Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385

www.elsevier.de/aeue

Computationally efficient methods for blind decision feedback equalizationof QAM signals

Kevin Banovica, Esam Abdel-Raheemb,∗, Mohammed A.S. Khalidb

aDepartment of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, Ont., Canada M5S 3G4bDepartment of Electrical and Computer Engineering, 401 Sunset Avenue, Windsor, Ont., Canada N9B 3P4

Received 28 July 2006; accepted 13 May 2007

Abstract

This paper investigates computationally efficient methods for blind decision feedback equalization (DFE) that reduce thecomplexity and power requirements of blind equalization algorithms while maintaining their steady-state characteristicsfor quadrature amplitude modulation (QAM) signals. These include the power-of-two error (POT), selective coefficientupdate (SCU), and frequency-domain block (FDB) methods. A novel radius-directed stop-and-go (RSG) method is introduced,which selectively adjusts the equalizer tap coefficients based on the equalizer output radius. In addition, a new activation/de-activation method based on the equalizer output radius is utilized to control the feedback equalizer (FBE) of the DFEs. Sim-ulation studies and analysis are provided for empirically derived cable and microwave channels and Ricean fading channels.� 2007 Elsevier GmbH. All rights reserved.

Keywords: Adaptive filtering; Decision feedback equalizers; Blind equalization algorithms; Fading channels

1. Introduction

Adaptive equalizers compensate for signal distortioncaused by intersymbol interference (ISI), whereby symbolstransmitted before and after a given symbol corrupt the de-tection of that symbol. All physical channels tend to exhibitISI at high enough symbol rates [1,2]. Blind equalizationschemes improve the bandwidth efficiency of a communi-cation system by achieving equalizer tap adaptation withoutthe transmission of a training sequence [2–4]. Instead, blindequalization algorithms utilize known symbols statistics forequalizer tap adaptation until switching to the decision-directed mode after the symbol error rate (SER) has beensufficiently reduced.

∗ Corresponding author. Tel.: +1 519 253 3000; fax: +1 519 971 3695.E-mail addresses: banovic@eecg.toronto.edu (K. Banovic),

eraheem@uwindsor.ca (E. Abdel-Raheem), mkhalid@uwindsor.ca(M.A.S. Khalid).

1434-8411/$ - see front matter � 2007 Elsevier GmbH. All rights reserved.doi:10.1016/j.aeue.2007.05.008

Recently, quadrature amplitude modulation (QAM)-basedcommunication standards were adopted for satellite, cable,and very high-speed digital subscriber line (VDSL) applica-tions. Blind equalization is recommended for both the Pan-European satellite-based Digital Video Broadcast (DVB-S)[5] and cable-based (DVB-C) [6] standards. Broadband stan-dards for VDSL include provisions for both single- andmultiple-carrier modulation [7]. The latter uses carrierlessamplitude-phase (CAP) or QAM and requires the receiver tostartup blindly. Although the Advance Television SystemsCommittee (ATSC) adopted 8-vestigal side-band modulation(VSB) over 32-QAM for terrestrial high definition television(HDTV) broadcast [8], blind decision feedback equalization(DFE) was chosen over trained equalization. In field testsconducted by HDTV manufacturers, the blind DFE achieveda lower error rate and faster data acquisition than its trainedcounterpart in time-varying terrestrial channels [9].

In mobile communication channels, such as those for mi-crowave radio, high-order filters are needed to achieve chan-nel equalization. Equalizer tap adaptation is costly in terms

K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385 375

of power, memory, and computations and can be impracti-cal for mobile units. In this paper, we investigate compu-tationally efficient methods for blind DFEs, which reducethe complexity and power requirements of blind equaliza-tion algorithms while maintaining their steady-state charac-teristics for QAM signals. These include the power-of-two(POT) error [10,11], selective coefficient update (also par-tial update) [12–14], and frequency-domain block methods[15–18]. A novel radius-directed stop-and-go method is in-troduced, which selectively updates the equalizer tap coef-ficients based on the equalizer output radius. This conceptwas conceived by Banovic et al. in [19] for linear equal-izers, where it was termed the selective update method. Inthis reformulation for DFEs, criteria is given for selectionof the static bound parameter, analysis is provided for ad-justment probability and transient/steady-state performance,and a modified method is proposed to reduce hardware com-plexity. In addition, a new activation/de-activation methodbased on the equalizer output radius is utilized to control thefeedback equalizer (FBE) of the DFEs. Simulation studiesfor blind DFEs employing the discussed methods are per-formed over empirically derived cable and microwave chan-nels and for Ricean fading channels.

The rest of this paper is organized as follows: Section 2discusses the fractionally spaced channel model for single-input single-output (SISO) systems. Section 3 reviews theconstant modulus algorithm (CMA), multimodulus algo-rithm (MMA), and the decision-directed (DD) algorithm.Section 4 reviews the POT error, selective coefficient update,and frequency-domain block methods. Section 5 introducesthe radius-directed stop-and-go method and modificationsto simplify the multiplications in the coefficient update.Section 6 presents simulation results for static and fadingchannels. Finally, in Section 7, conclusions are drawn fromsimulation results.

2. Fractionally spaced system model

In this section, a signal model is constructed for the T/2-spaced SISO baseband communication system for a DFE,where T is the symbol period and 1/T is the baud rate.A multirate model of the system is illustrated in Fig. 1,where the index ‘n’ denotes T-spaced quantities while ‘k’denotes T/2-spaced quantities. A T-spaced source symbols(n) is transmitted through a pulse-shaping filter and modu-lated onto a T/2-spaced propagation channel, whose impulse

Fig. 1. Multirate system model for a decision feedback equalizer.

response is given by the finite series {ck}Nc−1k=0 , where Nc is

the channel length. This corresponds to the Nc × 1 channelimpulse response vector of c = [c0, c1, . . . , cNc−1]T where(·)T is the transpose operator and the channel is stationary (atime-varying channel can be used as long as it does not varyfaster than can be tracked by the equalization algorithm).The source symbol is a random variable that is independentand identically distributed (i.i.d.) with zero mean and vari-ance �2

s = E{|s(n)|2} and is drawn from a finite alphabet,which is given by the finite set {sm =sm,R + Esm,I}Mm=1 for anM-QAM constellation, where E{·} is the expectation oper-ator and the subscripts R and I denote the magnitude of thereal and imaginary quantities, respectively.

The received T/2-spaced input signal u(k) is corrupted byISI and the additive white Gaussian noise (AWGN) signalv(k). The baseband receiver consists of a Nf -tap T/2-spacedfeedforward equalizer (FFE) and a Nb-tap T-spaced FBE toform a nonlinear DFE, where the FFE removes the precursorISI and the FBE removes the postcursor ISI. The FFE andFBE tap-coefficients are characterized by the finite series

{fk}Nf −1k=0 and {bk}Nb−1

k=0 , respectively, which correspond tothe Nf ×1 vector f(n)=[f0(n), f1(n), . . . , fNf −1(n)]T andthe Nb × 1 vector b(n) = [b0(n), b1(n), . . . , bNb−1(n)]T,respectively. The output of the FFE is baud spaced and isformed by convolving the received T/2-spaced input signalsequence with the FFE tap coefficients. The T/2-spacedconvolution matrix is constructed from the channel impulseresponse vector and is defined as [2]

CFS =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

c0c1 c0... c1 c0

cNc−1... c1

. . .

cNc−1...

. . . c0cNc−1 c1

. . ....

cNc−1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(1)

where CFS is an (Nc + Nf − 1) × Nf matrix. The T-spacedconvolution matrix is formed by the odd rows of (1) and isdefined as

C=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

c1 c0c3 c2 c1 c0...

... c3 c2. . .

cNc−1 cNc−2...

.... . . c1 c0

cNc−1 cNc−2 c3 c2. . .

......

cNc−1 cNc−2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

(2)

where C is a P ×Nf matrix and P =�(Nc+Nf −1)/2�. Theregressor vector of FFE input samples is comprised of the

376 K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385

previous Nf received T/2-spaced samples and is defined as

u(n) = CTs(n) + v(n), (3)

where s(n) = [s(n), s(n − 1), . . . , s(n − P + 1)]T

is the P × 1 transmitted source symbol vector andv(n) = [v0(n), v1(n), . . . , vNf −1(n)]T is the Nf × 1 vec-tor of AWGN samples. The FFE output is decimated by afactor of two and is defined as

y(n) = uT(n)f(n) = sT(n)Cf(n) + vT(n)f(n). (4)

The DFE output signal is defined as

z(n) = xT(n)w(n) = uT(n)f(n) − sT(n)b(n), (5)

where the Nw×1 vectors x(n)=[uT(n), sT(n)]T and w(n)=[fT(n), −bT(n)]T are the combined DFE input regressor andtap coefficient vectors, respectively, while s(n)=[s(n), s(n−1), . . . , s(n − Nb + 1)]T is the Nb × 1 regressor vector ofpast estimated symbol points and Nw = Nf + Nb.

The FBE does not exhibit noise enhancement since itutilizes past symbol estimates, which are assumed to becorrect [1]. When an incorrect symbol estimate is fed back tothe feedback tapped delay line, there is a greater likelihoodof error propagation. Therefore, the SER must be sufficientlylow before the FBE is activated. Initially, the FFE is utilizedto reduce the SER, while the FBE tap coefficients are fixed atzero. After the SER is sufficiently low, the FBE is activatedto reduce the postcursor ISI.

3. Blind equalization algorithms

3.1. Constant modulus algorithm

The CMA [20,21] achieves channel equalization by penal-izing the dispersion of the squared output modulus, |z(n)|2,from the constant �2

c . The cost function minimized by CMAis defined as

J cma = 14E{(|z(n)|2 − �2

c)2}, (6)

where �2c = E{|sm|4}/E{|sm|2} is the dispersion constant.

A gradient-descent equalizer tap adjustment algorithm thatminimizes J cma is defined as

w(n + 1) = w(n) + �(−∇wJ cma)

= w(n) + � z(n)(�2c − |z(n)|2)︸ ︷︷ ︸ecma(n)

x∗(n), (7)

where � is a positive stepsize, ∇w is the gradient operatorwith respect to the elements of vector w, ecma(n) is the CMAerror signal, and (·)∗ denotes complex conjugation.

3.2. Multimodulus algorithm

The MMA [22,23] achieves channel equalization bypenalizing the dispersion of zR(n) and zI(n) components

squared from the constant �2m, where z(n) = zR(n) + EzI(n).

The cost function minimized by MMA is defined as

J mma = 14E{(z2

R(n) − �2m)2 + E(z2

I (n) − �2m)2}, (8)

where �2m = E{s4

m,R}/E{s2m,R} is the dispersion constant.

A gradient-descent equalizer tap adjustment algorithm thatminimizes J mma is defined as

w(n + 1) = w(n) + �(−∇wJ mma)

= w(n) + �

( emmaR (n)︷ ︸︸ ︷

zR(n)(�2m − z2

R(n))

+ E zI(n)(�2m − z2

I (n))︸ ︷︷ ︸emma

I (n)

)x∗(n), (9)

where emmaR (n) and emma

I (n) are the real and imaginary com-ponents of the MMA error signal, respectively.

3.3. Decision-directed algorithm

The cost function minimized by the DD algorithm [24]utilizes the instantaneous error across the slicer and is de-fined as

J dd = 12E{(|z(n) − s(n)|)2}, (10)

where s(n)=sR(n)+EsI(n) is the estimated QAM symbol. Agradient-descent equalizer update algorithm that minimizesJ dd is defined as

w(n + 1) = w(n) + �(−∇wJ dd)

= w(n) + � (s(n) − z(n))︸ ︷︷ ︸edd(n)

x∗(n), (11)

where edd(n) is the DD error signal. The DD algorithmrequires the mean-squared error (MSE) to be lower than aspecified threshold [25] and cannot be applied at the onsetof equalization.

4. Computationally efficient methods

This section discusses computationally efficient methodsthat can be applied to both trained and blind adaptive equal-izers. As illustrated in Fig. 2, adaptive equalization can begeneralized into two operations: convolving the receivedsymbol sequence with the equalizer tap coefficients and up-dating the equalizer tap coefficients. One method to im-prove computational efficiency is to simplify or reduce thenumber of multiplications needed to realize the equalizer.Signed-error [26–28] and POT error [10,11] are methodswhich simplify the multiplications in the equalizer tap ad-justment to shift operations when a POT step size is applied.The selective coefficient update method [12,13] reduces the

K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385 377

Fig. 2. Direct form complex DFE with tap update portions indicated by dashed-boxes.

number of multiplications by updating only a subset of thetotal taps during an iteration, while frequency-domain blockalgorithms [15–18] perform time-domain convolution for ablock of samples in the frequency-domain.

4.1. Power-of-two error method

The most common method of reducing the complexity ofan adaptive algorithm is to retain only the sign of the errorsignal [26–28]. Signed-error algorithms simplify the multi-plications in the equalizer tap adjustment portion to shift op-erations when a POT step size is applied. However, signed-error algorithms are characterized by rough convergence andhigh steady-state MSE. An alternative method that avoidsthese characteristics is the POT error method [10,11], whichquantizes the error signal of the respective algorithm to aPOT. The general equalizer tap adjustment algorithm forPOT algorithms is defined as

w(n + 1) = w(n) + �Q{e(n)}x∗(n), (12)

where e(n) is the error signal of the respective algorithmand Q{·} is a nonlinear POT quantizer, which can be definedas [11]:

Q{x} =

⎧⎪⎨⎪⎩

csgn(x), |x|�1,

2�log2|x|�sgn(x), 2−L+2 � |x| < 1,

� csgn(x), |x| < 2−L+2,

(13)

where csgn(·) is the complex sign operator, L is the dataword length including the sign bit and � is typically set toeither 0 or 2−L+1.

The computational requirements for the POT methodequalizer tap update are specified in Table 1. When coupledwith a POT step size, the multiplications are reduced toshift operations while the equalizer tap adjustment becomesshift and add operations.

4.2. Selective coefficient update method

The complexity of an adaptive filter is proportional tothe number of its tap coefficients. By partially updating the

tap coefficients, the processor capacity can be utilized moreefficiently while reducing the power consumption [12–14].The general equalizer tap adjustment algorithm for selectivecoefficient update (SCU) algorithms is defined as

w(n + 1) = w(n) + �e(n)AIP (n)x∗(n), (14)

where e(n) is the respective error signal and AIP (n) is a di-agonal matrix having P elements equal to one in the posi-tions indicated by IP (n) and zeros elsewhere, where IP (n)

is the Nw ×1 update constraint vector. The update constraintvector is determined through information evaluation, whichcan be accomplished using a number of methods. We willconsider the fixed and time-varying set-membership crite-ria discussed in [14]. The first method is when P tap co-efficients are updated during each iteration, where P is afixed value between 0 < P < Nw. The equalizer input vec-tor x(n) is sorted and the index positions that correspond tothe largest P input samples are set to one in IP (n), while allother positions are set to zero. There is no restriction on theselection of P as long as the stability or convergence is notcompromised.

An alternate method is to let P vary with time, suchthat Pmin �P(n)�Pmax. Initially, P(n)=Pmin and is incre-mented by 1 until P(n)=Pmax or the regressor power meetsthe following condition:

‖AIP (n)x(n)‖2 ��p‖x(n)‖2, (15)

where ‖x‖ =√∑

i |xi |2 is the two norm and �p is a fixedconstant that ranges from 0 < �p < 1.

The computational requirements for the fixed and vari-able SCU methods are specified in Table 1. These figures donot include the overhead processing for the fixed and time-varying cases. One possible implementation of the equalizerinput power calculation is to immediately square the inputsample and apply the result to a separate tapped delay line(TDL). The square of the latest input sample would be addedto an accumulator while the square of the first sample toleave the TDL would be subtracted to obtain the current re-gressor power. This costs two real multiplications and fourreal additions per iteration while doubling the storage ele-ments for the TDLs.

378 K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385

Table 1. Number of real arithmetic operations for the equalizer tap adjustment of a direct form complex DFE when � = 2−n

Method Multiplications Additions Barrel shifts FFTs/IFFTs Adjustment percentage

None 4Nw 4Nw 2 0 1FDB (FFE, FBE) 8Nf , 4Nb 8Nf , 4Nb 4Nf , 2 3, 0 1/Nf , 1SCU 4P(n) 4P(n) 2 0 1POT 0 4Nw 4Nw + 2 0 1RSG 4Nw 4Nw 2 0 0 < F < 1RSG-DPOT 0 6Nw 8Nw + 2 0 0 < F < 1

4.3. Frequency-domain block method

Frequency-domain block (FDB) algorithms [15–18] usea block of input samples and instantaneous error samples toupdate the equalizer taps once every B input samples, whereB is the block length. Significant reductions in complexityare obtained by performing time-domain convolution in thefrequency-domain, which is due in part to the efficiency ofthe fast Fourier transform (FFT) and the inverse fast Fouriertransform (IFFT) algorithms. For DFE architectures, an ex-act frequency-domain implementation does not exist sincethe FBE would require future symbol estimates. Therefore,in typical frequency-domain DFE implementations, the FFEis implemented in the frequency domain while the FBE isimplemented in the time-domain [29–31]. FDB algorithmscan be realized using the overlap-save or overlap-add sec-tioning methods. The FDB implementation considered hereis the overlap-save method with a block size of B = N

(where in this section Nf = N ) since this is the most effi-cient value for the FFT algorithms [15]. This corresponds to2N frequency-domain equalizer taps. The general equalizertap adjustment algorithm for FDB algorithms is defined as

F(nN + N) = F(nN) + F{gF−1{�UH(nN)E(nN)}},(16)

where here uppercase letters denote frequency-domain quan-tities, g is the 2N × 2N gradient constraint matrix, and Fand F−1 are the FFT and the IFFT, respectively, while (·)H

is the Hermation transform (complex conjugation and trans-pose). The input signal matrix U(nN) is comprised of twoblocks of N input samples and is defined as

UT(nN)=D{F [u(nN − N), . . . , u(nN+N−1)]︸ ︷︷ ︸u(nT )

}, (17)

where the D{·} operator transforms the 2N ×1 vector into a2N × 2N diagonal matrix. The frequency-domain equalizeroutput is Y(nN) = U(nN)F(nN), while the N × 1 time-domain equalizer output vector is defined as

y(nN) = kF−1U(nN)F(nN)

= [y(nN − N + 1), . . . , y(nN)]T, (18)

where k is the N × 2N constraint matrix that en-sures the output result is a linear convolution [17]. The

Fig. 3. Realization of the FDB method using the overlap-savesectioning procedure.

frequency-domain error signal vector is defined as

E(nN) = F[0TN,1, eT(nN)]T, (19)

where e(nN) = [e(nN − N + 1), . . . , e(nN)]T is the time-domain error signal vector and 0N,1 is the N ×1 zero vector.The gradient and convolution constraint matrices, g and k,respectively, are defined as [17]

g =[ IN,N 0N,N

0N,N 0N,N

]k = [0N,N IN,N ], (20)

where 0N,N is an N ×N zero matrix and IN,N is an N ×N

identity matrix.The computational requirements for the FDB method

equalizer tap update are specified in Table 1. The FDBmethod is applied to the FFE while the FBE is implementedin the time-domain. As illustrated in Fig. 3, a total ofthree 2N -point FFTs and two 2N -point IFFTs are neededto implement the FFE utilizing the FDB method, wheretwo FFTs and one IFFT are utilized specifically for thefrequency-domain equalizer tap adjustment.

K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385 379

5. Radius-directed stop-and-go method

The radius-directed stop-and-go (RSG) method for QAMsignals selectively updates the equalizer tap coefficientsbased on the equalizer output radius, r(n) = |z(n) − s(n)|,which is the Euclidean distance between the equalizer out-put and estimated symbol. The equalizer tap coefficientsare only adjusted during iterations when r(n) > rs , where rsis a constant bound. The general equalizer tap adjustmentalgorithm for RSG algorithms is controlled by the flag �(n)

and is defined as

w(n + 1) = w(n) + ��(n)e(n)x∗(n), (21)

where e(n) is the error signal of the respective algorithmand �(n) is defined as

�(n) ={

1 if r(n) > rs,

0 otherwise,(22)

where the constant rs=�sd/2, �s is a user-defined parameter,and d is the distance between symbol points. This parameteris to be chosen between the following limits

2Nw�max < �s � 23 , (23)

where �max is the maximum adjustment error that can occurwhen the equalizer output is a constellation point (i.e. whenz(n) ∈ {sm}Mm=1 for a square M-QAM constellation) andthe upper limit of �s corresponds to the minimum level ofMSE required for transfer to the DD algorithm, which isdenoted dd [32]. While there is no adjustment error for theDD algorithm in steady-state operation, this is not the casefor statistical mean algorithms such as CMA and MMA,which have non-zero updates for z(n) ∈ {sm}Mm=1. Thesealgorithms accentuate the “bottom of the bowl” scenario ofclassical gradient search methods, where the equalizer tapcoefficients bounce around the optimal solution. As a result,these fluctuations cause the steady-state MSE to increase.

At the uppermost limit of (23), the number of equalizer tapupdates will be minimized at the expense of a high steady-state MSE. As �s approaches the lowermost limit, the steady-state MSE will be equivalent to the original algorithm withslightly fewer updates. When selecting �s , it is important tonote that when the MSE level is below dd, the steady-stateMSE for the original algorithm, ss, can be approximated asthe error across the slicer as follows:

ss�E{|s(n) − z(n)|2} = E{r2(n)}. (24)

The relationship between the steady-state MSE for the se-lective update method, rsg

ss , and ss for equalizers in thedecision-directed mode of operation, can be expressed as

rsgss

{ � ss if (ss �r2s and r2

s /?ss) or ss?r2s ,

> ss if ss �r2s and r2

s?ss.(25)

This can be explained as follows: if r2s �ss, then

r2s �E{r(n)2}, which will cause a significant reduction in

equalizer tap updates. This can effect rsgss constructively or

destructively, depending on the selection of rs . If r2s?ss,

the steady-state MSE will approach r2s since the equalizer

tap coefficients will only be updated once rsgss is degraded.

However, as r2s → ss, the rsg

ss will decrease to the pointwhere rsg

ss �ss. At first this may seem counterintuitive,however, recall the concept of adjustment error introducedby the equalizer tap update. Halting adaptation when theequalizer output is near symbol points can mitigate the un-necessary equalizer tap updates that increase the steady-stateMSE for statistical mean algorithms.

At the onset of equalization the equalizer output will bea random i.i.d. value, which will result in the followingequalizer update probability:

Pr[update] = d2 − (�sd/2)2

d2= 1 − �2

s

4. (26)

This probability decreases as the equalizer adapts andreaches a minimum when the equalizer is in steady-state op-eration. During the initial stages of adaptation, E{r(n)}?rscausing the equalizer taps to be updated frequently. Thisallows the respective algorithm to maintain its transientcharacteristics. As the E{r(n)2} → ss ∨ E{r(n)} → rs ,the equalizer is in steady-state operation and the number ofequalizer tap updates will be at a minimum. If the channelshould experience sudden changes, the MSE will increaseand the process will repeat.

While the RSG method reduces the number of equalizertap adjustments, there are no reductions in the hardware re-sources needed for implementation. The RSG method can bemodified to reduce the hardware complexity by combiningit with methods that simplify the multiplications in the tapcoefficient update, such as the POT method. Here we pro-pose retaining the first two leading ones of the error signal,which we term the double power-of-two (DPOT) method.This is proposed to improve the accuracy of the error signalestimate over the POT method, while minimizing the addedhardware complexity over that method. The general equal-izer tap adjustment algorithm for RSG-DPOT algorithms isdefined as

w(n + 1) = w(n) + ��(n)edpot(n)x∗(n), (27)

where edpot(n) = Q{e(n)} + Q{e(n) − Q{e(n)}}, e(n) isthe error signal of the respective algorithm and Q{·} is thenonlinear POT quantizer that was defined in (13) for thePOT method.

The computational requirements for the RSG and RSG-DPOT methods are specified in Table 1. The RSG methodmaintains the same hardware complexity but reduces thenumber of equalizer tap adjustments and hence, the numberof computations. However, in addition to reducing the num-ber of equalizer tap adjustments, the RSG-DPOT methodreduces the equalizer tap adjustment to shift and add oper-ations when a POT step size is applied. The calculation ofr(n) requires two real multiplications, three real additionsand one real square root. However, if the operand precision

380 K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385

is sufficient, r2(n) can be utilized to determine whether toadjust the equalizer taps, which eliminates the square rootfunction. Alternatively, a look-up-table (LUT) could be uti-lized to implement the square root function.

6. Simulation study

This section presents simulation results for computation-ally efficient methods applied to CMA- and MMA-basedDFEs. The algorithms are simulated over empirically de-rived cable and microwave channels from the Signal Process-ing Information Base (SPIB, located: http://spib.rice.edu/)and multi-tap Ricean fading channels. The simulation envi-ronment consists of a T/2-spaced channel in cascade witha 54-tap DFE consisting of an 18-tap T/2-spaced FFE anda 36-tap T-spaced FBE, where the channel, FFE, and FBEare modeled as complex FIR filters. The source symbol se-quence is randomly generated using an i.i.d. process and isdrawn from a normalized square QAM constellation. Thereceived equalizer input samples are generated by convolv-ing the source sequence with the channel impulse responseand adding AWGN.

The DFE is controlled by the activation/de-activationmethod illustrated in Fig. 4, which utilizes r(n) to determinewhether to activate or de-activate the FBE of the DFE. Atthe onset of equalization, the FFE is initialized with a dualcenter spike of 1/

√2 and the FBE is fixed at zero, while the

variable count is set to zero. The FFE is adapted blindlyusing CMA or MMA. During an iteration, if r(n) < d/3,count will increment by one while less than the userdefined threshold value rth, where d/3 corresponds to dd[32]. Once count reaches rth, the FBE will be activatedand count will saturate at rth. The adaptation of the DFEis switched to the DD algorithm. However, if r(n) > d/3,count will decrement by one while greater than zero. Ifthe DFE is in the active state, once count reaches zero,the DFE will deactivate. The FFE will be adapted blindlyusing CMA or MMA and the FBE will be fixed at zero,while count will saturate at zero.

The first set of simulation studies compare the standardcomputationally efficient methods of FDB, SCU and POTwith RSG for CMA- and MMA-based DFEs. In these sim-ulations, the algorithms are compared with the FDB algo-rithm since its performance is equivalent to that of the origi-nal DFE. The second set of simulations compare RSG-basedmethods that reduce the number of computations and hard-ware complexity of the original RSG method. The RSGmethod is combined with the SCU, POT, and DPOT meth-ods for CMA- and MMA-based DFEs, where each methodis compared to the original RSG method. The simulationparameters for each method are as follows: �p = 0.875 forSCU, L = 16 and � = 0 for POT, and �s = 2 × 10−3/2d−1

for RSG, while rth was set to 63 for all DFEs. Quanti-tative simulation results are presented for the steady-stateMSE, the average time-to-convergence (TTC), the equalizer

Calculate r(n)

r(n) < d/3

count < rth

count > 0

count=count-1

Yes

No

Yes

Turn off DFE

Yes

No

No

count=count+1 Turn on DFE

Fig. 4. DFE activation and de-activation flow chart.

adjustment percentage for RSG-based DFEs in transient andsteady-state operation (TR/SS), and the tap coefficient up-date (TCU) percentage for SCU-based DFEs. The MSEcurves are obtained by averaging the instantaneous squared-error across the slicer over 300 realizations. The TTC is cal-culated as the number of symbols needed to reach 90% ofthe steady-state MSE while ss is the average MSE over thefinal 10% of the estimated symbol sequence.

Microwave channel simulations were conducted forCMA- and MMA-based DFEs over SPIB microwave chan-nels # 1, 2, 4, 5, 8 −10 and # 1, 2, 4 −6, 8 −10, respectively,for 64- and 16-QAM signals with a signal-to-noise ratio(SNR) of 35 dB, where each channel consists of 208–300complex T/2-spaced taps. These modulation schemes werechosen since they are the largest schemes applied to thedata carriers in an orthogonal frequency division multiplex(OFDM) frame for terrestrial DVB. Step sizes of � = 2−11

and 2−10 were applied to the equalizer tap adjustment for64- and 16-QAM signals, respectively. Simulation resultsare illustrated for channel #8 in Fig. 5 for CMA- andMMA-based DFEs with 64-QAM, which are representativeof the typical results obtained. Quantitative results averagedover all channels are given in Table 2. On average, in com-parison with the FDB method, the RSG method achieves aslightly longer TTC, while the SCU and POT methods re-quire significantly more symbols to converge for CMA- andMMA-based DFEs. All methods are able to achieve similarsteady-state MSE values. As expected, in transient oper-ation, the equalizer adjustment percentage of RSG-basedDFEs is high at above 91% for both 64- and 16-QAM,respectively, while in steady-state operation, the percentageis below 67% and 56%, respectively. Less than 73% of the

K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385 381

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

-30

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RSG-SCU-CMA

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0 0.25 0.5 0.75 1 1.25 1.5 1.75 2

RSG-POT-MMA

RSG-SCU-MMA

RSG-SCU-MMA

RSG-MMA

Fig. 5. Comparison of efficient methods applied to CMA/MMA-based DFEs for SPIB microwave channel #8 with 64-QAM signals.Efficient methods for: (a) CMA-based DFEs, (b) MMA-based DFEs, (c) RSG-CMA-based DFEs and (d) RSG-MMA-based DFEs.

DFE tap coefficients are updated for SCU-based DFEs. Ofthe RSG-based DFEs, the RSG-DPOT method obtains aslightly longer TTC while maintaining the properties of theRSG-based DFEs. The results for CMA- and MMA-basedDFEs for microwave channels indicate similar transient andsteady-state characteristics for the efficient methods underconsideration. Therefore, without loss of generality, theremaining simulations will be for MMA-based DFEs.

Cable channel simulations were conducted for MMA-based DFEs over SPIB cable channels #1, 2 for 256- and64-QAM signals with an SNR of 40dB, where each channelconsists of 128 complex T/2-spaced taps. These modu-lation schemes were selected since they are the largestsquare schemes utilized for DVB-C. Step sizes of � = 2−12

and 2−11 were applied to the equalizer tap adjustment for256- and 64-QAM signals, respectively. Simulation re-sults are illustrated for channel #1 in Fig. 6 for 256-QAM,which are representative of the typical results obtained.

Quantitative results averaged over all cable channels aregiven in Table 3. On average, the RSG method achievesa slightly longer TTC, while the SCU and POT methodsrequire significantly more symbols to converge. All meth-ods are able to achieve similar steady-state MSE values.The equalizer adjustment percentage for RSG-based DFEsin transient operation is above 81% and 90% for 256- and64-QAM, respectively, while in steady-state operation, thepercentage is below 38% and 30%. For SCU-based DFEs,64% of the DFE tap coefficients were updated. Of the RSG-based DFEs, the RSG-DPOT method obtains a slightlylonger TTC while once again maintaining the properties ofthe RSG-based DFEs.

Fading channel simulations were conducted for MMA-based DFEs over a 5-tap Ricean fading channel as illustratedin Fig. 7 for 16-QAM and quadrature phase shift keying(QPSK) signals with a SNR of 40 dB, where g(k) is theRayleigh fading process and K is the rice factor which was

382 K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385

Table 2. Quantitative results averaged over SPIB static microwave channels

DFE method 64-QAM 16-QAM

ss (dB) TTC (T) TR/SS TCU ss (dB) TTC (T) TR/SS TCU

FDB-CMA −26.68 11,246 – – −28.24 3,769 – –SCU-CMA −26.24 12,627 – 0.72 −28.24 5,123 – 0.67POT-CMA −26.55 15,827 – – −28.24 5,236 – –RSG-CMA −26.67 11,284 0.92/0.63 – −28.22 3,813 0.92/0.53 –RSG-SCU-CMA −26.33 13,619 0.93/0.63 0.72 −28.25 5172 0.93/0.54 0.67RSG-POT-CMA −26.67 16,324 0.94/0.66 – −28.19 5291 0.94/0.55 –RSG-DPOT-CMA −26.59 12,633 0.93/0.64 – −28.20 4194 0.93/0.54 –

FDB-MMA −26.96 9,764 – – −28.36 3,664 – –SCU-MMA −26.91 15,449 – 0.65 −28.40 4,865 – 0.65POT-MMA −26.93 13,739 – – −28.36 5,046 – –RSG-MMA −26.95 9,898 0.93/0.62 – −28.34 3,686 0.93/0.52 –RSG-SCU-MMA −26.77 15,322 0.93/0.64 0.65 −28.39 4,916 0.93/0.52 0.65RSG-POT-MMA −26.80 13,880 0.94/0.65 – −28.35 5,063 0.93/0.52 –RSG-DPOT-MMA −26.91 10,836 0.93/0.63 – −28.35 4,016 0.93/0.52 –

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

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x 104Symbols

RSG-POT-MMA

RSG-SCU-MMA

RSG-DPOT-MMA

RSG-MMA

Fig. 6. Comparison of efficient methods applied to MMA-based DFEs for SPIB cable channel #1 with 256-QAM signals. Efficient methodsfor: (a) MMA-based DFEs and (b) RSG-MMA-based DFEs.

Table 3. Quantitative results averaged over SPIB static cable channels

DFE method 256-QAM 64-QAM

ss (dB) TTC (T) TR/SS TCU ss (dB) TTC (T) TR/SS TCU

FDB-MMA −30.98 11,298 – – −31.79 4,805 – –SCU-MMA −31.11 13,298 – 0.64 −31.94 5,429 – 0.64POT-MMA −30.87 16,539 – – −31.71 6,959 – –RSG-MMA −30.92 12,099 0.83/0.32 – −31.71 4,910 0.94/0.28 –RSG-SCU-MMA −31.02 13,980 0.82/0.32 0.64 −31.83 5,558 0.91/0.26 0.64RSG-POT-MMA −30.72 17,629 0.84/0.37 – −31.60 7,043 0.91/0.29 –RSG-DPOT-MMA −30.90 13,616 0.83/0.32 – −31.68 5,493 0.91/0.27 –

K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385 383

set to K = 1 dB, K = 2.5 dB, and K = 5 dB. The Rayleighfading processes applied to the channel taps were indepen-dent, randomly initialized, and simulated using the normal-ized low-pass fading process of Jakes’ simulator [33]. Thefading parameters were selected using the IMT-2000 eval-uation methodology for 3 G wireless communications [34],where the carrier and symbol frequencies were set to 2 GHzand 4.096 MBaud, respectively. The mobile transceiver wasmoving at a velocity of 12 km/h to represent a person jog-ging. Step sizes of � = 2−9 and 2−6 were applied to theequalizer tap adjustment for 16-QAM and QPSK signals,

Fig. 7. Ricean fading channel model used for simulations.

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5

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Symbols

RSG-POT-MMA

RSG-SCU-MMA

RSG-DPOT-MMA

RSG-MMA

Fig. 8. Comparison of efficient methods applied to MMA-based DFEs for Ricean fading channels with K = 1 dB for 16-QAM signals.Efficient methods for: (a) MMA-based DFEs and (b) RSG-MMA-based DFEs.

Table 4. Quantitative results for Ricean fading channels

DFE method 16-QAM QPSK

ss (dB) TTC (T) TR/SS TCU ss (dB) TTC (T) TR/SS TCU

FDB-MMA −25.23 5116 – – −28.86 888 – –SCU-MMA −24.62 5972 – 0.42 −27.19 1000 – 0.48POT-MMA −24.16 6386 – – −28.92 1126 – –RSG-MMA −25.03 5148 0.94/0.65 – −28.65 876 0.84/0.38 –RSG-SCU-MMA −24.44 5926 0.94/0.67 0.42 −26.99 968 0.88/0.43 0.31RSG-POT-MMA −24.01 6472 0.95/0.70 – −28.75 1150 0.86/0.38 –RSG-DPOT-MMA −24.77 5430 0.95/0.66 – −28.71 908 0.89/0.39 –

respectively. Simulation results are illustrated in Fig. 8 for16-QAM with K = 1dB. Quantitative results are given inTable 4. On average, for 16-QAM signals, the RSG methodachieves nearly identical TTC and steady-state MSE val-ues, while the SCU and POT methods obtain a significantlylarger TTC and a higher steady-state MSE, where the per-formance of the POT method is the worst for both. ForQPSK, the steady-state MSE of the SCU method is nearlyidentical to that of the original while the TTC propertiesremain the same for all methods. The equalizer adjustmentpercentage for RSG-based DFEs in transient operation isabove 93% and 83% for 16-QAM and QPSK, respectively,while in steady-state operation, the percentage is below 71%and 44%. For SCU-based DFEs, 42% of the DFE tap co-efficients were updated for 16-QAM and 48% for QPSK.Of the RSG-based DFEs, the RSG-DPOT method obtains aslightly longer TTC and slightly higher steady-state MSE,while the RSG-SCU and RSG-POT methods obtain a signifi-cantly higher steady-state MSE and longer TTC for 16-QAMsignals.

384 K. Banovic et al. / Int. J. Electron. Commun. (AEÜ) 62 (2008) 374–385

7. Conclusion

This paper investigated computationally efficient meth-ods for blind DFEs that reduced the complexity and powerrequirements of blind equalization algorithms while main-taining their steady-state characteristics for complex signals.New computationally efficient methods were proposed thatreduced the number of computations and hardware require-ments for blind DFEs, mainly the RSG and RSG-DPOTmethods. Simulation results for static cable and microwavechannels and for Ricean fading channels indicate that thenew methods maintain the transient and steady-state perfor-mance of the original algorithm, while reducing their com-plexity.

Acknowledgement

The authors wish to thank Harb Abdulhamid and Ray-mond Lee for their comments and suggestions during dis-cussions on fading channels and the proposed methods.

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Kevin Banovic received his B.A.Sc.and M.A.Sc. degrees in Electrical En-gineering from the University of Wind-sor, Ontario, Canada, in 2003 and 2006,respectively. He is a candidate in Elec-trical and Computer Engineering Ph.D.program at the University of Toronto,Ontario, Canada. His research interestsinclude the design and applicationsof signal-processing microsystems,

adaptive signal processing, high-performance VLSI design andfield-programmable logic.

Esam Abdel-Raheem received hisB.Sc. and M.Sc. degrees from AinShams University, Cairo, Egypt, in1984 and 1989, respectively, and Ph.D.degree from the University of Victoria,Canada in 1995, all in Electrical Engi-neering. Currently, he is an AssociateProfessor at the University of Windsor,Ontario, Canada and an Adjunct As-sociate Professor at the University of

Victoria, BC, Canada. From 1999 to 2001, he was a Senior DesignEngineer at the Network Product Division of AMD in Sunnyvale,California. Dr. Abdel-Raheem’s research fields of interests are indigital signal processing, signal processing for communications,and VLSI signal processing. He has authored and co-authoredover 50 refereed journal and conference papers, and one publishedUS/world patent. He is a Senior Member of the IEEE and a memberof the IEEE SPS tech. committee on Signal Processing Educationand IEEE CAS tech. committee on VLSI systems & applications.He has served as the technical program co-chair for IEEE ISSPIT2004 & 2005. He is the general co-chair for IEEE ISSPIT 2007. Heserves as an Associate Editor for the Canadian Journal of Electrical& Computer Engineering. Dr. Abdel-Raheem is a Member of theAssociation of Professional Engineers of Ontario.

Mohammed A.S. Khalid received hisPh.D. degree in Computer Engineer-ing from the University of Toronto in1999. He is an Assistant Professor inElectrical and Computer EngineeringDepartment at the University of Wind-sor. From 1999 to 2003, he was a Se-nior Member of Technical Staff in theVerification Acceleration R & D Group(formerly Quickturn), of Cadence

Design Systems, based in San Jose, California. His research anddevelopment interests are in architecture and CAD for field pro-grammable chips and systems, reconfigurable computing, digitalsystem design and hardware description languages.