10_04_678_688

678 MD. MASUD RANA, J. KIM, W. K. CHO, LMS BASED ADAPTIVE CHANNEL ESTIMATION FOR LTE UPLINK

LMS Based Adaptive Channel Estimation for LTE UplinkMd. Masud RANA, Jinsang KIM, Won-Kyung CHO

Dept. of Electronics and Radio Engg., Kyung Hee Univ., 1 Seocheon, Kihung, Yongin Gyeonggi, 446-701, South Korea

[email protected]

Abstract. In this paper, a variable step size based leastmean squares (LMS) channel estimation (CE) algorithm ispresented for a single carrier frequency division multiple ac-cess (SC-FDMA) system under the umbrella of the long termevolution (LTE). This unbiased CE method can automati-cally adapt the weighting coefficients on the channel con-dition. Therefore, it does not require knowledge of channel,and noise statistics. Furthermore, it uses a phase weightingscheme to eliminate the signal fluctuations due to noise anddecision errors. Such approaches can guarantee the conver-gence towards the true channel coefficient. The mean andmean square behaviors of the proposed CE algorithm arealso analyzed. With the help of theoretical analysis and sim-ulation results, we prove that the proposed algorithm outper-forms the existing algorithms in terms of mean square error(MSE) and bit error rate (BER) by more than around 2.5dB.

KeywordsSC-FDMA, PAPR, LMS, NLMS, VSS-LMS, RLS.

1. IntroductionThe further increasing demand on high data rates in

wireless communication systems has arisen in order to sup-port broadband services. The 3rd generation partnershipproject (3GPP) members started a feasibility study on the en-hancement of the universal terrestrial radio access (UTRA)in December 2004, to improve the mobile phone standardto cope with future requirements. This project was calledlong term evolution (LTE) [1], [2]. First 3GPP LTE specifi-cation is being finalized within 3GPP release-9. 3GPP LTEuses single carrier frequency division multiple access (SC-FDMA) for uplink transmission and orthogonal frequencydivision multiplexing access (OFDMA) for downlink trans-mission [3]. SC-FDMA is a promising technique for highdata rate transmission that utilizes single carrier modulationand frequency domain equalization. Single carrier transmit-ter structure leads to keep the peak to average power ratio(PAPR) as low as possible that will reduce the energy con-sumption. SC-FDMA combines the desirable characteris-tics of OFDMA, such as simple implementation, robustness

against frequency-selective channels, and relative easinessto employ multiple antenna transmission techniques as wellas similar throughput performance [4], [5]. The transmitterof a SC-FDMA system first groups the modulation signalsinto blocks, then performs a DFT operation to produce a fre-quency domain representation of the input signals. It thenmaps each of the DFT outputs to one of the orthogonal sub-carriers that can be transmitted. Finally, IDFT transformsthe subcarrier amplitudes to a complex time domain signal[6]. Therefore, SC-FDMA is an important technique forbroadband multimedia access, anywhere, and anytime wire-less communication systems.

Since the radio channel is highly dynamic, the transmit-ted signal travels to the receiver by undergoing many detri-mental effects that corrupt the signal and often place limita-tions on the performance of the system. Channel estimation(CE) techniques allow the receiver to approximate the im-pulse response of the channel and explain the behavior ofthe channel. It can be employed for the purpose of detectingreceived signal, improve signal to noise ratio (SNR), chan-nel equalization, reduced intersymbol interference (ISI), mo-bile localization, and improved system performance [7], [8].In general, CE techniques can be divided into two majorcategories such as the trained and blind. The former CEalgorithm requires probe sequences that occupies valuablebandwidth whereas the latter uses the received data only.Due of course to their self-sufficiency in training, blind CEtechniques are considered more attractive than trained basedtechniques [9], [10]. The training aided CE literature is rich,but, as our most important interest lies with blind CE meth-ods, we will not track its presentation any more.

Several CE techniques have been proposed to mitigateinter-channel interference (ICI) in the uplink direction of3GPP LTE system [11]. Most of the CE algorithms are de-rived from the minimization of the mean square error (MSE)between the output of the adaptive filter and noisy receivedsignal. In [12], Wiener filtering based iterative CE has beeninvestigated. But this scheme requires accurate knowledgeof second order channel statistics, which is not always feasi-ble at a destination. This scheme also requires high compu-tational complexity and knowledge of channel correlations.Probably the simplest and most popular CE algorithm be-longing of adaptive group is the standard least mean squares(LMS) algorithm which has the lead of low computationalcomplexity, memory load, and simplicity of practical imple-

RADIOENGINEERING, VOL. 19, NO. 4, DECEMBER 2010 679

mentation [13]. Moreover, its performance and fast conver-gence speed are inversely related through a single parameter,step size. For large values of step size, the convergence ofthe LMS filter coefficients is very fast, but the steady stateMSE is large and vice versa. In order to ensure the algo-rithm to be convergent, the range of step size is specifiedbut the choice of optimal step size has not been properly ad-dressed. Therefore, the existing LMS CE algorithm is notpossible to obtain fast convergence and small steady stateMSE at the same time. One of the important concerns in allpractical realistic situations is to develop algorithms whichgive fast convergence of the filter coefficients and good MSEperformance [14]. In order to increase MSE performanceas well as fast convergence, normalized LMS (NLMS) CEalgorithm is proposed which takes into account the varia-tion in the signal level at the filter input and selects a nor-malized step size parameter. A drawback of NLMS algo-rithms is a higher computational complexity and misadjust-ment i.e., the mismatch between the true and estimated co-efficients [15]. When a constant scalar step size is used inthe LMS/NLMS algorithm, there is a trade off between thesteady state error and the convergence speed, which preventsa fast convergence when the step size is chosen to be smallfor small output estimation error. In order to deal with thisproblem, one important idea is to use varying step size dur-ing adaptation.

Variable step size (VSS) methods are commonly soughtafter to provide steady state MSE performance. This methoduses larger step size at the start of the iteration to speed upthe convergence rate of the algorithm, and smaller step sizewhen the algorithm is convergent [16]. Several VSS-LMStype CE techniques have been proposed in the literature [17],[18], [19]. But these algorithms are not adaptive to trackthe optimum step size parameter in a nonstationary environ-ment. The existing VSS-LMS CE algorithms [20] cannotprovide the minimum MSE in the tracking problem, sincethey cannot acquire and track the optimum step size. Theymay even cause worse steady state results, when the algo-rithm parameters are not appropriately adjusted. In [21], anadaptive time-varying step size LMS is proposed where thestep size is adjusted using the energy of the instantaneouserror. But due to the presence of the estimation error andmeasurement noise, the step size update is not an accuratereflection of the state of adaptation before or after conver-gence. This degrades the significant performance of theseadaptive approaches. Furthermore, in [19], [22] proposeda time-varying step size LMS method that gives improvedperformance compared with standard LMS and NLMS algo-rithm. But when the channel is fast time-varying then thisalgorithm cannot accurately measure the autocorrelation be-tween estimation error to control step size update. There-fore, the performance is reduced significantly. To combat thechannel dynamics, the recursive least squares (RLS) basedCE algorithm is frequently used for rapid convergence andimproved MSE performance [9]. But it requires optimumforgetting factor such that the estimator error is minimized.

Although a lot of modified CE algorithms have been stud-ied on employing adaptive forgetting factor and parallel for-getting factor, the CE performance is severely degraded inhighly dynamic fading channel even when the forgetting fac-tor is well optimized [23]. However, this scheme also re-quires high computational complexity that is the major ob-stacle for practical base station (BS) as well as tiny mobileterminal implementation. Therefore, an efficient CE algo-rithm superior to the existing methods is necessary whichprovides both rapid convergence to the true channel coeffi-cient and smallest steady state MSE.

In this paper, we propose an adaptive algorithm in theuplink direction of LTE systems which estimates the channelimpulse response (CIR) from received block of data withoutany prior knowledge of the channel. To the best of the au-thors knowledge, this is the first time such scheme is pro-posed for LTE uplink systems. Specifically, we are goingto develop recursive LMS type variable step size algorithmthat can automatically adapt the optimum weighting coeffi-cients on the channel condition. This time-varying step sizemethod is re-selected at each iteration to minimize the sumof the squares of the prior estimation errors up to that cur-rent time point. Next, the mean and mean square behav-iors as well as complexity of the proposed CE algorithm arepresented. The following advantages are gained by usingthis proposed scheme. Firstly, this CE algorithm uses phaseweighting scheme such that the algorithm is less vulnerableto signal variations owing to noise as well as estimation er-rors. With such approaches, convergence towards the truechannel vector is guaranteed. Secondly, the proposed CEuses time-varying step size parameter such that larger stepsize at the beginning of the iteration to accelerate the con-vergence rate of the algorithm, and uses smaller step sizewhen the algorithm is convergent. Hence, this CE algorithmdoes not have the convergence speed toward the true channelcoefficients-MSE trade off problem. Thirdly, the proposedestimate h(m) is an unbiased estimate of the tap weight vec-tor w(m). An unbiased estimate indicates that its mean valueis identical to the true parameter value. Consequently, asthe number of observation increases, the estimate is assuredto converge to the true parameter. Ideally, we would like ourestimator to be unbiased and to have the smallest possible er-ror variance. Fourthly, the proposed method does not requiremeasurements of the relevant channel correlation functions,nor does it require matrix inversion. Finally, the proposedscheme outperforms conventional methods with respect tothe MSE and bit error rate (BER) of the estimated channelby more than 2.5 dB.

The following notations are adopted throughout the pa-per: bold face lower and upper case letters are used to rep-resent vectors and matrices respectively. Superscripts xT de-note the transpose of the x , tr[X] denote the trace of the ma-trix X, I is the identity matrix, and the symbol E(.) denoteexpectation.

The rest of the paper is organized as follows. In Sec. 2,


we describe the system model of the LTE uplink channel.In Sec. 3, the iterative CE method is reviewed for the con-venience of the introduction of the proposed method. Theproposed CE scheme is presented in Sec. 4. In Sec. 5, wecompare the proposed method with the existing CE schemesvia simulations. Finally, conclusions are made in Sec. 6.

2. LTE Uplink System DescriptionIn this Section, we briefly explain LTE SC-FDMA sys-

tem model, subcarrier mapping, fading channel statistics,and received signal model which is really imperative for de-signing efficient channel estimator.

2.1 Baseband System ModelAn equivalent baseband block diagram for the commu-

nication system under investigation is shown in Fig. 1.

CE and

equalization

Bit stream

Modulation

Add cyclic

prefix (CP) Synchronization

and remove CP

M-point

DFT

N-point IDFT

X +

Multipath

fading

channel

AWGN

noise

N-point DFT

Mapping

Demodulation

Bit stream

M-point IDFT

m1(q)

s1(n)

s(m)

Mapping

restore

Fig. 1. Simplified block diagram of a LTE SC-FDMA wirelesssystem.

At the transmitter side, a baseband modulator trans-mits the binary input to a multilevel sequences of complexnumbers m1(q) in one of several possible modulation for-mats including binary phase shift keying (BPSK), quater-nary PSK (QPSK), 8 level PSK (8-PSK), 16-QAM, and64-QAM. These modulated symbols perform an N-pointdiscrete Fourier transform (DFT) to produce a frequencydomain representation [1]. The DFT followed by IDFTin a distribution-FDMA (DFDMA) or localization-FDMA(LFDMA) subcarrier mapping setup operates as efficient im-plementation to an interpolation filter. In distributed subcar-rier mode, the outputs are allocated equally spaced subcarri-ers, with zeros occupying the unused subcarrier in between.While in localized subcarrier mode, the outputs are confinedto a continuous spectrum of subcarriers. Except the above

two modes, interleaved subcarrier mapping mode of FDMA(IFDMA) is another special subcarrier mapping mode [24],[25]. The difference between DFDMA and IFDMA is thatthe outputs of IFDMA are allocated over the entire band-width, whereas the DFDMAs outputs are allocated everyseveral subcarriers. An example of SC-FDMA transmit sym-bols in the frequency domain for 2 users, 3 subcarriers peruser and 6 subcarriers in total is illustrated in Fig. 2.

(x)

(y)

(z)

Time domain Frequency domain

User 3

User 2

User 1

Complex weight

Zeros

s1(3)s1(2)

s1(1)

Fig. 2. Multiple access scheme of SC-FDMA: (x) IFDMAmode, (y) DFDMA mode and (z) LFDMA.

Finally, the IDFT module output is followed by a cyclicprefix (CP) insertion that completes the digital stage of thesignal flow. A cyclic extension is used to eliminate ISI andpreserve the orthogonality of the tones. Assume that thechannel length of CP is larger than the channel delay spread.

2.2 Channel ModelChannel model is a mathematical representation of the

transfer characteristics of the physical medium. These mod-els are formulated by observing the characteristics of the re-ceived signal. According to the documents from 3GPP, inthe mobile environment, a radio wave propagation can be de-scribed by multipaths which arise from reflection and scat-tering. If there are l distinct paths from transmitter to thereceiver, the impulse response of the wide-sense stationaryuncorrelated scattering (WSSUS) fading channel can be rep-resented as [26]:

w(, t) =l1j=0

w j(t)( j) (1)

where fading channel coefficients w j(t) are the wide sensestationary i.e. w j(t) = w(m, j), uncorrelated complex Gaus-sian random paths gains at time instant t with their respec-tive delays j, where w(m, j) is the sample spaced channelresponse of the lth path during the time m, and (.) is theDirac delta function. Based on the WSSUS assumption, thefading channel coefficients in different delay taps are statis-tically independent. In time domain, fading coefficients arecorrelated and have Doppler power spectrum density mod-


eled in Jakes and has an autocorrelation function given by[27]:

E[w(m, j)wT (n, j)] = 2w( j)J0[2pi fdTf (mn)] (2)

where w(n, j) is a response of the lth propagation path mea-sured at time n, 2w( j) denotes the power of the channel coef-ficients, fd is the Doppler frequency in Hertz, Tf is the sym-bol duration in seconds, and J0(.) is the zero order Besselfunction of the first kind [26].

2.3 Received Signal ModelThe transmitted symbols propagating through the radio

channel can be modeled as a circular convolution betweenthe channel impulse response (CIR) and the transmitted datablock i.e., [s(m) w(, t)]. Since the channel coefficient isusually unknown to the receiver, it needs to be efficiently es-timated while maintain little computational complexity. Atthe receiver, the opposite set of the operation is performed.After synchronization, cyclic prefix samples are discardedand the remaining N samples are processed by the DFT toretrieve the complex constellation symbols transmitted overthe orthogonal sub-channels. The received signals are de-mapped and equalizer is used to compensate for the radiochannel frequency selectivity. After IDFT operation, thesereceived signals are demodulated and soft or hard values ofthe corresponding bits are passed to the decoder. The de-coder analyzes the structure of received bit pattern and triesto reconstruct the original signal. In order to achieve goodperformance the receiver has to know the impact of the chan-nel. Thus, an accurate and efficient CE algorithm is neces-sary to coherently demodulate the received data [28].

3. A Review of the CE AlgorithmsAdaptive CE is one the most important current research

interests in the wireless communications where the channelis rapidly time-varying. An adaptive algorithm is a processthat changes its parameter as it gains more information ofits possibly changing environment. This method tries to ad-just the filter parameter in such a way that minimizes theMSE between the output of the filter and the desired signal.Therefore, the adaptive filter parameters are entirely known,replicates the system in question whose parameters are un-known. In other words, the parameters of the adaptive filtergive a good approximation of the parameters of the unknownscheme. The performance of this type CE algorithm is de-pendent on the convergence towards the true channel coeffi-cients , computational complexity as well as minimum MSEperformance [29].

3.1 Least Mean Squares (LMS) AlgorithmThe LMS algorithm is based on the stochastic gradient

and is given by [13]

e(m) = ST (m)w(m)+ z(m)ST (m)h(m)h(m+1) = h(m)+S(m)e(m)

where is step size, S(m) is the transmitted diagonal ma-trix at sampling time m, h(m) is the adaptive filter coeffi-cient, and e(m) is the estimation error. The filter coefficientsare updated using an estimate of the cost function gradi-ent, [S(m)e(m)]. In all practical applications, the signalsinvolved might be corrupted by noise. When the noise ispresent in the received sequence, interference will also in thecoefficients adaption process through the term [S(m)e(m)].As a result, where the distribution of the noise is highly im-pulsive, the LMS scheme might have low convergence andlower steady state MSE performance. The step size param-eter, determines the convergence rate of the algorithm andhigher value provides faster convergence. However, if ex-ceeds certain bound then the algorithm will diverge. As thebound on is not known a priori and is dependent on the var-ious statistics. In practice, a somewhat conservative scalarvalue of is used. Also a higher value of results in highervariations in the tap weight vector estimate after the initialconvergence phase. Such variations result in increased dis-tortion in the combiner output which in turn results in anincreased MSE and BER [9], [13].

3.2 Normalized LMS (NLMS) AlgorithmThe main problem of the LMS CE algorithm is that it is

sensitive to the scaling of its input signals. This makes it veryhard to choose that guarantees stability of the algorithm.The NLMS is a variant of the LMS algorithm that solves thisproblem by normalizing with the power of the input signal.The NLMS algorithm can be summarized as [29]:

e(m) = ST (m)w(m)+ z(m)ST (m)h(m)h(m+1) = h(m)+e(m)[ST (m)S(m)]1S(m) (3)

when a constant scalar step size is employed in theLMS/NLMS algorithm, there is a trade off among the steadystate error-convergence towards the true channel coeffi-cients, which avoids a fast convergence when the step sizeis preferred to be small for small output estimation error. Inorder to guarantee the algorithm to be convergent, the rangeof step size is specified but the choice of optimal learningstep size has not been appropriately addressed. In order todeal with these troubles, one key idea is to exploit varyingstep size during adaptation.

3.3 Variable Step Size (VSS)-LMS AlgorithmThe VSS-LMS algorithm involves one additional step

size update equation compared with the standard LMS algo-rithm. The VSS algorithm is [30], [19]

(m+1) = (m)+ p2(m)

p(m) = p(m)+(1)eT (m)e(m1) (4)


where 0 < < 1, 0 < < 1, and > 0. When the channelis fast time-varying then algorithm cannot accurately mea-sure the autocorrelation between estimation error to controlstep size update. So, this CE algorithm cannot provide theminimum MSE in the tracking problem, since it cannot ac-quire and track the optimum step size. It may even causeworse steady state results, when the algorithm parametersare not appropriately adjusted. In addition, control parame-ters and need to be adjusted for a better performance. Ascan be seen here, a general characteristic of these VSS CEmethods is that predetermined control parameters are neces-sary to improve the performance. Though, in most of them,rules to choose control parameters are not specified. Thoseparameters are always selected from extensive simulations,or from experience. It is clear that the choice of parame-ters would significantly influence the performance of theseschemes.

3.4 Recursive Least Squares (RLS) AlgorithmTo combat the channel dynamics, the RLS based CE

algorithm is frequently used for rapid convergence and im-proved MSE performance [9]. The standard RLS algorithmis

e(m) = ST (m)w(m)+ z(m)ST (m)h(m)R(m) = B(m1)S(m)[+ST (m)B(m1)S(m)]1B(m) = 1B(m1)1R(m)ST (m)R(m1)

h(m+1) = h(m)+S(m)e(m)R(m) (5)

where is the exponential forgetting factor with 0 < < 1.The smaller value of leads to faster convergence rate aswell as larger fluctuations in the weight signal after the ini-tial convergence. On the other hand, too small value makesthis algorithm unstable. Subsequently, it requires best pos-sible forgetting factor such that the estimator error is de-creased. Although a lot of modified CE algorithm has beenstudied on employing adaptive forgetting factor and parallelforgetting factor, the CE performance is severely degradedin highly dynamic fading channel even when the forgettingfactor is well optimized [23]. However, this scheme alsohas computational complexity-performance trade off prob-lem that is the major obstacle for practical mobile terminalas well as base station (BS) implementation [31], [32]. Con-sequently, an efficient CE algorithm better than existing al-gorithms is required which gives both fast convergence andminimum steady state MSE.

4. Proposed Adaptive CE AlgorithmThe simplified block diagram of a proposed CE algo-

rithm in LTE SC-FDMA system is illustrated in Fig. 3 andits linear transversal filter in Fig. 4.

The signal S(m) is transmitted via a time-varying chan-nel w(m), and corrupted by an observation noise z(m) before

being detected in a receiver. The reference signal h(m) is es-timated by using any kind of CE algorithm such as LMS,NLMS, RLS. The noisy observation at time index m is

r(m) = s1(m1)w1(m)+ .....+ sl(m l)wl(m)+ z(m)

=l

j=1

s j(m j)w j(m)+ z(m)

= ST (m)w(m)+ z(m), (6)

where s j(m j), j = 1,2, ....., l are transmitted signal vectorsat time m, S(m) = diag[s1(m1),s2(m2), ......,sl(m l)],l is the distinct paths from transmitter to the receiver, w(m)is the channel coefficients at time m, and z(m) is the observa-tion noise with zero mean and variance 2. After processingsome intermediate steps (synchronization, remove CP, DFT,and demapping), the decision block reconstructs the detectedsignal to an approximate modulated signal and its phase. Theoutput y(m) of the adaptive filter is expressed as

y(m) = d1(m1)h1(m)+ .....+dl(m l)hl(m)

=l

j=1

d j(m j)h j(m)

= DT (m)h(m) (7)

where d j(m j), j = 1,2, ....., l are detected signal vectors attime m, D(m) = diag[d1(m 1),d2(m 2), ......,dl(m l)].In this problem formulation, the ideal adaptation procedurewould adjust w j(m) such that w j(m) = h j(m) as m . Inpractice, the adaptive filter can only adjust w(m) such thaty(m) closely approximates desired signal over time. There-fore, the instantaneous estimated error signal needed to up-date the weights of the adaptive filter is

e(m) = r(m)y(m)= r(m)DT (m)h(m). (8)

This priori error signal, e(m) is used for tuning the adaptivefilter weights to minimize the estimator error. The error sig-nal is fed into a procedure which alters or adapts the parame-ters of the filter from time m to time (m+1) in a well-definedmanner. As the time index m is incremented, it is hoped thatthe output of the adaptive filter becomes a better and bettermatch to the desired response signal through this adaptationprocess, such that the magnitude of e(m) decreases over timeas shown in Fig. 5.

Now define the proposed cost function j(m) for theadaptive filter structure that minimizes the square distancebetween the received signal and its estimate. This objectfunction also includes a phase discriminate weighting se-quence p(m) such that algorithm is less vulnerable to signalsvariations owing to noise as well as estimation errors. Theweighting sequence p(m) is the distance between the initialmodulated carrier phase and carrier synchronization phase i.e.,


Transmitted

sequence

+

-

r(m)

y(m)+

S(m)

Unknown

channel

model w(m)+

Observation

noise z(m)

e(m)x

Remove CP,

FFT, and

Demap

D(m)

x

Decision

(compute modulated

phase and signal)

Adaptive update

of weight

coefficients h(m)Time varying

step size

Phase discriminate

weight p(m)

p(m) e(m) ( )m( )m

Synchr-

onization

-p(m)

Fig. 3. Modeling an unknown time-varying system.

Delay Delay Delay

+ + +

w1(m) w2(m) w3(m) wl(m)

Input

signal S(m-2) S(m-3) S(m-l)

+Error

e(m)

r(m)

Adaptive weight

control

mechanism h(m)

-

+

y(m)

S(m)

Intermediate

steps

Phase weighted

signal p(m)

D(m)

p(m) e(m)

x

( )m

Fig. 4. Transversal adaptive filter with time-varying tap weights.

Filter

output

Observation

Prediction

Estimation

Error

Detected symbols

Fig. 5. Residual error.

p1(m) =min|(m)(m)|

pi/Mp(m) =p1(m)p1(m1).........p1(m l) (9)

where M is the alphabet size i.e. M = 2 for BPSK, M = 16for 16-QAM etc, (pi/M) is the normalized factor. So, theproposed cost function j(m) is

j(m) = p(m)eT (m)e(m)

= p(m)[rT (m)r(m) rT (m)DT (m)h(m)r(m)hT (m)D(m)+DT (m)D(m)hT (m)h(m)]. (10)

To minimize the cost function in (10), we take the gradient

with respect to filter coefficient which yields

h j(m) = p(m)[2r(m)D(m)+2D(m)DT (m)h(m)]. (11)The steepest descent method is used to adjust adaptive pa-rameters in order to search the quadratic MSE performancefunction for its minimum. According to this method, a se-quence of change is made to the weight vector along the di-rection of the negative gradient. Therefore, the next weightvector, h(m+1), is made equal to the present weight vector,h(m), plus a change proportional to the negative gradient atthe mth iteration:

h(m+1) = h(m)1/2(m)h j(m)= h(m)+ p(m)(m)D(m)[r(m)DT (m)h(m)]= h(m)+ p(m)(m)D(m)e(m) (12)

where h(m+ 1) denotes the weight vector to be computedat iteration (m+ 1), and (m) is the time-varying step sizeparameter which is related to the rate of convergence andcontrol stability. The term [(m)p(m)D(m)e(m)] representsthe correction factor or adjustment that is applied to the cur-rent estimate of the tap weight vector. Its observed that co-efficients of the adaptive filter are update using an estimatecost function gradient, priori error e(m), phase discriminateweighting sequence p(m), and time-varying step size param-eter (m). Using this phase weighting scheme to eliminatethe signal fluctuations more due to noise and decision er-rors.


In order to accelerate the convergence rate to the truechannel coefficient, we now derive the expression of time-varying step size for the proposed LMS algorithm. For doingthis, taking the gradient in (10) with respect to (m) as

j(m) = p(m)[eT (m)(m)

e(m)+e(m)(m)

eT (m)]

=p(m)[DT (m)c(m)e(m)+DT (m)c(m)e(m)]=2p(m)e(m)DT (m)c(m), (13)

for the sake of simplicity, assume that c(m) = h(m)(m) . From(12), differentiating h(m) with respect to (m), we obtain

c(m+1) = c(m)+b(m)(m)(m)

+(m)p(m)D(m)e(m)(m)

= c(m)+b(m)+(m)p(m)D(m)e(m)(m)

= c(m)+b(m)+(m)p(m)D(m)[DT (m)c(m)]= b(m)+ [I(m)p(m)DT (m)D(m)]c(m)= b(m)+c(m), (14)

for the sake of simplicity, assume that b(m) =p(m)D(m)e(m), and = [I (m)p(m)DT (m)D(m)]. Todecrease the computational complexity without sacrific-ing convergence speed and MSE performance, assume isa scalar positive constant close to, but less than unity, andc(m) is initial zeros vector. Similar to (12), the gradienttime-varying adaptive step size update equation can be writ-ten as

(m+1) = (m)1/2 j(m)= (m)+(m)DT (m)e(m)c(m) (15)

where is the learning rate parameter. This time-varyingstep size is re-selected at each iteration to minimize the sumof the squares of the prior estimation errors up to that recenttime point. So, this algorithm is able to sense the conver-gence rate at which the best possible tap weight coefficientsare changing. At the beginning of our CE algorithm an initialCIR and step size is given to start the iteration process. Thealgorithm is kept on iterative until the channel estimator con-verges towards the true channel vectors. The detailed stepsof this proposed algorithm is shown in Fig.6. A significantfeature of this iterative algorithm is its simplicity; it does notrequire measurements of the relevant correlation functions,nor does it require matrix inversion, and avoids convergencespeed-MSE trade off problem. Next, we compute the meanand mean square behaviors as well as valid range of step-sizeto achieve convergence of the proposed algorithm.

Update weight

and step size

No

Initialized tap weight, and step size

Compute phase weighted signal

End adaptive

algorithm

YesIs the

estimation

error zero?

Compute combiner output from received

signal, phase weighted, and tap weight

Compute estimation error between received

signal and combiner output

Save estimated

weight

Start

Set channel and LTE uplink parameters

Read noisy received signal

Synchronization, remove CP, FFT, and

subcarrier demap

Fig. 6. Flow chart of the proposed CE technique for LTE SC-FDMA system.

5. Simulation Results and DiscussionThe error performance of the aforementioned iterative

estimation algorithms is explored by performing extensivecomputer simulations. A Rayleigh fading channel is consid-ered for this purpose. More specifically, we use a Jakes-likemodel, proposed in [27], to simulate fading. All simulationparameters of the LTE SC-FDMA system are summarized inTab. 1.

Recall that the CE weight update recursion of the pro-posed algorithm is given in (12), where e(m) is illustrated inequation (8), and (m) is described in (15). The performanceof the proposed CE algorithm is compared with the fixed stepsize LMS algorithm [14], NLMS algorithm [29], AboulnasrVSS-LMS algorithm [22], and RLS algorithm [9]. In prac-tice, the ideal channel coefficient w is unavailable, so esti-mated reference signal h must be used instead. The moreaccurate h is, the better MSE performance of the CE willachieve. The performance is measured using MSE betweenthe actual and the estimated channel response. MSE versusSNR for different recursive CE techniques in slow (Dopplerfrequency, fd = 100 Hz), medium ( fd = 500 Hz), and fast


System parameters AssumptionsSystem bandwidth 5 MHz

Sampling frequency 7.68 MHzSubcarrier spacing 9.765 kHz (5 MHz/512)

Modulation data type BPSKFFT size 16

Subcarrier mapping schemes IFDMAIFFT size 512

Data block size 32 (512/16)Cyclic prefix 4 or 20 samples

Channel Rayleigh fadingForgetting factor 0.99

LMS gain 0.4Equalization ZF

Doppler frequency 100, 500, and 1000 HzNumber of iteration 300

Tab. 1. The major system parameters used for simulations.

0 5 10 15 20 25 30105

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103

102

101

100Channel estimation error as a function of SNR

SNR [dB]

MSE

Existing LMSExisting NLMSVSSLMS [Aboulnasr]Existing RLSProposed VSSLMS

Fig. 7. Comparison of MSE behaviors of all CE algorithms forfd = 100 Hz.

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102

101

100


SNR [dB]

MSE



( fd = 1000 Hz) time-varying channels are presented in Figs.7, 8, and 9, respectively.

From the Figs. one can see that proposed adaptive CEcan always achieve better performance than other existingLMS, NLMS, Aboulnasr, and RLS method. It is inevitablethat the MSE performance gets inferior as the fading in-creases i.e., fd increases from 100 Hz to 1000 Hz. The pro-posed CE algorithm outperforms the RLS and Aboulnasr es-timator by approximately 2.5 dB and 4.5 dB in the range ofthe tested SNRs. This proposed CE algorithm uses phaseweighting scheme such that algorithm is less susceptible toboth signals fluctuations due to noise and decision errors.Furthermore, the presence of the estimation error and mea-surement noise, the step size update is an accurate reflectionof the state of adaptation before or after convergence. Thisis due to the drastic decrease in noise level which increasesthe performance of this algorithm. Moreover, the proposedCE adapt optimal time varying step size parameter such thatlarger step size at the beginning of the iteration to acceleratethe convergence rate of the algorithm, and uses the smallerstep size when the algorithm is convergent. So, proposedalgorithm has a better anti-nose as well as tracking ability.Therefore, by modified LMS with time-varying step size CEalgorithm, we can obtain relatively exact reference signal.The BER is an another important performance parameter inwireless communication system for quality measurement ofrecovered data. Next, we evaluate the effect of the proposedCE in terms of BER performance. In Figs. 10, 11, and 12,we compare the BER performance of the several channelestimators. BER of the adaptive schemes is computed af-ter the algorithms have converged to their steady state. It isobserved that the proposed CE algorithm is the best choiceamong the others under all scenarios. The error performancedegrades with the increased Doppler frequency, i.e., when fdincreases from 100 Hz to 1000 Hz, as expected. The robust-ness of the estimation algorithms under consideration are

0 5 10 15 20 25 30104

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102

101

100


SNR [dB]

MSE




0 5 10 15 20 25 30105

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103

102

101

100BER as a function of SNR

SNR [dB]

BER


Fig. 10. BER performance of five algorithms as a function ofSNR in Rayleigh fading channel with fd = 100 Hz.

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102

101

100


SNR [dB]

BER



different with respect to the increase in Doppler frequency. Itconcludes that the proposed CE algorithm outperforms RLSand Aboulnasr estimator by around 2.5 dB and 4.5 dB in therange of the tested SNRs. Finally, we conclude that proposedCE technique provides the best results among all consideredestimators in both simulated scenarios.

We estimated the number of arithmetic operations bytaking into account the number of additions and multipli-cations. It is clear that the simplest estimator are the LMS[14] and Aboulnasr method [22] while proposed algorithmrequires slightly few more operations than LMS and Aboul-nasr method. On the other hand, NLMS [29] algorithm re-quires one real division but proposed approach doesnt re-quire matrix inversion for implementation. Furthermore, theRLS algorithm [9] requires one matrix inversion as well ashigher computational complexity than all others schemes.

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SNR [dB]

BER



However, the proposed scheme requires slightly more arith-metic complexity that is minor problem for practical basestation (BS) implementation because the SC-FDMA CE isdone by the information received by the BS. In BS, CE ac-curacy and data estimation is the most important.

6. ConclusionAn accurate CE is one of the most important issues

for reliable future wireless communication systems such asLTE. To combat the channel dynamics and support broad-band multimedia access, we proposed a time-varying stepsize LMS CE scheme for wireless systems. This estima-tor automatically adapts the weighting coefficients on thechannel condition so that information of channel, and noisestatistics are not essential. The proposed algorithm is ex-ploited to further eliminate signals fluctuations due to noisedecision errors by the phase weighting scheme. With suchapproaches, convergence towards the true channel vector isassured. The mean, mean square behaviors, unbiased esti-mator, and computational complexities of the proposed CEalgorithm are analyzed. Even though, the proposed CE tech-nique requires little bit high computational complexity, theadvantage in the MSE and convergence towards true chan-nel coefficient as well as BER performance may be signif-icant useful for future mobile communications which allowbroadband multimedia access, anywhere, and anytime wire-less communication.

Acknowledgment

This research was supported by Basic Science Research Pro-gram through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science andTechnology (2010- 0017118).


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