The Multimodulus Blind Equalization and Its Generalized Algorithms

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 5, JUNE 2002 997

The Multimodulus Blind Equalization and ItsGeneralized Algorithms

Jian Yang, Member, IEEE, Jean-Jacques Werner, Fellow, IEEE, and Guy A. Dumont, Fellow, IEEE

AbstractThis paper presents a new blind equalization algo-rithm called multimodulus algorithm (MMA). This algorithmcombines the benefits of the well-known reduced constellationalgorithm (RCA) and constant modulus algorithm (CMA). In ad-dition, MMA provides more flexibility than RCA and CMA, andis better suited to take advantage of the symbol statistics of certaintypes of signal constellations, such as nonsquare constellations,very dense constellations, and some wrong solutions.

Index TermsAdaptive equalizer, blind equalization, carrier-less amplitude and phase modulation, least mean square methods,quadrature amplitude modulation.

I. INTRODUCTION

THE CONCEPT of blind equalization has been knownsince the publication of Satos original work on thissubject, in 1975 [1]. Satos algorithm was subsequently gener-alized, and other types of blind equalization algorithms wereproposed and analyzed [2][11]. In spite of all these earlycontributions of significance, until recently, blind equalizationhad only found limited applications. The renewed interest inthis topic has been triggered by applications such as asyn-chronous transfer mode (ATM) local area network (LAN) andbroadband access on copper in fiber-to-the-curb (FTTC) andvery high-rate digital subscriber line (VDSL) networks, forwhich blind equalization provides major benefits [12][14].

We now briefly discuss one of these applications. Thepoint-to-multipoint arrangement shown in Fig. 1 is usedin FTTC networks, which provide broadband access to thehome using standard unshielded twisted pair (UTP) telephonewiring in the network and coaxial cable, or UTP wiring in thehome. Details on the characteristics of the communication linkbetween the optical network unit (ONU) in the cable plantand the various set-top boxes and personal computers (PCs)in the home are given in [14] and will not be repeated here.The downstream channel, from the ONU to the home, usesa 51.84 Mb/s 16-carrierless amplitude and phase modulation(CAP) signal, which is broadcast to the various terminationpoints inside the home. The upstream channel, from the hometo the ONU, uses a 1.62 Mb/s quadrature phase-shift keying(QPSK) burst modem. Our interest here is in downstream

Manuscript received March 30, 2001; revised December 17, 2001. The workof J. Yang and J.-J. Werner was supported by Bell Laboratories. This paper waspresented in part at DSP97, Santorini, Greece, 1997.

J. Yang is with Bell Laboratories, Holmdel, NJ 07733 USA (e-mail:[email protected]).

J.-J. Werner (deceased) was with Bell Laboratories, Holmdel, NJ 07733 USA.G. A. Dumont is with the University of British Columbia, Vancouver, BC

V6T 1Z4, Canada (e-mail: [email protected]).Publisher Item Identifier S 0733-8716(02)05381-7.

transmission. A simplified diagram showing the signal flow inthe downstream direction is shown on the bottom right in thefigure. Assume now, for example, that the two set-top boxes areoperational and that the PC is suddenly turned on. The 16-CAPreceiver in the PC needs to be trained before it can deliver validdata. Conceptually, this can be done in a variety of ways.

1) The transmitter in the ONU could interrupt its transmis-sion of data to the set-top boxes for some time intervaland transmit instead a known training sequence for thereceiver in the PC. Such an interruption of data transmis-sion to the set-top boxes is obviously not desirable.

2) The transmitter in the ONU could send a periodic trainingsequence, as is done in some broadcast applications, suchas high definition television (HDTV). However, the over-head incurred with such an approach cannot be justifiedfor the FTTC application considered here because of thesmall number of termination points in the home.

3) The best solution is to blindly train the receiver that isbeing turned on. That is, the receiver is trained withoutthe help of a known training sequence and uses insteadthe (unknown) sequence of data that is being sent to theother termination points in the home.

The most complex and time-consuming task during blindstartup of a receiver is the convergence of the equalizer, whichis done with a blind tap updating algorithm. The two bestknown blind equalization algorithms for two-dimensionalmodulation schemes, such as quadrature amplitude modulation(QAM) and CAP, are the reduced constellation algorithm(RCA) and the constant modulus algorithm (CMA). RCAis very simple to implement, but does not provide reliableinitial convergence. CMA provides reliable convergence, butincreases the complexity of implementation of the receiver insteady-state operation because of the need to add a rotator atthe output of the equalizer.

The multimodulus algorithm (MMA) presented here com-bines the benefits of RCA and CMA. It provides reliable ini-tial convergence and does not need the addition of a rotator insteady-state operation. The latter property seems to have beendiscovered independently in [15] and [16], [17]. In addition,MMA provides much more flexibility than RCA and CMA andis better suited to take advantage of the symbol statistics of cer-tain types of signal constellations, such as nonsquare and verydense constellations [16][18]. RCA and CMA are not very ef-fective in handling these types of signal constellations. MMAalso suitable for CAPQAM dual mode reception [19].

The rest of the material is organized as follows. A briefreview of CAP transceivers and equalizer structures is providedin the next section. Various commonly used cost functions

0733-8716/02$17.00 2002 IEEE

998 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 20, NO. 5, JUNE 2002

Fig. 1. Broadband access on copper in FTTC networks.

and tap updating algorithms are discussed in Section III.In Section IV, we introduce the basic MMA algorithm forsquare signal constellations, and show how it can be mod-ified to accommodate nonsquare constellations. Section Vpresents a generalized MMA (GMMA) algorithm, which iswell suited to very dense signal constellations. The combinedCMAMMA and dual-mode CAPQAM receiver will bepresented in Sections VI and VII separately. The issue of con-vergence to wrong solutions during blind startup is discussedin Section VIII. Then experimental results obtained in thelaboratory with a 51.84 Mb/s 16-CAP and other DSP setupare presented in Section IX. Finally, we summarize paper inSection X.

II. CARRIERLESS AM/PM MODULATION TRANSCEIVER

A. Transceiver Structure

CAP is a bandwidth-efficient two-dimensional passbandtransmission scheme, which is closely related to the morefamiliar QAM transmission scheme. The block diagram of adigital CAP transmitter is shown on the top of Fig. 2. The bitstream to be transmitted is first passed through a scrambler(not shown in the figure). The scrambled bits are then fed toan encoder, which maps blocks of bits into one ofdifferent complex symbols . A CAP linecode that uses different complex symbols is called a -CAPline code. The two-dimensional display of the discrete valuesassumed by the symbols and is called a signal constella-tion. Examples of 16-CAP and 32-CAP signal constellationsare shown in Fig. 3. After the encoder, the symbols andare fed to digital shaping filters. The outputs of the filter aresubtracted and the result is passed through a digital-to-analog(D/A) converter, which is followed by an interpolating low-passfilter (LPF). The digital shaping filters and the D/A operate ata sampling rate where is a suitably choseninteger and is the symbol rate.

Fig. 2. A communication link using a CAP transceiver. (a) Transmitterstructure. (b) Receiver structure.

The signal at the output of the CAP transmitter in Fig. 2 canbe written as

(2.1)

where is the symbol period, and are discrete multilevelsymbols, which are sent in symbol period , and andare the impulse responses of in-phase and quadrature passbandshaping filters, respectively, and form a Hilbert pair. Details onthe design of the shaping filters can be found in [17].

The structure of a digital CAP receiver is shown on the bottomof Fig. 2. It consists of an analog-to-digital (A/D) converter fol-lowed by an adaptive equalizer. Examples of adaptive equalizerstructures are given in the next section. The A/D operates at asampling rate , which is typically the same as the

YANG et al.: THE MULTIMODULUS BLIND EQUALIZATION AND ITS GENERALIZED ALGORITHMS 999

(a) (b)Fig. 3. (a) 16-point signal constellation. (b) 32-point signal constellation.

sampling rate used for the D/A at the transmitter. The two out-puts of the adaptive equalizers are sampled at the symbol rate

and, in steady-state operation, the results are fed to a de-cision device followed by a decoder, which maps the symbolsinto bits. Let be the impulse response of the channel and let

be some noise added to the CAP signal in the channel. Thesignal at the input of the CAP receiver can then be written as

(2.2)

where denotes convolution, and and. It should be pointed out that and

still form a Hilbert pair. Thus, the distortion introduced by thechannel does not affect the generic form of the expression for aCAP signal that is given in (2.1).B. Adaptive Equalizer Structures

A great variety of adaptive equalizer structures can be usedfor CAP signals. For example, all the types of equalizers thatare used for QAM can be used for CAP as well. The differ-ence between CAP and QAM is the way to implement. QAMrequires a modulator/demodulator which is explicitly used inCAP within the passband shaping filters. CAP does not requirethe phase-recovery circuit that is usually used at the output of theequalizer in a QAM receiver. In this paper, we will only considerlinear adaptive equalizers. Blind convergence of decision-feed-back equalizers will be discussed in a forthcoming paper.

A fractionally spaced linear equalizer (FSLE) that is par-ticularly well suited to the applications considered here is thephase-splitting equalizer shown in Fig. 4 [20]. It consists of aparallel arrangement of two adaptive digital filters, which taketheir inputs directly from the A/D at the sampling rate andare implemented as finite-impulse-response (FIR) filters. In thesteady-state mode of operation, the two outputs of the filters aresampled at the symbol rate , and are then fed to a decisiondevice (slicer). Normally, is held.

We now make the following definitions with respect to Fig. 4:

vector of A/D samples in delay line (2.3)

vector of in-phase tap coefficients (2.4)

vector of quadrature phase tap coefficients (2.5)where the superscript denotes vector transpose, and the sub-script is a short notation for the symbol period . With

, we have . The outputs and of thedigital filters can then be written as

(2.6)If we define the following complex equalizer output andcomplex tap vector :

(2.7)We can rewrite (2.6) in the more compact complex form

(2.8)

III. COST FUNCTIONS AND TAP UPDATING ALGORITHMSIn most practical applications, the tap coefficients of an adap-

tive equalizer are adjusted by using a stochastic gradient algo-rithm, i.e., the complex tap vector is updated according to

(3.1)where is a small number called step size and is the gra-dient of some cost function ( ) with respect to the tap vector

. The subscript refers to the th tap updating iteration. For aconvex cost function, the tap updating algorithm has convergedwhen the gradient is zero, that is

(3.2)Two main types of cost functions are used in practice. In onecase, they are functions of the symbols used in the signalconstellation, and in the other case they are functions of statisticsof the symbols . These two cases are discussed in the nexttwo sections. In the last section, we show how some of thesealgorithms can be used in a blind startup.

A. Tap Updating Algorithms Based on Symbol ValuesMany modems use a known training sequence during initial

startup, so that the receiver knows a priori what sequence ofcomplex symbols is sent by the transmitter.The adaptive equalizer can then be converged with a so-calledideal reference. In this case, the cost function that is usually usedis the mean squared error (MSE) defined as

(3.3)where is the complex output of the equalizers and de-notes expectation. The tap updating algorithm in (3.1) then be-comes the least-mean-square (LMS) algorithm. Practical imple-mentations usually use the unaveraged gradient, so that the LMSalgorithm for the two equalizers in Fig. 4 can be written as

(3.4)where , and is thecomplex error in symbol period , as shown in Fig. 5. Letbe the complex noise sample at the output of the equalizer afterconvergence, so that and .


Fig. 4. Structure of phase-splitting FSLE with decision devices.

Fig. 5. Error used for various types of tap updating algorithms.

Thus, after convergence of the equalizer, the MSE in (3.3) issimply equal to the variance of the noise . Note that the aboveequalizer convergence is applied for a linear channel with re-quired whiteness of source data. In addition, due to the limi-tation of some factors such as FIR filter length and adaptationparameters, the convergence of an adaptive equalizer always ex-ists certain amount of excess mean square error and some mis-adjustment.

In the steady-state mode of operation, the slicers in Fig. 4make the right decisions most of the time by selecting thesymbol which is the closest in Euclidean distance to thereceived sample . In this case, the known symbols can bereplaced with the estimated (or sliced) symbols . The costfunction that is then minimized is given by

(3.5)and the tap updating algorithms are the same as in (3.4) exceptthat replaces , as shown inFig. 5. When these algorithms are used, the equalizer is said tobe adapted in the decision-directed mode [21]. With this algo-rithm, the MSE reduces again to the variance of the noise at theoutput of the equalizer in the steady-state mode of operation.

The MSE in (3.3) and (3.5) uses second-order statistics ofthe equalizers output samples. It can be shown that the use ofsecond-order statistics only is generally not sufficient for theblind deconvolution of nonminimum-phase channels [22], [23].

For this reason, most blind equalization algorithms have costfunctions which use cyclostationary second or higher order sta-tistics (HOS) of the signals. However, the second-order cyclo-stationarity is extensively used in blind equalization just becauseof the convergence problems and larger variance associated withHOS.

B. Tap Updating Algorithms Based on Symbol StatisticsDuring initial convergence of an adaptive equalizer, it usu-

ally is not possible to use a decision-directed tap updating algo-rithm, because the slicers make too many errors. If no trainingsequence is available, then the equalizer has to be convergedunder so-called blind conditions. In this case, the cost func-tion that is minimized does not depend on known or estimatedsymbols and , but on known statistics of the symbols .

The simplest blind tap updating algorithm is the so-called re-duced constellation algorithm (RCA). The cost function mini-mized by RCA is the MSE with respect to a reduced numberof symbol values, which usually are not a subset of the symbolvalues used in the signal constellation. When four symbol valuesare used, the cost function for RCA can be written as

(3.6)where is the complex signumfunction, and the constant is a function of the statistics ofthe symbols in the signal constellation. Closed-form expressionsfor are given in Appendix A. The quantity minimizedby the RCA cost function is shown in Figs. 5 and 6(a). The tapupdating algorithm for the equalizers in Fig. 4 is given by

(3.7)Another well-known blind equalization algorithm is the con-stant modulus algorithm (CMA), which minimizes the disper-sion of the equalizers output samples around a circle, asshown in Fig. 6(b). The cost function for CMA is given by

(3.8)Notice that this cost function uses fourth-order statistics of thesignals. The corresponding tap updating algorithm for the twoequalizers in Fig. 4 is given by

(3.9)Expressions relating the constant to the statistics of the sym-bols are given in Appendix A.


(a) (b) (c)Fig. 6. Principles of (a) RCA, (b) CMA, and (c) MMA.

It is shown in [24] that the CMA cost functions in (3.8) isminimized when the channel is perfectly equalized (and noise-less) and the kurtosis of the complex symbols is negative, that is

and(3.10)

The condition on the right is satisfied for all signal constella-tions of practical interest. It should be pointed out that, even inthe absence of noise, the minimum of the cost function is gen-erally not equal to zero, except for 4-CAP. Closed-form expres-sions for the minimum values of the RCA, CMA, and MMAcost functions are derived in Appendix B.

Both RCA and CMA are used in practice. However, these twoblind equalization algorithms do not fully take advantage of thestatistics of the set of symbols used in certain signal constel-lations, such as nonsquare and very dense constellations. TheMMA algorithm described in the following sections is muchmore flexible in this regard. In addition, it is quite efficient inreducing occurrence of wrong solutions. This issue is discussedin detail in Section VIII.

C. Blind StartupWe now briefly discuss how the various tap updating algo-

rithms can be incorporated in a typical blind startup of a receiver.Illustration of the procedure will be given in Section IX whenwe discuss experimental results. A typical blind startup of a re-ceiver consists of the following three main sequential steps.

1) Adjust the automatic gain control (AGC) and acquiretiming (synchronize the receivers clock to the clock ofthe far-end transmitter).

2) Adapt the tap coefficients of the equalizer with a blindequalization algorithm until the eye of the signal con-stellation is open.

3) Switch to a decision-directed tap updating algorithmwhen the eye is open.

The eye of the signal constellation is considered to be openwhen the slicer makes the right decisions most of the time or,equivalently, when the MSE measured across the slicer is smallenough. A probability of symbol error of 10 is usually con-sidered acceptable to guarantee a safe switch between the blindequalization and decision-directed tap updating algorithms. Inreal-time DSP prototype, when the measure of MSE or symbol

error is not suitable, we observe experimentally and manuallyswitch the equalizer from blind startup to LMS algorithm.

IV. BASIC MMA

In this section, we discuss the simplest version of the MMAand its application to small and medium-sized ( ) squareand nonsquare signal constellations. The generalization of thealgorithm to dense constellations and other applications are pre-sented in the next three sections.

A. Square ConstellationsWe first consider square constellations. The cost function

minimized by MMA is then given by

(4.1)where is a positive integer. In practice, a value usuallyprovides the best compromise between performance and com-plexity of implementation. This cost function is similar to thecost function for CMA in (3.8), except that the termis missing. As a result, the MMA cost function is not a trulytwo-dimensional cost function. It can be considered as the sumof two one-dimensional cost functions, which minimize the dis-persion of the output samples and of the equalizer aroundseparate straight contours (or moduli), as shown in Fig. 6(c).This should be more obvious if we rewrite (4.1) as

(4.2)where is the cost function for the in-phase samples and

is the cost function for the quadrature samples . It isshown in Appendix C that a condition similar to (3.10) appliesto the minimum of the MMA cost function, except that the con-dition applies independently to the real and imaginary symbols.For the symbols we have

and (4.3)We now consider the phase-splitting equalizer in Fig. 4. Takingthe gradient of and with respect to the tap vectorsand , respectively, we get

(4.4)(4.5)


Fig. 7. MMA moduli for 128-point signal constellation.

The derivation of the expression for the constant is pro-vided in Appendix A, where it is shown that

(4.6)

The equality on the right in (4.6) holds in the usual case wherethe symbols and have the same statistics. The stochasticgradient tap updating algorithms for the tap vectors andare given by the following expressions:

(4.7)(4.8)

where we have incorporated the factor in the step size . For, we get

(4.9)(4.10)

(4.11)

Similar to the conditions required for an adaptive updating al-gorithm, MMA is applied for a linear channel and requires thewhiteness of the source data.

B. Nonsquare ConstellationsWe now show how the MMA algorithm can be modified to

take advantage of the statistics of the symbols used in non-square constellations. Fig. 7 shows a 128-point signal constel-lation, which is used to transmit blocks of 128 7 bits.This constellation is obtained from a square constellation with12 12 144 points by removing the four outer points in eachcorner. We will assume that the symbol levels used along eachdimension are taken from the set 1, 3, 5, 7, 9, 11and that all the 128 complex symbols are sent with the sameprobability 1/128. In this case, the discrete values taken by thereal and imaginary symbols and do not all have the sameprobability of occurrence. Specifically, the largest values 9,

11 occur less often than the smaller values 1, 3, 5, and7. Thus, the statistics of the symbols and vary along

their respective dimensions.MMA takes advantage of the variation of statistics along each

dimension by using piecewise linear contours (or moduli) for

nonsquare constellations, rather than the straight contours usedfor square constellations [see Fig. 6(c)]. The piecewise linearcontours used for the 128-point constellation are shown on theleft in Fig. 7. The dotted lines are used for the in-phase dimen-sion and the solid lines are used for the quadrature dimension.

The in-phase and quadrature cost functions for square con-stellations in (4.2) are modified as follows for nonsquare con-stellations having two different sets of statistics along each di-mension:

if

if (4.12)if

if (4.13)

where is a constant that is a function of the signal constellationunder consideration. Generalization to the case where there aremore than two sets of statistics along each dimension is straight-forward. Notice that two different constants are used in (4.12)and (4.13). We now show how these constants are computed forthe 128-point constellation.

The constants are always evaluated from the expressionin (4.11), which requires the computation of the second- andfourth-order moments of the symbols. We now show how tocompute , for example. We can compute this moment byconsidering the first quadrant only. Consider the subset of 24complex symbols in this quadrant that applies to in Fig. 7.For these symbols 1, 3, 5, 7, 9, 11 and 1, 3, 5, 7 so thateach value of occurs with probability 4/24 1/6. Similarly,the subset has 8 symbols for which 1, 3, 5, 7 and9, 11 so that each value of occurs with probability 2/8 1/4.Thus, the variance of the symbols becomes

for (4.14)

for (4.15)

Other moments for the symbols are computed in a similarfashion, and we find that the two moduli for the 128-point con-stellation are given by 9.2 and 6.1. The separateconstant moduli for 32-CAP and 128-CAP are listed in Table II.A single modulus could also be used, but this would increasethe probability of converging to the so-called 144-point wrongsolution, which will be discussed in Section VIII.

V. GENERALIZED MMA (GMMA)The RCA and CMA blind equalization algorithms discussed

previously are not very effective in providing a good eyeopening when the number of different symbols in the signalconstellation becomes very large. The basic MMA algorithmalso has difficulties with very dense constellations, but, becauseof its flexibility, it can be modified to ease the eye opening ofthese constellations. This is achieved by dividing the complexplane of in-phase and quadrature output samples of the equal-


izer into several disjoint regions, which all have their own costfunctions and moduli. This modified MMA algorithm is calledgeneralized MMA (GMMA).

A. The Problem With Dense ConstellationsIn this section, we provide a brief intuitive explanation of

why RCA, CMA, and basic MMA have difficulties in openingthe eye of very large signal constellations. From (3.7), (3.9),(4.7), and (4.8) we see that the stochastic gradient tap updatingalgorithms used for the phase-splitting equalizer, for example,all have the following generic form:

(5.1)where depends on the type of algorithm that is being used.Assume now that we are in steady-state and that the complextap vector has converged in the mean, so that

(5.2)Thus, in steady-state the mean of the correction termin the tap updating algorithm is zero, but its variance is gen-erally not equal to zero. This results in tap fluctuations, whichcontribute tap adaptation noise to the output signal of the equal-izer. When a decision-directed algorithm is used in steady-stateoperation, the quantity in the correction term becomes theerror across the slicer, which is also equal to the noise atthe output of the equalizer, as was shown in Section III-A. Themagnitude of this noise is comparable to the magnitude of theadditive noise in the channel, and is small compared with thespacing between the points in the signal constellation. Multipli-cation by the small step size further decreases the variance ofthe correction term. As a result, the tap fluctuations due to theadaptation algorithm are very small and do not contribute sig-nificantly to the MSE at the output of the equalizer.

Assume now that we use RCA, for example. The quantityused in the correction term is then the error with respect to

one of four points, as shown in Fig. 7. It should be apparentfrom the figure that this error cannot become small in steady-state operation, even in the absence of noise. It should also beapparent that the variance of increases compared with thesquared distance between symbols when the number of points

in the signal constellation is increased. This increases the tapfluctuations and associated tap adaptation noise, and makes itincreasingly difficult to open the eye. The only way to keep thetap fluctuation noise low when is increased is to decrease thestep size . However, this decreases the speed of convergenceof the equalizer. In addition, finite precision effects become afactor in a practical implementation if is too small. It has beenfound that, for the applications discussed here, RCA, CMA, andbasic MMA become impractical when 64, 128, and

256, respectively.Rather than decreasing the step size when is increased,

one can, instead, try to keep the variance of the correction termsmall by keeping small. This is the approach that

is used for GMMA. The idea is to divide the complex planeof equalizer output samples into smaller regions, which containsubsets of symbols of the main constellation. Each subset is thentreated as a constellation with a reduced number of symbols, and

Fig. 8. GMMA sample subsets and moduli for 256-point signal constellation.

has its own MMA cost function and modulus. With such an ap-proach, the variance of and the corresponding tap adaptationnoise remain manageable when the number of points in the mainsignal constellation increases.

B. Principle of GMMAThe principle of GMMA will be discussed with respect to

the 256-point signal constellation shown in Fig. 8, where, forsimplicity of notation, we are showing the in-phase dimension( ) along the vertical axis and the quadrature dimension ( )along the horizontal axis. We will again treat the in-phase andquadrature dimensions independently, as was done for the basicMMA. The dotted lines in the figure represent boundaries forvarious subsets of in-phase output samples of the equalizer.In this example, there are three subsets (1), (2), and (3), whichinclude the symbol levels 1, 3, 5, 7 , 9, 11 , and

13, 15 , respectively. The solid lines represent the moduli,which are used for each subset, and we have 6.08,10.25, and 14.17. Similar subsets and moduli are alsodefined for the quadrature samples . Multiple moduli (but notmultiple sample subsets) had previously been used in [26] for adifferent cost function than the one proposed here.

We now describe the GMMA tap updating algorithm as it ap-plies to the in-phase tap vector of the phase-splitting equal-izer shown in Fig. 4. The equation used for updating the tapvector is always the one given in (4.9). However, the valueof the constant use in the computation is a function of thevalue of the in-phase sample . For example, if 8 12in Fig. 8, then the modulus corresponding to this subset has to beutilized, i.e., 10.25. Similarly, when belongs toanother subset, then the corresponding modulus has to be usedin the computation of the tap updating algorithm in (4.9).

C. Computation of the GMMA Design ParametersThe definition of the various subsets of equalizer output sam-

ples and corresponding moduli used by GMMA is not straight-forward, and has to be made carefully if one wants to get the ben-efits provided by this algorithm. Here, to address this problem,


we present an iterative algorithm, which uses an equal energytype of principle and leads to a good choice of subsets of equal-izer output samples and corresponding moduli.

An ideal choice of sample subsets and moduli would guar-antee that the tap fluctuation noise introduced by each subsetis the same and is as low as possible. Practical designs, suchas the one described below, can only approximate this ideal de-sign. Rather than dealing directly with tap fluctuation noise, wewill instead use a closely related quantity, which is the minimumof the MMA cost function. The main idea behind the iterativealgorithm is to design sample subsets in such a way that thisminimum is (roughly) the same for all the subsets. Assume nowthat we have in-phase subsets and that represents theth sample subset, which includes the subset of symbols .

The cost function minimization is done with respect to eachsample subset , i.e., we minimize

(5.3)

The minimum of this cost function is again obtained whenor . If the minimum value of the cost function

is the same for all the sample subsets, then the cost function ofthe dense constellation has the same minimum value.

For simplicity, we will restrict our discussion of the itera-tive algorithm to square constellations, but it can also be usedfor nonsquare constellations. We will assume that the symbols

and take values 1 3 2 1 , so that themaximum number of symbol levels (in magnitude) is . Theexpressions for the modulus and minimum of the MMA costfunction for sample subset are the same as the ones used forbasic MMA, except that they are only evaluated over the subsetof symbols . The three main steps of the iterative algo-rithm are given below.

Step 1: We first choose a targeted value for the minimumof the GMMA cost function, and we choose it insuch a way that it corresponds to the minimum of asquare constellation that is easily handled by basicMMA. For the 256-point constellation in Fig. 8,we have chosen 592, which isthe minimum value of the MMA cost function fora 64-point constellation, as computed from (B-3)with 4. This choice also defines the designparameters for the first sample subset andwe have

(5.4)Step 2: We now outline the procedure used to find the

design parameters of the second subset of samples. First, we define a square constellation

using a number of symbol levels .This signal constellation is obtained from the64-point constellation by adding newsymbol values, so that the symbols inthe second subset take the following values

.

Equation (B-9) in Appendix B gives a generalexpression for the minimum of the cost function of

sample subset . Specializing (B-9) to the secondsample subset, we get

(5.5)

There are two different ways to use this equationto find . One approach is to compute the costfunction for various values of . The value ofthe cost function that is closest to the targeted

592 is obtained for 6, sothat the design parameters for the second samplesubset are

(5.6)The try-and-choose approach used above to deter-mine the value of the parameter (or in gen-eral) is adequate for most practical applications, butis not very rigorous. A more systematic and math-ematically pleasing approach is to replace the costfunction on the left in 5.5 with its targeted value of592 and then solve the equation for . This canbe done with commercially available software pack-ages. We find that the only solution that is in the de-sired range is 6.24, which has to be roundedto the closest integer, i.e., 6, for a practical imple-mentation.

Step 3: To get the design parameters for the third subset ofsamples, , we again use the procedure givenin Step 2, except that the subscripts 2 and 3 replacethe subscripts 1 and 2 in (B-9), respectively. Wefind that the design parameters for the third samplesubset are

(5.7)The same procedure is then iterated until the com-puted is larger than the number of symbollevels used by the main constellation. For 256-CAP,this number is eight and the iterative algorithm stopsafter three subsets of samples have been defined.

VI. CMAMMAAnother generalized MMA is proposed in this paper, called

CMAMMA. Rather than to create new cost functions, theCMAMMA jointly uses the two cost functions of CMA andMMA for in-phase and quadrature phase filters separately in aphase-splitting filter structure shown in Fig. 4.

The main aspect of the algorithm of CMAMMA introducesasymmetry into the tap updating algorithm of an equalizer,whereas other blind equalization algorithms use symmetricalalgorithms for the two-filter equalizer. Referring to Fig. 4, thecost functions of CMAMMA are proposed as

(6.1)(6.2)


The gradients of the cost functions derived from (6.1) and (6.2)as follows:

(6.3)(6.4)

Then we have the following stochastic gradient tap updatingalgorithms:

(6.5)(6.6)

Note that the constants are different for two filters. For in-stance with 16-CAP, we use 3.6 for CMA and2.86 for MMA.

CMAMMA algorithm jointly uses the CMA and MMA costfunctions. Now we compare CMAMMA and CMA. CMA isa truly two-dimensional (2-D) algorithm and CMAMMA is apseudo (2-D) algorithm. Due to the two-dimensional feature,both algorithms do not converge to diagonal solutions, whichwill be discussed in Section VIII. However, CMAMMA ro-tates a constellation in a right position, but CMA remains thephase offset of the constellation. This means that CMA requiresa rotator at blind startup and remains in steady-state, which re-sults in an increase of complexity.

The reason why CMA cannot rotate the constellation is dis-cussed in [17]. Now, we show why CMAMMA can rotate theconstellations. Assume a rotated constellation by some angle ,so that . The in-phase cost function isthen given as

(6.7)

We want to show that this cost function takes its minimum valuefor

(6.8)

This is equivalent to the following in (6.7):

(6.9)and

(6.10)

It is clear that the condition in (6.10) is true for any . We nowsimplify the conditions in (6.9). Equation (6.9) can be written

(6.11)

Fig. 9. Dual-mode CAPQAM receiver.

So that (6.7) becomes

(6.12)The quantity emphasized above is the negative of the so-calledkurtosis. It can be shown that the kurtosis is always negativefor typical signal constellation, so that this quantity is alwayspositive, so that (6.8) is satisfied for any .

CMAMMA takes advantages of the strength of both CMAand MMA. CMA cannot rotate a constellation and MMA mayconverge to some wrong solutions. Without an additional cost,the new algorithm achieves more reliable convergence duringinitial startup.

VII. DUAL MODE CAPQAM RECEIVERRecently, in some broadband excess standards require that the

transmitter can use either CAP or QAM line codes [28]. Thereceiver, on the other hand, must accommodate both line codes,blindly startup, then blindly deciding which line code was sent,and finally decoding the sent symbols. In a straight forward way,we can use a parallel receiver, separately using CAP or QAMthat may increase complexity. Another way to implement it is tostart in CAP mode and switch to QAM mode if CAP mode doesnot produce satisfactory results. Now, we propose a dual-modeCAPQAM receiver capable of demodulating both CAP andQAM-modulated data by using a single equalizer. In particular,this receiver applies to xDSL-type channels where no frequencyoffset is introduced.

The detailed CAP and QAM transmitter can be found in [19].The receiver structure of the dual-mode CAPQAM is illus-trated in Fig. 9. Now we propose a dual-mode CAPQAM re-ceiver for MMA. The MMA cost function for a CAP receiver isgiven as

(7.1)

If (7.1) is applied to the QAM signal in Fig. 3, the cost functionneeds to be adjusted as

(7.2)

We see from (7.2) that the constant needs to be computedfor QAM receiver. Note that the computation of for the costfunction in (7.1) with a CAP receiver can be found in Section IV


and Appendix A. Now we derive the constant for QAM. Thepartial derivative of (7.2) for the in-phase dimension is given by

(7.3)

where is the carrier frequency and is the symbol period.With perfect equalization, or , (7.3) be-comes

(7.4)

Setting (7.4) to zero, we obtain

(7.5)

With the assumptions of , and, seeing in [17], (7.5) can be simplified in the

steps shown in (7.6) at the bottom of the page.From (7.6), we see that for a QAM receiver, the constant

is not only a function of the transmitted symbols, but also ofthe angle . Then the constant can be numericallycomputed from (7.6). It can also be expressed as a function ofthe symbol level number . The expression of for a standardMMA for CAP constellation can be found in Appendix A. Wecan rewrite (7.6) in as shown in (7.7) at the bottom of the page.Defining , (7.7) canbe rewritten as

(7.8)

From (7.8), we see that is a function of and with, this leads (7.8) become

, which is the same as for MMA with a CAPreceiver, see Appendix A.

The constant can be computed from either (7.6) or (7.8),Because is dependent on , can be different even forthe same constellation. For instance, for 16-QAMwith 15.55 MHz and 25.92 MHz, we obtain

. The following example computesthe constant from (7.8)

(7.9)

Above shows a basic dual-mode CAPQAM blind equalizer.More developed algorithms of this type equalizer can be foundin [19]. RCA and CMA can be also applied for CAPQAM re-ception. However, for some applications, MMA achieves betterperformance.

VIII. CONVERGENCE TO WRONG SOLUTIONS AND INCOMPLETESOLUTIONS

One of the major problems with blind equalization is the pos-sibility to converge the equalizer to so-called wrong solutions.These wrong solutions should be distinguished from the localminima discussed in [26]. When the equalizer converges to alocal minimum, the cost function of the blind equalization al-gorithm is not minimized. Wrong solutions, on the other hand,can, potentially, minimize the cost function, as will be shownlater.

The probability of converging to a wrong solution duringblind startup is a function of the type of blind equalization al-gorithm and equalizer structure being used. The characteristicsof the channel can also be a major factor. Fig. 10 shows some

(7.6)

(7.7)


Fig. 10. Wrong solutions obtained with RCA.

wrong solutions, which were obtained in the laboratory with aphase-splitting equalizer when it was converged with RCA.

In Fig. 10, the two solutions on the top were obtained witha 16-CAP transceiver using the 16-point constellation shown inFig. 3(a) and the two solutions on the bottom were obtained witha 32-CAP transceiver using the 32-point constellation shownin Fig. 3(b). The diagonal solution for 16-CAP that is shownon the top left is the most frequently observed wrong solutionwhen RCA is used to blindly converge a phase-splitting equal-izer. This solution occurs when the in-phase and quadrature fil-ters of the equalizer synthesize the same transfer function. Itshould be pointed out that convergence to the diagonal solutionis not possible with a cross-coupled equalizer. The two wrongsolutions shown on the right produce signal constellations whichare rotated by 45 with respect to the original constellation. Thistype of wrong solution can also be observed with the cross-cou-pled equalizer. Finally, the wrong solutions shown in Fig. 10 canalso be obtained with MMA, but much less frequently than withRCA.

CMA cannot converge a phase-splitting equalizer to the diag-onal solutions shown on the left in Fig. 10. However, it systemat-ically produces rotated solutions, as shown in Fig. 11. This typeof incomplete solution can be obtained with both phase-splittingand cross-coupled equalizers, or any other type of equalizer thatuses CMA during blind startup. The use of differential codingcan only correct 90 phase ambiguity wrong solutions, but itcannot converge an equalizer with arbitrary phase offset intro-duced in the channel.

CMA produces these rotated solutions because it cannot com-pensate for a fixed phase offset introduced by the channel.Consider the CMA cost function in (3.8) and assume that theequalizers complex output samples are rotated by an angle

. The cost function can then be written as

(8.1)Thus, the value of the CMA cost function is not affected by arotation of the equalizers complex output samples . As aresult, CMA cannot compensate for such a rotation. A rotation

Fig. 11. Rotated solutions obtained with CMA.

of the equalizers output samples can be obtained by rotatingthe complex tap vector , as should be apparent from (2.8).Thus, if the tap vector minimizes the CMA cost functionwhen , then all its rotated versions alsominimize the cost function and . RCA andMMA cannot produce rotated solutions with an arbitrary angle

. However, they can, occasionally, produce the 45 rotationshown in Fig. 10.

We conclude with one last type of wrong solution, which isthe so-called offset solution. This wrong solution is peculiar tothe phase-splitting equalizer and can be observed with RCA,CMA, and MMA. It occurs when the in-phase and quadraturefilters synthesize tap vectors which are offset in time by an in-teger number of symbol periods. Fig. 12 shows the effect of sucha solution on a 128-point signal constellation. In this example,the transmitter uses the nonsquare 128-point signal constella-tion shown in Fig. 7. The signal constellation obtained at theoutput of the equalizer is shown in Fig. 12. This constellation issquare and has 144 points instead of 128 points. To understandhow this can happen, assume, for example, that the followingsuccessive real and imaginary symbols have been transmitted:

and . If theequalizer had converged to the right solution, the complex out-puts of the equalizer would be

and(8.2)

which correspond to valid points in the 128-point constella-tion. Assume now that the quadrature filter of the phase-split-ting equalizer introduces a propagation delay that is one symbolperiod larger than the delay introduced by the in-phase filter.The complex output of the equalizer in symbol period willthen be

(8.3)

This is a corner point of the 144-point constellation and is not avalid symbol of the 128-point constellation. Offset solutions areeasily detected with nonsquare constellations by simply looking


Fig. 12. 144-point offset solution for 128-CAP.

at the signal constellation at the output of the equalizer. How-ever, such a detection scheme is not possible when the signalconstellation used at the transmitter is square.

Coping With Wrong Solutions: There are many techniquesthat can be used to handle wrong solutions during initial startup.One possibility is to monitor the bit stream at a higher layer andsend a signal to the receiver if no valid data (such as ATM cells)are detected after a certain amount of time. The receiver canthen initiate another blind startup of the equalizer with a newset of initial tap coefficients, for example. This process is re-peated until proper convergence is achieved. This approach hasbeen found useful in the applications discussed in [12] and [13],which have to deal with mild channel characteristics. The tech-nique is somewhat less effective when the channel introducesvery severe linear distortion.

The rotated solutions obtained with CMA can be handled byusing a rotator at the output of the equalizer. This rotator canbe implemented as a carrier recovery loop of the type used forvoiceband modems, for example [4], [19], and [27]. However,with such an approach, the rotator must be used all the timeand must operate at the symbol rate in steady-state operation.This adds unnecessary complexity to the steady-state operationof the receiver when the rotator is not required for other pur-poses, such as tracking of frequency offset and carrier fluctu-ations introduced by the channel. The applications consideredhere do not have to deal with this kind of channel impairment.

One way to handle wrong solutions is to modify the basic costfunctions given in previous sections, and incorporate some con-straints which do not allow the occurrence of wrong solutions.We provide one example here, which has been found to be effec-tive in preventing the convergence of the phase-splitting equal-izer to various wrong solutions, such as the diagonal solution.The constraint that is used is based on the fact that the impulseresponses of the in-phase and quadrature filters of a phase-split-ting equalizer form a Hilbert pair after convergence to the rightsolution. It is well known that functions that form a Hilbert pairare orthogonal and have the same energy. For a passband struc-

ture, this leads to the following conditions on the tap vectorsand :

and (8.4)We now make the following definitions:

and (8.5)These quantities can be used to modify either the RCA, CMA,or MMA cost functions. For example, the MMA cost functionin (4.1) is modified in the following way (with ):

(8.6)and the tap updating algorithms in (4.9) and (4.10) become

(8.7)

(8.8)where , , and are different step sizes, which are best deter-mined empirically.

Another way to handle wrong solutions is to use theCMAMMA algorithm which is proposed in Section VI. In-stead of modifying the cost functions, we simply use CMA forone channel and MMA for the other. This combined algorithmtakes advantages of both CMA and MMA, where the formerdoes not converge to diagonal solutions and the latter rotatesthe constellation to the right positions.

IX. EXPERIMENTAL RESULTS

The experimental results presented in this paper were ob-tained in the laboratory with the setup pictured in Fig. 13. Thetransceiver prototype used in the experiments was initially de-veloped to implement the 51.84 Mb/s 16-CAP transceivers usedin the ATM LAN, and FTTC applications described in [12] and[14]. It has since been enhanced to accommodate other appli-cations, such as the various CAP transceivers considered forVDSL and the 64-CAP transceiver specified for ATM LAN at155.52 Mb/s [13]. The performance of the CAP transceivers hasbeen evaluated in the laboratory with a variety of communica-tion links using actual cables and connecting hardware. An ex-ample of such a cabling arrangement is the FTTC broadbandaccess network shown in Fig. 1.

The CAP receiver implemented in the prototype uses thephase-splitting equalizer shown in Fig. 4. This equalizer pro-vides the best trade-off between complexity and performancein steady-state operation for the type of applications consideredhere. However, it is more prone to convergence to wrongsolutions than some other equalizers, as was discussed in theprevious section. The equalizer consists of three main hardwarecomponents, a FIFO that collects blocks of A/D samples, fastFIR filters that compute the equalizer outputs at the symbolrate , and a programmable digital signal processor (DSP)chip that can implement any of the tap updating algorithmsdiscussed here. The tap updating algorithms are iterated in theDSP at a rate lower than . New tap coefficients are down-


Fig. 13. Laboratory experimental setup.

Fig. 14. CAP spectrum at the input and output of the channel.

loaded from the DSP to the fast FIR filters when computationof the tap updating algorithm is completed.

We now describe some experimental results obtained with the51.84 Mb/s 16-CAP transceiver described in [14]. The uppertrace in Fig. 14 shows the spectrum of the 16-CAP signal at theoutput of the transmitter. The lower trace shows the spectrumof the signal at the output of a VDSL communication link con-sisting of a 700 UTP loop with a 14 bridged tap. The deepnotch in the spectrum is due to the bridged tap. RCA cannotblindly converge the equalizer in a reliable fashion with this typeof channel impairment. Both CMA and MMA are much moreeffective in opening the eye. However, as was mentioned pre-viously, CMA requires a rotator operating at the symbol rate atthe output of the fast FIR filters in Fig. 13. Such a rotator hasnot been implemented in the prototype.

For the equalizer with DSP setup shown in Fig. 13, we use48 taps and initialize filter taps with the shaping filters of thetransmitter. Fig. 15 shows the various steps of a blind startupobtained with MMA over a VDSL channel. The picture on thetop left shows the signal constellation at the output of the equal-izer before any tap adaptation has started, but after the AGChas settled. The signal constellation obtained after a couple of

Fig. 15. Main steps of a blind startup using MMA.

Fig. 16. 32-CAP and 36-point constellations.

thousand tap updating iterations with MMA is shown on thetop right. In real-time DSP setup, we just manually control theswitch with observation. The picture on the bottom left showsthe eye opening after about ten seconds. Note that due to thelimitation of the DSP implementation, we can only record lab-oratory results within a few second duration. This eye openingis good enough to allow the receiver to switch from MMA tothe LMS algorithm, which is done at this point. The LMS al-gorithm quickly tightens the dots and produces the steady-statesignal constellation shown on the bottom right in the figure.

We tested MMA with different lengths of equalizer. A min-imum number of taps is required for a blind equalizer to obtaineye opening depending the applications. The equalizer used for16-CAP may not long enough for 256-CAP. Initially, a blindequalizer can converge faster with fewer taps, but with highererror rate. After initial equalization, taps can be added to im-prove error performance. Using longer equalizer during blindstartup will generate meaningless values for the extreme endtaps.

Fig. 16 shows the convergence for 32-point nonsquareconstellation. Using piecewise multiple moduli in separate dataspaces, MMA achieves better performance in terms of avoidingconverging to wrong mapping solutions. The picture on the leftshows the 32-point convergence when two moduli are used inMMA, and the picture on the right shows 36-point solution for32-CAP.


Fig. 17. Convergence performance with MMA and GMMA for 256-CAP.

TABLE IMINIMUM VALUES OF THE COST FUNCTION FOR GMMA

Fig. 17 shows that the simulation results of comparing MMAand GMMA with the dense constellation of 256-CAP. As dis-cussed in Section V, the use of multiple moduli gives better eyeopening. Table I shows minimum values of the cost functions for256 CAP when the separate data spaces are used. We see that theuse of two moduli does not significantly reduce minimum costfunctions. We use three moduli in this application. The com-putation of the three moduli can be found in Section V-C andillustrated in Fig. 8. Note that the subspace is divided after theAGC converges, i.e., with normalized gain. The picture on theleft side shows the convergence with MMA and the picture onthe right side shows that with GMMA. As shown in Fig. 17,GMMA exhibits better eye opening than MMA. By dividingcomplex plane into smaller regions, GMMA may increase therate of wrong decisions if the decision which region to use is notcorrect. It is true that the equalizer will not converge if the prob-ability of wrong decision is high. As discussed in Section III, theeye of the signal constellations is considered to be open whenthe slicer makes right decisions most of time, or when theMSE measured across the slicer is small enough. If subspacesare properly divided, the reduction of the minimum cost func-tion helps equalizer decrease MSE. Simulation results show thatwith properly chosen parameters, GMMA can achieve better eyeopening than MMA, particularly for some noisy channels.

The various steps of blind startup with CMMAMMA are il-lustrated in Fig. 18. CMAMMA first converges to a constel-lation with a phase-offset, and then rotates to the right posi-tion. In this case, the convergence rate to a diagonal solution istremendously reduced. In the laboratory experimental work, wetested CMAMMA with various channel lengths of 20 ft, 150 ft,300 ft, and 700 ft which introduce different phase-offset to thechannel. A few hundreds of tests with those channels show no

Fig. 18. Convergence of MMACMA algorithm.

convergence to diagonal solutions. At this point, CMAMMAalgorithm is more reliable than RCA and MMA.

Experimental results for CAPQAM are not covered in thispaper. Interested readers can find detailed information in [19].

X. SUMMARYIn this paper, the multimodulus blind equalization algo-

rithm and its generalized algorithms are presented. MMAcombines the benefits of RCA and CMA algorithms that areused extensively in practical applications. MMA providesmore flexibility than RCA and CMA. Along with GMMAand CMAMMA, MMA exhibits the capability to handlenonsquare constellation, dense constellation and certain wrong


solutions. MMA is also suitable in dual-mode CAPQAMreceiver.

Computer simulations and laboratory experimental resultsare provided. The results support the theoretical analysis forMMA. MMA shows reduced rate of convergence to wrongmapping solutions. In addition, for 256-CAP, GMMA exhibitsgood eye opening due to the reduction of minimum value ofthe cost function. Without additional cost, CMAMMA can bereliably used to avoid convergence to diagonal solutions.

In the appendices, we calculate the constant modulus . Theresults provide a useful tool to analyze the cost function andperformance.

APPENDIX ACOMPUTATION OF THE CONSTANTS

This appendix presents the derivation of closed-form expres-sions for the various constants used in the RCA, CMA, andMMA algorithms. As will be shown, these expressions can beconveniently expressed as a function of the number of symbollevels (in magnitude) used along each dimension of the signalconstellation.

The general approach used to compute the constant will beexplained for the MMA algorithm. The MMA cost function isgiven by

(A-1)where and represent the equalizers output samples, and

is a positive integer. For two-dimensional CAP systems,and represent the transmitted symbols for the in-phase andquadrature phase channels, respectively. When an equalizerconverges, and , and the cost functionbecomes

(A-2)Assuming the same statistics for and , we have

. In the following, only the analysis for thein-phase channel will be provided. The same analysis applies tothe quadrature phase channel. For the in-phase dimension, thegradient of the cost function with respect to the real tap vectorwas previously given in (4.4) as

(A-3)The constant can now be evaluated by assuming perfectequalization, i.e., , and by setting the gradientto zero [2][4]. Also, if we assume that symbols in differentsymbol periods are uncorrelated, we get ,where is a fixed vector whose entries are a function of thechannel. We then get

(A-4)Solving for in (A-4), we get

(A-5)

Using the same method, we obtain the following expression forthe constant used for CMA:

(A-6)

and for RCA, we get

(A-7)

Square Constellations: The expressions derived previouslyfor the constants are functions of the moments of the symbols

and . These moments can be computed on an individualbasis, although this can be tedious. For the usual case wherethe symbols have odd integer values it is possible to derivesimple closed-form expressions for the constants as a functionof the number of symbol levels. We will assume that the symbolstake the following values ,where indicates the number of symbol levels (in magnitude).The following summations can be found in [29], for example:

(A-8)

(A-9)

(A-10)

(A-11)

These summations do not apply directly to sums of powers ofodd integer, but can be used to derive closed-form expressionsfor these types of summations. For example, we can write

(A-12)

where the two sums in the middle have been evaluated from(A-8). Similar summation manipulations can be used for othersums of powers of odd integers, and we get

(A-13)

(A-14)

(A-15)

(A-16)For square constellations, the probability of occurrence of eachsymbol level is the same, i.e., , so that the moments of thesymbol levels become

(A-17)


TABLE IITHE CONSTANT R FOR BLIND EQUALIZATION ALGORITHMS

For CMA, the constant is a function of the moments of thecomplex symbols . Assuming that the symbols and areuncorrelated, it is easily verified that

(A-18)

Using the above results and , the constants for the threeblind algorithms can be expressed in the following simple waysas a function of :

(A-19)

(A-20)

(A-21)

For nonsquare constellations, MMA uses several modulialong each dimension. The various constants are then com-puted by evaluating the summations in (A-14) and (A-16) forvarious symbol subsets rather than the whole set of symbols.An example of the procedure is given in Section IV-B.

The values of the constants for the RCA, MMA, and CMAare listed in Table II for square and nonsquare CAP applications.

APPENDIX BMINIMUM VALUES OF THE COST FUNCTIONS

In this appendix, we derive closed-form expressions for theminimum values of the various cost functions. We will onlyconsider square constellations. For the in-phase cost functionof MMA, for example, we have

(B-1)

Using the value of in (A-19), we can rewrite the minimumof the cost function as follows:

(B-2)

Using the results in (A-16), (A-17), and (A-19) we then get

(B-3)

We can compute the minima of the CMA and RCA in a similarfashion, and we get

(B-4)

(B-5)

It should be pointed out that the CMA cost function in (B-4) isfor both dimensions and that the MMA and RCA cost functionsin (B-3) and (B-5) apply to one dimension only. Notice that theminimum of all three cost functions is zero for 1, whichcorresponds to 4-CAP. However, the minimum is nonzero when

1.The expressions for the modulus and cost function of the th

in-phase sample subset used by GMMA are given by

(B-6)

(B-7)

where is the subset of symbols belonging to samplesubset . These expressions are functions of the moments

and , which can be written

(B-8)

where and are the number of symbol levels in thesquare constellations corresponding to sample subsets and

, respectively. Using the results in (A-14) and (A-16) we get

(B-9)


APPENDIX CISI OPTIMIZATION FOR MMA

The input and channel vectors are defined as follows:

(C-1)(C-2)

where is the number of samples. So the second-orderexpectation of the sample is

(C-3)

With the assumption that different symbols are uncorrelated weobtain

(C-4)

With a perfectly equalized channel and no ISI at the input of theslicer, the channel vector has only one nonzero entry equal toone and can be written as

(C-5)

In order to show that minimization of the cost function resultsin zero ISI, we can show that

(C-6)

and that the minimum is achieved when the channel vectorsatisfies (C-5). We now provide two such proofs, one for a con-strained problem and one for a nonconstrained problem. A resultwhich will be useful in the following analysis is:

(C-7)

Equality in (C-7) holds if and only if one the terms is nonzeroand all the other terms are zero.

Constrained Problem: In this scenario, we assume that thechannel vector can be written as

(C-8)

That is, one of the entries of the vector is constrained to remainequal to one, and the other entries can take any value. This as-sumption is a good approximation of what is done in practice

when the gain of AGC and the initial values of the equalizer areproperly chosen. It is easily verified that, with the constraint in(C-8), the following holds:

(C-9)

and the equal sign holds if and only if the channel vector satisfiesthe ISI free condition in (C-5).

The condition in (C-6) can be written as

(C-10)

(C-11)

Replacing by its value and using (C-4) in (C-11), we canfurther simplify the equation to

(C-12)

From the inequality in (C-9), we then have

(C-13)

and equality can only hold if and only if the channel vector sat-isfies the ISI free condition in (C-5).

Unconstrained Problem: We now remove the constraint in(C-8) of keeping one entry of the channel vector equal to one.Using results given in [24], it is possible to show that

(C-14)(C-15)

where and are called the kurtosis of the equalizeroutput samples and symbols, respectively. We can use (C-4) and(C-15) to solve for in (C-14), and we obtain

(C-16)

Using (C-7) on the right side in (C-16), we obtain

(C-17)


Replacing by its value and using (C-15), the one-dimensionalcost function of MMA in (4.7) can be written as

(C-18)

Using (C-17) in (C-18), we get

(C-19)

Replacing in the above equation by its value on the rightin (C-7), we have

(C-20)

From (C-20), we conclude that the following holds:

(C-21)

if

and

(C-22)The left condition in (C-22) is always true. The right conditionsays that the kurtosis of the symbols has to be always negative.Equality can only hold if and in (C-20).From (C-7), this implies that only one can be nonzero and

implies that the nonzero term has to be one, sothat the channel vector has to have the ISI free form given in(C-5).

To show that the kurtosis of the symbols used in typical signalconstellation is always negative we can use the results obtained

in Appendix A. The second- and fourth-order moments of thesymbols are obtained from (A-14) and (A-16). Then, we have

(C-23)

It is obvious that the values in the brackets are always larger thanzero for . As a result

for (C-24)

This completes the proof that minimization of the MMA costfunction leads to ISI free output samples of the equalizer [17].

ACKNOWLEDGMENT

The authors would like to thank V. Lawrence for his supportof this work, D. Harman and R. L. Cupo for their help with theexperimental results, and L. M. Garth and K. Balemarthy formany useful comments on the material in this paper.

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[19] L. Garth, J. Yang, and J. J. Werner, Blind equalization algorithms fordual-mode CAPQAM reception, IEEE Trans. Commun., vol. 49, pp.455466, Mar. 2001.

[20] K. H. Mueller and J. J. Werner, A hardware efficient passband equalizerstructure for data transmission, IEEE Trans. Commun., vol. COM-30,pp. 538541, Mar. 1982.

[21] R. D. Gitlin, J. F. Hayes, and S. B. Weinstein, Data CommunicationsPrinciples. New York: Plenum Press, 1992.

[22] J. K. Tugnait, Identification of linear stochastic systems via second andfourth-order cumulant matching, IEEE Trans. Inform. Theory, vol. 33,pp. 393407, May 1987.

[23] S. Haykin, Blind Deconvolution. Englewood Cliffs, NJ: Prentice-Hall,1994.

[24] O. Shalvi and E. Weinstein, New criteria for blind deconvolution ofnonminimum phase systems (channels), IEEE Trans. Inform. Theory,vol. 36, pp. 313320, Mar. 1990.

[25] W. A. Sethares, G. A. Rey, and C. R. Johnson, Jr., Approaches toblind equalization of signals with multiple modulus, in Proc. IEEE Int.Conf. Acoustics, Speech, and Signal Processing, Glasgow, Scotland,May 2326, 1989, pp. 972975.

[26] Y. Li, J. K. Liu, and Z. Ding, Length- and cost-dependent local minimaof unconstrained blind channel equalizers, IEEE Trans. Signal Pro-cessing, vol. 44, pp. 27262735, Nov. 1996.

[27] N. K. Jablon, Joint blind equalization, carrier recovery, and timing re-covery for high-order QAM signal constellations, IEEE Trans. SignalProcessing, vol. 40, pp. 13831398, June 1992.

[28] V. Oksman, Ed., VDSL Draft Specification, June 1998, ANSI StandardsContribution T1E1.4/98 045R1.

[29] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Prod-ucts. New York: Academic, 1994.

Jian Yang (M98) received the M.S. and Ph.D. de-grees in the department of electrical engineering fromthe University of British Columbia, Vancouver, BC,Canada, in 1992 and 1997, respectively.

In 1995, she joined Bell Laboratories of LucentTechnologies, Holmdel, NJ, where she is currentlya Member of the Technical Staff. From 1995 to1999, she worked on the theory and practice ofdigital signal processing for communications, withapplications to single and multicarrier (DMT)modulation, particularly using blind equalization

techniques. Her research interests also include carrier detection, carrier andtiming recovery, echo cancellation, and single/multirate adaptive filtering.She holds eight patents, mostly on blind equalization algorithms. She iscurrently working on wireless and fiber-optical communications, focusing onmodulation, equalization, and echo cancellation.

Jean-Jacques Werner (F92) was born on February9, 1943, in Rountzheim in the Alsace region ofFrance, and grew up in the nearby village ofHageneau. He received the engineering degree inelectronics from the Institut National des SciencesAppliquees, Lyon, in 1965. He immigrated toCanada and received the Masters degree, in 1967,from Laval University, Quebec, Canada. He subse-quently moved to the United States and received theSc.D. from Columbia University, NY, in 1973.

He joined Bell Labs, Lucent Technologies (formerAT&T), Holmdel, NJ, in 1973, where he immediately began making his notablecontributions to both the theory and implementation of voiceband modems. Hepioneered the use of software for real-time digital processing and conducted re-search in general areas of data communication, digital signal processing, andbroadband access. His most outstanding contribution was the successful devel-opment of echo cancellation for full-duplex operation as is used in V.32 andV.34 modems. Over the course of his career, he has published extensively inarchived journals and conferences and has been issued 45 patents. In addition,he has made several key contributions in areas of voice-band modems, DSL,and home-networks at various standards organizations.

Dr. Werner, in recognition of his work, became a Fellow of the IEEE in 1992,and a Bell Labs Fellow in 1995. He was corecepient of two best paper awards,the 1996 Leonard G. Abraham Prize Paper Award and the best paper award atInterface 1984.

Guy A. Dumont (M78SM84F99) received theDiplme dIngnieur from ENSAM, Paris, France, in1973, and the Ph.D. degree in electrical engineeringfrom McGill University, Montreal, Canada, in 1977.

From 1973 to 1974, and then again from 1977 to1979, he worked for Tioxide France. From 1979 to1989, he was with Paprican where he headed the Con-trol Engineering Section, first in Montreal and then inVancouver, BC, Canada. In 1989, he joined the De-partment of Electrical and Computer Engineering atthe University of British Columbia, Vancouver, BC,

Canada, where he is a Professor and leads the Process Control Group, and is cur-rently Associate Dean of Applied Science. His current research interests are inadaptive control and filtering, distributed parameter system control, control loopperformance monitoring, predictive control, with applications to the process in-dustries, mainly pulp and paper, and more recently to biomedical engineering.

Dr. Dumont was awarded the 1979 IEEE TRANSACTIONS ON AUTOMATICCONTROL Honorable Paper Award; the 1985 Paprican Presidential Citation; the1990 UBC Killam research Prize; the 1995 CPPA Weldon Medal; the 1998 Uni-versal Dynamics Prize for Leadership in Process Control Technology; and theIEEE Control Systems Society 1998 Control Systems Technology Award. He isa Fellow of the BC Advanced Systems Institute, Vancouver, BC, Canada.

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The Multimodulus Blind Equalization and Its Generalized Algorithms