FRACTIONALLYSPACEDCONSTANTMODULUS EQUALIZATION … › PIERB › pierb04 › 17.08010802.pdf · 240 Kundu and Chakrabarty For BPSK γ is taken as 1 and for 8 PAM γ is taken as 9/5

Progress In Electromagnetics Research B, Vol. 4, 237–248, 2008

FRACTIONALLY SPACED CONSTANT MODULUSALGORITHM FOR WIRELESS CHANNELEQUALIZATION

A. Kundu and A. Chakrabarty

Kalpana Chawla Space Technology Cell (KCSTC)Department of Electronics and Electrical Communication EngineeringIndian Institute of TechnologyKharagpur-721 302, West Bengal, India

Abstract—Wireless channel identification and equalization is oneof the most challenging tasks because broadcast channels are oftensubject to frequency selective, time varying fading and there areseveral bandwidth limitations. Furthermore, each receiver channelhas vastly different types of channel characteristics and signals tonoise ratio. Here in this paper we consider channel equalization andestimation problem from trans-receiver perspective, specifically wetry to estimate blind equalization schemes particularly using constantmodulus Algorithm (CMA). We try to estimate a linear channel modeldriven by a QAM source and adapt a FSE (T/2) using CMA. It hasbeen shown CMA-FSE successfully reduces the cluster variance so thattransfer to a decision directed mode is possible and simultaneouslyerror is reduced.

1. INTRODUCTION

One approach to remove inter-symbol interference in a communicationchannel is to employ adaptive blind equalization [12, 13]. Themost popular class of algorithms used for blind equalization isthose that minimize the constant modulus criteria. Here we haveused CMA algorithm [4] to estimate the channel variation andcompare the performance of the Constant Modulus Algorithm witha more computationally efficient signed-error version of CMA (SE-CMA) [1, 3, 5, 6]. For channel estimation we used to take a FIR channelwith fractionally spaced equalizer (FSE) with sampling interval T/2where T is taken as symbol period. The source symbols are chosen

238 Kundu and Chakrabarty

from a finite M array real alphabet of zero mean means all symbolsare equally probable.

Figure 1. T/2 spaced multirate system model with added interference.

Here in Fig. 1, Sn implies baud spaced source symbol atsample index n, c is the vector representing the fractionally spacedchannel impulse response. w is the additive white Gaussian channelnoise, f represents vector containing the fractionally spaced equalizercoefficients. yn is the baud spaced equalizer output. No ofcoefficients in the channel and equalizer response vectors are Nc andNf respectively. The system output can be expressed as

yn = sT (n)Cf + wT (m)f (1)

where s(n) = [sn, sn−1, . . . , sn−Ns+1]T is the length Ns =

[(Nc +Nf − 1)/2] vector of baud spaced source symbols w(m) =[wm, wm−1, . . . , wm−Nf+1]T is the vector of additive zero mean whiteGaussian noise with variance σω and C is the Ns∗Nf decimated channelconvolution matrix given by

C =

c1 . . . . . . . . . c0. . . . . . . . . . . . . . . . . . c1 . . . . . . c0. . . . . . . . . . . . . . . . . .cNc−1cNc−2 . . . . . .. . . . . . . . . . . . . . . . . . cNc−1 · cNc−2

2. CONSTANT MODULUS ALGORITHM

The CMA criterion may be expressed by the non negative cost functionJCMAp, q parameterized by positive integer p and q.

JCMAp,q =1pqE {||ya|p − γ|q} (2)

where γ is a fixed constant. This is a gradient based algorithm [19] andwork on the premise that the existing interference causes fluctuation

Progress In Electromagnetics Research B, Vol. 4, 2008 239

in the amplitude of output that otherwise has a constant modulus. Forsimplest case we put p = 2 & q = 2. It updates weights by minimizingthe cost function. The steepest gradient descent algorithm [13–15, 19]is obtained by taking the instantaneous gradient of JCMAp, q whichresults equation which updates the system.

f(n+ 1) = f(n) − µg(w(n)) (3)g(w(n)) = r∗(n)ψ(yn) (4)

ψ(yn) = −∇yn

14

(|yn|2 − γ

)2= yn

(γ − |yn|2

)(5)

where f is the length Nf is equalizer coefficient vector, r(n) is thelength-Nf receiver input vector, µ is small step size and ψ(yn) is theCM error function for CMA2, 2. Where γ is taken as

γ = E{|Sn|4

}/E

{|Sn|2

}(6)

To improve computational efficiency a signed error algorithm used tomodify the update equation of the channel equalizer. Here we took onlythe sign of the error function there simplified by eliminating multiplyoperation. The Equation (3) modified as

f(n+ 1) = f(n) − µr(n)sgn(ψ(yn)) (7)

This modified CMA known as SE-CMA is equivalent to CMA1, 1. Nowwe consider CMA1, 1 cost function

JCMA1, 1 = E {||yn| − β|} (8)

and corresponding update equation

f(n+ 1) = f(n) − µr(n)sgn(yn(β − |yn|)) (9)

When β =√γ, the CMA1, 1 update equation is identical to SE-CMA

so (8) becomes

JSE−CMA = E {||yn| −√γ|} (10)

Selection of γ is important because of the systems convergence. Thedissipation constant γ should be chosen such that the γ = a2

v whereav is the vth positive member of the source alphabet and integer vsatisfies

v = arg mink

(k − 1/2

(1 +

√M2/2 − 1

))2

k − 1/2(11)


For BPSK γ is taken as 1 and for 8 PAM γ is taken as 9/5 forsatisfactory result. The cost function of SE-CMA depends on thefollowing terms s, C, f , γ, M , Ns, σω. S is the set of all MNs sourcesymbol possibilities. For a system with BPSK source, a channel lengthof 6 and an equalization length of 2 (M = 2; Nc = 6; Nf = 2; Ns = 3).The set of possible symbol combination is

S =

{[ 111

],

[ 11−1

],

[ 1−11

] [ −111

] [ −11−1

] [ −1−11

],

[ −1−1−1

]}

We know

erf(x) =2√π

x∫0

exp(−t2

)dt (12)

and

Q(t) =1√2π

∞∫x

exp(− t2

2

)dt (13)

JSE−CMA = E {||yn| −√γ|} = E

{∣∣∣∣sTCf + wT f∣∣ −√

γ∣∣} (14)

After full expansion the expression comes as

JSE−CMA = M−Ns∑n∈s

−√γ − 4

(√γ + sTCf

)Q

(√γ + sTCf

σω ‖f‖2

)

+2sTCfQ

(sTCf

σω ‖f‖2

)+

√2πσω ‖f‖2[

2 exp

(−

(√γ + sTCf

)2

2σ2ω ‖f‖2

2

)− exp

(−

(sTCf

)2

2σ2ω ‖f‖2

2

)(15)

It can be seen that the cost function is complex enough that fromgeneral conceptual study it is not possible the rigorous analysis.Convergence analysis of fractional spaced equalizers draws twoimportant conclusions, 1. a finite length channel satisfying a length& zero condition allows CMA-FSE to be globally convergent [2, 5–7]and 2. The linear FSE filter length need not be longer than the channeldelay spread. For some channel with deep spectral nulls, the CMA FSEdoesn’t require a large number of parameters so that it can convergefaster.


3. CHANNEL MODEL FOR A QAM SOURCE

Channel output of a QAM communication system described byequation

x(t) =∞∑

n=0

anh(t− nT − t0) + w(t) (16)

Figure 2. Channel model for a QAM.

Figure 2 shows a sequence of independent identically distributedcomplex data {an} is sent by the transmitter over a LTI channelwith impulse response h(t). The receiver attempts to recover theinput data sequence {an} for measurable channel output x(t) in whichT is the symbol period. The channel output may be corrupted byw(t) channel noise which is zero mean stationary, white and complexGaussian with variance σ2 and is independent of the channel input an.Assume the complex data and noise both satisfy symmetric propertyE

{a2

n

}= E

{w2

t

}= 0 in addition E

{|an|4

}− 2E2

{|an|2

}< 0, i.e.,

the kurtosis K(an) of an is negative as is often the case of QAMsystem. When the distortion caused by channel (LTI [10, 11], nonideal) is significant, equalization is needed to remove the ISI [20] atthe sampling instants (t = nT ). Due to the presence of ISI, therecovery of input signal sequence an requires that the channel impulseresponse h(t−T0) be identified either explicitly or in decision feedbackequalization (DFE) [9, 20] or implicitly as in linear equalization. Intraditional blind equalization system the channel output sampled atthe known baud rate 1/T . The sampled channel output

x(nT ) =∞∑

k=0

akh(nT − kT − t0) + w(nT )

= an h(nT − t0) + w(nT )

(17)


is a stationary process, using this notations xn∆= x(nT ); wn

∆=w(nT ); hn

∆= h(nT − t0) the previous equation can be written asdiscrete convolution with noise

xn =∞∑

k=−∞akhn−k + wn (18)

Based on this relationship traditional linear TSE are designed as FIR

filter θ(z−1) =N∑

k=0

θkz−k to be applied on xn to remove the ISI from

the equalizer output yn =N∑

k=0

θkxn−k. Most of the equalizer algorithm

proposes to adjust the parameter {θk}Nk=0. CMA aims to minimize the

cost function given in (19).

J2(θ) =14E

{(|yn|2 −R2

)2}

R2 =E

{|an|4

}E

{|an|2

} (19)

If we wish to maximize |K(yn)| it requires E∣∣y2

n

∣∣ = E∣∣a2

n

∣∣ where K(yn)is kurtosis of signal yn.

|K(yn)| ∆= E{|yn|4

}− 2

(E

{|yn|2

})−

∣∣E {y2

n

}∣∣2 (20)

From the Fig. 3 it is apparent that x(t) is a continuous timecyclostationary process with period T as long as the channel bandwidthis greater than the minimum BW 1/2T . Let sampling interval ∆ = T

P ;

sampled channel output x(k∆) =∞∑

n=0anh(k∆−np∆−t0) for p > 1,

the over sampled channel output x(k∆) can be divided into p sub-sequences. x(i)

k∆= x[(kp+ i)∆] = x(kT + i∆), i = 1, . . . , p. By defining

the sub-channel impulse response as h(i)k

∆= h(pk∆+ i∆− t0) = h(kT +

i∆− t0); the p sub-sequence can be written as x(i)k =

+∞∑n=0

anhik−n +w

(i)k ,

these p sub-sequences can be viewed as stationary outputs of p discrete

FIR channels. Hi(z) =K∑

k=0

hikz

−k with a common input sequence ak.


H1(Z)

H2(Z)

Hp(Z)

2 ( )zθ

( )p zθ

na

Yn1( )zθ

CMA

pny

1ny

an

Descision

Figure 3. Multichannel vector representation of blind adaptive FSE.

One adjustable filter [8] is provided for each sub-sequences x(i)k , thus

the actual equalizer is a vector of filters. θi(z) =N∑

k=0

θ(i)k z−k, i =

1, . . . , p. The p stationary filter output{y

(i)n

}are summed to form

the stationary equalizer output

yn =p∑

i=1

y(i)n (21)

Define FSE parameter as θ ∆= [θ(1)0 , θ

(1)N , θ

(p)0 , . . . , θ

(p)N ]. To adaptively

adjust θ without a training sequence, CMA can be implemented tojointly update the p filters to minimize cost function. From costfunction we obtain stochastic gradient algorithm as follows

θ(i)k (n+ 1) = θ

(i)k (n) − µx

(i)n−kyn

(|yn|2 −R2

), i = 0, 1, . . . , p− 1;

(22)

where µ is small step size and θ(i)k is the kth coefficient of the ith

filter at the nth iteration. This combination of CMA & FSE used toimplement blind adaptive algorithm for equalizer [18].

4. BLIND ADAPTIVE FSE

Figure 4 shows the frequency response of the microwave channel takenfor simulation [16, 17]. Fig. 5 shows noisy received signals from a linearchannel model driven by a QAM [15] source constellation and adaptsa FSE (T/2) using CMA. User selectable variables are in the Input


0 50 100 150 200 250 300

100

Magnitude of Impulse Response Coefficients

0 100 200 300 400 5000

1

2

3Frequency Response Magnitude

Figure 4. Frequency responsecharacteristics of channel.

0 1000 2000 3000 4000 5000

10-0.31

10-0.29

10-0.27

iteration vs MSE plot

iteration

mse

e

Figure 5. CMA blind equalizerperformance with 16 QAM, stepsize = 0.001, SNR at equalizerinput 35 dB.

Variables section, and the channel is selected in the Channel sectionof the code. The CMA-FSE successfully reduces the cluster varianceso that transfer to a Decision Directed mode is possible, further errorrate reduction is desired. Simulated results demonstrate CMAs abilityto adapt a FSE blindly from a received sequence synthesized froma shortened version (length-16). The source is 16 QAM, white andequiprobable. The equalizer is length-16 and initialized with a unitycenter spike and all other taps zero, and the step-size is 0.001. The

f0

f 1

CMA Error Surface

MSE ellipse axes

-3 -2 -1 0 1 2 3

-1

-0.5

0

0.5

1

Figure 6. CM Cost Surface plotsover equalizer plane for length-2real-valued channels.

SE-CMA cost surface

f0

f 1

-4 -3 -2 -1 0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Figure 7. Cost contours for SE-CMA of 2-tap fractionally spacedequalizers given the following In-put parameters: SNR = 15, alpha-bet size = 8 source variance = 1.


SNR at the equalizer input is 35 dB. The received signal is normalizedto (near) unity power.

The convergence of equalizer parameter vector under CMA canbe viewed as a transverse of the CMA cost surface [2, 5, 6, 8] withaverage movement in the direction of steepest descent, so from theFig. 6 dynamical behavior of CMA can be estimated.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-30

-20

-10

0

10

dB

normalized to baud frequency

Channel, equalizer frequency response magnitudes

-0.5 0 0.5-1.5

-1

-0.5

0

dB

normalized to baud frequency

System frequency response magnitude

channeleq

mmse-glob

sysmmse-glob

Figure 8. Frequency re-sponse analysis of 6 tap, T/2fractionally-spaced, 16 QAMsource, equalizer length 2, SNR50 dB, step size 5e−3, no ofiteration 5000.

0 1000 2000 3000 4000 5000

-6-4-202

Smoothed CM-error history

iteration number

dB

0 1000 2000 3000 4000 5000

-10

-5

0

Squared-error history

iteration number

dB

MSEmmse-loc

MSEmmse-glob

Figure 9. Plot of smoothedCM error history and variation ofsquare error with iteration.

0 1000 2000 3000 4000 5000

0

0.5

1Equalizer coefficient histories

iteration number

real

0 1000 2000 3000 4000 5000

-0.1

0

0.1

iteration number

imag

Figure 10. Plot of CMA equalizercoefficient history with iteration.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-0.6-0.4-0.2

00.20.40.60.8

Equalizer coefficient histories

iteration number

real

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-0.4-0.2

00.20.40.6

iteration number

imag

Figure 11. SE-CMA equalizercoefficient history with iteration.

Figure 7 shows FSE-CMA cost surface which is multimodal, FSEis able to perfectly equalize the channel. Here multiple minimaexist. It can be shown that multiple minima occur very near to thewiener solutions corresponding to particular combination of systempolarity and system delay. CM minima with better MSE stay closer to


corresponding wiener solutions since CM criterion operates purely onthe magnitude of the equalizer output not the phase shift of adaptivefilter.

Figure 12. Transmitted symbol,received and equalized symbol,convergence plot with iterationfor CMA.

Figure 13. Transmitted symbol,received and equalized symbol,convergence plot with iterationfor FSE-CMA.

5. CONCLUSION

Study of convergence of blind equalizer CMA-FSE is based on simplelength and zero condition. The requirement of no common zero amongsub-channels may sometime be restrictive. When common zeros existCMA-FSE may not be able to estimate the common factor among sub-channels, in this case an additional linear filter may be added after thevector equalizer to minimize the remaining ISI. Another option is tosimply increase the length of all filters. The vector blind equalizerstructure of FSE can be simply applied to spatial channel diversitybecause the sub channels of Fig. 3 can be replaced by physical channelsin an array of wide band sensors. Hence xn(t) can be treated as theoutput of each channel sampled at baud rate as long as the length andzero condition is satisfied, the convergence will hold for spatial andspectral diversity channel. Finally conclusion can be drawn that a finitelength channel which satisfy the ‘length and zero’ conditions will beconvergent and the linear FSE filter length need not be longer than thechannel delay spread for perfect equalization. Constant modulus costfunction describing the deformation of the CM error surface. Figs. 10and 11 estimate the proximity of equalizer coefficient history withiteration number.


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FRACTIONALLYSPACEDCONSTANTMODULUS EQUALIZATION … › PIERB › pierb04 › 17.08010802.pdf · 240 Kundu and Chakrabarty For BPSK γ is taken as 1 and for 8 PAM γ is taken as 9/5