# adaptive blind channel equalisation

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Adaptive blind channel equalisation in chaoticcommunications by using nonlinear predictiontechnique

B.-Y. Wang and W.X. Zheng

Abstract: In chaotic communications, an ideal channel is often assumed. In practice, channeldistortion is inevitable. In particular, in wireless chaotic communications, the channel distortionmay be serious and must be compensated. An adaptive blind equalisation algorithm is proposed.The aim of the algorithm is to recover the chaotic signal transmitted through a finite impulseresponse (FIR) channel. The inherent characteristic of the chaotic signal, that is the high sensitivityto initial conditions, is exploited to formulate the criterion used in deriving the algorithm. Theanalysis of stability of the proposed algorithm is also provided.

1 Introduction

A chaotic signal is generated by a deterministic dynamicsystem, but shows some randomness in its behaviour. Itsspectrum has a continuous, broadband nature. In addition,most chaotic signals exhibit highly sensitive dependenceon initial conditions. Given two distinct initial conditionsarbitrarily close to one another, the trajectories of thechaotic signal starting from these initial conditionsdiverge and finally become uncorrelated [1].

In many chaotic communication schemes, it is assumedthat the transmitter is connected to the receiver by anideal channel [2]. However, in a realistic communicationsystem, the distortion due to the propagation channel maycorrupt the transmitted signal and make it difficult for thereceiver to directly recover the transmitted signal. In suchsituations, channel equalisation is required to compensatethe channel distortion [3–6].

In the work of Sharma and Ott [6], a synchronisation-based approach was proposed to compensate for thechannel distortions. However, the fact that the conditionalLyapunov exponents of the receiver depend on both thechaotic system and the channel causes some difficulties indetermining the coupling parameters to reach an approxi-mate synchronisation. Recently, Zhu and Leung [4] pro-posed an identification approach called minimumnonlinear prediction error (MNPE). The estimation accu-racy of the MNPE approach is limited by the truncationerror in approximating the infinite-order autoregressive(AR) model by a finite-order AR system. In addition tothe above batch algorithms, an extended Kalman filter(EKF)-based adaptive equalisation algorithm has been pre-sented [5]. However, in Leung et al. [7], it was observed that

# The Institution of Engineering and Technology 2006

IEE Proceedings online no. 20060009

doi:10.1049/ip-vis:20060009

Paper first received 17th January and in revised form 17th April 2006

B.-Y. Wang is with the College of Communications and InformationEngineering, Nanjing University of Posts and Telecommunications, Nanjing210003, People’s Republic of China

W.X. Zheng is with the School of Computing and Mathematics, University ofWestern Sydney, Penrith South, DC, NSW 1797, Australia

E-mail: [email protected]

810

the stability of the latter algorithm cannot always beguaranteed.

In this paper, an adaptive algorithm is proposed to blindlyequalise the channel distortion in chaotic communications.The proposed algorithm is developed by exploiting theinherent characteristic of the chaotic signal, that is theorbits of a chaotic system are very sensitive to initial con-ditions. The stability of the proposed algorithm is analysedas well.

2 Problem statement

Consider a chaotic communication system, in which theinformation-bearing chaotic signal is transmitted througha non-ideal channel before it reaches the receiver.Suppose that the chaotic transmitter system is described by

xn ¼ f ðxn�1; . . . ; xn�dÞ; xn [ Rmð1Þ

The signal of transmission st is assumed to be stored in thebifurcation parameter of the chaotic system (1). By keepingthe bifurcation parameter in the chaotic region, the output ofthe chaotic system is chaotic and hence occupies a widebandwidth.

In practice, an ideal channel does not exist. The trans-mitted signal passes through a propagation channel beforearriving at the receiver. It was found in the work of Iltisand Fuxjaeger [8] that a typical fading and multipathchannel could be modelled by a finite impulse response(FIR) filter. Thus, the received signal would be

yn ¼XL�1

l¼0

alxn�l þ vn ð2Þ

where a ¼ [1 a1 � � � aL21]T is the channel coefficient vectorand vn is zero-mean white Gaussian noise with variance sn

2.The objective of blind identification/equalisation is toestimate the channel parameters ak and recover the trans-mitted signal xn from the received signal yn.

In many blind algorithms that deal with non-chaoticsignals, a priori information about the statistics of theinput signal is exploited. However, instead of the statisticsof the chaotic signal, we will exploit the knowledge aboutthe nonlinearity of the chaotic system. That is, the algorithm

IEE Proc.-Vis. Image Signal Process., Vol. 153, No. 6, December 2006

to be developed assumes the knowledge of the chaotic mapand its parameters.

3 Adaptive blind identification of FIR systems

3.1 Related work

3.1.1 EKF-based approach [5]: In order to apply EKF,the equalisation problem has to be reformulated as a non-linear state-space estimation problem. The channel coeffi-cients will be taken as state variables and modelled aszero-order AR systems. This will lead to the followingstate equation and measurement equation

X n ¼ FðX n�1Þ þ wn�1

yn ¼XL�1

l¼0

aðnÞl xn�l þ vn

ð3Þ

where

X n ¼ xn xn�1 � � � xn�Lþ1 aðnÞ1 � � � a

ðnÞL�1

h iT

wn�1 ¼ 01�ðLþ1Þ wðn�1Þ1 � � � w

ðn�1ÞL�1

h iT

FðX n�1Þ ¼

hf ðxn�1Þ xn�1 � � � xn�Lþ2

aðn�1Þ1 � � � a

ðn�1ÞL�1

iT

The EKF algorithm [5] for the system in (3) is summarisedas follows

Xnjn�1 ¼ FðXn�1Þ

Pnjn�1 ¼ Fn�1Pn�1FTn�1 þ Qn

Kn ¼ Pnjn�1GTn�1 Gn�1Pnjn�1G

Tn�1 þ s2

n

� ��1ð4Þ

Xn ¼ Xnjn�1 þ Kn yn � H Xnjn�1

� �� �Pn ¼ ðI2L�1 � KnGn�1ÞPnjn�1

where

Fn�1 ¼ @F=@X jX¼Xn�1

Gn�1 ¼ @H=@X jX¼Xn�1

Qn ¼ EfwnwTn g

and H(Xn) ¼P

l¼0L21al

(n)xn2l is the transfer function of thechannel.

In applying EKF, the instability problem must bechecked, because the condition for the stability cannot besatisfied in some scenarios [7]. In the work of Zhu andLeung [5], a thorough analysis of the dynamics of EKFwas conducted.

3.1.2 Synchronisation method [6]: The coupling syn-chronisation method receiver of Pecora and Carrol [6] isexpressed as

xn ¼ f ðxn�1Þ þ K yn �XL�1

j¼0

aðnÞj xn�j

!ð5Þ

IEE Proc.-Vis. Image Signal Process., Vol. 153, No. 6, December 2006

where the estimates of the coefficients are updated by usingthe least mean squares (LMS) algorithm

aðnÞ ¼ aðn�1Þ� m

@SðaÞ

@a

����a¼aðn�1Þ

SðaÞ ¼XN

n¼d

e2nðaÞ ¼

XN

n¼d

yn �XL�1

j¼0

ajxn�j

!2 ð6Þ

The coupling vector K should be selected to ensure thesynchronisation between the transmitter and the receiver.But a suitable choice would require the knowledge ofthe unknown channel. Moreover, the slower convergencerate of the synchronisation method is also an issue.

3.2 Proposed approach

As the driving signal considered in this paper is notGaussian, the least square criterion cannot produce anoptimal estimator. As is well known, the initialisation-sensitivity peculiar to a chaotic signal could be used todesign an effective criterion for identifying a linear system.

For the accurate value of the input sequence xt, thefollowing equation

xt � f ðxt�1; . . . ; xt�dÞ ¼ 0 ð7Þ

holds for any instant t. Hence, we define the nonlinear pre-diction error (NPE) criterion as follows.

Definition: The NPE criterion is defined as

VtðaÞ ¼ e2t ðaÞ ¼ ðxtðaÞ � f ðxt�1ðaÞ; . . . ; xt�dðaÞÞÞ

2ð8Þ

where xt(a) is used to show the dependence of the estimatedsignal xt on the vector a.

A vital problem in seeking the optimal solution of (8) ishow to represent xt21, . . . , xt2d as functions of the knownquantities. It has a very great effect on the performance ofthe derived algorithm. In the work of Zhu and Leung [4],xt21, . . . , xt2d are expressed as functions of the obser-vations. The main idea is that a moving average (MA)system is approximated by an AR system of finite order.Evidently, the truncation error is unavoidable. The simu-lation results therein indicated that the performance oftheir approach levels-off and saturate at no more than40 dB. Although that algorithm achieved a very high accu-racy in AR cases, the behaviour of the algorithm in identify-ing MA systems is not satisfactory.

To avoid the performance saturation of the NPEapproach, we attempt to find another way to expressxt21, . . . , xt2d. From (2), it follows that

xt ¼ yt �XL�1

l¼1

alxt�l ð9Þ

However, only yt is known in the above equation. A tech-nique used in the recursive nonlinear least square algorithm[9] is very helpful here. During the progress of a recursivealgorithm, the estimate of xt can be formed as

ut ¼ yt � ðut�Lt�1Þ

Taðt�1Þ; ut�Lt�1 ¼ ½ut�1; . . . ; ut�L�

T

xt ¼ ut � ðxt�Lt�1Þ

Taðt�1Þ; xt�Lt�1 ¼ ½xt�1; . . . ; xt�L�

Tð10Þ

where uts are intermediate variables. More simply, we canreplace (10) with

xt ¼ yt � ðxt�Lt�1Þ

Taðt�1Þ; xt�Lt�1 ¼ ½xt�1; . . . ; xt�L�

Tð11Þ

811

The derivative @Vt(a)/@a is

@VtðaÞ

@a¼ 2ðxtðaÞ � f ðxt�1ðaÞ; . . . ; xt�dðaÞÞÞ

� �xt�Lt�1 �

@

@af ðxt�1ðaÞ; . . . ; xt�dðaÞÞ

� �ð12Þ

Considering the case where the driving chaotic signal isgenerated by

xt ¼ f ðxt�1Þ

Equation (12) can be expanded as

@VtðaÞ

@a¼ 2ðxtðaÞ � f ðxt�1ðaÞÞÞ �x

t�Lt�1 �

@

@af ðxt�1ðaÞÞ

� �

¼�2ðxtðaÞ � f ðxt�1ðaÞÞÞ xt�Lt�1 þ

@

@af ðxt�1ðaÞÞ

� �ð13Þ

With the derived gradient, it is straightforward to implementan LMS-like algorithm. The proposed algorithm is summar-ised in Table 1.

This algorithm requires about 4L multiplication oper-ations at each iteration, so its computational complexity isO(L). Note that the EKF-based algorithm has a compu-tational complexity of O(L2) [5].

To analyse the stability of the proposed algorithm, asdone in Mandic [10] and Mandic and Krcmar [11], wedefine an a posteriori error

eðapÞt ¼ xtða

ðtÞÞ � f ðxt�1ðaÞÞ ð14Þ

Using (11), we have

xtðaðtÞÞ ¼ yt � ðx

t�Lt�1Þ

TaðtÞ ¼ xtðaðt�1ÞÞ

� ðxt�Lt�1Þ

TðaðtÞ � aðt�1Þ

Þ ð15Þ

After substitution of (13) and (15) into (14), we have

jeðapÞt j ¼ jet � metðx

t�Lt�1Þ

Tðxt�L

t�1 � f 0ðxt�1Þxt�L�1t�2 Þj ð16Þ

Therefore the step-size m should satisfy

j1� mðxt�Lt�1Þ

Tðxt�L

t�1 � f 0ðxt�1Þxt�L�1t�2 Þj , 1 ð17Þ

so that jet(ap)j , jetj.

Equation (17) imposes a constraint on the step size. Onemay be concerned about if the second term in the left side of(17) is positive. In order to explain this point, we rewritethat term as

ðxt�Lt�1Þ

Tx

t�Lt�1 � f

0ðxt�1Þðx

t�Lt�1Þ

Tx

t�L�1t�2 ð18Þ

The expression in (18) can be approximated by

rð0Þ � f 0ðxt�1Þrð1Þ ð19Þ

where r(0) and r(1) denote the correlation of the estimatedchaotic sequence. Generally, r(1) is much smaller thanr(0). Therefore it can be reasonably expected that the termin (19) is positive and thus the condition (17) is non-trivial.

Table 1: Proposed approach for identifying MA systems

Initialisation: a(0) ¼ 0L�1; x2L ¼ . . . ¼ x21 ¼ 0;

For t ¼ 1, 2, . . . , do

x t ¼ yt 2 (x t2Lt21 )Ta(t21), x t2L

t21 ¼ [xt21, . . . , xt2L]T

a(t) ¼ a(t21)2m(@Vt/@a)ja¼a(t21)

where m is the step size

812

Remark 1: As a gradient-type algorithm, its convergencewill depend on the initial value. The proposed algorithmcan be initialised with a least square solution based on avery short data record in a blind but batch fashion.

4 Simulation results

In this section, simulation results are presented to evaluatethe efficacy of the proposed algorithm. In initialisation ofthe proposed algorithm, the first block with ten samples isused to produce an initial estimate for the relevant par-ameters. Moreover, at the beginning of the algorithmimplementation, the step size in the adaptive algorithm isset at a small positive number. As the iteration increases,the step size will be changed into another much smaller posi-tive number so as to ensure the best steady-state performance.

Example 1: Consider a chaotic communication system, inwhich the chaotic transmitter is the logistic map

xt ¼ lxt�1ð1� xt�1Þ

where the bifurcation parameter l is controlled in thechaotic region [3.7 4.0]. The signal of transmission xt isassumed to pass through a third-order FIR channela ¼ [1, 0.45, 20.22]. The mean-squared error (MSE)between the true channel coefficients and the estimatedchannel parameters is exploited to measure the identifi-cation performance. Fig. 1 shows the steady-state MSE ofthe proposed approach under various signal-to-noise ratios(SNRs). For four different SNRs, the algorithm can offer avery accurate data recovery with 200 samples. Fig. 2 pro-vides a comparison between the proposed approach andthe synchronisation-based approach [6]. In a large rangeof SNRs, the proposed algorithm can achieve a 1–2.5 dBimprovement in the equalisation error. As the channelorder is often unknown, it is necessary to investigate therobustness of the algorithm to order overestimation. InFig. 3, the algorithm is evaluated when the channel orderis overestimated. The observed degradation is less than3 dB after 300 samples. As the bifurcation l in thechaotic dynamics may be unknown, it is interesting toinvestigate the impact of the mismatch L ¼ l 2 l on theperformance. We show the simulation results for the logisticand Chebyshev driving sequences in Fig. 4a and b,

Fig. 1 Error in the equalised data l ¼ 4.0

Four curves corresponding to four different SNRs, 20, 30, 40, and50 dB, respectively, with 200 data samples

IEE Proc.-Vis. Image Signal Process., Vol. 153, No. 6, December 2006

respectively, where the Chebyshev map is given by

xt ¼ cosðl cos�1ðxt�1ÞÞ

In the case of the logistic driving sequence, the algorithm isvery robust to the mismatch. An interesting observation isthat the proposed algorithm can still work very well evenwhen the mismatch is rather large. In contrast to theabove observation, the algorithm is very sensitive to themismatch in the bifurcation parameter of the Chebyshevmap, as a very small mismatch leads to a greater perform-ance degradation. The main reason for the difference inthese two chaotic sequences may be that their dynamicshave different sensitivities to the bifurcation parameter.In other words, the Lyapunov exponent of the Chebyshevmap is more sensitive to the bifurcation parameter.

Example 2: In the second example, we evaluate the per-formance of the algorithm when some different chaoticsequences are transmitted in the chaotic communicationsystem. Fig. 5 shows the performance of the algorithmwhen the transmitted signals are generated by the logisticmap with four different l. In Fig. 6, we show the perform-ance of the algorithm when the transmitter sends the logistic

30 40 50 60 70-75

-70

-65

-60

-55

-50

-45

-40

-35

-30

SNR (dB)

Err

or

in

Eq

ua

lize

d D

ata

(d

B)

our approachthe approach in [6 ]

Fig. 2 Error in the equalised data, l ¼ 4.0, 256 data samples

Solid line indicates our approach; dashed line indicates the approach inthe work of Sharma and Ott [6]

0 100 200 300 400 500 600-60

-50

-40

-30

-20

-10

0

Number of data samples

MS

E

in

dB

LL+1L+3

Fig. 3 MSE with the overestimated channel order, SNR ¼ 50 dB,l ¼ 4.0

IEE Proc.-Vis. Image Signal Process., Vol. 153, No. 6, December 2006

sequence and Chebyshev sequence. In all cases, the step-size is appropriately selected to guarantee a goodsteady-state performance. Clearly, the performance of thealgorithm relies mainly upon the SNR, rather than uponwhich type of driving chaotic sequence is transmitted.

Example 3: In this last example, we investigate theperformance of the algorithm under two time-varying

0 50 100 150 200 250 300 350 400-60

-50

-40

-30

-20

-10

0

Number of data samples

MS

E

in

dB

∆=0∆=0.2∆=0.4

a

0 100 200 300 400 500 600 700 800 900 1000-60

-50

-40

-30

-20

-10

0

Number of data samples

MS

E

in

dB

Λ=1e-4Λ=0

b

Fig. 4 Performance of the algorithm when there is a mismatch inthe bifurcation parameter

0 50 100 150 200 250 300-60

-50

-40

-30

-20

-10

0

Number of data samples

MS

E i

n d

B

λ=3.78

λ=3.86λ=3.94

λ=4.0

Fig. 5 MSE for different values of parameter l of the logisticmap, SNR ¼ 50 dB

813

channels:

Channel A: aðtÞ ¼½1 0:5 �0:22�T; if t , 200

½1 0:45 �0:22�T; if 200 � t , 400

½1 0:53 �0:19�T; if 400 � t

8<:

0 100 200 300 400 500 600 700 800 900 1000-60

-50

-40

-30

-20

-10

0

Number of data samples

MS

E i

n d

BChebyshev sequenceLogistic sequence

Fig. 6 MSE for different driving sequences, SNR ¼ 50 dB,l ¼ 4.0

0 100 200 300 400 500 600 700 800-60

-50

-40

-30

-20

-10

0

Number of data samples

MS

E i

n d

B

SNR=50dBSNR=30dB

a

0 100 200 300 400 500 600 700 800-60

-50

-40

-30

-20

-10

0

Number of data samples

MS

E i

n d

B

SNR=50dBSNR=30dB

b

Fig. 7 Performance of the proposed algorithm in the time-varying channels

a Channel Ab Channel B

814

and

Channel B: aðtÞ ¼ ½1 0:45þ 0:01 sinðp t=400Þ

�0:22þ 0:02 cosðpt1=3=300Þ�T

Fig. 7a shows the performance of the algorithm when thelogistic chaotic sequence is transmitted throughChannel A. Fig. 7b corresponds to the case for ChannelB. In both cases, the proposed algorithm can offer similarperformance to that in the time-invariant channel.

5 Conclusions

In this paper, an adaptive algorithm has been proposed toblindly identify and equalise an FIR channel in chaoticcommunications. The inherent characteristic of a chaoticsystem, that is the sensitivity to initial conditions, has beenexploited to develop our algorithm. With an appropriatestep size, the stability of the proposed algorithm can be guar-anteed. Compared with the algorithms given in the work ofZhu and Leung [4, 5], the algorithmic simplicity and high esti-mation accuracy make the proposed approach more attrac-tively applicable in the receiver design of chaoticcommunication systems. In contrast, however, the algorithmsgiven in Zhu and Leung [4, 5] may be more preferable in thecase where the nonlinear function is not differentiable.Finally, an interesting topic for future research is to rigorouslyshow that the expression in (19) is positive and subsequentlyto determine a constraint for stability of the proposed algor-ithm on the step size m through upper and lower boundvalues in terms of the Lyapunov exponent of the chaotic map.

6 Acknowledgment

The authors would like to thank the Editor and three anon-ymous reviewers for their valuable comments and sugges-tions that have significantly improved this paper. This workwas supported in part by a research grant from theAustralian Research Council and in part by the ResearchFoundation of the Education Department of JiangsuProvince, People’s Republic of China (No. 05KJB510089).

7 References

1 Parker, T.S., and Chua, L.O.: ‘Practical numerical algorithms forchaotic systems’ (Springer-Verlag, New York, 1989)

2 Cummo, K.M., and Oppenheim, A.V.: ‘Circuit implementation ofsynchronized chaos with application to communications’, Phys. Rev.Lett., 1993, 71, pp. 65–68

3 Wang, B.-Y., Chow, T.W.S., and Ng, K.T.: ‘Adaptive identificationalgorithm for AR system driven by chaotic sequence’,Int. J. Bifurcation Chaos, 2003, 13, (4), pp. 963–972

4 Zhu, Z., and Leung, H.: ‘Identification of linear systems driven bychaotic signals using nonlinear prediction’, IEEE Trans. CircuitsSyst. I, 2002, 49, (2), pp. 170–180

5 Zhu, Z., and Leung, H.: ‘Adaptive blind equalization for chaoticcommunication systems using extended Kalman filter’, IEEE Trans.Circuits Syst. I, 2001, 48, (8), pp. 979–989

6 Sharma, N., and Ott, E.: ‘Combating channel distortions incommunication with chaotic systems’, Phys. Lett. A, 1998, 248, (5/6),pp. 347–352

7 Leung, H., Zhu, Z., and Ding, Z.: ‘An aperiodic phenomenon of theextended Kalman filter in filtering noisy chaotic signals’, IEEETrans. Signal Process., 2000, 48, (6), pp. 1807–1810

8 Iltis, R.A., and Fuxjaeger, A.W.: ‘A digital DS spread-spectrumreceiver with joint channel and Doppler shift estimation’, IEEETrans. Commun., 1991, 39, (8), pp. 1255–1267

9 Porat, B.: ‘Digital processing of random signals’ (Prentice-Hall,Englewood Cliffs, NJ, 1994)

10 Mandic, D.P.: ‘NNGD algorithm for neural adaptive filters’, Electron.Lett., 2000, 36, (9), pp. 845–846

11 Mandic, D.P., and Krcmar, I.R.: ‘Stability of NNGD algorithm fornonlinear system identification’, Electron. Lett., 2001, 37, (2),pp. 200–202

IEE Proc.-Vis. Image Signal Process., Vol. 153, No. 6, December 2006

3000 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 11, NOVEMBER 1999

Adaptive Blind Channel Estimationby Least Squares Smoothing

Qing Zhao and Lang Tong,Member, IEEE

Abstract—A least squares smoothing (LSS) approach is pre-sented for the blind estimation of single-input multiple-output(SIMO) finite impulse response systems. By exploiting the iso-morphic relation between the input and output subspaces, thisgeometrical approach identifies the channel from a speciallyformed least squares smoothing error of the channel output. LSShas the finite sample convergence property, i.e., in the absenceof noise, the channel is estimated perfectly with only a finitenumber of data samples. Referred to as the adaptive least squaressmoothing (A-LSS) algorithm, the adaptive implementation has ahigh convergence rate and low computation cost with no matrixoperations. A-LSS is order recursive and is implemented inpart using a lattice filter. It has the advantage that when thechannel order varies, channel estimates can be obtained withoutstructural change of the implementation. For uncorrelated inputsequence, the proposed algorithm performs direct deconvolutionas a by-product.

Index Terms—Adaptive least squares method, blind channelidentification.

I. INTRODUCTION

BLIND channel equalization has the potential to increasedata throughput for communications over time varying

channels. To achieve this goal, several requirements must besatisfied. First, blind channel estimation and equalization al-gorithms must converge quickly. An important property is theso-calledfinite sample convergenceproperty, i.e., the channelcan be perfectly estimated using a finite number of samplesin the absence of noise. This property is especially criticalin packet transmission systems where only a small numberof data samples are available for processing. Second, theadaptivity of the algorithm is important in tracking the channelvariation and maintaining the communication link. Third, lowcomplexity in both computation and VLSI implementation isdesired.

Although many blind channel estimation and equalizationalgorithms have been proposed in recent years, few cansimultaneously satisfy requirements in convergence speed,adaptivity, and complexity. Deterministic batch algorithmssuch as the subspace (SS) algorithm [14], the cross relation(CR) algorithm1 [5], [22], the two-step maximum likelihood

Manuscript received June 2, 1998; revised May 11, 1999. This workwas supported in part by the National Science Foundation under ContractCCR-9804019 and by the Office of Naval Research under Contract N00014-96-1-0895. The associate editor coordinating the review of this paper andapproving it for publication was Dr. Phillip A. Regalia.

The authors are with the School of Electrical Engineering, Cornell Univer-sity, Ithaca, NY 14853 USA (e-mail: [email protected]).

Publisher Item Identifier S 1053-587X(99)08320-8.1This is also referred to as the least squares algorithm, or EVAM.

(TSML) approach [7], the linear prediction-subspace (LP-SS)algorithm [16], and the joint order detection and channelestimation algorithm (J-LSS) [20] converge quickly. Withoutassuming a specific stochastic model of the input sequence,these methods have the finite sample convergence property.Unfortunately, they suffer from high computation cost, whichis usually associated with eigenvalue decomposition, and theiradaptive implementations are often cumbersome. On the otherhand, recently proposed linear prediction (LP) based algo-rithms [1], [4] are attractive for their efficient adaptive im-plementations. The key component of these algorithms is theclassical linear predictor that can be implemented recursivelyboth in time and in filter order using, for example, lattice filters.The modular structure of lattice filters makes them good can-didates for VLSI implementation. Perhaps the most importantshortcoming of these LP-based algorithms is the relatively lowconvergence speed. Relying on the statistical uncorrelation ofthe input sequence, these LP-based algorithms fail to havethe finite sample convergence property. Consequently, theydemand a relatively large sample size for accurate channelestimation, which limits their application in the small datasize scenarios.

Aiming to satisfy the three design requirements at the sametime, we present in this paper aleast squares smoothing(LSS)approach to blind channel estimation. Recognizing the isomor-phic relation between the input and output subspaces, we firstconsider estimating the channel from the inputsubspace. Byprojecting the channel output into a particular input subspace

, the channel is obtained from the least squares projectionerror. When is constructed from the channel output byexploiting the isomorphism between the input and outputsubspaces, this projection leads to least squares smoothing.This geometrical approach to blind channel estimation alsoprovides a simple and unified derivation of different LP-basedchannel estimators and a clear explanation for the loss offinite-sample convergence property in LP-based approaches.

Based on the LSS approach, a newadaptive least squaressmoothing(A-LSS) channel estimation algorithm is proposed.Like all deterministic methods, A-LSS preserves the finitesample convergence property with high convergence rate.Similar to linear prediction-based algorithms, the least squaressmoothing approach naturally leads to an order and timeadaptive implementation that accommodates a wide range ofchannel variation, both in channel parameters and channellength. In wireless communications, for example, when thechannel order suddenly changes due to the addition or loss ofreflectors, A-LSS simply selects signals from different parts of

1053–587X/99$10.00 1999 IEEE

ZHAO AND TONG: ADAPTIVE BLIND CHANNEL ESTIMATION BY LEAST SQUARES SMOOTHING 3001

Fig. 1. Single-inputP -output linear system.

the filter with neither structural change nor extra computation.Implemented with commonly used basic modules in classicallattice filters, the structure of A-LSS is highly parallel andregular with only scalar operations.

This paper is organized as follows. Section II presents thesystem model and two assumptions used in this paper. Thegeometrical approach to linear least squares smoothing channelestimation is introduced in Section III. A batch least squaressmoothing (B-LSS) algorithm and its connections with existinglinear prediction-based approaches are obtained. In Section IV,we present the data structure for the unknown channel ordercase followed by the adaptive least squares smoothing channelestimation and order detection algorithm. Simulation resultsare presented in Section V to demonstrate the convergence andperformance of A-LSS in channel estimation and tracking.

II. PROBLEM STATEMENT

A. Notations and Definitions

Notations used in this paper are mostly standard. Upper-case and lowercase bold letters denote matrices and vectorswith , , and denoting transpose, Hermitian, andMoore–Penrose pseudoinverse operations, respectively. Givena matrix , ( ) is the row (column) space of

. For a matrix having the same number of columnsas , ( )is the projection of into (the orthogonal complement of)the row space of . We define as theprojection error matrix of into . For a set of vec-tors , denotes the linear subspacespanned by . For a vector and a linear subspace

, is the orthogonal projection of

into , and is its projection error. Finally,and denote the 2-norm and Frobenius norm, respectively.

B. The Model

The identification and estimation of a single-input-outputlinear system shown in Fig. 1 is considered in this paper. Thesystem equation is given by

(1)

where is the (noiseless) channeloutput, is the additive noise, is the received signal,

is the channel impulse response, and is the inputsequence. Given samples of the system input and output,define the row vector and the block row vector as

(2)

With and similarly defined, we have, from (1)

(3)

Our goal is to estimate from .

C. Assumptions and Properties

Two assumptions are made in this paper. The first one isabout channel diversity.

• A1: Channel Diversity: The subchannel transfer functionsdo not share common zeros, i.e., are co-prime.

A1 is shared by all deterministic blind channel estimationmethods. The co-primeness of the subchannel transfer func-tions ensures that the channel is fully specified (up to ascaling factor) by the noncommon zeros of the channel output.If A1 is not satisfied, although the noncommon zeros ofthe subchannel transfer functions can still be identified fromthe output, the common zeros cannot be distinguished fromzeros of the input. Therefore, the channel cannot be identifiedwithout knowing the input sequence. The following property,which has also been exploited in [13] and [21], reveals theequivalence between the input and output subspaces under thechannel diversity assumption. First, define the input and outputsubspaces spanned by consecutive row (block row)vectors as

(4)

Note that for , we similarly define and as thesubspaces spanned by consecutive future data vectors. Thefollowing property results directly from A1.

Property 1: Under A1, there exists a (smallest)such that for any , we have the isomorphic

relation between the input and output subspaces

(5)

Proof: Let

(6)

From (3), we have

(7)

where the matrix is thefiltering matrixwith the following block Toeplitz structure:

...... (8)

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It has been shown in [9] and [19] that A1 implies the existenceof a smallest such that for any ,

has full column rank. Thus, from (7), we have

(9)

which leads to (5).Property 1 plays an important role in the smoothing and

linear prediction approaches to blind channel estimation. Itis this isomorphic relation between the input and outputsubspaces that makes it possible to identify the channel fromthe inputsubspacewithout the direct use of the input sequence.

To ensure channel identifiability, the input sequence mustbe sufficiently complex in order to excite all modes of thechannel. This requirement is imposed on the linear complexity[2] of the sequence.

• A2: Linear Complexity: The input sequence haslinear complexity greater than .

With the definition of linear complexity2 [2], A2 implies thatfor any , we have

rank ... Toeplitz

(10)

This implication of A2 will be exploited in Section III aswe discuss the necessary number of input symbols for thechannel identification by the proposed algorithm. It has beenshown in [18] that when , the necessary and sufficientcondition for the unique identification of the channel and itsinput is A1 and that has linear complexity greater than

. Because both future and past data are required in thesmoothing approach, a stronger condition is assumed in A2in this paper.

III. L EAST SQUARES SMOOTHING—AGEOMETRICAL APPROACH

In this section, we introduce a geometrical approach to linearleast squares smoothing channel estimation by exploiting theisomorphic relation between the input and output subspaces.The same approach also leads to a least squares derivation ofthe LP-based algorithms.

A. Channel Identification from the Input Subspace

The isomorphism between the input and output subspacesgiven in Property 1 implies that the input subspace can beconstructed from the output. The following question arisesfrom this property:Can we identify the channel from the inputsubspace?If so, by constructing the input subspace from theoutput, the channel is obtained from the output alone. Ananswer to this question is presented below.

2The linear complexity of the vectorv with componentsv0; � � � ; vn�1is defined as the smallest value ofc for which a recursion of the formvi = �

cj=1 ajvi�j(i = c; � � � ; n � 1) exists that will generatev from

its first c components.

Fig. 2. Projection ofxt+i(i = 0; � � � ; L) ontoZ .

Consider consecutive output block row vectors. From (3), we have

......

.. .

(11)

Suppose that we want to identify up to a commonscaling factor from . One way to achieve this isto eliminate in all other terms except the onesassociated with . Consider projectinginto a “punctured” input subspacethat satisfies the followingtwo conditions:

• C1: .• C2: .

Note that A2 ensures. Thus, this punctured input subspace

exists. As illustrated in Fig. 2, is the summation of(a vector outside ) and (a vector inside

). The two shaded right triangles immediately suggest

(12)

Because is independent of, we have the projection errormatrix

...... (13)

Note that is a rank-1 matrix whose column and row spacesare spanned by and , respectively. From , the channel

can be directly identified. One approach is the least squaresfitting of the column space of

(14)

The above optimization can be solved by the singular valuedecomposition of either or the sample covariance of theprojection error sequence. A simpler approach is to take therow ( ) with the maximum 2-norm in as an estimate of

. Then, from (13), we have

(15)

ZHAO AND TONG: ADAPTIVE BLIND CHANNEL ESTIMATION BY LEAST SQUARES SMOOTHING 3003

Fig. 3. Isomorphism between input and output subspaces.

To gain a better understanding of this geometrical approachto channel estimation, we make the following remarks.

1) When is orthogonal to , which is asymptoticallytrue for an uncorrelated zero-mean input sequence, theprojection error converges to (see Fig. 2). Inthis case, the rank-one decomposition of the projectionerror matrix provides the input sequence as wellas the channel vector, i.e., this geometrical approachto channel estimation performs direct deconvolution asa by-product.

2) The projections of into for different are inde-pendent of each other and can be carried out in parallel.This property is attractive in algorithm implementation.

B. Channel Identification from the OutputSubspace—Least Squares Smoothing

1) Construction of the Input Subspace from the Output:Ithas been shown in Section III-A that the channel can beidentified from the projection errors of into theinput subspace satisfying C1 and C2. To avoid the directuse of the input sequence, we constructfrom the channeloutput by exploiting the isomorphic relation between the inputand output subspaces given in Property 1.

Using the definition in (4), conditions C1 and C2 can bewritten in the form

(16)

for any . With the isomorphic relation between the inputand output subspaces given in (5), we have, for

(17)

The isomorphism between the input and output subspaces isillustrated in Fig. 3. The projection of intois converted into the smoothing [6], [8] of the current data

by the past data and the futuredata . The projection error used toidentify the channel [see (13)] now becomes the smoothingerror, which can be obtained from the output alone. Becauseof the least squares criterion used in the projection, thisgeometrical approach to blind channel estimation is referredto as theleast squares smoothing(LSS) algorithm. Here, we

introduce two terms:smoothing window sizeand smoothingorder. The smoothing window size is defined as the numberof symbols in the current data, and the smoothing order is thenumber of symbols in the past data. For the case discussedabove, the smoothing window size and the smoothing orderare and , respectively.

The price we paid for avoiding the direct use of inputsequence is that more input symbols are required to identify thechannel. From Fig. 3, we can see that the projection subspace

and the current data span a -dimensional inputsubspace denoted as, as shown in

(18)

... Toeplitz

(19)

The construction of imposes the full-row-rank condition onthe matrix in (19). As a result, we require inputsymbols ( observation symbols) to be availableand the linear complexity of the input sequence to be greaterthan [see (10)].

2) Batch Least Squares Smoothing Algorithm:We considerhere the problem of estimating a channel with orderfrom abatch of observation symbols. The data structure is speci-fied, followed by some useful properties. Then, the batch leastsquares smoothing channel estimation algorithm is presented.

Given observation symbols, for a fixed smoothing order and the known

channel order , define the overall data matrix as

...

...

...

(20)

from which we have specified

• the past data matrix ;

• the current data matrix ;

• the future data matrix ;

• future-past data matrix .

In the absence of noise, the overall data matrixhas thefollowing relation with the input signal:

... Toeplitz

(21)

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Fig. 4. Batch LSS algorithm.

As direct consequences of (21) and Property 1, the relationamong the data matrices in (20) and various subspaces issummarized in Property 2. The rank conditions given beloware useful in finding the least squares approximation of thenoisy data matrices.

Property 2: Suppose that the input sequence has linearcomplexity greater than , , .We have the following properties in the noiseless case:

1) Overall Data Matrix :

rank

2) Past Data Matrix :

rank

3) Future Data Matrix :

rank rank

4) Future-Past Data Matrix :

rank

As stated in the above property, the row vectors in thefuture-past data matrix span the -dimensionalprojection space . Therefore, the projection error in (13)can be obtained as the least squares smoothing error of thecurrent data by the future-past data . Specifically, in theabsence of noise, we have

......

(22)from which the batch least squares smoothing channel estima-tion algorithm (B-LSS) can be derived directly. One possibleimplementation is summarized in Fig. 4.

3) Stochastic Equivalence:The least squares smoothing ap-proach presented above is based on the deterministic modelingof the input sequence. In order to obtain the consistent estimateof the channel in the presence of noise, we consider astochastic equivalence of the LSS approach. After replacingthe projection of data vectors ( ) into a Euclidean space bythe projection of corresponding random variables ( )into a Hilbert space, the geometrical approach presented inSection III-B1 can be adapted to the stochastic framework.Specifically, consider the input sequence as a whiterandom process with zero mean and unit variance. Define theoutput random vectors

(23)

The error sequence of the linear minimum mean squareerror (MMSE) estimator of based on is then givenby

(24)

where and are the covariance matrices defined as

(25)

Based on the principle of orthogonality, the error sequencecan be considered as the projection error of into

the Hilbert space spanned by the random variables in .Similarly to (13), we then have, for the white input sequence

,

(26)

With and denoting the covariance of and ,respectively, we have, from (24) and (26)

(27)

This implies that can be directly obtained from . Fora noise sequence independent of and with knownsecond-order statistics, the covariance matrices, , and

can be estimated consistently from the observation. Theconsistent estimate of the channel can then be obtained.

C. Connections with Linear Prediction-Based Approaches

The main difference between the LP-based approaches andthe LSS approach is the definition of the subspaceintowhich the projection is made. Naturally, LP-based approachesinclude only the past data in the projection space. We nowdraw connections with existing LP-based methods by usingthe same geometrical approach to rederive them under thedeterministic model of the input sequence. We comment thatthe LP-based algorithms presented here are not identical to thestochastic versions presented in the literature, although they areclosely related. LP-based channel estimators can be classifiedinto one-step and multistep linear prediction algorithms.

ZHAO AND TONG: ADAPTIVE BLIND CHANNEL ESTIMATION BY LEAST SQUARES SMOOTHING 3005

1) One-Step LP Algorithm:We consider here the LP-based approach proposed by Abed-Meraimet al. in [1]and [3] to which we refer as the linear prediction-leastsquares (LP-LS) algorithm. Instead of projecting into

, let us consider projecting into, which contains only the past data. We then have,

again using the isomorphic relation

(28)

from which can be obtained directly. By treatingas an estimate of , the problem becomes conven-

tional channel estimation with known channel input. The leastsquares criterion can then be employed to estimate the wholechannel, which is the reason that we refer to this one-step LPalgorithm as LP-LS. Note that if , which isasymptotically true for uncorrelated zero-mean sequence, wehave (see also Fig. 2). Hence, this approachprovides consistent estimates for white inputs. Unfortunately,for a finite sample size, , which causes theloss of the finite sample convergence property in the LP-LSapproach.

3) Multistep LP Algorithm: In contrast to the LP-LS ap-proach, the multistep LP approach proposed by Gesbert andDuhamel [4] works simultaneously on predictions of severalfuture data samples. By projecting into , we havethe prediction error equations

(29)

For the uncorrelated zero-mean input sequence, we have,asymptotically

(30)

Interestingly, the above equation was also used by Slock andPapadias in their extension of one-step prediction to-stepprediction [17]. However, these equations were not exploitedjointly in [17]. Gesbert and Duhamel treated the above as atriangular system and constructed the error differentials

... (31)

The identification of from becomes a familiar problem.Note that unlike the one-step approach, the entire channelparameter is identified at once.

The multistep LP approach again relies on the uncorrelate-ness of the input sequence, which, for the same reason as in theLP-LS approach, causes the loss of finite-sample convergenceproperty.

IV. A DAPTIVE LEAST SQUARES

SMOOTHING CHANNEL ESTIMATION

In this section, we develop an adaptive LSS algorithm withunknown channel order. The data structure for a variablesmoothing window size is first specified. The adaptive least

squares smoothing channel estimation and order detectionalgorithm is then presented.

A. Data Structure and Order Detection

In contrast with the data structure given in (20) where thesmoothing window size is fixed, the data structure for anarbitrary smoothing window size is now considered. Supposethat an upper-bound of the channel order is known. Fora fixed smoothing order and a variable smoothing windowsize , define the overall data matrix as

...

...

...

(32)

(33)

from which we have defined the future data matrix ,the current data matrix , and the past data matrix .The future-past data matrix and the future-current datamatrix are also defined. Compared with the ones definedin (20), we can see that except for and , the data matricesdefined here are functions of the variable smoothing windowsize . We emphasize that in this particular data structure,

and are independent of. This property leads to aconvenient implementation of A-LSS, which will be discussedin Section IV-B.

Similar to the case presented in Section III-B2, whenand has linear complexity greater than

, in the noiseless case, we have the followingrelation among the matrices defined in (32) and the variousspaces:

(34)

(35)

In parallel to the known channel order case in (22), weconsider channel identification by a rank-1 decomposition of

. Since

(36)

the rank-1 condition on requires that the orthogonalcomplement of in has the dimension 1. Comparingthe input subspaces in (34) and (35), we can see that thisrequirement on can only be met when . When

, equals ; there is no projection (smoothing)

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Fig. 5. Isomorphism between input and output subspaces forl 6= L + 1.

error. The isomorphism between the input and output spacesin this case is shown in Fig. 5(a). The case whenis shown in Fig. 5(b). Here, are not in .Since each of these input vectors not lying in contributesto the smoothing error, we have the formulation of , asgiven in the following theorem.

Theorem 1: Let be the least squares smooth-ing error. With the data structure specified in (32), we have,in the absence of noise

......

.. ....

...

(37)

Theorem 1 holds the key to the adaptive least squaressmoothing channel estimation and order detection algorithm.For order detection, we consider the energy of the

smoothing error defined as . Theorem 1 impliesthat in the noiseless case, is zero when . When

, jumps to a value related to the power of thechannel and it increases (asymptotically) with, as shown inFig. 6. Hence, the channel ordercan be detected by varying

from 1 to and comparing the energy of the smoothingerror at each with a certain threshold.

B. Adaptive Least Squares Smoothing

1) Key Components:When the channel order is unknown,there are three key components involved in the A-LSS ap-proach to channel estimation. First, the smoothing errorsat each smoothing window size need tobe calculated. Then, based on Theorem 1, the energy of these

Fig. 6. Energy of the smoothing error at different smoothing window sizel.

Fig. 7. Main structure of A-LSS.

smoothing errors is compared with a threshold to detect thechannel order. Finally, with the detected channel order, thechannel is obtained from .

Our goal here is to develop an algorithm to implement theabove three components while simultaneously satisfying therequirements in convergence speed, adaptivity, and complex-ity. Since the LSS approach has the finite sample convergenceproperty and adaptivity is a characteristic feature of linear leastsquares based algorithms, we next concentrate on the efficientimplementation of A-LSS.

The main computation cost of the A-LSS algorithm comesfrom the calculation of the smoothing errors at all . Thus,it is desirable to obtain the smoothing errors efficiently, whichcan be achieved by

a) calculating recursively;b) decomposing the smoothing into multistep predictions

followed by linear projections;

ZHAO AND TONG: ADAPTIVE BLIND CHANNEL ESTIMATION BY LEAST SQUARES SMOOTHING 3007

Fig. 8. Lattice filter for multistep prediction.

c) sharing the same multistep predictions among the calcu-lation of the smoothing errors at all.

Implementing smoothing by multistep predictions enables theuse of lattice filters that are modular, computationally efficient,and robust to round-off errors. Details are presented as follows.

2) Key Decomposition:In A-LSS, smoothing is decom-posed into multistep predictions and linear projections basedon the following basic lemma.

Lemma 1: For a matrix and a partitioned matrixwith , the least squares projection error ofinto

is given by

(38)

In words, the projection error of into is equal to theprojection error of into , where and areprojection errors of and into , respectively.

Proof: In , is the orthogonal complementof . Therefore

(39)

which leads to

(40)

Based on the partition of given in (33), weapply Lemma 1 to the smoothing error

(41)

where and are the multistep prediction errorsof and by the past data , respectively. Thesmoothing error is then obtained by projectinginto the row space of , which implies that smoothing isdecomposed into multistep predictions and linear projections.Although (41) appears to imply that separate linear predictorsare required for different, it turns out that with the special data

structure defined in (32), only a single predictor is necessary.This is because from (32), we have

(42)

where both and are independent of.3) Main Structure: Based on the decomposition presented

above, A-LSS identifies the channel in three steps, as shownin Fig. 7. Specifically, the multistep predictor generates the

prediction errors , which are independent of.In the second step, the smoothing errors at all possibleareobtained recursively, and the channel order is detected. Finally,with the detected channel order, the channel is estimatedfrom the smoothing error with the approach given in(15) to meet the requirement in adaptivity. The implementationof these three steps is discussed below.

a) Multistep predictions:A lattice filter shown in Fig. 8is used to obtain the prediction errors of the future and currentdata by the past data . Various adaptive algorithms forlattice filters can be applied here; see [10], [12], and [15].Besides the appealing properties mentioned in Section IV-B1,the order-recursive property of lattice filters can be very usefulin saving computation cost for the unknown channel-ordercase. Specifically, when the channel orderis unknown, thesmoothing order (also the prediction order here) has to bedetermined based on the upper-bound. For example, when

, we choose . If is a poor bound of the chan-nel order, may be much larger than necessary, which leadsto a higher computation cost. Because of the order-recursiveproperty of the lattice filter, for a detected channel order

, only the first prediction errors ( )

at the th lattice stage are required. This implies that thecomputation involved in the latter stages and the

joint estimation ( ) is saved withno structural change.

b) Projections and order detection:From the predictionerrors , the smoothing errors at each smoothing

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Fig. 9. Recursive projection and order detection.

window size are obtained in this step forthe order detection and channel estimation.

An approach to obtaining recursively (both in andin time) by applying the recursive modified Gram–Schmidt(RMGS) algorithm [11] is presented with a simple exampleshown in Fig. 9. Suppose that the channel order is upperbounded by , and we choose the smoothing order

. From the prediction errors obtainedby the multistep predictor, the smoothing errors at

are calculated recursively. As shown in Fig. 9, byapplying Lemma 1 first to and ,we have

(43)

where denotes the partitioned matrix .

Applying Lemma 1 next to and, we have

(44)

which is the th row block of . Similarly, andcan be obtained. In general, we have the following recursion

in :

(45)

where has been partitioned into withdefined as the first row block in . This

recursion in arises naturally from the data structure givenin (32).

Once all are obtained, the channel order is detected byan energy detector. Since the energy of is calculatedfor the channel order detection, the index () of the row withthe maximum 2-norm in [see (15)] can be easilyobtained. This information will be used in the next step toestimate the channel from .

Finally, we want to emphasize that the structure shown inFig. 9 is particularly attractive for time-varying channels. Forexample, when the true channel order changes from 2 to 1, theenergy detector switches from to . The channel withthe new order 1 can be identified without structural change orextra computation. In addition, the modular structure shownin Fig. 9 is suitable for implementation using systolic arrayprocessors.

c) Recursive channel estimation:With obtained as theby-product of the energy detector in the second step, the

ZHAO AND TONG: ADAPTIVE BLIND CHANNEL ESTIMATION BY LEAST SQUARES SMOOTHING 3009

Fig. 10. Recursive channel estimation.

TABLE IALGORITHMS COMPARED IN THE SIMULATION AND THEIR CHARACTERISTICS

channel is estimated from the smoothing error bythe approach given in (15). In addition, is an asymptoticalestimate of the uncorrelated input sequence. The structure ofthis recursive channel estimator is illustrated in Fig. 10.

V. SIMULATION EXAMPLES

A. Algorithm Characteristics and Performance Measure

Simulation studies of the proposed A-LSS algorithm as itis compared with existing techniques listed in Table I arepresented in this section. We remark that only A-LSS haseasy adaptive implementations while still preserving the finitesample convergence property.

Algorithms were compared by Monte Carlo simulationusing the normalized root mean square error (NRMSE) as aperformance measure. Specifically, NRMSE was defined as3

NRMSE (46)

where was the estimated channel from theth trial,and was the true channel. Noise samples were generatedfrom i.i.d. zero mean Gaussian random sequences, and thesignal-to-noise ratio (SNR) was defined as

SNR (47)

The input to a single-input and 2-output linear FIR systemwas generated from an i.i.d. binary phase shift keying (BPSK)sequence.

B. A Second-Order Channel

A second-order channel first used by Hua in [7] is consid-ered here. There are several reasons to consider this channel.

3The inherent ambiguity was removed before the computation of NRMSE.

First, this channel allows us to study the location of zeros andhow they affect the performance. Second, Hua showed that theCR algorithm [22] and SS algorithm [14], along with the two-step maximum likelihood (TSML) algorithm [7], approach tothe Cramer–Rao lower bound.4 By comparing with CR(SS),we can evaluate the efficiency of the algorithms listed inTable I.

The second-order channel used by Hua in [7] is specifiedby a pair of roots on the unit circle. The channel impulseresponse is given by

(48)

where are the angular positions of zeros on the unit circle.The relation between the zeros of the two subchannels isspecified by , where is the angular distancebetween the zeros of the two subchannels (common zerosoccur when is zero).

C. Performance and Robustness Comparison

The left side of Fig. 11 shows the performance of algorithmslisted in Table I against SNR. A well-conditioned channel wasused with , as in [7]. All the deterministicalgorithms (CR, SS, B-LSS, and A-LSS) had comparableperformance. In fact, it was shown in [7] that CR and SSapproached the CRB even at a very low SNR for this channel.The LP-based algorithms (MSP and LP-LS) leveled off asSNR because of the loss of finite sample convergenceproperty. We noticed that B-LSS had the same performanceas CR (SS), but there was a gap between the performanceof A-LSS and B-LSS at low SNR. This gap was causedby the prewindowing problem in A-LSS; several symbolswere discarded in the transient stage. Thus, the number ofthe effective symbols used by A-LSS was less than thatused by the batch algorithms, which led to the performancedegradation. This performance gap will diminish as SNRor more observation samples become available.

An ill-conditioned channel was used to compare the ro-bustness of different algorithms with respect to the loss ofchannel diversity. With , zeros of thetwo subchannels were very close to each other. In fact, thecondition number of the filtering matrix was around 1.6104

in this case. The right side Fig. 11 shows that CR andSS performed rather poorly for this ill-conditioned channel.Evidently, B-LSS and A-LSS performed considerably betterthan all other algorithms. This improvement in robustness isprobably because the input subspace may still be well ap-proximated by the output subspace when the channel diversityassumption does not hold.

D. Tracking of the Channel Order and Parameters

In this simulation, A-LSS was applied to a case when bothchannel order and channel parameters had a sudden change.The initial channel was given in (48) with

. At time , both the channel order and channelparameters were changed by adding zeros

4In [7], the CRB was derived based on a normalization that differs fromthe one used in this paper.

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Fig. 11. Performance and robustness comparison (100 Monte Carlo runs, 200 input symbols).

Fig. 12. Channel order and parameter tracking performance (SNR= 30 dB).

to the two subchannels, respectively. The left side of Fig. 12shows the energy of the smoothing error before and afterthe channel variation (the energy of the smoothing error wascalculated every 10 symbols). From Fig. 12(a), we can see thatbefore the channel changed, the energy of the smoothing errorwas around zero at and was relatively largeat , as predicted by Theorem 1. Fig. 12(b)–(d)are the snapshots of the smoothing error energy at 160,170, and 180, respectively. We can see that the energy of

decreased to zero within 30 samples. At 180, thenew channel order can be accurately detected. Consequently,the channel is estimated from instead of . Neitherstructural change nor extra computation is involved in thisprocess. The NRMSE convergence of A-LSS is shown in theright side of Fig. 12, where A-LSS tracked the channel orderand parameters nicely.

D. Order Detection for Multipath Channels

In order to evaluate the applicability of the proposed orderdetection and channel estimation algorithm, we present here

the simulation study of the performance of A-LSS with amultipath channel. The channel we used is a fifth-order 4-rayraised-cosine channel as shown in the top left of Fig. 13 withthe even and odd samples corresponding to the two subchan-nels. The upperbound of the channel order was assumed to be6. The energy of the smoothing error at different smoothingwindow size is shown in the top right of Fig. 13. We cansee that the energy of the smoothing error jumped to a largevalue at the smoothing window size . Based on this fact,the energy detector estimated the channel order as 1. In thebottom left of Fig. 13, we also plotted the relative incrementof the smoothing error energy at two consecutive smoothingwindow sizes, and we chose maximizing the relative incrementas an alternative criterion for the order detection. The sameestimated channel order was obtained. As shown inthe bottom right of Fig. 13, among all the possible channelorders (from 0 to ), A-LSS gave the best estimate of thechannel at the detected order. In fact, as pointed out in [20],it is perhaps not wise to estimate the small head and tailtaps in the multipath channel. Instead, it is better to find thechannel order as well as its impulse response that matches the

ZHAO AND TONG: ADAPTIVE BLIND CHANNEL ESTIMATION BY LEAST SQUARES SMOOTHING 3011

Fig. 13. Order detection and channel estimation with multipath channel (SNR= 30 dB, 100 Monte Carlo runs, 200 input symbols).

data in some optimal way. In the case here, with the channelorder detected as 1, A-LSS captured the four major taps ofthe channel impulse response while ignoring the small headand tail taps.

VI. CONCLUSION

A least squares smoothing (LSS) approach to blind channelestimation based on the isomorphic relation between the inputand output subspaces is presented. This approach convertsthe channel estimation into a linear least squares smoothingproblem that can be solved (order and time) recursively. Sincethe input sequence is modeled as a deterministic signal, thisapproach preserves the finite-sample convergence property notpresent in LP-based approaches.

Based on the LSS approach, an adaptive channel estimation(A-LSS) algorithm has been proposed. Compared with thebatch algorithms such as SS and CR, the A-LSS algorithmis efficient in both computation and VLSI implementation.Because smoothing is computationally more expensive thanprediction, A-LSS has higher complexity than the LP-LS andthe MSP approaches, which is a price we paid for the finitesample convergence property. The A-LSS algorithm does notassume the channel order asa priori information. The orderdetector and the recursive property enable A-LSS to detectand track the channel order without structural change or extracomputation. Although separate order detection techniques canbe applied to the deterministic batch algorithms such as SS andCR, their ability to track the channel order variation demandshigh implementation cost.

The future work involves the theoretical determination ofthe threshold for the channel order detection. Maximizingthe relative increment in the smoothing error energy at twoconsecutive smoothing window sizes may be considered as analternative criterion.

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[9] S. Y. Kung, T. Kailath, and M. Morf, “A generalized resultant matrixfor polynomial matrices,” inProc. IEEE Conf. Decision Contr.,1976,pp. 892–895.

[10] H. Lev-Ari, “Modular architectures for adaptive multichannel latticealgorithms,”IEEE Trans. Signal Processing,vol. 35, pp. 543–552, Apr.1987.

[11] F. Ling, D. Manolakis, and J. G. Proakis, “A recursive modifiedGram–Schmidt algorithm for least-squares estimation,”IEEE Trans.Acoust., Speech, Signal Processing,vol. ASSP-34, pp. 829–836, Aug.1986.

[12] F. Ling and J. G. Proakis, “A generalized multichannel least squareslattice algorithm based on sequential processing stages,”IEEE Trans.Acoust., Speech, Signal Processing,vol. ASSP-32, pp. 381–390, Apr.1984.

[13] H. Liu and G. Xu, “Closed-form blind symbol estimation in digital com-munications,”IEEE Trans. Signal Processing,vol. 43, pp. 2714–2723,Nov. 1995.

[14] E. Moulines, P. Duhamel, J. F. Cardoso, and S. Mayrargue, “Subspace-methods for the blind identification of multichannel FIR filters,”IEEETrans. Signal Processing,vol. 43, pp. 516–525, Feb. 1995.

[15] A. H. Sayed and T. Kailath, “A state-space approach to adaptive RLSfiltering,” IEEE Signal Processing Mag.,vol. 11, pp. 18–60, July 1994.

[16] D. Slock, “Blind fractionally-spaced equalization, perfect reconstruc-tion filterbanks, and multilinear prediction,” inProc. ICASSP Conf.,Adelaide, Australia, Apr. 1994.

[17] D. Slock and C. B. Papadias, “Further results on blind identificationand equalization of multiple FIR channels,” inProc. Int. Conf. Acoust.Speech Signal Process.,Detroit, MI, Apr. 1995, pp. 1964–1967.

[18] L. Tong and J. Bao, “Equalizations in wireless ATM,” inProc. 1997Allerton Conf. Commun., Contr. Comput.,Urbana, IL, Oct. 1997, pp.64–73.

[19] L. Tong, G. Xu, B. Hassibi, and T. Kailath, “Blind identification andequalization of multipath channels: A frequency domain approach,”IEEE Trans. Inform. Theory,vol. 41, pp. 329–334, Jan. 1995.

3012 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 11, NOVEMBER 1999

[20] L. Tong and Q. Zhao, “Joint order detection and channel estimation byleast squares smoothing,”IEEE Trans. Signal Processing, vol. 47, pp.2345–2355, Sept. 1999.

[21] A. van der Veen, S. Talwar, and A. Paulraj, “A subspace approachto blind space-time signal processing for wireless communication sys-tems,” IEEE Trans. Signal Processing,vol. 45, pp. 173–190, Jan. 1997.

[22] G. Xu, H. Liu, L. Tong, and T. Kailath, “A least-squares approach toblind channel identification,”IEEE Trans. Signal Processing,vol. 43,pp. 2982–2993, Dec. 1995.

Qing Zhao received the B.S. degree in electri-cal engineering in 1994 from Sichuan University,Chengdu, China, and the M.S. degree in 1997 fromFudan University, Shanghai, China. She is nowpursuing the Ph.D. degree at the School of ElectricalEngineering, Cornell University, Ithaca, NY.

Her current research interests include wirelesscommunications and array signal processing.

Lang Tong (S’87–M’91), for a photograph and biography, see this issue, p.2999.

IEEE Transactions on Consumer Electronics, Vol. 50, No. 4, NOVEMBER 2004

Contributed Paper Manuscript received August 1, 2004 0098 3063/04/$20.00 © 2004 IEEE

1026

Blind Adaptive Channel Equalization Using Multichannel Linear Prediction-Based Cross-Correlation Vector Estimation

Kyung Seung Ahn, Student Member, IEEE, Juphil Cho, and Heung Ki Baik, Member, IEEE

Abstract — Blind equalization of transmission channel is important in communication areas and signal processing applications because it does not need training sequences, nor does it require a priori channel information. In this paper, an adaptive blind MMSE channel equalization technique based on second-order statistics is investigated. We present an adaptive blind MMSE channel equalization using multichannel linear prediction error method for estimating cross-correlation vector. They can be implemented as RLS or LMS algorithms to recursively update the cross-correlation vector. Once cross-correlation vector is available, it can be used for MMSE channel equalization. Unlike many known subspace methods, our proposed algorithms do not require channel order estimation. Therefore, our algorithms are robust to channel order mismatch. Performance of our algorithms and comparison with existing algorithms are shown for real measured digital microwave channel.1

Index Terms — Blind channel equalization, decision delay, intersymbol interference, prediction error filter.

I. INTRODUCTION

Multipath propagation appears to be a typical limitation in mobile digital communication where it leads to severe intersymbol interference (ISI). The classical techniques to overcome this problem use either periodically sent training sequences or blind techniques exploiting higher order statistics (HOS) [1]. Adaptive equalization using training sequence wastes the bandwidth efficiency but in blind equalization, no training is needed and the equalizer is obtained only with the utilization of the received signal. A well-known HOS-based algorithm is constant modulus algorithm (CMA). A main disadvantage of this algorithm is the possibility of local convergence [2]. The fractionally-spaced CMA (FS-CMA) was shown in [3] to guarantee global convergence with finite length equalizers. Since the seminal work by Tong et al. the problem of estimating the channel response of multiple FIR channel driven by an unknown input symbol has interested many researchers in the signal processing areas and communication fields [4], [5].

Kyung Seung Ahn is with the Department of Electronic Engineering,

Chonbuk National University, Jeonju, Korea (e-mail: [email protected]). Juphil Cho is with the Electronics and Telecommunications Research

Institute (ETRI), Daejeon, Korea. Heung Ki Baik is with the Division of Electronics & Information

Engineering, Electronics & Information Advanced Technology Research Center, Chonbuk National University, Jeonju, Korea.

For the most part, algebraic and second-order statistics (SOS) techniques have been proposed that exploit the structural techniques (Hankel, Toeplitz matrix, et al.) of the single-input multiple-output (SIMO) channel or data matrices. The information on channel parameters or transmitted data is typically recovered through subspace decomposition of the received data matrix (deterministic method) or that of the received data correlation matrix (stochastic method). Although very appealing from the conceptual and signal processing techniques point of view, the use of the aforementioned techniques in real world applications faces serious challenges. Subspace-based techniques lay in the fact that they relay on the existence of numerically well-defined dimensions of the noise-free signal or noise subspaces. Since these dimensions are obviously closely related to the channel length, subspace-based techniques are extremely sensitive to channel order mismatch [6], [7].

The prediction error method (PEM) offers an alternative to the class of techniques above. PEM, which was first introduced by Slock et al. and later refined by Meraim et al. exploited the independent, identically distributed (i.i.d.) property of the transmitted symbols and applies a linear prediction error filter on the received data [7], [8]. The PEM offers great practical advantages over most other proposed techniques. First, channel estimation using the PEM remains consistent in the presence of the channel length mismatch. This property guarantees the robustness of the technique with respect to the difficult channel length estimation problem. Another significant advantage of the PEM is that it lends itself easily to a low-cost adaptive implementation such as adaptive lattice filters. But the delay cannot be controlled with existing one-step linear prediction method [8]−[11]. These algorithms calculate ZF or MMSE equalizers based on channel identification.

In this paper, we propose adaptive blind MMSE channel equalization algorithm that is less sensitive to the equalizer order length. This paper makes three results. First, we estimate the multichannel prediction-based cross-correlation vector between the channel output and the transmitted signal. Estimated cross-correlation vector can then be used to calculate the MMSE blind equalizer. Second, we develop an efficient algorithm to perform this cross-correlation vector tracking. A simple iterative algorithm can be used to find the cross-correlation vector. Third, we derive LMS and RLS type adaptive algorithms for MMSE blind equalization. Most notations are standard: vectors and matrices are boldface small and capital letters, respectively; the matrix transpose, the

K. S. Ahn et al.: Blind Adaptive Channel Equalization Using Multichannel Linear Prediction-Based Cross-Correlation Vector Estimation 1027

h0(n)

s(n)

hP-1(n)

...

Σ

v0(n)

Σ

vP-1(n)

x0(n)

xP-1(n)

...

a0(n)

aP-1(n)

g0(n)

gP-1(n)

Σ )(ˆ Dns −

Fig. 1 The multichannel representation of a T P -spaced equalizer.

Hermitian, and the Moore-Penrose pseudoinverse are denoted by ( )T⋅ , ( )H⋅ , and ( )⋅ + , respectively; PI is the P P× identity matrix; [ ]E ⋅ is the statistical expectation. This paper is organized as follows. In section II, we present models for the channel and blind MMSE equalizer, and we introduce the concepts of signal vectors and matrices. In section III, we discuss multichannel linear prediction problem for cross-correlation vector estimation. Simulation results and performance comparisons of our proposed algorithms with some well-known existing algorithms are presented in section IV. We conclude our results in section V.

II. PROBLEM FORMULATION Let ( )x t be the signal at the output of a noisy

communication channel

( ) ( ) ( ) ( )k

x t s k h t kT v t∞

=−∞

= − +∑ (1)

where ( )s k denotes the transmitted symbol at time kT , ( )h t denotes the continuous-time channel impulse response, and

( )v t is additive noise. The fractionally-spaced discrete-time model can be obtained either by time oversampling or by the sensor array at the receiver. The oversampled single-input single-output (SISO) model results SIMO model as in Fig. 1. The corresponding SIMO model is described as following

1

0( ) ( ) ( ) ( )

( ) ( ), 0, , 1

L

i i ik

i i

x n h n s n k v n

a n v n i P

−

=

= − +

= + = −

∑L

(2)

where P is the number of subchannel, and L is the maximum order of the P subchannel.

Let

0 1

0 1

0 1

( ) [ ( ) ( )]

( ) [ ( ) ( )]

( ) [ ( ) ( )] .

TP

TP

TP

n x n x nn h n h nn v n v n

−

−

−

=

=

=

xhv

L

L

L

(3)

We represent ( )ix n in a vector for as

1

0( ) ( ) ( ) ( ).

L

kn s k n k n

−

=

= − +∑x h v (4)

Stacking N received vector samples into an ( 1)NP × vector, we can write a matrix equation as ( ) ( ) ( )N Nn n n= +x Hs v (5) where H is a ( )NP N L× + block Toeplitz matrix, ( )ns is

( ) 1N L+ × , ( )N nx and ( )N nv are 1NP × vectors as following.

( ) [ ( ) ( 1)]( ) [ ( ) ( 1)]

( ) [ ( ) ( )]

T

T T TN

T T TN

n s n s n L Nn n n Nn n n

= − − += − +

=

sx x xv v v

L

L

L

(6)

and

(0) ( 1)

.(0) ( 1)

L

L

− = −

h h 0H

0 h h

L LM O M O M

L L (7)

We assume the following in this paper.

A1) The input sequence ( )s n is zero-mean and white with variance 2

sσ . A2) The additive noise ( )v n is stationary with zero-mean

and white with variance 2vσ .

A3) The sequence ( )s n and ( )v n are uncorrelated. A4) The matrix H has full rank, i.e., the subchannels ( )ih n

have no common zeros to satisfy the Bezout equation [4].

A5) The dimension of H obeys 1NP N L> + − .

Consider an FIR linear MMSE equalizer shown in Fig. 1, where ( )ig n for 0, 1, , 1i P= −L is the order N equalizer of the ith subchannel and the symbol is estimated from ˆ( ) ( ).H

Ns n d n− = g x (8) According to (5)−(7), ( )N nx has nonzero correlation with only ( ), , ( 1)s n s n L N− − +L . Therefore, decision delay d is usually in the interval [0, 1]N L+ − . For finite SIMO channels, MMSE equalizer of the finite length can be found if assumption A4) holds and the equalizer length N L≥ [8]. An MMSE equalizer minimizes the cost function

2MMSE ˆ( ) ( ) .J E s n d s n d = − − − (9)

MMSE blind equalizer with delay d is given by

2

MMSE arg min ( ) ( ) .HNE n s n d = − − g

g g x (10)

The minimum MSE filter to estimate ˆ( )s n d− is the solution to the Wiener-Hopf equation [12]

MMSE( ) ( ) ( ) ( ) .HN N NE n n E n s n d∗ = − x x g x (11)

Using assumptions A1)−A3), we can write the exact correlation matrix ( )N nx as

2 2( ) ( )H HN N s vE n n σ σ = = + R x x HH I (12)

where the cross-correlation vector equals

2( ) ( ) ( ).N sE n s n d dσ∗ − = x H (13)

where ( )dH denotes the ( 1)thd + block column of the channel convolution matrix H [12]. Based on assumption that

( )s n the unit variance, the blind MMSE equalizer with decision delay d is given by MMSE ( ).d= +g R H (14)

IEEE Transactions on Consumer Electronics, Vol. 50, No. 4, NOVEMBER 2004

1028

The minimum MSE filter to estimate ˆ( )s n d− is the solution to the Wiener-Hopf equation [12]

III. PROPOSED ALGORITHMS

A. Multichannel Linear Prediction Consider the noise-free case. For convenience, we can

rewrite (5) as

[ ]1

1 2 3 2

3

( )( ) ( ) ( )

( )N

nn n n

n

= =

sx Hs H H H s

s (15)

where 1H is of dimension NP d× , the 1NP × vector 2H is the ( 1)thd + column of H , and the last part of H is denoted by 3H with dimension ( 1)NP N P d× + − − [13].

An 1NP × multichannel linear prediction error vector can be obtained as [13]

1 1 1 1( )( ) ( )( )

N

M

nn nn d = − = −

xf I P H sx (16)

where 1P is an NP MP× matrix. ( )N nx is defined in (6), and ( )M n d−x is defined as

( )

( ) .( )

M

n dn

n d

− = −

xx

xM (17)

The optimal 1P is obtained by minimizing as following cost function

1 1 1tr ( ) ( ) .HJ E n n = f f (18)

Letting the partial derivative of (18) with respective to 1P equals to zero as following

11

1

( ) ( ) ( ) ( ) =0.HN M M M

J E n n d n d n d∂ = − − + − − ∂x x P x x

P (19)

We get as

{ } [ ]1 ( ) ( ) ( ) ( )HM M N ME n d n d E n n d = − − ⋅ −

+P x x x x (20)

Consider another multichannel linear prediction error problem

12 2 1 2

2

( ) ( )( ) .( 1) ( )N

M

n nn n d n = − = − −

x sf I P H Hx s (21)

The proof is again provided in [13]. Compared with (18), (19), and (20), we can know that the optimal 2P is obtained as following

{ } [ ]2 ( 1) ( 1) ( ) ( 1)HM M N ME n d n d E n n d = − − − − ⋅ − −

+P x x x x

(22) In order to consider 2H , we can compute

2 1 2 2( ) ( ) ( ) ( ).n n n n= − =f f f H s (23)

The transmitted signal with decision delay d is written as

[ ]22

2 2

( ) ( ).H

Hs n d n− =H 0 H 0 s

H H (24)

Substitute (8) and (24) back to (9), we get

MMSE MMSE 2 2 MMSE MMSE MMSE1 H H HJ = − − +g H H g g Rg (25) Then, minimizing MMSEJ yields

MMSE 2 ( )d= =+ +g R H R H (26) From (23), we know that

2 22 2( ) ( ) ( ) ( )H H H

s sE n n d dσ σ = = f f H H H H (27)

Then, the rank of the matrix [ ( ) ( )]HE n nf f is one. Therefore, ( )dH is the singular vector corresponding to the largest

singular value of matrix [ ( ) ( )]HE n nf f . Substituting this estimated cross-correlation vector into (26), MMSE blind equalizer is obtained. If the decision delay d equals channel length, then we can obtain channel coefficient since 2H is just the channel coefficient vector.

B. Adaptive Implementation We propose the adaptive algorithms for updating the multichannel linear prediction error filter coefficients. Two multichannel linear prediction problems are required in estimating the MMSE blind equalizer. We are required to compute the multichannel prediction matrices and to estimate the multichannel prediction errors and in (16) and (20). In order to fast convergence, we can use the RLS algorithm to update the multichannel linear prediction as following

Compute multichannel linear prediction error vector:

1 1

2 2

( ) ( ) ( 1) ( )

( ) ( ) ( 1) ( 1).N M

N M

n n n n dn n n n d

= − − −

= − − − −

f x P xf x P x

(28)

Compute Kalmna gain:

11

1 11

12

2 12

( 1) ( )( )

1 ( ) ( 1) ( )

( 1) ( 1)( ) .

1 ( 1) ( 1) ( 1)

MHM M

MHM M

n n dnn d n n d

n n dnn d n n d

λλ

λλ

−

−

−

−

− −=

+ − − −

− − −=

+ − − − − −

Q xKx Q x

Q xKx Q x

(29)

Update inverse of the correlation matrix:

1

1 1 1 1

12 2 2 2

( ) ( 1) ( ) ( ) ( 1)

( ) ( 1) ( ) ( 1) ( 1) .

HM

HM

n n n n d n

n n n n d n

λ

λ

−

−

= − − − − = − − − − −

Q Q K x Q

Q Q K x Q (30)

Update multichannel linear prediction coefficients matrix:

1 1 1 1

2 2 2 2

( ) ( 1) ( ) ( )

( ) ( 1) ( ) ( ).

H

H

n n n n

n n n n

= − +

= − +

P P f K

P P f K (31)

Compute another prediction error vector: 1 2( ) ( ) ( ).n n n= −f f f (32)

The term (0 1)λ λ< ≤ is intended to reduce the effect of past values on the statistics when the filter operates in non-stationary environment. It affects the convergence speed and the tracking accuracy of the algorithm [14]. From the covariance matrix of ( )nf in (27), its estimation of adaptive manner is given by ( ) ( 1) ( ) ( ).Hn n n nλ= − +F F f f (33)

K. S. Ahn et al.: Blind Adaptive Channel Equalization Using Multichannel Linear Prediction-Based Cross-Correlation Vector Estimation 1029

To update the correlation estimates recursively, we can use the sample estimator

1

0( ) ( ) ( ) ( ) ( )

( 1) ( ) ( ).

nn k H H

N N N Nk

HN N

n k k n n

n n n

λ

λ

−−

=

= +

= − +

∑R x x x x

R x x (34)

Defining †( ) ( )n n=B R and using the matrix inversion lemma (e.g., [14], p. 565A) with (34), we can write 1 1( ) ( 1) ( ) ( ) ( 1)H

Nn n n n nλ λ− −= − − −B B K x B (35) and

1

1

( 1) ( )( ) .

1 ( ) ( 1) ( )M

HN N

n nnn n n

λλ

−

−

−=

+ −B xK

x B x (36)

Notice that (35) and (36) do not require a matrix inverse problem and thereby provide a recursive method for computing ( ).nB From (26), we can use the following equation to obtain the MMSE-equalizer estimation ( ) ( ) ( ).n n d=g B H (37) The above multichannel linear prediction problems can be computed by an LMS algorithm. The first one can be updated by

1 1

2 2

( ) ( ) ( 1) ( )

( ) ( ) ( 1) ( 1).N M

N M

n n n n dn n n n d

= − − −

= − − − −

f x P xf x P x

(38)

The second one is updated by

1 1 1 1

2 2 2 2

( ) ( 1) ( ) ( )

( ) ( 1) ( ) ( 1).

HM

HM

n n n n d

n n n n d

µ

µ

= − + −

= − + − −

P P f x

P P f x (39)

C. Cross-Correlation Vector Tracking Algorithm A simple iterative algorithm known as the power method

can be used to find the largest eigenvector and its associated eigenvalue [15], [23]. Let F be a matrix with possibly eigenvalues ordered as 1 2 3 ,NPλ λ λ λ> ≥ ≥ ≥L with corresponding eigenvectors 1 2, , , .NPe e eL Let (0)e be a normalized vector that is assumed to be not orthogonal (1)e . The vector (0)e can be written in terms of the eigenvector as 1 1 2 2(0) NP NPa a a= + + +e e e eL (40) for some set of coefficient ia , where 1 0a ≠ . We define the power method recursion by

( )( 1) .( )nnn

+ = FeeFe

(41)

Then

( ) ( )( )

( ) ( )( )

( ) ( )( )

2 2

1 1 1 1

2 2

1 1 1 1

2 2

1 1 1 1

1 1 1 2

22

1 1 1 2

1 1 1 2

(0)(1)(0) (0)

(1)(2)(1) (1)

( 1)( ) .( 1) ( 1)

NP NP

NP NP

NP NP

aaNPa a

aaNPa a

nn aaNPa a

a

a

annn n

λλλ λ

λλλ λ

λλλ λ

λ

λ

λ

+ + += =

+ + += =

+ + +−= =− −

e e eFeeFe Fe

e e eFeeFe Fe

e e eFeeFe Fe

L

L

M

L

(42) Because of the ordering of the eigenvalues, as n → ∞ 1 1( ) a∞ =e e (43) which is the eigenvector of F corresponding to the largest eigenvalue. The eigenvalue itself is found by a Rayleigh quotient,

1( ) ( ) .

( )

H n nn

λ→e Fee

(44)

It should be noted that the MMSE equalizer is designed for transmitted symbol recovery at specific decision delay. Thus, different decision delay can result in different performance. A recursive form to get best decision delay is discussed in [12] and [22]. To get best decision delay choice, [12] proposes the minimizing MSE is given by MMSE ( ) 1 ( ) ( ).HJ d d d= − +H R H (45) If the transmitted symbols have constant modulus (CM) property, which is practical case in digitally modulated signal such as QAM or PSK, the best decision delayed blind equalizer can be determined by the following CM index [22]:

22

CM ( ) ( ) 1 .Hd NJ d g x n = − ∑ (46)

where dg is d-delay blind equalizer. The blind equalizer having the smallest MMSEJ or CMJ value will be considered as the best decision delayed blind equalizer. In many practical channels, it has been observed [1] that selecting

( ) 2d N L≈ + results in good performance.

IV. SIMULATION RESULTS

In this section, we use computer simulations to evaluate the performance of the proposed algorithm. We compare the performance of the proposed algorithm with some existing algorithms. The SNR is defined to be at the input to the equalizer as shown in Fig. 1.

21

0

21

0

( )SNR .

( )

Pjj

Pjj

E a n

E v n

−

=

−

=

=

∑∑

(47)

As a performance index, we estimate the residual ISI over 50 independent Monte Carlo runs. The residual ISI is defined as

IEEE Transactions on Consumer Electronics, Vol. 50, No. 4, NOVEMBER 2004

1030

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2Real part

0 5 10 15 20-0.1

0

0.1

0.2Imaginary part

Fig. 2 Real and imaginary part of the shortened real measured channel impulse response.

0 2 4 6 8 10 12 1410

-2

10-1

100

MSE and CM curves

Decision Delay

Nor

mal

ized

Val

ue

MSECM

Fig. 3 The best decision delay choice rule.

0 2 4 6 8 10 12 14 16-0.2

0

0.2

0.4

0.6

0.8

Tap

Mag

nitu

de

SNR=20dB

0 2 4 6 8 10 12 14 16-0.2

0

0.2

0.4

0.6

0.8

Tap

Mag

nitu

de

SNR=30dB

Fig. 4 Magnitude of the estimated channel under SNR = 20dB and 30dB at 50 trials, the original channel (solid line), the averaged estimated ±±±±standard deviation (dotted line), and the mean value of 50 estimates (square symbol)

0 500 1000 1500 200010

-3

10-2

10-1

100

101 ISI curves, SNR=25dB

Samples

ISI

LMS, µ=0.001LMS, µ=0.003RLS, λ=0.950RLS, λ=0.995

Fig. 5 Comparison of residual ISI for the proposed algorithm under SNR = 25dB.

212 2

0

2

( ) max ( ) ( )ISI

max ( )

Pjn jn

n

q n q n E a n

q n

−

= − =

∑ ∑ (48)

where

1 1

0 0( ) ( ) ( ).

P L

i ii j

q n g j h n j− −

= =

= −∑∑ (49)

The source symbols are drawn from a 16-QAM constellation with a uniform distribution. The noise is drawn from white Gaussian distribution at a varying SNR. The simulated channel is a length-16 version of an empirically measured 2

T -spaced ( 2)P = digital microwave radio channel with 230 taps, which we truncated to obtain a channel with 8L = . The Microwave channel chan1.mat is founded at http://spib.rice.edu/spib/ microwave.html. The shortened version is derived by linear decimation of the FFT of the full-length 2T -spaced impulse response and taking the IFFT of the decimated version (see [16] for more details on this channel). The overall channel impulse response ( )h n is shown in Fig. 2.

We consider the performance of the proposed RLS and LMS algorithms for blind MMSE equalizer. Let the equalizer order 8,N = and let the delay be 7d = . In Fig. 3, we show the MMSEJ and CMJ versus decision delay d under SNR =

30dB. Fig. 4 shows 50 estimates of the cross-correlation vector under SNR = 20dB and 30dB, respectively. We have discussed earlier that we can obtain channel coefficient vector if delay equals to channel length. From this figure, we can

K. S. Ahn et al.: Blind Adaptive Channel Equalization Using Multichannel Linear Prediction-Based Cross-Correlation Vector Estimation 1031

0 500 1000 1500 200010

-3

10-2

10-1

100

101

Samples

ISI

ISI curves

SNR=15dBSNR=20dBSNR=25dB

Fig. 6 ISI curves of the proposed RLS algorithm for different SNRs

4 5 6 7 8 9 10 11 1210

-3

10-2

10-1

100

101 Residual ISI versus different equalizer order

Equalizer order

Res

idua

l IS

I

SNR = 20dBSNR = 25dB

Fig. 7 Residual ISI curves versus the different equalizer order under SNR = 20dB and SNR = 25dB.

-2 0 2

-2

0

2

Unequalized

-1 0 1

-1

0

1

After equalization

Fig. 8 Scatter plots before and after equalization for the proposed RLS algorithm under SNR = 25dB.

0 500 1000 1500 200010

-3

10-2

10-1

100

101 ISI curves, SNR = 25dB

Samples

ISI

CMAFEPFBPEFHalfordOur LMSOur RLS

Fig. 9 ISI comparisons of existing algorithms and the proposed algorithm under SNR = 25dB.

obtain the channel estimation from the cross-correlation vector with delay corresponding to the channel length. The performance of the proposed RLS and LMS algorithms are shown in Fig. 5. It is found from the results that the proposed RLS and LMS algorithms and achieve sufficiently low ISI after 1000 samples. Fig. 6 shows the ISI curves of the proposed RLS algorithm under SNR = 15dB, 20dB, and 25dB. Fig. 7 presents the performance of the proposed RLS

algorithm after 2000 samples for several equalizer orders under SNR = 20dB and 25dB. From this figure, we can conclude that the exact order estimation is not needed in the proposed algorithm. Fig. 8 shows the received constellation and the equalized constellation at SNR = 25dB for 1000 samples. We compare the performance of the proposed algorithm with some existing algorithms: the CMA of [17], least-squares lattice (LSL) one-step forward prediction of [9] (denotes FPEF), LSL one-step backward prediction of [9] (denotes BPEF), and RLS equalizer of [18] (denote Halford). Let the all equalizer orders be 8N = for fair comparison. Fig. 9 shows the ISI curves for the proposed LMS and RLS algorithms and existing algorithms at SNR = 25dB. It is shown that the proposed algorithms perform better than the others.

V. CONCLUSION

This paper presents blind MMSE channel equalization using multichannel prediction-based cross-correlation vector estimation. A block-type algorithm is developed using the correlation matrices of the received signal. Then, we have developed the RLS and LMS type algorithm for obtaining the multichannel linear prediction error. Furthermore, we have used the iterative power method for tracking the cross-correlation vector. Our proposed algorithms do not require the exact channel order estimation and are robust to channel order mismatch in nature of linear prediction characteristics. Moreover, cross-correlation vector estimation algorithm can be used for partial or complete channel estimation via delay control. Simulation results show that the proposed algorithms outperform many existing algorithms.

REFERENCES [1] J. G. Proakis, Digital Communication, 4th ed. New York: McGraw Hill,

2000. [2] Z. Ding, R. A. Kennedy, B. D. O. Anderson, and C. R. Johnson, Jr., “Ill-

convergence of Godard blind equalizers in data communication systems,” IEEE Trans. Signal Processing, vol. 39, no. 9, pp. 1313−1327, Sept. 1991.

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[3] Y. Li and Z. Ding, “Global convergence of fractionally-spaced Godard (CMA) adaptive equalizers,” IEEE Trans. Signal Processig, vol. 44, no. 4, pp. 818−826, Apr. 1996.

[4] L. Tong, G. Xu, and T. Kailath, “Blind identification and equalization based on second-order statistics: A time domain approach,” IEEE Trans. Inform. Theory, vol. 40, no. 2, pp. 340−349, Mar. 1994.

[5] L. Tong, G. Xu, B. Hassibi, and T. Kailath, “Blind channel identification based on second-order statistics: A frequency-domain approach,” IEEE Trans. Inform. Theory, vol. 41, no. 1, pp. 329−334, Jan. 1995.

[6] E. Moulines, P. Duhamel, J.F. Cardoso, and S. Mayrargue, “Subspace methods for the blind identification of multichannel FIR filter,” IEEE Trans. Signal Processing, vol. 43, no. 2, pp. 516−525, Feb. 1995.

[7] K. Abed-Meraim, E. Moulines, and P. Loubaton, “Prediction error method for second-order blind identification,” IEEE Trans. Signal Processing, vol. 45, no. 3, pp. 694−704, Mar. 1997.

[8] C. B. Papadias and D. T. M. Slock, “Fractionally spaced equalization of linear polyphase channels and related blind techniques based on multichannel linear prediction,” IEEE Trans. Signal Processing, vol. 47, no. 3, pp. 641−653, Mar. 1999.

[9] J. Mannerkoski and D. P. Taylor, “Blind equalization using least-squares lattice prediction,” IEEE Trans. Signal Processing, vol. 47, no. 3, pp. 630−640, Mar. 1999.

[10] J. Mannerkoski, V. Koivunen, and D. P. Taylor, “Performance bounds for multistep prediction-based blind equalization,” IEEE Trans. Signal Processing, vol. 49, no. 1, pp. 84−93, Jan. 2001.

[11] D. T. M. Slock and C. B. Papadias, “Further results on blind identification and equalization of multiple FIR channels,” in Proc. ICASSP, 1995, pp. 1964−1967.

[12] J. Shen and Z. Ding, “Direct blind MMSE channel equalization based on second order statistics,” IEEE Trans. Signal Processing, vol. 48, no. 4, pp. 1015−1022, Apr. 2000.

[13] X. Li and H. Fan, “Direct estimation of blind zero-forcing equalizers based on second-order statistics,” IEEE Trans. Signal Processing, vol. 48, no. 8, pp. 2211−2218, Aug. 2000.

[14] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.

[15] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins Univ. Press, 1996.

[16] J. Endres, S. D. Halford, C. R. Johnson, and G. B. Giannakis, “Simulated comparisons of blind equalization algorithms for cold startup applications,” Int. J. Adaptive Contr. Signal Process., vol. 12, no. 3, pp. 283−301, May 1998.

[17] T. R. Treichler, I. Fijalkow, and C. R. Johnson, “Fractionally spaced equalizers: how long should they be?,” IEEE Signal Processing Mag., vol. 13, no. 3, pp. 65−81, May 1996.

[18] G. B. Giannakis and S. D. Halford, “Blind fractionally spaced equalization of noisy FIR channels: direct and adaptive solutions,” IEEE Trans. Signal Processing, vol. 45, no. 9, pp. 2277−2292, Sept. 1997.

[19] B. Yang, “Projection approximation subspace tracking,” IEEE Trans. Signal Processing, vol. 43, no. 1, pp. 95−107, Jan. 1995.

[20] P. Common and G. H. Golub, “Tracking a few extreme singular values and vectors in signal processing,” Proc. IEEE, vol. 78, no. 8, pp. 1327−1343, Aug. 1990.

[21] D. Gesbert and P. Duhamel, “Robust blind channel identification and equalization based on multi-step predictors,” in Proc. ICASSP, 1997, pp. 3621−3624.

[22] H. Luo and R. Liu, “Blind equalizers for multipath channels with best equalization delay,” in Proc. ICASSP, 1999, pp. 2511−2514.

[23] T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithms for Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 2000.

Kyung Seung Ahn (S’02) received the B.S., and M.S. degrees in the Department of Electronic Engineering from Chonbuk National University in 1996 and 1998, respectively. He is now working toward to Ph.D. degree in Department of Electronic Engineering from Chonbuk National University. His current interests, which include signal processing and communication theory, are currently focused on the field of blind channel identification, equalization, OFDM, broadband multiple antenna systems, and

wireless communications. Juphil Cho was born in Jeonju, Korea, in 1970. He received the B.S. degree in Information and Telecommunication Engineering from Chonbuk National University, Korea, in 1992, and M.S. and Ph.D degrees in Electronic Engineering from Chonbuk National University, in 1994 and 2001, respectively. Since December 2000, he has been with Electronics and Telecommunications Research Institute (ETRI). Currently, he is senior member of engineering staff at Mobile Telecommunication Research Division. His current research interests are 4G mobile communications, OFDM systems, and signal processing technology in communications.

Heung Ki Baik (S’80-M’83) received the B.S., M.S., and Ph.D. degrees in the Department of Electronic Engineering from Seoul National University in 1977, 1979, and 1987, respectively. From Jan. 1990 to Dec. 1990, he was with the Electrical Engineering, University of Utah as a visiting scholar. He has been with the Division of Electronics and Information Engineering, Chonbuk National University, Jeonju, Korea, since 1981 and is currently a professor there. He is a staff of

Electronics and Information Advanced Technology Research Center at Chonbuk National University. His research interests are in signal processing with emphasis on adaptive and nonlinear signal processing. He is a member of the KICS, IEEK, ASK, and IEEE.

182

Channel Estimation Standard and Adaptive Blind Equalizaion

Yuang Lou, Member, IEEE

Abstract - In this paper, we present a novel concept of channel estimation standard (CES) and apply a new CES error criterion to the process of an adaptive blind equalization. It is shown that the establishment of the CES contributes to the development of a practical communication scheme for approaching to the capacity of a high SNR band-limited channel without using a preamble signal training.

I. INTRODUCTION

HANNON’s capacity formula of a band-limited S G aussian channel indicates the possible perfor- mance gain of a digital communication system. Even though a coded-modulation scheme contributes to the significant improvements of a communication system, its basic requirement of an ideal Gaussian channel is hard to reach in practice. Thus, the technique of a combined adaptive equalization and coding scheme is highly desirable in order to combat the IS1 degrada- tion [l] and therefore to achieve a higher performance gain. It motivates our research on a new CES error criterion such that the higher performance gain of a communication system can be accessed through an IS1 environment without resorting to a preamble sig- nal training. The block diagram of a digital communication sys- tem is shown in Fig. 1. {Ik} stands for a transmit- ted data sequence. The overall communication sys- tem {s} consists of a transmission channel {h} for which the samples of an equivalent low-pass impulse response {hl} are unknown, and an adaptive blind equalizer { c } [5] with a coefficient set { c l } . {nk} rep-

Paper approved by Nikolms A. Zervos, the Editor for Trans- mission Systems of the IEEE Communications Society. Manuscript received April 13,1992; revised November 25, 1993; May 23, 1994. This research was supported in part by SGA Foundation for Re- search Development and in part by NASA Research Grant NAG

Yuang Lou was with the Department of Electrical and Com- puter Engineering, Northeastern University, Boston, MA 02115, USA, and is now with the Department of Radio Systems, GTE Laboratories, 40 Sylvan Road, Waltham, MA 02154, USA.

1-734.

IEEE Log Number 9410046.

0090-6778/95$4.00

resents a noise process and {yk} is an observable sig- nal sequence at the input port of {c}. {ik) is the sequence of equalized signal at the output port of the equalizer { c } . The adaptive equalizer { c } is followed by a decision device, which slices the equalized signal {fk} and makes the final decision {ik}. The purpose of an adaptive blind equalization process is to recap- ture the input data sequence { I k } with the minimum IS1 without resorting to a preamble signal training [2 - 51. Once the IS1 is minimized or in general the char- acteristics of the overall communication system (3) are under control, a combined coding technique can then be applied. Since the characteristics control of an overall communication system is a key factor to a system performance, we shall concentrate our dis- cussion on this topic. The remainder of this paper is organized as follows. A novel concept of channel es- timation standard (CES) is proposed in section 11. A convex CES cost function €1 is then constructed. In section 111, a new CES error criterion is developed, an- alyzed, and applied to an adaptive blind equalization process. Compared computer simulations of the CES adaptive blind equalizers are then presented. Section IV concludes this paper.

P I

S ( t ) = h(t) 63 c ( t )

Fig. 1. Block diagram of a communication system

11. CHANNEL ESTIMATION STANDARD

At the beginning of this section, the assumptions and definitions used in the CES development are presented.

0 1995 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995 183

(1) We assume that the input data { I k } to a trans- mission channel is a QAM signal sequence. The { I k }

sequence is ergodic with E(&} =E 0, m d E ( I p I z } = &bk kt . Here, the superscript * stands for the complex conjugate, & = E(I Ik 1') is the average power of transmitted signah, and 6 k k' denotes the Kronecker delta function. (2) The noise process nk is assumed to be additive, white Ganssisn with zero mean and variance U:. It 8s asmmed that {ak} is independent of { I k ) . (3) We assume that the transIlltssion chan- nel { h} is linear, and received signal samples { Y k } are elements of a stationary PFWBS. In the following developments, we define a time up- date average at the kth iteration such that

for k < Nob, where Nob is the finite length of an ob- servation window in time. When k 1 Nob, the time update average with a sliding window is defined as

* (2) { ' k ) - { 'k -N, ,c )

Nob < >k =< >k-1 +

Thus, the time update average over a random signal process is still a random signal process. Since the adaptive equalizer {c} and the transmission channel {h} are both linear, the overall communication sys- tem 3 = h @ c is also linear. Here, @ denotes the con- volution. In the follodring analysis, we assume that s is time invariant. Thus, the samples of a system characteristics under the 1/T sampling rate can be expressed as

(3)

where the summation indexed by I goes over the length of rui adaptive equalizer {e}. The concept of a novel estimation standard - channel estimation standard (CES) is illustrated as follows. The CES is the basis of an error measure to an adap- tive aystem, which i s d&ed by the difference between the ideal characteristics and the estimated character- istics, say the system impulse response, of an overall communication system. Thus, a CES cost function €1

can be constructcad as

In (4), (S(m)} is the ideal impulse response samples of the {SI system, N is the length of the { J } system

impulse response in the unit of input symbol duration T, and { $ k ( m ) } is the estimation set of the actual im- pulse response samples {S(m)} at the Cth iteration. Through the signal reeeption process, an adaptive equalizer is self adjusted in order to control the char- acteristics of the {$} system by minimizing the €1 cost function. Thus, a CES adaptive blidd equalization directly supports the basic requirements In the devel- opment of a TCM coded-modulation scheme. Under the situation where we can assume that E { . i i j k } is a reasonable approximation of E{I&} [5], we formu- late the estimation of {B(m)} at time t = kT such that

( 5 )

for m = 0, 1, . . .,N - 1. The {gk(m)} estimation of ( 5 ) is then substituted into (4) to complete the design of a CES cost function. The convvexity property of the CES cost function €1 is shown in [5].

III. CES ADAPTIVE BLIND EQUALIZATION

Since a CES error measure relies on the statistical properties of a transmitted signal instead of a reli- able signal reference, it forms the theoretical founda- tion of an adaptive blind equalization process. When we apply the CES concept and the €1 cost function to an adaptive blind equalization, we realize that, in order to minimize the noise influence and the resid- ual IS1 simultaneously, futither improvements on the €1 cost function is of significant importance [4, 51. Combining the CES cost function el directly with a decision-directed (D-D) MSE cost function, we have

N - 1

€2 = 2E{(1-0) cl s(m)-Sl,(m) 1' +a I . fk-fk 1'). m = O

(6) Parameter a in (6) is a pre-selected constant within the range of 0 5 Q 5 1. Even though €2 shows the merits in its simple design, it presents a weak point of using the D-D symbol { f k ) . The process_ of minimiz- ing €2 is affected by the e m r from the ( I&} sequence, which results in a larger SNR loss and a slower con- vergence speed. In order to improve the dwign of €2, we propose a new CES error criterion e such that

N - 1

€ = 2E{(1-a) I S(m)-8k(m) 1' +a I z k - f k 1') m = O

where a Statistical reference z k is set up as [5] (7)

184 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995

0 0

0

0

0 0 0

t Im Ik

0 0 IMPULSE RESPONSE OF TRANSMISSION CHANNEL I

Channel I Impulse Response M O ) -0.00467736-jO.0037418 h(1) 0.003968872+j0.028064163 h(2) -0,02245133-jO.09728 h(3) 0.798893185fj0.48644 h(4) -0.20393292+j0.25538 h(5) 0.045838-jO.06922 h(6) -0.01496+j0.0187

0 0 Euclidean Distance 0 Re

0 0

TABLE A

TABLE B

IMPULSE RESPONSE OF TRANSMISSION CHANNEL I1 Fig. 2. 16-point QAM constellation & its decision regions

for 0 < a 5 1. Several positive points of 21, are il- lustrated as follows. (1) Due to the application of Zk, the c error criterion is an efficient utilization of the CES error measure and the Euclidean distance measure. The minimization of c does not rely on the correctness of D-D symbol jk. (2) Whenever the c er- ror measure is minimized, i.e., > k ( m ) -+ S(m) , Z k is converged to f k . Thus, the mode transition is smooth between a converged CES process and the process of an MSE equalization. The convexity property of the c error criterion and the existence of its unique global minimum are investigated as follows. Define the gra- dient of a real quantity with respect to a complex vector C = ( C - L , . . . , q, . . . , CL) T as

(9) and

(10) for a finite length adaptive equalizer of 2L + 1 ad- justable complex coefficients. Here, the superscript stands for a vector transpose.

Lemma I: When a signal point ik falls far away from the predetermined decision boundaries, we have the decision symbol f k as a highly nonlinear functioc of the equalizer tap weight vector C with V c I k - = 0.

Proof: Since the D-D symbol jk is obtained by the rule of a minimum Euclidean distance on the basis of ik, the distance between any two adjacent reference points in the {fk} set makes the local property of {jk} to be a constant with respect to when Lemma I is true.

Q.E.D.

Theorem I: Under the condition of Lemma I, the c error criterion is a convex U [6] function in the vector C space. Its global minimum given at C = cpt is unique.

Proof: By Lemma I, we have the gradient of c with respect to C as

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995 185

-6. 00

Fig. 3(a) Eye pattern of received signal (Channel I with 16-point Q.4M at 30 dB/symbol)

Here, the superscript t stands for the conjugate trans- pose, and y k = ( y k + ~ , . . ., Yk,. . . , Y ~ - L ) ~ represents the vector of received signal samples. Since the ma- trix in v~[(o,c)'] is Hermitian and positive definite, the conv&ty of c with respect to C is shown. The global minimum of c can only be obtained at vcr - = 0 when C is equal to Copt.

Q.E.D. Least-mean square algorithms of the c and €2 adap- tive blind equalization are developed and discussed in [4, 51. Compared computer simulations between the r and €2 adaptive blind equalizers are shown here for a full response signaling system, i.e., S(0) = 1 and S(m) = 0 for all m # 0. The weighting factor is set at a = I and the constellation of a transmitted

2 '. QAM signal IS shown in Fig. 2. Characteristics of the two complex channel models are given in Table A and Table B, respectively. At SNR = 30 dB/symbol or equivalently at SNR = 23.9 dB/bit, the eye pat- terns of received signals from both channel I and chan- nel I1 are totally blurred. By the comparison of Fig. 3(a) and Fig. 3(b), we see that the IS1 distortions from channel I1 is much worse than that of channel I such that a severe IS1 results in a homogeneously blurred eye pattern but not only in a tilted square [5]. Fig. 4 shows the comparison of Monte Carlo simulations. The lower bound of the error rate is ob- tained from the theoretical calculation of a 16-point &AM data transmission over a Gaussian channel [l],

Fig. 3(b) Eye pattern of received signal (Channel I1 with 16-point QAM at 30 dB/symbol)

Three types of adaptive equalizers are under the sim- ulation. The triangle points are for a training MSE equalizer. The dotted and crossed points are from the processes of a CES adaptive blind equalization where the 6 and €2 error criteria are applied, respectively. The comparison shows that there are a small amount of SNR losses between the 6 and €2 processes over the channel I transmission where the IS1 distortion is relatively light. The differences shown in the output SNR become much larger between the c and €2 pro- cesses over the channel I1 transmission where the IS1 distortion is much more severe. From these compared computer simulations, we see that the convergence of a CES adaptive blind equalization sets the charac- teristics of an overall communication system 3 under control such that the average error rate in { f k } and the undesirable IS1 distortions axe minimized simul- t aneously.

w. SUMMARY A N D CONCLUSION

In this paper, a novel concept of channel estimation standard (CES) is illustratd which defines an error measure between the ideal chatacteristics and the es- timated characteristics of an overall communication system. It has been shown that a converged CES error measure supports the basic assumption in the development of a TCM coded-modulation scheme. In the paper, we illustrate the CES concept via its ap-

186 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 43, NO. 2/3/4, FEBRUARY/MARCH/APRIL 1995

L

0, G

4.00 8.00 12.00 16.00 20.00 24.00 28-00

Signal-to-Noise Ra t io ( S N R / b i t )

Fig. 4. Monte Carlo simulation on 16-point QAM systems

plication to an adaptive blind equalization process. A CES cost function is defined and its convexity is shown. The superior performance of a CES adaptive blind equalizer supports our theoretical arguments.

V. ACKNOWLEDGEMENT

The author would like to acknowledge the comments and support from John G. Proakis, M. Schetzen, C.L. Nikias and H. Raemer.

REFERENCES

[l] Proakis, John G., Digital Communications, Second Edition, McGraw-Hill Book Company, 1989.

[2] Godard, D.N., “Self Recovering Equalization and Carrier Track- ing in Two Dimensional Data Communication Systems,” ZEEE Transaction on Communications, vol. COM-28, No.11, pp.1867- 1875, November 1980.

[3] Lou, Y., “Adaptive Blind Equalization Criteria and Algo- rithms for QAM Data Communication Systems,” Proceedings of the 9th International Phoeniz Conference on Computers and Communications, pp.222-229, March 1990.

[4] Lou, Y., “Comparison of Adaptive Blind Equalizers,” ZEEE International Conference on Acoustics, Speech, And Signal Processing, San Francisco, CA, March 1992.

[5] Lou, Y., Adaptive Blind Equalization Technique for QAM Data Communications, Ph.D. Dissertation, Department of Elec- trical and Computer Engineering, Northeastern University, Boston, MA 02115, June 1992.

[6] Schetzen, M., The Volterra and Wiener Theories of Nonlinear Systems, Reprinted Edition, John Wiley & Sons, Inc., pp.504- 509, 1989.

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