International Telecommunications Symposium – ITS2002, Natal, Brazil

A PDF Estimation-Based Blind Criterion forAdaptive Equalization

Charles C. Cavalcante�, F. Rodrigo P. Cavalcanti

�and Joao Cesar M. Mota

��

UNICAMP, Campinas-SP, Brazil�UFC, Fortaleza-CE, Brazil

Abstract — A blind criterion for adaptive equalization based on proba-bility density function (pdf) estimation is proposed. The criterion measuresthe divergence of the pdf of an ideally equalized signal against the one froma parametric model resulting in a cost function that is a sort of entropy min-imization of the equalizer output signal. It is also shown a link between theconstant modulus (CM) criterion and the proposed one under certain cir-cumstances. Some convergence properties are studied and the performanceof the proposal is evaluated through simulations faced to a classical blindcriterion.

I. INTRODUCTION

�N THE DESIGN of receivers in digital communication sys-tems blind receivers are being more and more used. Since the

work of Sato [1] and lately the one of Godard [2] the study intoblind criteria has increased a lot.

The most part of proposed blind criteria take into account theBussgang methods, it means, a memoryless nonlinear functionis applied to the equalizer output aiming to produce a signal withthe same statistical properties of the transmitted ones in order tocompare it with the equalizer output [3].

Some recent works have proposed information-theoreticbased approaches for blind equalization criteria [4, 5]. Inthose papers, nonparametric estimation of the probability den-sity function (pdf) of the signal on the equalizer output is re-quired for the cost function construction, besides, a nonlinearstructure for the equalizer, such as neural networks, is used [5].Another characteristic of information-theoretical approaches istheir use in blind source separation, which has a strong link withblind equalization (deconvolution) [6]

The present proposal is information-theoretical based withparametric pdf estimation. The pdf of the ideally (perfectly)equalized signal is a priori determined through feasible assump-tions, making possible to use a parametric model that fits thesystem order. Hence, the resulting criterion is a sort of entropyminimization of the equalizer output signal.

A stochastic gradient type algorithm is derived for the up-dating of the filter equalizer. A brief analyze of the stochasticversion of the constant modulus (CM) criterion is done in orderto show a link between that criterion and the proposal.

C. C. Cavalcante is with the Digital Signal Processing for Communications(DSPCom) Lab., Dep. of Communications, Faculty of Electrical and Com-puter Engineering, Campinas State University (UNICAMP), Phone: +55 193788 3702, Fax: +55 19 3289 1395, C.P. 6101, CEP: 13083-970, Campinas-SP, Brazil. (email: charles@decom.fee.unicamp.br)

F. R. P. Cavalcanti and J. C. M. Mota are with the GTEL Lab., Dep. of Tele-informatics Engineering, Federal University of Ceara (UFC), C.P. 6007, CEP:60755-460, Fortaleza-CE, Brazil. (emails: � rod,mota � @gtel.ufc.br)

This work was supported by the Ericsson Research Brazilian Branch underthe ERBB/UFC Technical Cooperation Contract.

The used notation is given as follows. The discrete transmit-

ted sequence ��� ������� �� ������� � ������������� !�#"�$is assumed independent and identically distributed (i.i.d) andthe symbols � �� �&%�' which has cardinality ( . The chan-

nel is represented by a FIR filter )*�+�-,/. ����� ,1032 � "�$ .Additive noise is white, Gaussian, uncorrelated from thetransmitted sequence, its variance 4 �5 is given accordingthe signal-to-noise ratio (SNR) and will be denoted by6 �� �7� �389�� �:�����;89��<������ !� " $ . The equalizer,which has finite impulse response (FIR) denoted by =>�� �>��3? �� �:����� ? ��<������ !�@"A$ , is feed by the channels out-

puts BC�� �#�EDBF�� �G��89�� � where DBC�� �#� 032�H

I J . , I � ����LKM� are

the noiseless channel outputs. � and � are, respectively, thechannel and equalizer lengths. The equalizer output is denotedby NO�� �P��= $ �� �RQP�� � .

The rest of the paper is organized as follows: Section IIpresents the proposed criterion and related cost function; con-vergence properties are described in Section III; in Section IV itis shown a link between the proposed criterion and the CM ones,such as Sato and Godard; Section V presents simulation resultswith the performance evaluation of the proposed criterion and,finally, in Section VI the conclusions are stated.

II. PDF ESTIMATION-BASED BLIND CRITERION

Let = ideal an ideal linear equalizer, the output of the equalizercan be written as

NO�� �P��= $ideal QP�� �TS (1)

whereQP�� �G�VUW��� � � 6 �� � (2)

and U is the convolution matrix of the channel of dimension�YX��Z�[�&�\�L !� [7].

Then, using Equation (2) in (1), it is possible to write:

NO�� �P�*�ZUW��� � � 6 �� �]� $ = ideal

�V� $ �� �MU $ = ideal � 6 $ �� �R= ideal

�V� $ �� �^U $ = ideal_ `Ta bcideal

� 6 $ �� �R= ideal

�V� $ �� �Md ideal ��ef�� �� � �����gh� ��ef�� �TS

(3)

where d ideal is the ideal system response, g is a delay and ef�� �is a random variable (r.v.) assumed Gaussian1 [7].i

This assumption is the same in Bussgang algorithms.

International Telecommunications Symposium – ITS2002, Natal, Brazil

Equation (3) states that the pdf of the signal on the output ofthe equalizer is a mixture of equiprobable Gaussians (since thetransmitted symbols are i.i.d.) given by:�����

ideal ��N/�G� � ��� 4 � � ( � � I J � ����� � ��� NO�� �C� � I � �� 4 � � S (4)

where the � I are the possible values of � �����gh� that are alsosymbols of the transmitted alphabet ' .

Since the pdf of the equalized signal is known, it is desiredto construct a criterion that forces the adaptive filter to producesignals with the same (or almost) pdf than the ideal one. Thisleads to the well known measure of similarities between strictlypositive functions (as the pdfs), the Kullback-Leibler Divergence(KLD) [6].

In order to use the KLD it is constructed a parametric model,which is function of the filter parameters, to provide pdf estima-tion [7]. A natural choice is the same model of mixture of Gaus-sians like that one in Equation (4) where NO�� � � = $ �� �RQP�� � ,then �

��N9S 4 �� �P� � ��� 4 ��_ `Ta b� ��I J � ������� ��� NO�� �C� � I � �� 4 �� � S (5)

is the chosen parametric model where 4 �� is the variance of eachGaussian in the model.

In pattern classification field these kind of parametric func-tions, which are used to measure similarities against other func-tions, are called target functions [7].

Then, applying KLD to compare Equations (4) and (5) it has:����� �� "! ! #$� � � %�&' �)(*2 ( � ��N/�F��+ ,.-

� ��N/����N9S 4 �� ��/.0 N

�)(*2 ( � ��N/�F��+ , � � ��N/�]� 0 N �1(*

2 ( � ��N/�C�2+ ,43���N9S 4 �� �65 0 N9S (6)

where � ��N/�P� ����� ideal ��N/� .Minimize Equation (6) is equivalent to minimize only the

� 3 N9S 4 �� 5 dependent term, it means:7FP ��= �G� �98;:<+ ,<� � 3 N9S 4 �� 5 ">=

� �98@?A+ ,CBED�� � I J � ������� ��� NO�� �F� � I � �� 4 �� �GF9HJI (7)

The Fitting pdf (FP) criterion corresponds to minimize7FP ��= � (Equation (7)). Furthermore, it is known that minimize

Equation (7) corresponds to find the entropy of N if

� 3 N9S 4 �� 5 ������ideal ��N/� [8, p. 59].

A stochastic version for filter adaptation is given by:K47FP ��=>�� �]�G�

HI J � �����ML � ! �N� 5 2$OQP ! &� % &'SR ��NO�� �F� � I �4 �� � HI J � �����TL � ! �N� 5 2$OQP ! &� %�&' R QVU

=>�� �� !�P��=>�� �F��W X K47 FP ��= � I (8)

The adaptive algorithm which uses the proposed criterion willbe called Fitting pdf Algorithm (FPA). Equation (8) shows animportant property of the algorithm, it takes in account the phaseof the transmitted symbols.

The computational complexity of this algorithm is propor-tional to the computation of ( exponentials which are requiredby Equation (8). Thus, its complexity is a little higher than otherLMS-like algorithms.

Other important point is, although the ideal equalizer isknown to have infinity length, the use of the FP criterion doesnot requires a long filter to compensate the channel effect. It hasbeen observed, through simulations, the length of the equalizerfor this criterion has the same order of other blind criteria.Y The parameter 4 �� :

As shown in the previous section, the parametric model usedto update the filter coefficients is also 4 �� dependent. This pa-rameter plays an important role since it is the variance of eachGaussian in the parametric model.

Moreover, 4 �� is also important for convergence rate becauseit modifies the effective step size, it means, W eff �[Z�\%�&' . In classi-fication field this parameter is similar to the temperature one inannealing processes [7].

A numerical problem that arises with the use of the FPA isthe nonconvergence for very small values of 4 �� . This is due tothe Gaussians being very sharp and much more difficult to fit thedata on them. This model have also been observed in [9] wherethe ideal pdf of the received signal is assumed to be a mixtureof impulses and later a Gaussian mixture model is considered inorder to make the assumption more realistic and feasible.

Finding the optimum value for 4 �� parameter is still under in-vestigation.

III. CONVERGENCE PROPERTIES

The behavior study of the FP cost function is done briefly. Itis used a BPSK (Binary Phase Shift Keying) random sequencetransmitted through an AR (Autoregressive) and MA (MovingAverage) channels with one pole and one zero, respectively, overthe real axis which have the following transfer function:]

AR �_^ �P� �a`V^ 2 � S (9)

and ]MA �_^ �P� �a`V^ 2 � I (10)

For ` �;b I c , W eff � db 2 � , 4 �� �;b I and SNR � db dB, Fig. 1shows the cost function and the corresponding contour plot andtrajectories of 20 FPA simulations (with different initializations)for the AR channel. It was considered an equalizer with twocoefficients.

With the same parameters, Fig. 2 shows the cost functionand the corresponding contour plot and trajectories of 20 FPAsimulations (with different initializations) for the MA channel.Again, a two coefficients equalizer was considered.

One can easily note that the FP cost function presents globaland local minima and convergence behavior is highly initializa-tion dependent as in the constant modulus (CM) criterion. Thischaracteristic shows (as Figs. 1 and 2 illustrate) the phenomenon

International Telecommunications Symposium – ITS2002, Natal, Brazil

−2

−1

0

1

2 −2

−1

0

1

2−25

−20

−15

−10

−5

0

5

10

15

20

w1

w0

JF

P (

dB)

−2 −1.5 −1 −0.5 0 0.5 1 1.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

w0

w1

Fig. 1. (Up) FP cost function for AR channel. (Down) Corresponding contourplot and trajectories of 20 simulations ( � eff � ����������� � ��� �

).

of ill-convergence also present in the FP criterion. It worths tomention that the cost function for the CM criterion is pretty sim-ilar to the FP one. Nevertheless the presence of local minima inthe cost function, it has been observed that the global minimaare pretty close to those ones from the Wiener solution [10].

Next section is devoted to formalize mathematically the localconvergence property of the criterion observed in the simula-tions.Y Convergence in FP Criterion:

Inspired by the local convergence analysis presented in [11],which uses a mixed adaptation strategy switching from theCMA to a criterion based on a mixture of Gaussians, in thissection is presented a brief analysis that shows the local conver-gence property observed by computational simulations.

As previously stated (Section II), the equalizer inputs aregiven by Q ��UW��� � � 6 �� � . The aim is to obtain a filter as closeas possible of an optimum one represented by = opt which is con-sidered to provide = $opt UW��� �P� � �� �Wgh� where � �� �Wgh�f� � Iand ��*K�� ( . It is also considered some perturbation in the

−2

−1

0

1

2 −2

−1

0

1

2

−5

0

5

10

15

20

w0

w1

JF

P (

dB)

−2 −1.5 −1 −0.5 0 0.5 1 1.5−2

−1.5

−1

−0.5

0

0.5

1

1.5

w0

w1

Fig. 2. (Up) FP cost function for MA channel. (Down) Corresponding contourplot and trajectories of 20 simulations ( � eff � ����������� � ��� � ).optimum filter estimation, it means = � = opt ��� = , in orderto analyze its influence on the convergence when the solution isnear to a region of minimum (global or local).

In the cost function (7) it is retained only the term correspond-ing to ��� � � �� �Lgh� which the equalizer output is closer andthe other components are neglected. Thus,�

FP � ��� ������ � � !#"%$'&)(*+ ,i.-0/�132 �544 �76 896;:<� =���>#�76;?;� =�� �#@ + 44 ��A � BDCFEG ��� � � !#"%$'& -0/�1�2 �544 H � 6 896;:<� =���>#�76;?;� =�� 44 ��A � BDCFEG ��� � �544 H � 6 896;:<� =���>#�76;?;� =�� 44 ��A � > � ! � $ � EG �I H ��JK898)J H �L> �M ��JN��A � � � ! � $ �

(11)

where 4 �O indicates the average power of the transmitted se-quence and 4 �5 is the input noise variance. Hence, the gradient

International Telecommunications Symposium – ITS2002, Natal, Brazil

can also be approximate by:K47FP ��= ��� 4 �O� 4 �� -/UWU � � = � 4 �54 �O = / (12)

where% &�% &� is the inverse of the signal-to-noise ratio (SNR).

In the absence of noise (SNR ��� ) the following LMS adap-tation rule is obtained:

=>�� �� !� � =>�� �F��W 4 �O4 �� UWU � � =>�� �= opt �� �� !� � � =>�� �� !�P��= opt �� � � � =>�� �

� W 4 �O4 �� UWU � � =>�� � (13)

Since the optimum weight vector is not time instant-dependentit follows: � =>�� �� !�P��� =>�� � - � ��W 4 �O4 �� UWU � / S (14)

where�

is the identity matrix. This recursive rule converges if:b W� � 4 ��4 �O� max

(15)

where max is the greatest eigenvalue of UWU � .The result in (15) confirms our intuition, stated in Section II,

that the 4 �� strongly influences the convergence speed due to its“control” on the convergence factor.

Unfortunately, it cannot be assured the global convergenceindependently of the initialization, as in the CM criterion. Thisissue is still under study for a formal mathematical formulation.

The results presented in this section could indicate that CMand FP criteria have a link and this issue is addressed in nextsection.

IV. A LINK TO THE CM CRITERION

In the previous section it has been observed that the FP crite-rion has the same convergence characteristics than the CM one,then it seems reasonable to compare the FP criterion with it andfind some link or equivalence.

Thinking about measure of similarities between functions,other criteria have also a parametric function (target function,�

CM �M� � ) used to find the equalizer outputs. In that case, the CMcriterion is written as a measure of similarities of functions tofind some link between both criteria, the CM and FP ones.

Using the same approach to find the cost function, for the CMcriterion the following equation is given:7

CM ��= �G� �98��2+ , � � CM ��N9S]=WS � � ���PS (16)

where � chooses the criterion to Sato criterion (� � ) and Go-dard one (� � � ). But it is known that7

CM ��= �G�;8��F� � N � � ��� � � ��� S (17)

then to find the equivalent target function, Equations (16) and(17) have to be equal, resulting�

CM ��N9S]=WS � �G� ������� � � N � � ��� � � ���(�2 ( ����� � � � � � �L !� 0 � S (18)

where the term in the denominator is used to guarantee that�CM ��N9S]=WS � � has unity module. It can be then seen that the CM

criterion has the same structure for the parametric function thatthe FP one.

It has also been simulated the target functions for the FPCwith different values of 4 �� and compared them with those onesfrom the CM criterion (Sato and Godard), Fig. 3 shows the tar-get functions for BPSK modulation.

-1 ,5 -1 ,0 -0 ,5 0,0 0,5 1,0 1,5-4 ,0

-3 ,5

-3 ,0

-2 ,5

-2 ,0

-1 ,5

-1 ,0

-0 ,5

0,0

0,5

σ2

r = 0 .29

σ2

r = 0 .12

σ2

r = 0 .2

G odard

S ato

y

C M crite rion

FP crite rion

Fig. 3. Target functions for FP and CM criteria.

It is observed that for some choice of 4 �� the CM and FP targetfunctions are equivalent. It wonders that this value is modulationtype-dependent and, as mentioned in Section II, an strategy tofind the optimum value is still under investigation.

Moreover, for more complex modulation schemes, the FP cri-terion target function can better fits the idealized equalizer out-put pdf with ( Gaussian kernels. Thus, since Equation (8) cantake into account even complex symbol parts and multilevel al-phabets, it can be predicted an improved performance w.r.t. theCM criterion.

V. SIMULATION RESULTS

For performance evaluation, the FPA has been compared withthe CMA in the equalization context. A QPSK modulationtransmitted over a channel with the following impulse responseis used:

) �

!!!!"� �$# b I � I % �&#1 I ' I � �$#1 I (b I ' �&#1 I c

) ****+ I (19)

This channel has been used in some recent works to illustrate therobustness of algorithms facing a very distorsive channel [12].

As merit figure, it has been used the measure of residual in-terference (RI), defined as:

RI �� �P�

,,,, H � � - � �� � � ��.0/ �� � - � �� � � ,,,,.0/ �� � - � �� � � S (20)

where - � �� � is the 1 -th element of the vector dF�� �G�VU $ =>�� � .

International Telecommunications Symposium – ITS2002, Natal, Brazil

The simulation parameters were: filters with 30 taps for bothalgorithms with center-spiked initialization [6] and WVX&� db 2��for both algorithms and SNR = 30 dB. FPA has also used 4 �� �b I ��� . Fig. 4 shows the RI evaluation for both algorithms.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

0

2

4

6

8

10

12Residual Interference

Iterations

RI (

dB)

FPA

CMA

Fig. 4. Residual interference evolution for FPA and CMA.

One can easily note that FPA has faster convergence than theCMA, it also worths to mention that final performance is practi-cally the same for both of them with a little gain from the FPA.The faster convergence is due to the fact FPA uses more suitabletarget function that better fits the transmitted signals characteris-tics than the CMA that considers the modulus only. However, itshould be strongly emphasize that CMA requires a phase recov-ering device for correctly match the constellation with the trans-mitted one whereas the FPA could done it by itself, as shown inFigures 5 and 6.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5Constelation of FPA (last 5000 symbols)

Real part

Imag

inar

y pa

rt

Fig. 5. Equalizer output signal constellation for the FPA.

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

1

1.5Constelation of CMA (last 5000 symbols)

Real part

Imag

inar

y pa

rt

Fig. 6. Equalizer output signal constellation for the CMA.

VI. CONCLUSIONS AND PERSPECTIVES

In this paper it was proposed a new blind information-theoretic based criterion for adaptive equalization. The deriva-tion of the cost function is done by means of the estimation ofthe ideally equalized signal through a parametric model result-ing in a sort of entropy minimization of the equalizer outputsignal. Convergence properties are briefly studied in order toclarify some important points.

It has also been shown that the proposition has a link withthe constant modulus criterion when that criterion is studied asa measure of similarities between two functions.

Through simulations it has been observed that, for somecases, the FPA presents faster convergence than the CMA, thisproperty and the interesting issue of the phase recover ability ofthe FPA may indicate a plausible alternative for blind equaliza-tion.

As perspectives to this work, we note that the parameter 4 ��plays an important role on the FP cost function and deservesmuch more investigation on it. As well, the consideration ofmore complex modulations (nonconstant modulus) may providesome gain to the FPA since it considers a target function withthe number of Gaussian kernels equals the number of symbolsfrom the modulation alphabet. As another front, a multiuserversion of the algorithm is under investigation for space-timesignal detection.

ACKNOWLEDGEMENTS

The authors would like to thank Romis R. F. Attux by hisuseful comments on the convergence analysis issue.

REFERENCES

[1] Y. Sato, “A Method of Self-recovering Equalization for Multi-level Ampli-tude Modulation,” IEEE Trans. on Communications, vol. 23, pp. 679–682,June 1975.

[2] D. N. Godard, “Self-Recovering Equalization and Carrier Tracking inTwo-Dimensional Data Communication Systems,” IEEE Trans. on Com-munications, vol. COM-28, pp. 1867–1875, November 1980.

International Telecommunications Symposium – ITS2002, Natal, Brazil

[3] S. Haykin, ed., Blind Deconvolution. (Information and Systems SciencesSeries), NJ, USA: Prentice-Hall, Englewood Cliffs, 1994.

[4] J. Sala-Alvarez and G. Vazquez-Grau, “Statistical Reference Criteria forAdaptive Signal Processing in Digital Communications,” IEEE Trans. onSignal Processing, vol. 45, pp. 14–31, January 1997.

[5] A. J. Bell and T. Sejnowski, “An Information-Maximisation Approach toBlind Separation and Blind Deconvolution,” Neural Computation, vol. 7,pp. 1129–1159, November 1995.

[6] S. Haykin, ed., Unsupervised Adaptive Filtering, Vol. II: Blind Deconvo-lution. John Wiley & Sons, 2000.

[7] C. C. Cavalcante, “Neural Prediction and Probability Density FunctionEstimation Applied to Blind Equalization,” Master’s thesis, Federal Uni-versity of Ceara (UFC), Fortaleza-CE, Brasil, February 2001.

[8] C. M. Bishop, Neural Networks for Pattern Recognition. UK: Oxford

University Press, 1995.[9] R. Amara, Egalisation de Canaux Lineaires et Non Lineaires, Approche

Bayesienne. Ph.D. thesis (in French), Universite Paris XI Orsay, France,November 2001.

[10] S. Haykin, Adaptive Filter Theory. (Prentice-Hall Information and SystemSciences Series), Prentice-Hall, 3rd ed., 1996.

[11] S. Barbarossa and A. Scaglione, “Blind Equalization Using Cost FunctionMatched to the Signal Constelation,” in Proc. 31st Asilomar Conferenceon Signals, Systems and Computer, (Pacific Grove (CA) - USA), pp. 550–554, November 1997.

[12] J. Labat, O. Macchi, and C. Laot, “Adaptive Decision Feedback Equaliza-tion: Can You Skip the Training Period?,” IEEE Trans. on Communica-tions, vol. 46, pp. 921–930, July 1998.