Performance analysis of a family of adaptive blind equalization algorithms for square-QAM

Contents lists available atScienceDirect

Digital

Signal

Processing

www.elsevier.com/locate/dsp

Performance

analysis

of

a

family

of

adaptive

blind

equalization

algorithms

for

square-QAM

Ali

W. Azim

a

,

b

,

Shafayat Abrar

c

,

∗

,

Azzedine Zerguine

d

,

Asoke

K. Nandi

e a_{InstitutePolytechniquedeGrenobleSaintMartind’Hères,38400,France}

b_{COMSATSInstituteofInformationTechnology,WahCantt47040,Pakistan} c_{COMSATSInstituteofInformationTechnology,Islamabad44000,Pakistan} d_{KingFahdUniversityofPetroleum&Minerals,Dhahran31261,SaudiArabia} e_{BrunelUniversityLondon,Uxbridge,MiddlesexUB83PH,UnitedKingdom}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Availableonline25September2015 Keywords: Multimodulusalgorithm Blindequalization Adaptiveequalizers Steady-stateanalysis Receiverdesign Convergenceanalysis

Multimodulus algorithms (MMA) based adaptive blind equalizers mitigate inter-symbol interference and recover carrier-phase in communication systems by minimizing dispersion in the in-phase and quadrature components of the received signal using the respective components of the equalized sequenceinadecoupledmanner.Theseequalizersaremostlyincorporatedinbandwidth-efficientdigital receivers which rely onquadrature amplitude modulation(QAM) signaling. The nonlinearities inthe update equations of these equalizers tend to lead to difficulties in the study of their steady-state performance.Thispaper presentsoriginallythe steady-stateexcessmean-square-error(EMSE)analysis of different members of multimodulus equalizers MMAp–q in a non-stationary environment using energyconservationarguments,andthusbypassingtheneedforworkingdirectlywiththeweighterror covariancematrix.Indoingso,theexactandapproximateexpressionsforthesteady-state mean-square-error ofseveral MMA basedblind equalization algorithms are derived, includingMMA2–2, MMA2–1, MMA1–2, and MMA1–1.The accuracyofthederived analyticalresultsis validated usingMonte–Carlo experimentsandfoundtobeincloseagreement.

©2015TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Blindequalizers mitigate different types ofinterferences such asinter-symbolinterference (ISI), frequency selectivefading, etc., causedby non-idealtransformations performedby thedispersive channelsin a communicationsystem. A blind adaptive equalizer attemptstocompensateforthedistortionsofthechannelby pro-cessingthereceivedsignalsandreconstructingthetransmitted sig-nalup tosome indeterminaciesbythe useoflinearornonlinear filterswithoutanyknowledgeofthechannelimpulseresponseand withoutdirectaccesstothetransmittedsequenceitself.Thebasic idea behind an adaptive blind equalizer is to minimize or maxi-mizesomeadmissibleblindobjectiveorcostfunctionthroughthe choiceoffiltercoefficientsbasedontheequalizeroutput[1–3].

*

Correspondingauthor.Fax:+92-336-232-1845. E-mailaddresses:[email protected],

[email protected](A.W. Azim),[email protected](S. Abrar), [email protected](A. Zerguine),[email protected](A.K. Nandi).

The performance of an adaptive filter can be evaluated using transient andsteady-stateanalyses.Theformerprovides informa-tion about the stability and the convergence rateof an adaptive filter, whereas the latter provides information about the mean-square-errorofthefilteronceitreachessteadystate.Inthe steady-state analysisofadaptivefilters,one ofthe properties tobe con-sidered is their ability to track changes/variations in the signal statisticsofthereceivedsignal. Thispropertyisofsignificant im-portance,particularlyinmobile communicationssystemsand ap-plicationslikeacousticechocancellation,etc.

Blindadaptivefilters(orequalizers) arebasedon recursive al-gorithmsthatallowthefiltertoadaptandtrack(slow)variations in input statistics. Such adaptive filters start from certain initial conditions without any prior knowledge about the input signal statistics,thenthefiltercoefficientsareupdatedbasedonthe cho-sen adaptive algorithms and the sequence of the sampled data values.Instationaryenvironments,adaptivefiltersconvergeto op-timum Wiener solution [4–13]. However, in non-stationary envi-ronments, theoptimum Wienersolution takestime-varyingform that resultsinvariation ofsaddlepoint inerrorperformance

sur-http://dx.doi.org/10.1016/j.dsp.2015.09.002

face and consequently affecting the performance of filters, thus, tracking the variations in underlying signal statistics is consid-ered to be a useful and important property for adaptive filters. These variations in underlying signal statistics and consequently saddlepointcanbe trackedbyusingtrackingperformanceanalysis. Theperformancemetrictobeconsideredfortrackingperformance ofan adaptive filteris thesteady-state excessmean-square-error (EMSE).The EMSE can be defined asthe difference betweenthe mean-square-error(MSE)ofthefilterinsteady-stateandits mini-mumvalue.ThesmallertheEMSEofanadaptivefilter,thebetter itis[14].Iffilterparameters (likestep-size) are chosen correctly, thefiltercantrackvariationsinunderlyingsignal statistics. How-ever,trackingfastvariationsmightprovetobe achallenging task orattimesimpossibletoperform[14].

The widely adopted adaptive blind equalization algorithm is theso-calledConstantModulusAlgorithm(CMA2–2) [2,15–17].For quadratureamplitudemodulation(QAM)signaling,however,a tai-lored version of CMA2–2, commonly known asMultimodulus

Al-gorithm (MMA2–2) is considered more suitable. The MMA2–2 is

capable of jointly achieving blind equalization and carrier phase recovery,whereastheCMA2–2 requiresaseparatephase-lockloop forachievingcarrierphaserecovery.ThefamilyofMMA,MMAp–q, is associated with the minimization of the dispersion-directed cost-function with two degrees of freedom. By selecting appro-priate values of p andq, thegeneric split cost-function leads to therespectivecost-functionsofseveralexistingblindequalization algorithms[18–21].Interestedreadersarereferred to[22]for de-tailed discussion on MMAp–q. The update expressions of these algorithmsareinherentlynonlinearinnatureduetothepresence ofnonlinearerror-functions[20,23–27].

AlgorithmslikeCMA2–2/MMA2–2 haverecentlybeenemployed inoptical systems forpolarization mode demultiplexingandalso to mitigate the effects of other types of interferences like chro-maticandpolarizationmodedispersionsinopticalsystems.Since 2008[28],CMA2–2 anditsvariantshavebecomethemost exper-imentedalgorithmsforblind polarizationdemultiplexing[29–36]. In [37], authors have compared CMA2–2 with an independent component analysis (ICA) based algorithm to demultiplex the polarization adaptively. Recently in [38–43], authors have used MMA2–2 anditsvariantsasajointadaptivesolutionforblind de-multiplexingandcarrierphaserecoveryincoherentopticalsystem. Afterwards,

β

MMA (which is an optimized version of MMA2–1)

[44] has been employed in coherent optical receiver to demulti-plexpolarizationmodesignalsadaptively[45].

Inthispaper, theapproach that hasbeenadopted for steady-state tracking analysis of multimodulus equalizers exploits the studyofenergypropagationthrougheachiterationofanadaptive filterusingafeedbackstructure(whichconsistsofalossless feed-forwardblock anda feedbackpath), andit reliesonenergy con-servationarguments[14].Theconvenienceofthisapproachisthat itallowsustoavoidworkingwithnonlinearupdateequationsand thusbypassestheneedforworkingdirectlywiththeweighterror covariance matrix. In particular, using the fundamental variance relation arguments, we derive expressions forsteady-state EMSE ofMMA2–2,MMA2–1,MMA1–2 andMMA1–1 under the assump-tionthatthequadraturecomponentsofthesuccessfullyequalized signalareGaussiandistributedwhenconditionedontruesignal al-phabets.Ourobjectiveisnottostudytheconditionsunderwhich analgorithmwilltendtoconvergesuccessfully,rathertoevaluate its expectedsteady-stateperformance onceit hasconverged suc-cessfully.

1.1. Literaturereview

The nonlinearity of most of the adaptive equalizers, includ-ing CMA2–2 and MMA2–2, makes the steady-state analysis and

trackingperformance a difficulttasktoperform.As aresult, only a handful of results is available in the literature concerning the steady-stateperformance ofadaptiveequalizers.Afew resultsare available on EMSE analysis of CMA2–2 like Fijalkow et al. [46]

employed ingenioususeofLyapunovstability andaveraging anal-ysis,Shynket al.[47]usedGaussianregressionvectorassumption, andsomeexploitedthevariance relationtheorem[48,49]to eval-uatethesame.Steady-stateanalysesofadaptivefiltershavegained interestduetotheireaseinanalysis.Recently,Abraret al.[50] per-formedtheEMSEanalysisofCMA2–2 and

β

CMA[51]byassuming that the modulus of equalized signals are Rician distributed in the steady-state. In a recent work [52], we have performed the EMSE analysisofMMA2–2 and

β

MMA[44] by assumingthat the real and imaginary parts of equalized signals are Gaussian dis-tributed in the steady-state. Moreover, the approach of [14] has been employed to studythesteady-state performance ofa num-ber of adaptive blind equalization algorithms e.g., the so-called hybridalgorithm [53],thesquare contouralgorithm[54],the im-proved square contour algorithm [55], and the varying-modulus algorithms[56].

1.2. Notation

Unless otherwise mentioned, scalars are represented by italic letters (e.g., K). Lower-case boldface letters are used to denote vectorsandupper-case boldfacelettersareassociatedwith matri-ces, e.g., w and R, respectively. In addition, the symbol

⊗

and operators

(

·

)

∗,

(

·

)

T and

(

·

)

H respectively represents the convo-lution operation, complex conjugate operator, transpose operator and Hermitian (conjugate transpose) operator. The operator

·

when applied to a vector gives the Euclidean norm of the vec-tor, whereas, the operator

|

· |

gives the absolutevector. Further, E,

[·]

and I denotes the expectation operator, the real part of the complex entity, and identity matrix of appropriate dimen-sions, respectively. The operator Tr(

·

)

gives the trace of the ma-trix.

1.3. Paperorganization

The paper is organized as follows: In Section 2, we describe themathematicalmodelforthesystem.Section3providesabrief introduction of different members of MMA family that we aim to discusshere. Section 4introduces thenon-stationary environ-ment and the framework for EMSE analyses. Section 5 presents the analyticalexpressions evaluated forsteady-statetracking per-formanceanalysisforMMA2–2,MMA2–1,MMA1–2 andMMA1–1 equalizers. Section 6 compares the proposed approach with ex-isting state-of-the-art methods. Section 7 provides a number of computersimulationsonsteady-statetrackingperformance analy-sisofthealgorithms consideringdifferentscenarios:forequalized zero-forcing solution,equalizing atime-varyingchannel, studying the effect of filter-length on EMSE on a time-invariant channel, andadaptive opticaldemultiplexing ina coherentopticalsystem. In addition, it also compares the theoretical results predictedby ourexpressionswiththesimulatedvalues.Finally,Section8draws conclusions.

2. System model

Fig. 1 depicts a typical baseband communication system.

Consider that the channel response is given by a K-tap vector hn

= [

hn,0

,

hn,1

,

· · ·

,

hn,K−1

]

, then the full rank

(

N

+

K

−

1)

×

N channel convolution matrix

H

is given by following Toeplitz matrix

Fig. 1.A typical baseband communication system.

H

=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

hn,0 0

· · ·

0

· · ·

hn,1 hn,0

. .

.

0

. .

.

..

.

hn,1

. .

.

..

.

. .

.

hn,K−1

..

.

. .

.

hn,0

. .

.

0 hn,K−1

. .

.

hn,1

. .

.

..

.

..

.

· · ·

. .

.

. .

.

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

(1)

The received signal xn is the convolution of transmitted se-quence

{

an

}

= [

an,an−1

,

· · ·

,

an−K+1

]

T and channel impulse re-sponse hn as givenby xn

=

hnTan; the sequence

{

an

}

is indepen-dentandidenticallydistributed(i.i.d.),andtakesvaluesofequally likelysquare-QAM symbols. Thevector xn isfed to theequalizer tocombattheinterferenceintroduced bythepropagationchannel andestimate delayedversion ofthe transmittedsequence

{

an−δ

}

,

where

δ

denotesdelayparameter.

Letwn

= [

wn,0

,

wn,1

,

· · ·

,

wn,N−1

]

T betheimpulseresponseof equalizerandxn

= [

xn,xn−1

,

· · ·

,

xn−N+1

]

T be thevector of chan-nelobservations(the regressorvector)withinputcovariance ma-trix R

=

ExnxnH, where N is the number of equalizer taps. The output of equalizer is the convolution of regression vector and equalizer impulse response is given as yn

=

wnH−1xn. Let tn

=

hn

⊗

w∗n−1 betheoverallchannel-equalizerimpulseresponse. Us-ing (1), we obtain tn

=

hn

⊗

w_n∗₋₁

=

H

w∗_n−1. Under successful convergence, we have tn

=

e is ideally single-spike where e

=

[

0,

· · ·

,

0,1,0,

· · ·

,

0

]

T.

A generic stochastic gradient-based adaptive equalizer for whichtheupdatingalgorithmisgivenas[14]

wn

=

wn−1

+

μϕ(

yn

)

∗xn (2)

where

μ

isasmallpositivestep-size,governingthespeedof con-vergenceandthelevelofsteady-stateequalizerperformance,and

ϕ

(

yn) is complex-valued errorfunction. For multimodulus equal-izers,the errorfunction is non-analyticinnature, i.e., it isa de-coupled function of the quadrature components of deconvolved sequence yn,whichisexpressedas

ϕ(

yn

)

=

ψ (

yR,n

)

+

j

ψ (

yI,n

),

(3) sothat the real andimaginaryparts of

ϕ

(

yn

)

are obtained from thereal yR,n andimaginaryparts yI,n of yn,respectively.

3. The multimodulus equalizers

TheMultimodulusAlgorithm(MMA)isconsideredmoresuitable

forQAM signaling. A generalized dispersion-directed(split) cost-functionofgenericMMAp–qequalizersisgivenasfollows[22]:

JMMAp–q

=

E

|

yR,n

|

p

−

RpR

q

+

E

|

yI,n

|

p

−

RpI

q

(4)

where p andq are positive integers, and RR and RI are disper-sionconstants chosen inaccordance withthe sourcestatistics in orderto guarantee that the globalminima of JMMAp–q occurs at zero-forcingsolutions.Thecostfunctiondefinedin(4)canbe con-sideredasageneralizationofWesolowski’scost-function[23] with

twodegreesoffreedomorthesplitversionofLarimoreand Treich-ler(CM)cost-function[57].Thecorrespondingstochastic gradient-basedadaptivealgorithmis[22]

wn

=

wn−1

+

μ

|

y_Rp_,n

| −

Rp_R

q−2

|

yp_R−_,n2

|

Rp_R

− |

yp_R,n

|

yR,n

+

j

|

yp_I,n

| −

R_Ip

q−2

|

yp_I,−_n2

|

Rp_I

− |

yp_I,n

|

yI,n

∗ xn (5)

A multitude of algorithms can be obtained for different choices of p andq, providinga possibleflexibility in thedesign ofblind equalizers.Inthesequel,thealgorithmdefinedbyrecursion(5)is referred asMMAp–q and for the sake of simplicity,we will use subscript L to denote either R or I.Expression (5) generalizesa numberofexistingblind adaptiveequalizationalgorithms.Among them,thesearethefollowing:

1. For p

=

q

=

2, (4) reduces to following split cost function which was proposed independently by Wesolowski [19], Oh andChin[20]andYanget al.[24]:

JMMA2–2

=

min w

E

y2_R,n

−

R2_R

2

+

E

y2_I,n

−

R2_I

2

(6)

whereR2_L

=

Ea4_L

/E

a2_L.ThetapweightvectorofMMA2–2 is up-datedaccordingto

wn

=

wn−1

+

μ

(

R2R

−

y2R,n

)

yR,n

+

j

(

R2I

−

y2I,n

)

yI,n

_∗

xn

(7)

2. For p

=

2 andq

=

1,(4)resultsinMMA2–1 equalization algo-rithmthatemploysthefollowingcostfunction1

JMMA2–1

=

min w

E

y2_R,n

−

R2_R

+

E

y2_I,n

−

R2_I

(8)

ThetapweightvectorofMMA2–1 isupdatedtominimize(8)

usingagradient-descentadjustmentalgorithmaccordingto

wn

=

wn−1

+

μ

sgn

R2_R

−

y2_R,n

yR,n

+

jsgn

R2_I

−

y2_I,n

yI,n

∗_x n (9)

3. For p

=

1, q

=

2, (4) reduces to an equivalent form of Benveniste–Goursat cost–function[18].We denotethe result-ingalgorithmasMMA1–2,andultimately(4)resultsin

JMMA1–2

=

min w

E

|

yR,n

| −

RR

2

+

E

|

yI,n

| −

RI

2

(10)

where RL

=

Ea2L

/E

|

aL

|

. The tap weight vector of MMA1–2 is updatedtominimize(10)usingagradient-descentadjustment algorithmaccordingto wn

=

wn−1

+

μ

RRsgn

(

yR,n

)

−

yR,n

+

j

RI sgn

(

yI,n

)

−

yI,n

∗_x n (11)

4. For p

=

q

=

1, (4)reducestoan equivalent formofthe cost-functionindependentlyproposedbyWeerackodyet al.in1991

[58]andImet al.in2001[21].Wedenotetheresulting algo-rithmasMMA1–1 anditscostfunctionisgivenasfollows2_:

1 _{InMMA2–1,thedispersionconstantR}

L isobtainedas RL=2z −1,where

zisthesmallestpositiveintegergreaterthanorequaltoz[22].Theparameter zisgivenbyz=(z1/12)

+

(1/z1)wherez1isgivenas 3

108z2+12 81z2 2−12, z2=0.5 √

M(M−1)andMdenotesthesizeofconstellation.ItgivesRL=3,7,and

13for16-,64-,and256-QAM,respectively. 2 _{ThedispersionconstantR}

L for MMA1–1 isgivenas RL=2z −1[22]. For

M-pointconstellationwehavez=√M/8 whichgivesRL=3,5,and 11 for16-,

JMMA1–1

=

min w

E

|

yR,n

| −

RR

+

E

|

yI,n

| −

RI

(12) ThetapweightvectorofMMA1–1 isupdatedtominimize(12)

usingagradient-descentadjustmentalgorithmaccordingto

wn

=

wn−1

+

μ

sgn

RRsgn

(

yR,n

)

−

yR,n

+

jsgn

RI sgn

(

yI,n

)

−

yI,n

∗ xn (13)

4. Non-stationary environment and energy conservation relation

Weconsideranon-stationarysystemmodelinwhichthe vari-ationsintheWienersolution,wo_{,followusuallyafirst-order} ran-domwalkmodel[14]:

wo_n

=

wo_n−1

+

qn (14)

where the random vector q_n is an i.i.d. zero-mean random vector with positive definite covariance matrix given as Q

=

EqnqnH

=

σ

q2I. Weassumethatqn isindependentofboth

{

am

}

and

{

xm,wo₋₁

}

for all m

<

n [14]. Using the time-dependent Wiener solution,thedesireddataan canbeexpressedas

an

=

(

wno−1

)

Hxn

+

ϑ

n

,

(15)

where

ϑn

is the measurement or gradient noise and is uncorre-latedwithxn,i.e.,Eϑ_n∗xn

=

0[59].Definingtheweighterrorvector

˜

wn as w

˜

n

:=

won

−

wn, (2),for a non-stationary environment is expressedas

˜

wn

= ˜

wn−1

−

μϕ(

yn

)

∗xn

+

qn (16) Defining the so-called apriori and a posteriori estimation errors asea,n

:= ˜

wnH−1xn andep,n

:=

(

w

˜

n

−

qn

)

Hxn,respectively. Wecan rewrite(16)in termsoftheerrormeasures

{ ˜

wn,w

˜

n−1

,

ea,n,ep,n

}

alone. For this purpose, we note that if we multiply (16) by xn fromtheright,wefindthattheaprioriandaposteriori estimation errors

{

ea,n,ep,n

}

arerelatedvia

ea,n

=

ep,n

+

μ

xn

2

ϕ(

yn

)

(17)

Relation(17)revealsthatea,ndependsonchannelvariation, adap-tion, and gradient noise. Thus, the steady-state EMSE and the trackingperformanceofanadaptiveequalizercanbequantifiedby theenergyofea,n.From (17),we canassociatetheerror-function

ofanequalizerwiththeaprioriandtheaposterioriestimation er-rorsasfollows:

ϕ(

yn

)

=

ea,n

−

ep,n

μ

xn

2

(18)

Substituting(18)in(16)andrearrangingtheterms,weobtainthe energyconservationrelation

˜

wn

2

+

|

ea,n

|

2

xn

2

= ˜

wn−1

2

+

|

ep,n

|

2

xn

2 (19)

Itisimportanttonote that(19)holdsforanyadaptivealgorithm.

Fig. 2representsthephysicalinterpretationof(19)whichlinksthe energiesoftheweighterrorvectoraswellastheaprioriandthea

posterioriestimationerrorsbystatingthatmappingfromthe

vari-ables

w

˜

n−1

,

ep,n/

xn

tothevariables

w

˜

n

,

ea,n/

xn

isenergy preserving. The relation (19)characterizes the energy preserving propertyofthefeed-forwardpath,whereastherelation(17) char-acterizesthefeedbackpath.Thefunction

M

denotesthemapping betweenthetwo variablesandz−1 _{denotestheunit delay} opera-tor.Substitutingtheexpressionofep,n from(17)into(19),weget thefundamentalvariancerelationtheorem.

Fig. 2.Lossless mapping and feedback loop.

Theorem 1 (Variancerelation).(See[14].)Consideranyadaptive

fil-teroftheform(2),andassumefilteroperationinsteady-state.Assume furtherthatan

=

(

won−1

)

Hxn

+

ϑn

,where won−1 variesaccording to

therandom-walkmodel(14),whereqn isazero-meani.i.d.sequence

withcovariance matrix Q.Moreover,qn is independentof

{

am

}

and

{

xm,wo₋₁

}

forallm

<

n.Withyn

=

an

−

ea,n,itistruethat

2E

e∗_a,n

ϕ(

yn

)

=

μ

Tr

(

R)E

|

ϕ(

yn

)

|

2

+

μ

−1Tr

(

Q

)

(T1.1) Expression(T1.1)canbesolvedforsteady-stateEMSE,whichis definedas

EMSE

lim

n→∞E

|

ea,n

|

2 ₍₂₀₎

TheprocedureofevaluatingEMSEusing(T1.1)avoidstheneedfor explicit evaluation of E

˜

wn

2 or its steady-state value E

˜

w∞

2

whichcanbeaburdenespeciallyforadaptiveschemeswith non-linearupdateequations.Inthesequel,inadditiontothevariance relation,thefollowingjustifiedassumptionsareused:

A1) Insteady-statetheaprioriestimationerrorea,nisindependent ofboththetransmittedsequence

{

an

}

andtheregressorvector xn [14].

A2) Thenumberoffiltertapsislargeenoughsothatbyvirtueof the central limit theorem, ea,n is zero-mean complex valued Gaussian[59,60].

A3) Theoptimumfilterachievesperfectequalization(zero-forcing solution) an

≈

(

wno−1

)

Hxn; however,dueto channel variation andgradientnoise,theequalizerweightvectorisnotequalto wo

n eveninsteady-state[61].Additionally,noadditivenoiseis assumedinthesystem(see[48,49,62–67]).

Assumption A1is theorthogonality condition requiredfora suc-cessful convergence. Assumption A2, the Gaussianity of a priori

estimationerror,hasappearedinanumberofrecentpublications. For example, Bellini [68] discussed that the convolutional noise (which bears similar mathematical definition as that of a priori

estimationerror)maybeconsideredaszero-meanGaussian. More-over, [69] discussedthat theapriori estimation error(for a long equalizer)maybemodeledasazero-meanGaussianrandom vari-able. It has been shown that the steady-state apriori estimation erroriszero-meanGaussian,evenforthecasewherethe measure-ment noise is takento be uniformlydistributed. The assumption

A3 is basedon the understanding that CMA2–2 and similarly its multimodulusvariantsdivergeoninfinitetimehorizonwhennoise is unbounded.Interestedreaders mayrefer to[70] for adetailed discussion on thisissue.Note that the (total)mean square error, MSE ofanon-divergingequalizerinthepresenceofadditivenoise, however,can always be givenasMSE

=

σ

2

ϑ

+

EMSE,where

σ

ϑ2 is

thevariance ofmodelingerror/measurementnoise.The degreeof non-stationarity(DN)ofthedataisdefinedasDN

Tr(R Q

)/

σ

2

ϑ

[14]. DN

>

1 means that the statistical variations in the optimal weightvector aretoofastforthefiltertotrackthem.However,if DN

1,thenthefilterwouldgenerallybeabletotrackthe varia-tionsinweightvector[14].

Hereonwards,forthesakeofnotationalsimplicity,weuse

ζ

:=

EMSE, ea

:=

ea,n, y

:=

yn, a

:=

an,

ϕ

:=

ϕ

(

yn) and Pa

=

E

|

a

|

2

=

E(a2_R

+

a2_I

).

Also, the acronyms LHS andRHS are used to denote theleft-hand sideandtheright-hand side,respectively.

5. Steady-state EMSE analysis

We now apply the fundamental variance relation to differ-entMMA adaptivealgorithmsto obtainanalytical expressions for steady-state EMSE by evaluating the energy of error-function as wellasitscorrelationwithaprioriestimationerror.Duetospace limitations, we omit some trivial details and only highlight the mainstepsinthearguments.

5.1.TheEMSEofMMA2–2 equalizer

Usingthefundamentalvariancerelation(T1.1),wehavethe fol-lowingtheoremforthetrackingEMSEofMMA2–2 equalizer:

Theorem 2 (TrackingEMSEofMMA2–2).ConsidertheMMA2–2

recur-sion(7)withcomplex-valueddata.Considerthenon-stationarymodel

(14)withasufficientlysmalldegreeofnon-stationarity.ThenitsEMSE canbeapproximatedbythefollowingexpressionforasufficientlysmall step-size

μ

:

ζ

MMA2–2

(μ)

=

μ

c1

+

1 μTr

(

Q

)

c2

−

μ

c3

,

(T2.1)

μ

MMA2_opt –2

=

Tr

(

Q

)

c1c2₂

+

Tr

(

Q

)

2c2₃

−

Tr

(

Q

)

c3 c1c2

with

ζ

_minMMA2–2

=

2Tr

(

Q

)

μ

optc2 (T2.2) where,c1

:=

2Tr(R

)

Ea6 R

−

2R2REa4R

+

R4REa2R

,c2

:=

2(3Ea2R

−

R2R

)

, andc3

:=

Tr(R

)

3Ea4 R

+

R4R

.Substitutingtheexpressionfor

μ

optinto

theexpressionofEMSEwefindthecorrespondingoptimalEMSE.

Proof.In[52],weobtainedthefollowingpolynomial forEMSEof

MMA2–2: 15 4

ζ

3

μ

Tr

(R

)

+

ζ

2

μ

Tr

(R)

45 2Ea2R

−

3R2R

−

3

−

ζ

6Ea2_R

−

2R2_R

−

μ

Tr

(R

)

3Ea4_R

+

R4_R

+

μ

Tr

(

R)

2Ea6_R

+

2R4_REa2_R

−

4R2_REa4_R

+

μ

−1Tr

(

Q

)

=

0 (21)

Inordertoevaluatesomeclosed-formexpressionsof

ζ

MMA2–2, cer-tainapproximationshavetobemade,e.g.,byneglectingthecubic andquadratictermsin(21),weobtain

ζ

μ

Tr

(R)(

3Ea4_R

+

R4_R

)

−

6Ea2_R

+

2R2_R

+

μ

Tr

(

R)

2Ea6_R

+

2R4_REa2_R

−

4R2_REa4_R

+

μ

−1Tr

(

Q

)

=

0 (22)

whichyieldsthefollowingclosed-formsolution:

ζ

MMA2–2

=

μ

2_Tr

(R)

_2Ea6 R

+

2R4REaR2

−

4R2REa4R

+

Tr

(

Q

)

μ(

6Ea2 R

−

2R2R

)

−

μ

2Tr

(

R)(3Ea4R

+

R4R

)

(23)

2

5.2. TheEMSEofMMA2–1 equalizer

Undersimilarconditionsandassumptions,asmentionedin Sec-tion4,wehavefollowingtheoremforMMA2–1:

Theorem 3 (TrackingEMSEofMMA2–1).ConsidertheMMA2–1

recur-sion(9)withcomplex-valueddata.Considerthenon-stationarymodel

(14)withasufficientlysmalldegreeofnon-stationarity.ThenitsEMSE canbeapproximatedbythefollowingexpressionforasufficientlysmall step-size

μ

:

ζ

MMA2–1

(μ)

=

⎛

⎜

⎝

−

c3

+

c2₃

+

4

(

c1

−

c2

μ)(

Pac2

μ

+

_μ1Tr

(

Q

))

2

(

c1

−

c2

μ)

⎞

⎟

⎠

2

,

(T3.1)

μ

MMA2–1 opt

=

!

Tr

(

Q

)

Tr

(R)

Pa

with

ζ

_minMMA2–1

=

ζ

MMA2–1

μ

MMA2–1 opt

,

(T3.2)

wherec1

:=

2(√2_M

−

1),c2

:=

Tr(R

)

,c3

:=

√8_πRR_M,andM isthesizeof

square-QAM.Substitutingtheexpressionfor

μ

optintotheexpressionof

EMSEwefindthecorrespondingoptimalEMSE.

Proof. ForMMA2–1 equalizer, the error-function isgiven as

ϕ

=

fRyR

+

j fIyI

=

ϕ

R

+

j

ϕ

I.Forsimplicitywecanrepresentthereal andimaginarypartsoftheerror-functionas

ϕ

L,where

ϕ

L isequal to yL and

−

yL for

|

yL,n

|

<

RL and

|

yL,n

|

>

RL, respectively. We obtainE

|ϕ|

2 _{forMMA2–1 asfollows:}

E|

ϕ

|

2

=

E y2_R{|y R|<RR}

+

y 2 R{|yR|>RR}

+

y 2 I{|yI|<RI}

+

y 2 R{|yI|>RI}

=

E y2_R

+

y2_I

=

Ey2_R

+

Ey2_I (24)

SubstitutingEy2_L,itfollowsimmediatelythat E

|

ϕ

|

2

=

_P

a

+

ζ

.Thus theRHS of(T1.1)forMMA2–1 isthusevaluatedasfollows:

RHS

=

μ

Tr

(

R) (Pa

+

ζ )

+

μ

−1Tr

(

Q

)

(25) Next substituting the apriori error in (T1.1), and computing the correlationbetweenequalizererror-functionandconjugateofa pri-orierror,theLHS of(T1.1)forMMA2–1 isevaluatedas:

LHS

=

2E

e∗_a

ϕ

=

2E aRfRyR

−

fRy2R

+

aIfIyI

−

fIy2I

=

2E aRyR

−

y2R

{|yR|<RR}

−

aRyR

−

y2R

{|yR|>RR}

+

aIyI

−

y2I

{|yI|<RI}

−

aIyI

−

y2I

{|yI|>RI}

=

2E

aRyR

−

y2R

−

8E

aRyR

−

y2R

{yR>RR}

+

2E

aIyI

−

y2I

−

8E

aIyI

−

y2I

{yI>RI} (26)

ExploitingassumptionA2,weobtain

LHS

= −2

ζ

+

8E

"

RR 2

ζ

π

exp

−

(

aR

−

RR

)

2

ζ

+

ζ

4

1

+

erf

aR

−

RR

√

ζ

#

+

8E

"

RI 2

ζ

π

exp

−

(

aI

−

RI

)

2

ζ

+

ζ

4

1

+

erf

aI

−

RI

√

ζ

#

(27)

where erf(

·

),

the Gauss error function, is defined as erf(x

)

=

2 √ π

$

x 0exp

−

t2

dt. Owing to four quadrant symmetry of QAM constellation,themomentsevaluatedforin-phasecomponentare sameasthoseforquadraturecomponent.Simplifyingand combin-ing(25)and(27),weobtain

−

2

ζ

+

A

−

μ

Tr

(

R) (Pa

+

ζ )

−

μ

−1Tr

(

Q

)

=

0 (28) where A

:=

16E

RR 2

ζ πexp

−

(aR−RR)2 ζ

+

ζ 4

(1

+

erf

aR_√−RR ζ

)

. Since theargumentinsidetheexponentfunction,

(

aR

−

RR

)

2,is al-wayspositive,wehaveexp(

·

)

=

0 foraR

=

RR and

ζ

1.However, whenaR

=

RR,we have exp(

·

)

=

1 with probability Pr

[

aR

=

RR

]

. Similarly,undertheassumption

ζ

1,erf(

·

)

isequalto

−

1,and 0, respectively,forthecases(aR

<

RR),and(aR

=

RR).These consid-erationsyield A

≈

⎧

⎨

⎩

0

,

ifaR

=

RR

8RR

ζ π

+

4

ζ

Pr

[

aR

=

RR

]

,

ifaR

=

RR (29) Sincean M-pointconstellationisbeingconsidered,theprobability Pr

[

aR

=

RR

]

is equal to 1/

√

M. Denoting c1

:=

2(√2_M

−

1), c2

:=

Tr(R

),

c3

:=

√8RR

πM,andc4

:=

Tr(R

)

Pa,andbycombining(28)–(29), weobtain

(

c1

−

c2

μ)ζ

+

c3

(

ζ

−

(

c4

μ

+

_μ1Tr

(

Q

))

=

0

.

(30) Solving it by quadratic formulawe obtain (T3.1). Further, substi-tuting

ζ

=

v2 _{andtakingderivative withrespectto}

μ

_{,we obtain}

(2

vc1

+

c3

)

_dd_μv

−

v2c2

−

c4

+

μ

−2Tr(Q

)

=

0.Fortheoptimumvalue of

μ

,wehave _dd_μv

=

0;thisgives

μ

MMA2–1_opt

=

!

Tr

(

Q

)

c4

+

c2

ζ

_minMMA2–1

(31)

Sincec2

ζ

minMMA2–1

c4,thusignoringitweobtain(T3.2).

2

5.3. TheEMSEofMMA1–2 equalizer

Undersimilarconditionsandassumptions,asmentionedin Sec-tion4,wehavefollowingtheoremforMMA1–2:

Theorem 4 (TrackingEMSEofMMA1–2).ConsidertheMMA1–2

recur-sion(11)withcomplex-valueddata.Considerthenon-stationarymodel (14)withasufficientlysmalldegreeofnon-stationarity.ThenitsEMSE canbeapproximatedbythefollowingexpressionforasufficientlysmall step-size

μ

:

ζ

MMA1–2

(μ)

=

μ

c1

+

1 μTr

(

Q

)

2

−

μ

Tr

(

R)

,

(T4.1)

μ

MMA1_opt –2

=

!

Tr

(

Q

)

c1

with

ζ

_minMMA1–2

=

ζ

MMA1–2

μ

MMA1_opt –2

,

(T4.2)

wherec1

:=

2Tr(R

)

E(RR

− |

aR

|

)

2.Substitutingtheexpressionfor

μ

opt

intotheexpressionofEMSEwefindthecorrespondingoptimalEMSE.

Proof. ForMMA1–2 equalizer, we have

ϕ

=

ϕ

R

+

j

ϕ

I,where

ϕ

L

isequalto

(

RL

−

yL) and

(

−

RL

−

yL) for yL

>

0 and yL

<

0, re-spectively.Now,substituting theerror-functionin (T1.1)andthen pluggingintherequiredmoments,itfollowsimmediatelythat

E

|

ϕ

|

2

=

_E

(

_R Rsgn

(

yR

)

−

yR

)

2

+

(

RIsgn

(

yI

)

−

yI

)

2

=

E R2_R

+

y2_R

−

2RR

|

yR

| +

R2I

+

y2I

−

2RI

|

yI

|

=

R2_R

+

Ey2_R

−

2RRE|yR

| +

R2I

+

Ey2I

−

2RIE|yI

|

=

R2_R

+

Pa

+

ζ

−

2RRE

)

exp

*

−

a2R

ζ

+

ζ

π

+

aRerf

aR

√

ζ

,

+

R2_I

−

2RIE

)

exp

*

−

a2I

ζ

+

ζ

π

+

aIerf

aI

√

ζ

,

(32)

Using(32),theRHSforMMA1–2 equalizerbecomes:

RHS

=

μ

Tr

(R)

*

R2_R

+

Pa

+

ζ

−

2RRE

"

exp

*

−

a2R

ζ

+

ζ

π

+

aRerf

aR

√

ζ

#

+

R2_I

−

2RIE

)

exp

*

−

a2I

ζ

+

ζ

π

+

aIerf

aI

√

ζ

,+

+

μ

−1Tr

(

Q

)

(33)

TheLHSof(T1.1)forMMA1–2 canbeevaluatedas:

LHS

=

2E

e∗_a

ϕ

=

2RRE

(

aRsgn

(

yR

))

−

2E

(

aRyR

)

−

2RRE

|

yR

| +

2Ey2R

+

2RIE

(

aIsgn

(

yI

))

−

2E

(

aIyI

)

−

2RIE|yI

| +

2Ey2I (34) Exploiting assumption A2and aftersome straightforward mathe-maticalmanipulation,itfollowsthat

LHS

=

2

ζ

−

2RRE

)

exp

*

−

a2R

ζ

+

ζ

2

,

−

2RIE

)

exp

*

−

a2I

ζ

+

ζ

2

,

(35)

Owing tofourquadrantsymmetry ofQAMconstellation,the mo-ments evaluated for in-phase component are same as those for quadrature component. Simplifying the equality LHS

=

RHS, we obtain 2

ζ

−

4RRA

−

μ

Tr

(R

)

2R2_R

+

Pa

+

ζ

−

4RRB

−

μ

−1Tr

(

Q

)

=

0 (36) where A

:=

E

[

exp

−

a2R ζ

_ζ 2

andB

:=

E

exp

−

a2R ζ

_ζ π

+

aRerf

_aR √ ζ

. Since the argument inside the exponent function, a2

R, is always positive, and

ζ

1, thus we have exp(

·

)

=

0 for both (aR

>

0), and(aR

<

0).Similarly,undertheassumption

ζ

1,erf(

·

)

isequal to

+

1 and

−

1, respectively,for thecases (aR

>

0), and(aR

<

0). TheseconsiderationsyieldA

≈

0 andB

≈

E

|

aR

|

.So,wecanrewrite theequalityin(36)as 2

ζ

−

2

μ

Tr

(R

)

ζ 2

+

E

(

RR

− |

aR

|

)

2

−

μ

−1_Tr

(

_Q

)

=

_{0 (37)} Solving(37)for

ζ,

wedirectlyobtain

ζ

MMA1–2

=

2

μ

Tr

(R)

E

(

RR

− |

aR

|

)

2

+

μ

−1_Tr

(

_Q

)

2

−

μ

Tr

(

R) (38)

Denoting c1

:=

2Tr(R

)E(

RR

− |

aR

|

)

2, and c2

:=

Tr(R

)

we obtain

(T4.1).Further,substituting

ζ

=

v2 andtakingderivative with re-spectto

μ

,weobtain 2v_dd_μv

−

v2_c

2

−

c1

+

μ

−2Tr(Q

)

=

0.Forthe optimumvalueof

μ

,wehave _dd_μv

=

0;thisgives

μ

MMA1–2_opt

=

!

Tr

(

Q

)

c1

+

c2

ζ

_minMMA1–2

(39)

Wecan assume that the termc2

ζ

_minMMA2–1 is negligiblerelative to thefirsttermc1,thusignoringitweobtain(T4.2).

2

5.4.TheEMSEofMMA1–1 equalizer

Undersimilarconditionsandassumptions,asdiscussedearlier, wehavethefollowingtheoremforMMA1–1:

Theorem 5 (TrackingEMSEofMMA1–1).ConsidertheMMA1–1

recur-sion(13)withcomplex-valueddata.Considerthenon-stationarymodel (14)withasufficientlysmalldegreeofnon-stationarity.ThenitsEMSE canbeapproximatedbythefollowingexpressionforasufficientlysmall step-size

μ

:

ζ

MMA1–1

(μ)

=

*

2Tr

(R

)μ

+

_μ1Tr

(

Q

)

(

8

/

√

M

π

)

+

2

,

(T5.1)

μ

MMA1_opt –1

=

!

Tr

(

Q

)

2Tr

(R)

with

ζ

MMA1–1 min

=

ζ

MMA1–1

μ

MMA1_opt –1

.

(T5.2)

Substitutingtheexpressionfor

μ

optintotheexpressionofEMSEwefind

thecorrespondingoptimalEMSE.

Proof.FortheMMA1–1 equalizer,wehave

ϕ

n

=

ϕ

R

+

j

ϕ

I,where

ϕ

L isequalto

+

1 for

|

yL

|

<

RL and

−

1 for

|

yL

|

>

RL.Now, substi-tutingtheerror-functionin(T1.1),theenergyoftheerror-function E

|ϕ|

2_asfollows: E

|

ϕ

|

2

=

_E

(

+

₁

)

2 {|yR|<RR}

+

(

−

1

)

2 {|yR|>RR}

+

(

+

1

)

2_{|yI|<RI}

+

(

−

1

)

2_{|yI|>RI}

=

E

1_{−∞<yR<∞}

+

1{−∞<yI<∞}

=

2 (40)

AftertheevaluationofE

|ϕ|

2_{,itimmediatelyfollowsthat}

RHS

=

2

μ

Tr

(

R)

+

μ

−1Tr

(

Q

)

(41)

Substituting theconjugate apriori estimation errore∗_a,n in (T1.1), wecanobtaintheLHS of(T1.1)forMMA1–1 equalizerasfollows:

LHS

=

2E

e_a∗

ϕ

=

2E [aR

ϕ

R

−

yR

ϕ

R

+

aI

ϕ

I

−

yI

ϕ

I]

=

2E

(

aR

−

yR

)

_{|yR|<RR}

−

2E

(

aR

−

yR

)

{|yR|>RR}

+

2E

(

aI

−

yI

)

{|yI|<RI}

−

2E

(

aI

−

yI

)

{|yI|>RI}

=

2E

(

aR

−

yR

)

−

8E

(

aR

−

yR

)

{yR>RR}

+

2E

(

aI

−

yI

)

−

8E

(

aI

−

yI

)

_{yI>RI} (42)

ExploitingassumptionA2,weobtain

LHS

=

8E

)

1

√

2

π

exp

−

(

aR

−

RR

)

2

ζ

ζ

2

,

+

8E

)

1

√

2

π

exp

−

(

aI

−

RI

)

2

ζ

ζ

2

,

(43)

Owingtofourquadrant symmetryofQAMconstellation,the mo-ments evaluated for in-phase component are same as those for quadraturecomponent. Simplifying andcombining(41) and(43), weobtain A

−

2

μ

Tr

(R

)

−

μ

−1Tr

(

Q

)

=

0 (44) where A

:=

16E

√1 2πexp

−

(aR−RR)2 ζ

ζ 2

.Sincetheargument in-sidetheexponentfunction,

(

aR

−

RR

)

2,isalwayspositive,wehave exp(

·

)

=

0 for aR

=

RR and

ζ

1.However, when aR

=

RR, we haveexp(

·

)

=

1 withprobabilityPr

[

aR

=

RR

]

.Theseconsiderations yield A

≈

⎧

⎨

⎩

0

,

ifaR

=

RR

8

ζ π

Pr

[

aR

=

RR

]

,

ifaR

=

RR (45) SinceanM-pointconstellationisbeingconsidered,theprobability Pr

[

aR

=

RR

]

isequalto1/

√

M.Rewritingtheequality(44)as

8

ζ

π

M

−

2

μ

Tr

(R)

−

μ

−1_Tr

(

_Q

)

=

_{0 (46)}

Solving(46)for

ζ

,itfollowsdirectlythat

ζ

_minMMA1–1

=

*

√

π

M

2

μ

Tr

(R

)

+

μ

−1_Tr

(

_Q

)

8

+

2 (47) Denoting c1

:=

√8

πM and c2

:=

2Tr(R

),

we obtain (T5.1). Further, substituting

ζ

=

v2 _{andtaking derivative with respect to}

μ

_{, we} obtain c1_dd_μv

−

c2

+

μ

−2Tr(Q

)

=

0.For the optimum value of

μ

, wehave _dd_μv

=

0;solvingthisyields(T5.2).

2

6. Comparison with existing methods

Some state oftheartmethodsforEMSE analysisareavailable inliterature,see[65,66,69].In[66],GouptilandPalicotdeveloped ageometricalapproachtosteady-stateanalysisforBussgang algo-rithms, andderived aclosed-form analyticalexpressionforEMSE, whichwhenextendedtotrackinganalysisisgivenas

ζ

≈

μ

Tr

(R

)

E

|

ϕ

|

(a,a∗)

|

2

+

μ

−1_Tr

(

_Q

)

2E_∂y∂_∂2_y∗

[

ea∗

ϕ

]|

(a,a∗)

(48)

It is important to note that thisapproach could be extended to certain algorithms whichhavecontinuous error-functions (like MMA2–2 andMMA1–2)inastraightforwardmannertoobtain ap-proximate expressions which we have mentioned inTheorems 2 and 4. However, it becomes mathematically intractable to apply thisapproachforEMSEanalysisofalgorithmswithdiscontinuous error-functions like MMA2–1 and MMA1–1, due to the fact that therequiredderivativesdonotexit.

The EMSE expression by Gouptil and Palicot was based on circularity assumption for the a priori estimation error, ea, i.e., Ee2a

=

0. Without exploiting the circularity assumption, Linet al. in[65] derived steady-stateexpressionsutilizingTaylorseries ex-pansion.Theyobtainedthefollowingexpressions:

ζ

≈

μ

Tr

(R

)

E

ϕ

|

(a,a∗)

2

+

μ

−1_Tr

(

_Q

)

A1

−

μ

Tr

(R)

A2

,

(49) where A1

:= −2E

*

∂

ϕ

∂

y

(a,a∗)

+

(50) and A2

:=

E

∂

ϕ

∂

y

(a,a∗)

2

+

E

∂ϕ

∂

y∗

(a,a∗)

2

+

2E

*

ϕ

∗

∂

2

ϕ

∂

y

∂

y∗

(a,a∗)

+

.

(51)

Table 1

EMSEinanon-stationaryenvironmentforfourmembersofMMAp–q. MMA2–2 μTr(R)d1+μ−1Tr(Q) d2−μTr(R)d3 , where d1=Ea6R−2R2REa4R+R4REa2R,d2=6Ea2R−2R2R,d3=3Ea4R+R4R MMA2–1 * −d1+ d2 1+d2 μTr(R)+μ−1Tr(Q)−2Tr(Q)(μTr(R)+Tr(Q)) d2+4−μTr(R) +2 whered1=√8_πRR_M,d2=8 2 √ M−1 MMA1–2 2μTr(R)E(RR−|aR|)2+μ−1Tr(Q) 2−μTr(R) MMA1–1 πM 2μTr(R)+μ−1_Tr(Q)2 64 Table 2

Optimumstep-sizeinanon-stationaryenvironmentforfourmembersofMMAp–q.

MMA2–2 Tr(Q)Tr(R)d1d22+Tr(Q)2Tr(R)2d23−Tr(Q)Tr(R)d3 Tr(R)d1d2 , where d1=Ea6R−2R2REa4R+R4REa2R,d2=6Ea2R−2R2R,d3=3Ea4R+R4R MMA2–1 Tr(Q) Tr(R)Pa MMA1–2 _2Tr(R)Tr_E((_RQ) R−|aR|)2 MMA1–1 _2TrTr(₍Q_R)₎

Similar to (48), the expressions (49)–(51) involve the evalua-tionofderivatives.Incaseofcontinuouserror-functions,theEMSE analysiscan be carriedout by this approach,but is not applica-bletothecaseofdiscontinuous error-functions.Itisimportantto notethatwehaveoriginallyprovidedtheaccurateandclosed-form (undercertainassumptions)expressionsforsteady-stateEMSEfor different multimodulus equalizers. However, the approaches pro-posed by Gouptil and Palicot and Lin et al., only applicable for algorithmswithcontinuouserror-function,andthereforecannotbe extendedtoalgorithmswithdiscontinuouserror-functions.

In[69],NaffouriandSayedproposedaningeniousapproachfor theevaluationofEMSEby exploitingfundamental energy conser-vationrelationandPricetheorem.TheproposedEMSEexpression isgivenas

ζ

=

μ

Tr

(R

)

hU

(ζ )

+

μ

−1_Tr

(

_Q

)

2hG

(ζ )

,

(52) where hU

(ζ )

E|

ϕ

|

2 (53) and hG

(ζ )

E

[

e∗_a

ϕ

]

E

|

ea

|

2

.

(54)

Theresult(52)–(54) corroborate theexpressions that we have obtainedfordifferentmultimodulusequalizers.

7. Simulation results

In this section, we verify the tracking performance analyses forMMA2–2,MMA2–1,MMA1–2 andMMA1–1 (assummarizedin

Tables 1 and2).Theexperimentshavebeenperformedconsidering (i) comparison withstate of the artmethods, (ii) a time-varying channel(withaconstantmeanpartandanautoregressiverandom part), (iii)the effectof filter-length onequalization performance, and(iv)equalizinganopticalchannelforadaptivepolarization de-multiplexing.

7.1. ExperimentI:Consideringzero-forcingsolution

Inthisexperiment, theelements ofperturbation vectorqn are modeledaszeromeanwide-sensestationaryandmutually uncor-related. Thecorresponding positivedefiniteautocorrelationmatrix ofqn isobtainedas Q

=

σ

q2I (where

σ

q

=

10−3).3 Thesimulated EMSE have beenobtainedforequalizerlengthsN

=

7 andN

=

11 for16-QAM signals.ThevaluesofRR

=

RI areequalto8.2,3,2.5 and31forMMA2–2,MMA2–1,MMA1–2 andMMA1–1 for16-QAM signals, respectively.Eachsimulatedtraceisobtainedby perform-ing100independentrunswhereeachrunisexecutedfor5

×

103 iterations. Note that, dueto assuming an already equalized sce-nario, we do not haveto worry aboutthe iterations requiredfor successful convergence of the equalizer; thus the EMSE is com-puted for all iterations. The Monte-Carlo simulation requires to addtheperturbationqndirectlyintheweightupdateprocess.The weightupdate,inthisexperiment,isthusgovernedby

wo_n

=

wo_n−1

+

μϕ(

yn

)

∗xn

+

qn (55) Thetermscontainingthestep-size

μ

andq_ncontributetotracking andacquisitionerrors[14].Therule(55)hasbeenadoptedin[14, 48–50,61].

Since the steady-state EMSE of the MMA algorithms is com-posedof two(tracking andacquisition)errors.The trackingerror decreases with

μ

andincreaseswiththe systemnon-stationarity variance Tr(Q

).

The acquisition error increases with

μ

and the received signal varianceTr(R

),

thus,theresultingEMSE is a con-vex downward(bowlshaped)function ofstep-size

μ

.Noticeably, forall simulationcases,theanalyticallyobtainedminimumEMSE (ζ_minMMAp–q)andtheoptimumstep-size(

μ

MMA_opt p–q), aremarked, re-spectively, with markers

and

♦

. Refer to Figs. 3(a) and 4(a) forthecomparisonofanalyticalandsimulatedEMSE ofMMA2–2 equalizer. The legends ‘Numerical’ and ‘Closed-from’ refer to the solutions(21)and(T2.1),respectively.Itisevidentfromthisresult that,for16-QAM withsmallerfilterlength(i.e.,N

=

7),the numer-ical,theclosed-formandthesimulatedtracesconformeachother for all valuesof step-sizes. However, for larger filter length (i.e.,

N

=

11),tracesstartdeviatingfromeachotherforhighervaluesof EMSE.

Next, refer to Figs. 3(b) and 4(b) in which the analytical and simulatedEMSE ofMMA2–1 equalizerarecompared.The legends ‘Numerical’and‘Closed-from’refertothesolutions(28)and(T3.1), respectively.Itcanbeobservedfromtheresultsthat,for16-QAM, the numerical,theclosed-form andthesimulatedtracesconform eachother forallvaluesofstep-sizesforbothfilterlength(N

=

7 and N

=

11). Figs. 3(c)and4(c)compare the analytical and sim-ulated EMSE of MMA1–2 equalizer. The legends ‘Numerical’ and ‘Closed-from’ refer to the solutions (36) and (T4.1), respectively. It is evident from the results that both expressions (numerical andclosed-form)areingoodagreementwiththesimulatedtraces for 16-QAM for both filterlength (N

=

7 and N

=

11). Similarly, refertoFigs. 3(d)and4(d)whichcomparetheanalyticaland sim-ulated EMSE of MMA1–1 equalizer. The legends ‘Numerical’ and ‘Closed-from’ refertothesolutions(44)and(T5.1),respectively.It isevident fromthisresultthat,for16-QAM forboth filterlength (N

=

7 andN

=

11),thenumerical,theclosed-formandthe simu-latedtracesconformeachotherforallvaluesofstep-sizes.

Here we observe that the numerical results deviate from Monte-Carloresultswhenthestep-sizeisfarawayfromthe

opti-3 _{Notethatthismodeling(i.e.,zerooff-diagonalelementsin}

Q

_{)isjustifiedinthe} lightofouranalyticalfindingsinTheorems2,3,4 and5whichimplythattheEMSE dependsneitherontheindividualdiagonalelementsnortheoff-diagonalelements ofmatrix

Q

,butratherdependsonTr(Q).Inotherwords,giventhesumofthe meansquarefluctuationsoftheelementsof

q

n,theEMSE doesnotdependonthe

Fig. 3.EMSE traces forN=7 and 16-QAM.

Fig. 5.EMSE traces for four members of MMAp–qwith 16-QAM signaling onchannel-1. For fDT=0.01 and unit lag, we haveα=0.999.

mum step-size.We emphasize atthe fact that the EMSE expres-sions obtained(in thiswork) are validonly forsufficiently small step-sizes.Itisalsoimportanttonotethatthereareupperbounds on the step-sizes above which an adaptive filter cannot provide anyusefuloutput.The reasonwhyEMSEanalyticaltrace deviates fromthesimulatedonesatverysmallstep-sizeisnotcompletely understoodatthisstage.

7.2. ExperimentII:Equalizingtime-varyingchannel

Weevaluatetheperformanceanalysisoftheaddressed equaliz-ers in the presence of a time-varying (TV) channel. A TV chan-nel is usually modeled such that its autocorrelation properties correspond to wide-sense stationary and uncorrelated scattering (WSSUS) (as suggested by Bello [71]). However, as reported in

[72], a first-order (Gauss–Markov) autoregressive model is suffi-cientenoughtomodelaslow-varyingchannel,wherethechannel atindexnis givenashn

=

hconst

+

cn.The channel isacomplex Gaussian random process with a constant mean hconst (because of shadowing,reflections, andlarge scale path loss) and a time-variant part cn, which is a first-order Markov process as given by cn

=

α

cn−1

+

dn where

α

is a constant, andthe vector dn is a zero-mean i.i.d. circular complex Gaussian process with corre-lation matrix D.4 _{The channel taps varies fromsymbol to}

sym-4 _{ForanAR(1)system,}

α

_=J

o(2πfDT),whichmakestheautocorrelationofthe

tapsmodeledby

c

n=αcn−1+dnequalthetrueautocorrelationatunitlag(where

Joisthezero-orderBesselfunctionofthefirstkind,fDistheDopplerrateandT

isthebaudduration).Theparameter

α

determinestherateofthechannelvariation whilethevariances

σ

2

d,i oftheithentryof

d

n determinesthemagnitudeofthe

variation.So,

α

and

σ

2

d,i determineshow“fast”andhow“much”thetime-varying

partcn,i ofeachchanneltaphn,i varieswithrespecttotheknownmeanofthat

taphconst,i.Thevalue of

α

can beestimatedfromtheestimateof fD. Similarly,

giventheaverageenergyoftheithpartof

c

n,E|cn,i|2,thevalueof

σ

d,iisevaluated

as[72]σd,i= |hconst,i|2

√

1−α2-(_E|cn ,i|2.

bol and are modeled as mutually uncorrelated circular complex Gaussian randomprocesses.The time-varyingpartofthechannel canbe modeledbya pth-order autoregressiveprocess AR(p

).

The matrix D, due to WSSUS assumption,is diagonal andeach ofits diagonalelementis

σ

2

d.Inthepresentscenario,weconsider

σ

2 d

=

1

×

10−3

,

α

=

0.999,andhconst

= [

1

+

0.2j

,

−

0.2

+

0.1j

,

0.1

−

0.1j

]

T usinga7-tapbaud-spacedequalizerwith16-QAMsignaling.Refer

to Fig. 5 for the comparison of theoretical and simulated EMSE

ofMMA2–2,MMA2–1,MMA1–2 andMMA1–1 equalizers.The leg-end‘Analysis’referstothesolutions(T2.1),(T3.1),(T4.1)and(T5.1)

forMMA2–2,MMA2–1,MMA1–2 andMMA1–1 equalizers, respec-tively.Itisevidentfromtheresultsthatthetheoreticaland simu-latedEMSEtracesconformeachother.NotethatthefactorTr(Q

)

hasbeenreplacedwithTr(D

)

intheevaluationofanalyticalEMSE. Inthesequel,werefertothischannelas

channel-1

.

7.3. ExperimentIII:Effectoffilter-lengthonEMSE

Inthepreviousexperiment,weconsideredaTVchannelwhere the effectof filter-length onequalization capabilityhas not been taken into consideration. It is widely known that a reasonable filter-length is required to equalize successfully a propagation channel. An insufficientfilter-length introduces an additional dis-tortionwhichwehavenotconsidered inTheorems 2,3,4,and5. However, asmentioned in [3], thedistortive effect ofinsufficient filter-lengthmayeasilybeincorporated(intheEMSEexpressions) asanadditiveterm;thetotalEMSE,whichwedenoteasTEMSE,is thusgivenasfollows:

TEMSE

=

lim n→∞E

|

ea,n

|

2

.

/0

1

=:ζ

+

.

E

|

an

|

2

H

/0

wo∗

−

e

2

1

=:χ (56)

where

ζ

isEMSEaswe obtainedinTheorems 2,3,4 and5,and

Fig. 6.Transient EMSE traces for MMAp–qwith 16-QAM signaling onchannel-2.

Fig. 7.EMSEtracesconsideringtheeffectoffilter-lengthonchannel-2.(a) Theoptimalfilter-lengthforMMA2–2 isfoundtobe7and9for9.5

×

10−5_and4

.7

×

10−5 respectively,(b) Theoptimalfilter-lengthforMMA2–1 isfoundtobe9and11for2.8×10−3_and1.3×10−4_{,(c) Theoptimalfilter-lengthforMMA1–2 isfoundtobe7for} both6.8×10−4_and3.5×10−4_{,(d) Theoptimalfilter-lengthforMMA1–1 isfoundtobe7and9for9.5×10}−4_and4×10−4_{respectively.}

filter-length.Thevector wo _{iszero-forcingsolution,}

H

_ischannel matrix,ande isoverall idealistic (single-spike) channel-equalizer impulseresponseasdefinedinSection2.

Notethat theEMSE,

ζ

, isproportional to filter-length forthe given step-size. The parameter

χ

on the other hand decreases withfilter-length.5 _{Inoursimulation,thevalueofoptimalweight}

5 _{Theactualexpressionof}

χ

_{(asdenotedbyD}

f in[3,Eq. 4.8.24])containsan

equalizersolution(asdenotedbyθ)thatalsodependsonblindequalization error-function.However,wehaveobservedthatthetruevalueofθisveryclosetoH+e forallfouraddressedmembersofMMAp–qwhere(·)+denotespseudo-inverse.So, inthiswork,wehavereplacedthetrueexpressionofθwithitssimplifiedform wo∗_=H+_{eandoursimulationfindings (asdepictedinFig. 7)validatethatthis} simplificationisreasonable.

vector, wo_{,isobtainedas w}o

=

pinv

(

H

)

_{e where}

pinv

(

·

)

_{is the} MATLABfunctionfortheevaluationofpseudo-inverse.TheTEMSE, as expressed in (56), is a convex downward function of filter-length. Evaluating TEMSE for different filter lengths can provide us with the optimal value of filter-length required to equalize the given channel and given step-size. In this simulation, we have considered avoice-band telephone channel hn

= [−

0.005

−

0.004j

,

0.009

+

0.03j

,

−

0.024

−

0.104j

,

0.854

+

0.52j

,

−

0.218

+

0.273j

,

0.049

−

0.074j

,

−

0.016

+

0.02j

]

[73] and16-QAM signal-ing.Theeigenvaluespreadofthechannelis5.83andtheISI intro-ducedbythischannel is

−

8.44 dB.Inthesequel,werefertothis channelas

channel-2

.

Thetwo differentvaluesofstep-size(

μ

) arechosen suchthat theequalizersconvergetosteady-statearound1500iterationsand 3000 iterations, asdepicted in Fig. 6. All simulation points were

obtainedbyexecutingtheprogram10times(orruns)withrandom andindependentgenerationoftransmitteddata.Eachrunwas exe-cutedforasmanyiterationsasrequiredfortheconvergence.Once convergenceisacquired,theequalizerisrunforfurther5000 iter-ationsforthecomputationofsteady-statevalueofEMSE.InFig. 7, we depict analytical and simulated TEMSE obtained as a func-tionoffilter-lengthforthegivenstep-sizesforMMA2–2,MMA2–1, MMA1–2 andMMA1–1.Both analyticalandsimulatedTEMSEare foundtobeincloseagreement.

7.4. ExperimentIV:Adaptivepolarizationdemultiplexing

Inthisexperimentweconsideranadaptiveoptical demultiplex-ing scenario. A key part of the digital signal processingreceiver unit is to demultiplexthe received signal to recover thetwo or-thogonal polarization tributaries sent from the transmitter end. ThiscanbedoneusingblindadaptiveFIRfilters,updatedusingthe stochastic gradient algorithm (employing only the demultiplexed sequence)asproposedin[28].Thefiltersarearrangedina butter-flystructure[74]asshowninFig. 8andarecontinuouslyupdated. NotethatthemultiplexingphenomenoncanbemodeledasaJones

matrix.Giventheazimuthrotationangle2θ andtheelevation

rota-tionangle

φ,

theunitary2

×

2 (Jones)matrix

R

,whichrepresents thebasebandmodeloftwomultiplexed opticalchannels, isgiven by[29]

Fig. 8.Optical butterfly equalizer.

R

(θ, φ)

=

"

cos

(θ )

sin

(θ )

exp

(

−

j

φ)

−

exp

(

j

φ)

sin

(θ )

cos

(θ )

#

.

(57)

Notethatthetworowsrepresentmultiplexedchannelswhich ro-tatethehorizontalandverticalstatesofpolarizedtransmitteddata and convert them into a new but arbitrary pair of orthogonal states.Supposexnandyn arethetransmittedpolarizationdivision multiplexed QAM (PDM-QAM) signals, using the channel model, thereceivedpolarizedsignals(whichbecomeinputtothe demul-tiplexer)are

"

xin_n y_nin

#

=

R

"

xn yn

#

(58)

Ithastobenotedthatthetwoinputsignalsoftheblock,xinn and

yin

n,are a mixture of the two signals emitted along the two or-thogonal statesof polarization oflight. Therefore the taskof the adaptiveequalizeristoestimatetheinverseoftheJonesmatrixso astoreversethe effectsinducedby thechannelpropagation. The adaptiveequalizer(demultiplexer) wn isan adaptive2

×

2 matrix andisdefinedaswn

=

wxx_n wnxy

;

w yx n w