shown in [3, Table II]. The values of Table IV have been obtainedwith very high precision and double-checked.
8Their pulse shapes are not presented here due to space limitation.
TABLE IVSEPS OF BPSK SYMBOLS AT SNR= 15 DB FOR α = 0.35
ε/T=0 ε/T=0.05 ε/T=0.1 ε/T=0.2RC 9.4649e-9 6.0723e-8 1.4081e-6 3.9553e-4
BTRC 9.7795e-9 4.1499e-8 5.8010e-7 1.0987e-4ASECH 1.0600e-8 4.0767e-8 4.9969e-7 8.2339e-5POLY 1.2398e-8 4.1961e-8 4.4397e-7 6.3506e-5
DJ 1.8025e-8 5.1400e-8 4.6900e-7 6.1700e-5
NewEo,RC0 9.3610e-9 7.0508e-8 2.0082e-6 6.3979e-4
NewEo,ASECH0.05 9.8471e-9 3.6670e-8 4.8376e-7 9.5148e-5
NewEo,POLY0.1 1.3867e-8 4.0665e-8 4.3208e-7 7.1265e-5
NewEo,DJ0.2 4.3866e-7 5.1049e-7 1.2023e-6 5.1288e-5
TABLE VSEPS OF BPSK SYMBOLS AT SNR= 15 DB FOR α = 0.25
ε/T=0 ε/T=0.05 ε/T=0.1 ε/T=0.2RC 9.7487e-9 8.5457e-8 2.9232e-6 9.9941e-4
BTRC 1.1976e-8 7.2063e-8 1.5338e-6 3.8806e-4ASECH 1.1402e-8 6.5179e-8 1.3051e-6 3.1988e-4POLY 1.3840e-8 6.5912e-8 1.0979e-6 2.3587e-4
DJ 1.4983e-8 6.5667e-8 1.0311e-6 2.1786e-4
NewEo,RC0 9.3633e-9 8.6280e-8 3.1496e-6 1.1135e-3
NewEo,ASECH0.05 1.0281e-8 6.2763e-8 1.3735e-6 3.6615e-4
NewEo,DJ0.1 1.3741e-8 5.7636e-8 8.6802e-7 1.7698e-4
NewEo,DJ0.2 3.9959e-7 6.7817e-7 3.4375e-6 1.9920e-4
pulse optimized for the value by using the proposed method.In another scenario, we may assume that an approximatedistribution of timing offsets is given by Pr[ε/T = el] = pl
(l = 0, · · · , L−1). Then, we can extend the proposed methodstraightforwardly by considering the MSEs averaged overtiming offsets, that is, MSE =
∑L−1l=0 MSE(el)pl, instead of
MSE(ε0/T ) for a specific ε0.
V. CONCLUSION
Novel time-limited transmit pulses represented by aweighted sum of cosine functions were proposed, where theweighting coefficients were numerically optimized to mini-mize the mean square error of received symbols while main-taining the restricted out-of-band energy. The proposed pulseswere confirmed to outperform the time-truncated versions ofconventional square-root Nyquist pulses in terms of symbolerror probability as well as the mean square error.
REFERENCES
[1] J. G. Proakis, Digital Communications, 4th ed., ser. Electrical Engineer-ing. McGraw-Hill, 2000.
[2] N. C. Beaulieu, C. C. Tan, and M. O. Damen, “A “better than” Nyquistpulse,” IEEE Commun. Lett., vol. 5, no. 9, pp. 367–368, Sept. 2001.
[3] A. Assalini and A. M. Tonello, “Improved Nyquist pulses,” IEEE Com-mun. Lett., vol. 8, no. 2, pp. 87–89, Feb. 2004.
[4] S. Chandan, P. Sandeep, and A. K. Chaturvedi, “A family of ISI-freepolynomial pulses,” IEEE Commun. Lett., vol. 9, no. 6, pp. 496–498,June 2005.
[5] L. E. Franks, “Further results on Nyquist’s problem in pulse transmis-sion,” IEEE Trans. Commun. Technol., vol. COM-16, no. 2, pp. 337–340,Apr. 1968.
[6] N. C. Beaulieu and M. O. Damen, “Parametric construction of Nyquist-Ipulses,” IEEE Trans. Commun., vol. 52, no. 12, pp. 2134–2142, Dec.2004.
[7] S. S. Rao, Optimization: Theory and Applications, 2nd ed. John Willey& Sons, 1983.
[8] N. C. Beaulieu, “The evaluation of error probabilities for intersymboland cochannel interference,” IEEE Trans. Commun., vol. 39, no. 12, pp.1740–1749, Dec. 1991.
