variableforgettingfactorlsalgorithmfor …correspondence should be addressed to amit kumar...

International Scholarly Research NetworkISRN Signal ProcessingVolume 2011, Article ID 915259, 4 pagesdoi:10.5402/2011/915259

Research Article

Variable Forgetting Factor LS Algorithm forPolynomial Channel Model

Amit Kumar Kohli,1 Amrita Rai,2 and Meher Krishna Patel3

1 Department of Electronics and Communication Engineering, Thapar University, Punjab 147004, Patiala, India2 Electronics and Communication Engineering Department, Lingaya’s University, Faridabad 121002, India3 TBRL, DRDO, Chandigarh 160020, India

Correspondence should be addressed to Amit Kumar Kohli, [email protected]

Received 3 November 2010; Accepted 15 December 2010

Academic Editors: V. Chandran, E. Ciaccio, X. Jiang, and C.-M. Kuo

Copyright © 2011 Amit Kumar Kohli et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Variable forgetting factor (VFF) least squares (LS) algorithm for polynomial channel paradigm is presented for improved trackingperformance under nonstationary environment. The main focus is on updating VFF when each time-varying fading channel isconsidered to be a first-order Markov process. In addition to efficient tracking under frequency-selective fading channels, theincorporation of proposed numeric variable forgetting factor (NVFF) in LS algorithm reduces the computational complexity.

1. Introduction

Time-varying frequency-selective fading wireless channelscan be modeled by using the tapped-delay-line filter, inwhich each channel tap coefficient is considered to be anindependent autoregressive process [1]. The analytical andsimulation results presented in [2] manifest that the first-order Markov channel provides a mathematically tractablemodel for the time-varying channels. Under such Rayleighfading environment, the linear least squares algorithm (linearpolynomial model-based approach [3]) using variable for-getting factor (LSn-VFF) is developed for channel estimationin [4]. The lag error variance due to time variations andadditive white Gaussian noise (AWGN) variance in channelestimation error have a tradeoff relation with each other. TheVFF can be determined with the degree of nonstationarityand signal-to-noise ratio by using the well-known least meansquare (LMS) algorithm. The LSn-VFF algorithm is reportedto perform well only at high signal-to-noise ratios [4].However, it increases the computational complexity.

Based on an extended estimation error criterion, whichaccounts for the nonstationarity of signal, a method fordetermining the numeric variable forgetting factor (NVFF)is presented in [5]. When the signal experiences nonstation-arity, NVFF decreases automatically to estimate the globaltrend quickly using the extended estimation error criterion.

On the contrary, NVFF increases under stationary conditionsby increasing the memory for accurate estimation. In thiscorrespondence, we propose a channel estimation methodusing NVFF least squares algorithm combined with polyno-mial time-varying channel paradigm.

This correspondence is organized as follows. In Section 2,we first describe the time-varying frequency-selective wire-less system model. Details about the presented second-orderpolynomial model-based least squares algorithm using VFF(LSn2-VFF) are given in Section 3, and we also introducemathematical formulation of NVFF based on the extendedestimation error criterion. In Section 4, simulation resultsare presented to compare the performance of VFF and NVFFunder nonstationary environment. Finally, conclusions aregiven in Section 5.

2. System Model

Let the received signal be

y(n) = cH(n)x(n) + v(n), (1)

where, x(n) = [x(n)x(n − 1) · · · x(n − L + 1)]T is thetransmitted data symbol vector with x = ±1/

√L, c(n) =

[c0(n)c1(n) · · · cL−1(n)]H is the frequency-selective time-

2 ISRN Signal Processing

varying channel coefficient vector, which changes after eachsymbol period Ts, and v(n) is the zero-mean AWGN withvariance σ2

v . The (·)H is conjugate transposition of thematrix, and L is the length of multipath fading channel. Eachchannel coefficient cl(n) is an independent stationary ergodicfirst-order Markov process with correlation coefficient α =J0(2π fdTs), where fd is the maximum Doppler frequency, andJ0(·) is the Bessel function of first-kind and zeroth order [1].It follows that

cl(n) = αcl(n− 1) + ql(n), (2)

where, ql is the zero-mean process noise with varianceσ2q . Using estimated channel coefficient vector w(n) =

[w0(n)w1(n) · · ·wL−1(n)]H , the estimated received signal is

y(n) = wH(n)x(n). (3)

The estimation error is e(n) = y(n) − y(n) with zero-meanand variance σ2

e .By invoking Taylor’s theorem, the time variations of

each channel coefficient is explicitly represented in termsof the polynomial paradigm [3]. It results in cl(n) =∑∞

i=0(ni/i!)cl,i(n), where cl,i is the time-variation parameterfor lth coefficient such that

c(n) = c0(n) +n

1!c1(n) +

n2

2!c2(n)+ · · · , (4)

where, ci(n) = [c0,i(n)c1,i(n) · · · cL−1,i(n)]H . The leastsquares algorithm using above second-order channel modelis called LSn2 estimation algorithm, which is a modificationof least squares algorithm using the first-order channelmodel (LSn) in [4]. The tracking capability of LSn2algorithm in the time-varying environment can be furtherimproved by incorporating the variable forgetting factor(LSn2-VFF) without the explicit knowledge of process noisevariance.

3. LS Algorithms with PolynomialChannel Model

3.1. LSn2-VFF Algorithm for Time-Varying Channels. Thepolynomial channel model-based LS-VFF algorithm usesforgetting factor λ(n) to update the channel state after eachsymbol duration. The adaptive weight vector w(n) = w0(n)+nw1(n)+(n2/2)w2(n) in LSn2-VFF algorithm is implementedby exploiting the least squares algorithm proposed in [4] bySong et al.; it follows that

X(n) =[

xT(n)nxT(n)n2

2xT(n)

]T

,

W(n− 1) =[

wH0 (n− 1)wH

1 (n− 1)wH2 (n− 1)

]H,

(5)

where wi(n) = [w0,i(n)w1,i(n) · · ·wL−1,i(n)]H ,

ε(n) = y(n)−WH(n− 1)X(n),

k(n) = P(n− 1)X(n)λ(n)σ2

v + XH(n)P(n− 1)X(n),

W(n) = W(n− 1) + k(n)ε(n)∗,

P(n) = 1λ(n)

[P(n− 1)− k(n)XH(n)P(n− 1)

].

(6)

The minimum mean square error (MMSE) in channelestimation is J = E[|c(n)−w(n)|2]. For unknown processnoise variance, VFF is updated as

λ(n) ={

λ(n− 1) +μ

σ2v

Re[

DH(n− 1)X(n)ε(n)∗]}λmax

λmin

,

(7)

D(n) =[

I− k(n)XH(n)]

D(n− 1) + M(n)X(n)ε(n)∗

σ2v

,

M(n) = λ(n)−1[

I− k(n)XH(n)]

M(n− 1)[

I−X(n)kH(n)]

+(λ(n)σ−2

v

)−1k(n)kH(n)− λ(n)−1P(n),

(8)

where μ is the step size [6], which controls convergence andstability of the LMS algorithm in (7). The variation rangefor VFF is [λmin, λmax] to ensure the bounded nonnegativevalue of VFF, which increases the computational complexityof LSn2-VFF algorithm. However, it requires the knowledgeof σ2

v at receiver.

3.2. LSn2-NVFF Algorithm for Time-Varying Channels.Equations (5)–(6) are used in combination with NVFF todevelop LSn2-NVFF algorithm. The speed of adaptationis proportional to the asymptotic memory length N =(1− λ)−1 [5]. The memories corresponding to λmax and λmin

are denoted by Nmax and Nmin, respectively. If process noisevariance is small in comparison to variance of AWGN thatis σ2

q � σ2v (see the appendix), then σ2

e ≈ σ2v . The extended

estimation error is determined by

Z(n) = 1M

M−1∑

m=0

|e(n−m)|2. (9)

To ensure that the averaging in above equation is notobscuring the nonstationarity introduced by time-varyingchannel, M is kept smaller than minimum asymptoticmemory length, that is, M� Nmin. The NVFF is determinedby using the extended estimation error in (9). It follows that

N(n) = σ2e Nmax

Z(n)≈ σ2

v Nmax

Z(n),

λ(n) = 1− (N(n))−1.

(10)

ISRN Signal Processing 3

600 650 700 7500

0.02

0.04

0.06

0.08

0.1

0.12

Number of iterations

Ch

ann

elco

effici

ent

valu

e

TRUELSnLSn2

LSn-VFFLSn2-VFF

Figure 1: Tracking performance of LS-VFF algorithms.

It takes relatively long time for accurate parameterestimation for a value of NVFF close to unity, when thesignal experiences stationarity [5]. Therefore, Nmax controlsthe speed of adaptation. However, a small value of NVFFappears beneficial under nonstationary environment, whichis bounded (lower) by λ(n) ≥ λmin to guarantee positivenonzero values of NVFF.

4. Simulation Results

For simulations, the BPSK independent and identicallydistributed data is considered to be input. The presentedresults are based on the ensemble average of 250 independentsimulation runs. Note that we have kept σ2

q = 0.001,σ2v = 0.01 and M = 3. The channel tracking performances

of LSn2-VFF and LSn-VFF algorithms are compared inFigure 1 for L = 3 and fdTs = 0.001, where λmax, λmin

and step-size μ are empirically chosen as 0.99, 0.75 and0.005, respectively, for all cases (see [4]). The actual channelcoefficient is denoted as “TRUE” in Figures 1 and 2. Thetracking performance results presented in Figure 1 depictthat LSn2 algorithm (with second-order channel model)combats lag noise more efficiently than LSn algorithm, but atthe cost of increased computational complexity. It is apparentfrom the simulation results shown in Figures 1 and 2 thatLSn-VFF and LSn-NVFF algorithms supersede LSn2-VFFand LSn2-NVFF algorithms.

Both variable forgetting factors overwhelm the loss intracking capability caused due to first-order channel model.The simulation results are in good agreement with previ-ous studies [4]. For smoothly fading channels, LSn-NVFFalgorithm reduces tracking weight error relatively more ascompared to LSn-VFF algorithm (as shown in Figure 3). AtfdTs = 0.01, the tracking performances of both adaptive

325 330 335 340 345 350 355 360 365 370 3750

0.02

0.04

0.06

0.08

0.1

0.12


Ch

ann

elco

effici

ent

valu

e

TRUELSn-VFFLSn2-VFF

LSn-NVFF

LSn2-NVFF

Figure 2: Tracking performance of LS-NVFF algorithms.

0 50 100 150 200−36

−35.5

−35

−34.5

−34

−33.5

−33

−32.5

−32

−31.5

−31


MM

SE(

dB)

LSn-VFFLSn-VFF

LSn-NVFFLSn-NVFF

fdTs = 0.01

fdTs = 0.001

Figure 3: Tracking weight error comparison of presented algo-rithms.

algorithms are observed to be approximately equal. Howeverfor fast fading channels, LSn-VFF outperforms LSn-NVFFalgorithm due to the inaccurate extended estimation errorused in determining NVFF.

5. Concluding Remarks

The LSn-NVFF algorithm not only performs better thanLSn-VFF algorithm, but also precludes the need of LMSalgorithm in variable forgetting factor updating at eachiteration, which in turn reduces computational burden.

4 ISRN Signal Processing

It is inferred from simulation results that the higher-order polynomial model-based LS algorithms (e.g., LSn2) inconjunction with NVFF are not providing any additionaladvantage. However, the linear least squares algorithm usingNVFF is found to be efficient under slow and smoothly time-varying fading channels.

Appendix

Using (1) and (3), the estimation error is

e(n) = y(n)− y(n)

= (c(n)−w(n))Hx(n) + v(n).(A.1)

The above equation can be simplified using (2) as

e(n) = (αc(n− 1) + q(n)−w(n))Hx(n) + v(n). (A.2)

Under optimum conditions, it is assumed thatw(n) ∼= αc(n− 1). It leads to

e(n) ∼= qH(n)x(n) + v(n). (A.3)

Therefore, the estimation error variance is

σ2e = σ2

v

(

1 +σ2q

σ2v

)

. (A.4)

However, σ2e ≈ σ2

v for σ2q /σ

2v � 1.

References

[1] L. M. Chen and B. S. Chen, “A robust adaptive DFE receiverfor DS-CDMA systems under multipath fading channels,” IEEETransactions on Signal Processing, vol. 49, no. 7, pp. 1523–1532,2001.

[2] H. S. Wang and P. C. Chang, “On verifying the first-ordermarkovian assumption for a rayleigh fading channel model,”IEEE Transactions on Vehicular Technology, vol. 45, no. 2, pp.353–357, 1996.

[3] D. K. Borah and B. D. Hart, “Frequency-selective fading chan-nel estimation with a polynomial time-varying channel model,”IEEE Transactions on Communications, vol. 47, no. 6, pp. 862–873, 1999.

[4] S. Song, J. S. Lim, S. J. Baek, and K. M. Sung, “Variable forget-ting factor linear least squares algorithm for frequency selectivefading channel estimation,” IEEE Transactions on VehicularTechnology, vol. 51, no. 3, pp. 613–616, 2002.

[5] Y. S. Cho, S. B. Kim, and E. J. Powers, “Time-varying spectralestimation using AR models with variable forgetting factors,”IEEE Transactions on Signal Processing, vol. 39, no. 6, pp. 1422–1426, 1991.

[6] A. K. Kohli and D. K. Mehra, “Tracking of time-varying chan-nels using two-step LMS-type adaptive algorithm,” IEEE Trans-actions on Signal Processing, vol. 54, no. 7, pp. 2606–2615, 2006.

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