A Novel Blind Equalizer Based onDual-Mode MCMA and DD Algorithm�

Seokho Yoon1, Sang Won Choi2, Jumi Lee2,Hyoungmoon Kwon2, and Iickho Song2,��

1 School of Information and Communication Engineering,Sungkyunkwan University, 300 Cheoncheon Dong,

Jangan Gu, Suwon 440-746, Koreasyoon@ece.skku.ac.kr

2 Department of Electrical Engineering and Computer Science,Korea Advanced Institute of Science and Technology,

373-1 Guseong Dong, Yuseong Gu, Daejeon 305-701, Korea{swchoi, jmlee, kwon}@Sejong.kaist.ac.kr, i.song@ieee.org

Abstract. We address a new blind equalizer incorporating both thegood initial convergence characteristic of the dual-mode modified con-stant modulus algorithm (MCMA) and the low residual error character-istic after convergence of the decision-directed (DD) algorithm. In theproposed scheme, a convergence detector is employed to help switchingfrom the dual-mode MCMA to the DD algorithm. We have observed thatthe proposed scheme exhibits a good overall performance in comparisonwith the CMA, MCMA, and dual-mode MCMA.

1 Introduction

In many modern communication systems including digital mobile and digitalTV systems, data is often transmitted through unknown channels and is thussubject to intersymbol interference (ISI), mostly due to the channel dispersioncharacterized by the non-ideal nature of the channel. The ISI is a primary causedegrading the performance of digital communication systems. Thus minimizingISI in digital communication channel is crucial for the improvement of systemperformance at high speed transmission rate.

Being an efficient tool to extract the transmitted symbol sequence by counter-acting the effects of ISI, an equalizer increases the probability of correct symboldetection. Data-aided algorithms initialize and adjust equalizer coefficients witha known training sequence from the transmitter before information-bearing data

� This research was supported by the Ministry of Science and Technology (MOST)under the National Research Laboratory (NRL) Program of Korea Science and En-gineering Foundation (KOSEF), for which the authors would like to express theirthanks.

�� Corresponding author.

Y.-S. Ho and H.J. Kim (Eds.): PCM 2005, Part II, LNCS 3768, pp. 711–722, 2005.c© Springer-Verlag Berlin Heidelberg 2005

712 S. Yoon et al.

transmission. Use of a training sequence, however, reduces the bandwidth effi-ciency and may become impractical when updating of the coefficients should beperformed frequently at the receiver end.

It is therefore desirable to equalize a channel without the aid of a trainingsequence, resulting in the self-recovering, blind, or non-data aided equalization[1]–[3]. Among the major advantages of blind equalization techniques is thatno training sequence is necessary to start-up or restart the equalization sys-tem when the communication breaks down unpredictably. Blind equalizationmethods also offer potential improvement in system capacity by eliminating thetraining overhead.

Numerous studies on blind equalization can be found in the literature. For ex-ample, normalized sliding-window constant-modulus and decision-directed (DD)algorithms have been proposed in [4], establishing a link between blind equal-ization and classical adaptive filtering. A minimum-disturbance technique wasproposed in [5] to avoid the gradient noise amplification problem and achieveimproved stability and robustness with low computational complexity. The mul-timodulus algorithm (MMA) introduced in [6] takes advantage of the symbolstatistics of such signal constellations as nonsquare and very dense constellations.

Among the various adaptive blind equalization algorithms, the Godard al-gorithm [1] is one of the best known and simplest adaptive blind equalizationalgorithms. This algorithm was also developed independently and extended asthe constant modulus algorithm (CMA) in [2]. Since the CMA is phase-blind,the equalizer output has an arbitrary phase rotation after convergence. Someperformance improvement of DD algorithm has also been achieved in [7] by con-trolling the step size parameter according to the regions in which the equalizedoutput lies in a constellation.

A particular problem of the CMA and modified CMA (MCMA) is that theresidual mean square error (MSE) in the steady state is sometimes not sufficientlysmall for the system to exhibit adequate performance. Minimizing the residualMSE and/or speeding up the convergence rate, the dual-mode algorithms such asthose considered in [8] are possibly a plausible solution to improving the overallperformance. The dual-mode algorithms possess a faster convergence rate andlower residual MSE than the CMA and MCMA at a small additional complexityto detect the convergence and/or an open eye pattern. The concurrent CMAand soft DD adaptation proposed in [9] has lower computational requirementsthan the concurrent CMA and DD algorithm.

In this paper, we propose a new blind equalization algorithm employing thedual-mode MCMA with modified parameters in the blind mode to improve theconvergence rate and the DD algorithm in the steady state mode to reduce theresidual MSE.

2 The Channel and Equalizer Models

Assume that the transmitted data {an} are an independent and identically dis-tributed (i.i.d.) zero-mean sequence with independent real and imaginary parts

A Novel Blind Equalizer Based on Dual-Mode MCMA and DD Algorithm 713

derived from a quadrature amplitude modulation (QAM) constellation. Let thecausal and linear time-invariant channel has coefficients {h(0), h(1), · · · , h(L−1)}with L the length of the channel impulse response, channel memory, or channelorder. Then, the received signal at time index n is

xn =L−1∑

k=0

h∗(k)an−k + vn, (1)

where vn is an i.i.d. additive white Gaussian noise (AWGN) and ∗ denotes com-plex conjugate.

To recover {an}, the received signals {xn} are passed through an equalizermodelled as an N -tap FIR filter with coefficients {w(0), w(1), · · · , w(N − 1)}.The output is then

yn = WHn Xn, (2)

where Xn = [xn, xn−1, · · · , xn−N+1]T is the vector of the received signals, Wn =[w(0), w(1), · · · , w(N − 1)]T is the vector of the equalizer tap weights (coeffi-cients), and the superscript H denotes the complex conjugate transpose.

3 A Novel Scheme for Blind Equalization

3.1 Dual-Mode MCMA

Let the error signal of the dual-mode MCMA be

en = γn · eMCMAn + βn · eDD

n , (3)

where γn and βn are adaptive parameters,

eMCMAn = yn,R(y2

n,R − R2,R) + iyn,I(y2n,I − R2,I), (4)

and

eDDn = yn − an (5)

with i =√

−1. Here, yn,R and yn,I are the real and imaginary parts of theequalizer output yn, respectively, and the hard decision output an of yn is anestimate of an. The real quantities R2,R and R2,I in (4) are obtained as

R2,R =E[a4

n,R]E[a2

n,R](6)

and

R2,I =E[a4

n,I ]E[a2

n,I ](7)

714 S. Yoon et al.

by setting the derivative of a non-convex cost function with respect to the equal-izer tap weights to be zero to minimize the cost function.

Using a stochastic gradient algorithm as the updating rule, the vector Wn

is adapted by

Wn+1 = Wn − µ · e∗nXn, (8)

where µ is the step size. The dual-mode MCMA is known to have good perfor-mance in terms of the convergence rate and residual MSE when it uses a sigmoidfunction in the construction of the relation between γn and βn. Note that thesigmoid function (11) is the cumulative distribution function (cdf) of the logisticpdf [10], one of the well-known heavy-tailed pdf’s.

To illustrate simply and clearly a drawback of the dual-mode MCMA, let usconsider the adaptive parameters

γsigmoidn = g(|eDD

n |) (9)

and

βsigmoidn =

|eMCMAn ||eDD

n | {1 − g(|eDDn |)} (10)

of the dual-mode MCMA with the sigmoid function

g(x) =1

1 + e−a(x−0.5) , a > 0. (11)

From (3), (9), and (11), it is obvious that the component γsigmoidn eMCMA

n of theerror signal (3) will not be zero in the steady state since γsigmoid

n is not zeroeven when the channel is perfectly equalized. This results in a large output errorlevel (relative to the case where en = βsigmoid

n eDDn ) in the steady state after the

equalizer has converged completely. In addition, the parameter a in (11) hassome restriction on its range due to a tradeoff relationship that a large value ofa increases the convergence rate but results in a large error signal in the steadystate, and vice versa.

3.2 The Proposed Algorithm for Blind Equalization

To overcome the drawback of the dual-mode MCMA, we propose a method inwhich only the DD algorithm operates in the steady state thereby improvingthe residual MSE performance of the dual-mode MCMA in the steady state.The proposed equalization algorithm consists of the dual-mode MCMA, DDalgorithm, and a convergence detector as shown in Figure 1. The combinationproposed in Figure 1 basically attempts to utilize the advantages of the dual-mode MCMA and DD algorithm, thereby improving both the convergence rateand the residual MSE in the steady state.

A Novel Blind Equalizer Based on Dual-Mode MCMA and DD Algorithm 715

nv

na

nyn

xn

aChannel

Equalizer

Convergence

Detector

Dual-Mode

MCMA

DD Algorithm

-

nW

+

Fig. 1. A block diagram of the proposed system

Blind Mode: At the beginning of the equalization process, the blind modeerror eDD

n will tend to be large. Thus, based on the simplification 11+exp(−x) ≈

1 − exp(−x) for x sufficiently large in (9) and (10), we propose to use

γpropn = 1 − e−α1|η′

n| (12)

and

βpropn =

|eMCMAn ||eDD

n | · e−α2|η′n|, (13)

where α1 and α2 are positive numbers,

η′n = ηn − ηn−1, (14)

and

ηn = (1 − ψ)ηn−1 + ψ|eDDn |2, 0 < ψ < 1 (15)

with η0 = 0.When the present output error level (represented approximately by ηn) has a

larger value with respect to the previous output error level (represented approx-imately by ηn−1), γprop

n has a large value while βpropn is small, and vice versa.

Since we consider only the convergence rate but not the residual MSE in theblind mode, we have more flexibility at the expense of some more complexityfor the tuning of the parameters α1, ψ, and α2. Because of this advantage, wecan use larger values of both γprop

n and βpropn in the blind mode. Consequently,

the convergence is expected to be faster in the proposed method than in thedual-mode MCMA using (9) and (10).

Steady State Mode: In our context, the steady state means a state afteran initial convergence has been attained during the training period with thedual mode MCMA. Once the proposed equalizer begins to converge, the dual-mode MCMA is switched into the DD algorithm by a convergence detector. Theconvergence rate after the shift from the dual-mode MCMA to the DD algorithm

716 S. Yoon et al.

gets lower and lower while the residual MSE gets smaller and smaller. Since onlythe DD algorithm is employed in the steady state mode, the residual MSE isexpected to be reduced more compared with the dual-mode MCMA.

Let the error signal of the proposed algorithm in the steady state mode be

epropn = λn · (yn − an), (16)

where the adaptive gain λn is given by

λn = |eMCMAn |. (17)

Note that epropn = |eMCMA

n | · eDDn . Normally a larger λn results in a faster

convergence at the cost of less residual MSE reduction, and vice versa; theubiquitous tradeoff between the convergence rate and residual MSE reduction.We are to select λn considering this tradeoff between the convergence rateand residual MSE reduction. The choice (17) of λn essentially allows us tochange smoothly from the dual-mode MCMA to the DD algorithm and hasbeen found to cope effectively with variant SNR and channels in simulationsalso.

Convergence Detector: If the DD algorithm is triggered too early before theproposed scheme converges, the convergence will be slow, and if it starts toolate after the convergence, the equalizer may converge to a different state. It istherefore highly important to adequately determine the instant of the switchingto the DD algorithm.

To derive a measure for the detection of convergence, let us consider Figure 2,where St denotes a time interval of length C and

dt =1C

⎧⎨

⎩∑

St

|eDD| −∑

St−1

|eDD|

⎫⎬

⎭

=1C

C∑

l=1

(|eDD

C∗(t−1)+l| − |eDDC∗(t−2)+l|

), t = 2, 3, · · · . (18)

We assume d1 = 1 and the superscripts ‘before’ and ‘after’ refer to before andafter the convergence, respectively. Since the time interval Sbefore

t would be lo-cated on the steep slope and Safter

t on the convergence floor, we can find the(approximate) time instant of the convergence by computing the value of dt andcomparing it with a reference value.

It is clear that dt is small also at the beginning of the blind mode wherethe output error level is large. If we judge the convergence solely on the basisof the value of dt, we might therefore end up with an undesirable result. In[8], by noting that an open eye condition can be expressed as |eDD| = |yn −an| < D′

2 , a detection scheme |eDD| < D′

4 is proposed, where D′ is the minimumdistance between the symbols in the constellation. To detect the convergencemore correctly by smoothing the effects of the fluctuation of the errors {|eDD|}

A Novel Blind Equalizer Based on Dual-Mode MCMA and DD Algorithm 717

0 200 400 600 800 1000 12000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Err

or le

vel

Iteration

dtbefore

dtafter

Stbefore

St−1before

Stafter

St−1after

Fig. 2. A graph for the detection algorithm

and consequently making the detector less dependent on the problem, we proposeto use the average

Et =1C

∑

St

|eDD| (19)

of the output errors in St in addition to dt in the decision of the convergence.In summary, the steps of the detection algorithm are as follows:

i) Let t = 2.ii) Check if |dt| < d and Et < D′

4 , where d is a positive constant.iii) If the results in ii) are both positive, the convergence detector switches dual-

mode MCMA into DD algorithm at 50 ∗ t + 1. Otherwise, we repeat ii) andiii) with t = 3, 4, · · ·.

The DD algorithm is known [4], [8] to converge surely when an initial con-vergence has already been obtained during a training period, which occur forexample when the eye pattern of the signal is initially open. By using (19) withthe threshold guaranteeing the eye to open (i.e., D′

4 < D′

2 ) except for a severedistortion case, the convergence detector can be used to switch from the dualMCMA to DD algorithm.

Step Size of the Proposed Algorithm: Using the methods similar to thenormalized least mean square (NLMS) algorithm [11], we can derive the stabilitycriterion for the proposed algorithm. The NLMS adjusts the step size µ such thatthe updated filter coefficients would produce zero error with the current datavector.

718 S. Yoon et al.

First, let us adjust the step size µ in (8) such that the updated coefficientswould achieve the desired modulus when applied to Xn. That is, we select µMCMA

n

such that

WHn+1Xn =

√R2,R + i

√R2,I . (20)

From (8), we have

WHn+1Xn = WH

n Xn − µMCMAn · eMCMA

n · ||Xn||2 (21)

after some algebraic manipulations. Substituting (2) and (20) into (21) gives

yn,R + iyn,I − µMCMAn · eMCMA

n · ||Xn||2 =√

R2,R + i√

R2,I . (22)

The solution to the equation (22) can be shown to be

µMCMAn,R =

1yn,R(yn,R +

√R2,R) · ||Xn||2

(23)

and

µMCMAn,I =

1yn,I(yn,I +

√R2,I) · ||Xn||2

(24)

for the real and imaginary parts of µMCMAn , respectively.

Similarly, the step size for the DD algorithm is obtained to be

µDDn =

1||Xn||2 (25)

from

WHn+1Xn = an (26)

and

yn,R + iyn,I − µDDn · eDD

n · ||Xn||2 = an. (27)

Finally, let us derive the step size of the proposed algorithm using the results(23)–(25). Multiplying both sides of (22) by γprop

n · µDDn and both sides of (27)

by βpropn · µMCMA

n , and then adding the results, we get

yn,R + iyn,I − 1γprop

n

µMCMAn

+ βpropn

µDDn

· epropn · ||Xn||2

=γprop

n · µDDn · (

√R2,R + i

√R2,I) + βprop

n · µMCMAn · an

γpropn · µDD

n + βpropn · µMCMA

n

(28)

after some algebraic manipulations using (3). Comparing (22), (27), and (28), itlooks reasonable at the first glance to choose the step size as

µn =1

γpropn

µMCMAn

+ βpropn

µDDn

. (29)

A Novel Blind Equalizer Based on Dual-Mode MCMA and DD Algorithm 719

Unfortunately, we have observed in preliminary simulations that the step size(29) is sometimes too large and the algorithm may fail to converge when ||Xn||2is too small. One simple solution is clearly to use the modified step size

µpropn =

1

σ + γpropn

µMCMAn

+ βpropn

µDDn

, (30)

where σ is a small positive number. The proposed algorithm converges in themean-square sense if the step size µ satisfies the condition

0 < µ < min(µpropn ). (31)

4 Simulation Results

In all the simulations herein, we have assumed that the clock of the received sig-nal is perfectly recovered and a carrier phase offset does not affect the equalizer.A simple 16-QAM constellation has been chosen and the signal to noise ratio

−5 −4 −3 −2 −1 0 1 2 3 4 5−5

−4

−3

−2

−1

0

1

2

3

4

5

Real

Imag

inar

y

(a)

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Real

Imag

inar

y

(b)

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Real

Imag

inar

y

(c)

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

Real

Imag

inar

y

(d)

Fig. 3. The constellations of the equalized 16-QAM signals when SNR=30dB. (a)CMA, (b) MCMA, (c) dual-mode MCMA, (d) proposed algorithm.

720 S. Yoon et al.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

−30

−25

−20

−15

−10

−5

0

5

MS

E (d

B)

Sample Index

CMA

Dual−mode MCMA

MCMA

proposed

Fig. 4. The MSE of various schemes when SNR=40dB

(SNR) at the input of the equalizer is defined as SNR = 10 log10E[|an∗h(n)|2]

σ2n

indB, where σ2

n is the variance of the AWGN. As a measure of the performance,the residual MSE defined as

MSE = 10 log10 E[|yn − an|2] [dB] (32)

is used. All the simulated residual MSE values are averaged over 200 trials withthe step size µ = 0.00008.

The proposed algorithm has four parameters, α1, α2, ψ, and d which needto be set only at the initial stage. We have chosen the value of the parametersas α1 = 20, α2 = 1.5, ψ = 0.9, D′/4 = 0.5 and d = 0.008 and used these valuesin all the following simulations to make the residual MSE smaller than −10dB:the values of the parameters can be chosen appropriately in other channels also.

For a 22-tap channel impulse response adopted from [9], Figure 3 showsthe constellations of the equalized outputs, and Figures 4 and 5 depict the MSEtrajectories when SNR=30dB and SNR=40dB, respectively. The CMA exhibits aslow convergence while the MCMA converges faster with less residual MSE thanthe CMA. It is evident that the dual-mode MCMA achieves some performanceimprovement in terms of the convergence rate and residual MSE compared withCMA and MCMA. The proposed algorithm presents the best performance (theconvergence rate and residual MSE) irrespective of the SNR.

It is clearly observed that the proposed algorithm achieves the best residualMSE, although the proposed algorithm sometimes exhibits a slightly slower con-vergence than the dual-mode MCMA. This is due to the inherent characteristicsof the DD algorithm which, after the convergence, reduces the residual MSE

A Novel Blind Equalizer Based on Dual-Mode MCMA and DD Algorithm 721

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

−30

−25

−20

−15

−10

−5

0

5

MS

E (d

B)

Sample Index

CMA

MCMA

Dual−mode MCMA

Proposed

Fig. 5. The MSE of various schemes when SNR=50dB.

more while converges slowly with a very small adaptive gain when the SNR ishigh. The proposed algorithm always has a lower residual MSE than the dual-mode MCMA and the difference of convergence rate between the two algorithmsin the blind mode is negligible.

5 Conclusion

In this paper, we have proposed a new blind equalizer allowing reduced resid-ual error level and faster convergence compared with the CMA, MCMA, anddual-mode MCMA. The proposed algorithm makes use of both the good initialconvergence characteristic of the dual-mode MCMA and the low residual errorcharacteristic of the DD algorithm after convergence. The proposed algorithm, inexchange for a slightly higher complexity when compared to other conventionalalgorithms, offers improved equalization performance. Simulation results havesupported the performance advantages of the proposed algorithm in general.

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