Modi!ed Constant Modulus Algorithm for jointblind equalization and synchronization

Ali A. Nasir, Salman Durrani and Rodney A. KennedySchool of Engineering, CECS, The Australian National University, Canberra, Australia.

Email: {ali.nasir, salman.durrani, rodney.kennedy}@anu.edu.au

Abstract—The development of low-complexity blind techniquesfor equalization, timing and carrier offset recovery is of enormousimportance in the design of high data rate wireless systems. Inthis paper, we propose a practical solution for blind equalization,timing recovery and small carrier offset correction in slowly-fading frequency selective wireless communication channels. Weextend the Modi ed Constant Modulus Algorithm (MCMA) tohandle the timing offset parameter. We propose a single objectivefunction to achieve equalization, timing and carrier recoverywithout the need of any training sequences. Our algorithmachieves 5-10 times faster convergence, compared to previousresearch, for the Mean Square Error (MSE) at the equalizeroutput due to the nely tuned step sizes and updating theequalizer taps and timing offset jointly to minimize the meandispersion and inter symbol interference. Our results show thatthe proposed technique can successfully handle synchronizationand combat the frequency selectivity of the wireless channel.

I. INTRODUCTION

Adaptive equalization techniques are a topic of immensetheoretical and practical value with growing applications inareas of digital and wireless communications [1]. The useof initial training sequences in adaptive equalization givesrise to signi!cant overhead with data rate reduction andsometimes may become unrealistic or impractical. For in-stance, no training signal may be available to receivers inmilitary communication scenarios and defence applications. Ina multicast or broadcast system, it is highly undesirable for thetransmitter to engage in a training session for a single user bytemporarily suspending its normal transmission to a numberof other users. Consequently, there is a strong and practicalneed for blind equalization without training [1].There are two main approaches to blind equalization. The

!rst category is stochastic gradient algorithms which processone data sample or a small block of data at a time anditeratively minimize a chosen cost function [2]. The secondcategory is statistical methods which use suf!cient stationarystatistics collected over a large block of received data [2].A limitation of statistical methods is that they require thechannel to be constant over the block of received data. Thisrequirement is generally not ful!lled by real-world wirelesschannels which cannot be guaranteed to be static over longtime intervals. Hence stochastic gradient algorithms can beused to adapt in slowly fading wireless channels while workingon a symbol by symbol basis. The issue of slow convergencein stochastic gradient algorithms is well known and has beenaddressed by many authors recently to make it faster andpractical in various scenarios [3]–[7].

Due to its simplicity, Constant Modulus Algorithm (CMA)is the most commonly used stochastic gradient algorithmfrom practical implementation point of view [8]. In CMA,a separate loop is required for tracking the channel and thecarrier phase offset respectively, which can be avoided usingModi!ed CMA (MCMA) cost function [9]. The convergencespeed of MCMA is, however, slower than CMA. This can beimproved in a number of ways, e.g. by using a combinationof MCMA and Super Exponential algorithm [4], by joiningMCMA and Decision Directed (DD) tracking with variablestep size adjustment [3], [6], by using decision feedbackequalizer [7] and also by appending DD-Least Mean Squarealgorithm with CMA [5]. All of these algorithms assumeperfect timing recovery at the receiver, which is never possiblein realistic communication systems.To the best of our knowledge, the important problem of

joint equalization, carrier and timing recovery for wirelesscommunications has only been targeted by a few authors [10],[11]. In [10], an algorithm is presented for joint equalization,carrier and timing recovery which is suited for cable modemtransmission and cannot be directly applied to wireless com-munication scenarios. In [11], the task is accomplished byusing digital phase-locked loop (PLL). However the acquisi-tion speed of a PLL is very slow, which makes it impracticalfor time varying wireless channels. Most of the previous workin [2]–[9] focuses on the subset of these problems, either jointequalization and carrier recovery or blind equalization alone.Hence there is a need for the development of practical, low-complexity, blind techniques for equalization, timing recoveryand carrier offset correction.In this paper, we propose a technique for blind timing re-

covery, equalization and carrier offset recovery jointly withoutany aid of training, which is applicable for slowly-fading,frequency selective wireless channels. We modify MCMAalgorithm of Oh et al. [9] to handle timing offset parameter.We propose a modi!ed single objective function to achieveequalization, carrier and timing recovery. Our simulation re-sults show very fast convergence compared to the resultsof all the above mentioned papers, i.e., [3]–[5]. The majorcontributions of this paper, in comparison to previous research,are as follows:-

• We propose an algorithm, based on joint minimisation ofa single objective function, to accomplish equalization,timing recovery and small carrier offset recovery withoutthe need of any training sequences and phase locked loop.

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Digital

Modulator

Pulse Shaping

gT (t)

Channel

c(t)! +

Matched Filter

gR(t)

sourcebits {bk} s(t){ak}

ej(!!)t

noise w(t)

r(t, !)

Interpolator Equalizer SlicerDemodulation

& Decoding

r(t, !)

r(nTs, !)

1Ts u(kT, !) y" (k) {ak} {bk}

Blind Equalization

& Timing Adjustment

!

1T

1TW

Fig. 1. Baseband Communication System for Blind Synchronization and Equalization.

• We show that the proposed algorithm has 5 ! 10 timesfaster convergence of Mean Square Error compared to theprevious research, while providing additional capabilityof handling synchronization. This is due to the !nelytuned step sizes and updating the equalizer taps andtiming offset jointly to minimize the mean dispersion andinter symbol interference.

Finally, we outline the algorithm in a step-wise manner foran ease of practical implementation.

II. SYSTEM MODEL

The baseband model of the proposed system for a singlesource and a single destination is shown in Fig. 1. At thetransmitter, the sequence of information bits bk is applied toconstellation mapper, which converts bk into a complex valuedsequence ak of data symbols, taken from two dimensionalconstellation and the two components (in-phase and quadra-ture) are transmitted by amplitude modulating two quadraturecarrier waves. Such a modulation scheme can be treatedin a concise manner by combining in-phase and quadraturecomponents into complex valued symbols. The data symbolsenter the transmit !lter with the impulse response gT (t) andthe resulting transmit signal s(t) is given by

s(t) =!!

k="!akgT (t ! kT ) (1)

where 1/T is the symbol rate, i.e., the rate at which the datasymbols are applied to the transmit !lter, gT is the basebandpulse at the transmitter which is matched to the receiver !ltergR, i.e., gT (t) = gR(!t). The signal at the output of thereceive !lter is given by

r(t, !) =!!

k="!akh(t ! !T ! kT )ej!(t) + v(t), (2)

where h(t) = gT (t) " c(t) " gR(t) represents the overallbaseband impulse response, c(t) is the baseband multipathchannel impulse response which introduces attenuation andphase distortion in the signal, "(t) = (##)t represents

the frequency offset between the transmitter and receiveroscillators, which can be visualized as the time varying phaseoffset and results in continuous spinning of constellation withtime and ‘"’ represents the convolution operator. The constantphase offset between the transmitter and receiver carriers canbe merged with the channel induced phase offset "o whichrotates the transmitted constellation, ! , normalized by thesymbol duration T , is the fractional unknown timing offset(|! | $ 1

2 ) between the transmitter and receiver !lters and v(t)is the complex !ltered noise v(t) = gR(t)"w(t) with variance$2

v , where w(t) is the zero mean stationary, white, and complexGaussian process.After pulse shaping, the signal is sampled with some timing

offset since the receiver does not know the exact samplingpoint corresponding to maximum Signal to Noise Ratio (SNR).The receive !lter output is oversampled by the factor Q suchthat the oversampling period Ts = T/Q, so (2) becomes

r(nTs, !) =!!

k="!akh(nTs ! !T ! kT )ej!(nTs) + v(nTs),

(3)where n is the sampling index, "(nTs) = 2%(#f)nTs =2%(#f/Fs)n, #f is the frequency offset in Hz and #f/Fs

is the digital frequency offset in cycles/sample [12]. The noisesamples v(nTs) are assumed to be statistically independent ofthe input symbols ak. Sampling the received signal at wronginstant (not at the maximum eye opening) or any jitter in thesampler introduces Inter Symbol Interference (ISI) and resultsin the reduction of noise margin. Zero ISI can be achievedonly by sampling at exact instant kT , for kth symbol appliedto the transmit !lter.The problem of joint synchronization, equalization and

small carrier offset recovery using blind algorithms is to:1) Estimate the timing offset ! .2) Estimate the equalizer taps to equalize the multipath

channel c(t).3) Mitigate of constant channel induced phase offset "o.4) Mitigate small time varying phase offset "(k).

The proposed solution is presented in the next section.

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III. PROPOSED RECEIVER

In this section, we present the receiver design, which isdedicated to compensate for all the distortions (timing offset,frequency and phase offset and multipath channel distortion)jointly without any aid or known preamble.

A. Interpolator

Simple linear interpolation is used, which works on esti-mated timing offset ! provided by the Blind Equalization andTiming Adjustment block of Fig. 1. Instead of linear inter-polators, Lagrange or higher degree polynomial interpolatorscan also be used [13]. However the resulting performanceimprovement over linear interpolators is very minute andcomes at a cost of greater computational complexity. Hencewe use linear interpolators as a good compromise betweenperformance and complexity. Linear interpolator has a delayof Q! 1 samples which can be stored in its delay line beforepreforming interpolation. Depending on ! and Q, the linearinterpolation [13] between two appropriate samples is givenby

u(nTs, !) = r(nTs, !) + µ[r((n + 1)Ts, !)! r(nTs, !)], (4)

where µ = ! % Q. The sampling rate after interpolation isreduced to 1/T for baud spaced equalization, i.e., interpolationis performed on interleaved samples.

B. Blind Equalization and Timing Adjustment

For multipath channels c(t), each received signal sampler(nTs, !) depends on multiple transmitted symbols ak and theresulting ISI is mitigated through multitap equalizer W . Letus denote the equalizer output y(kT, !) as y" (k) as shownin Fig. 1. The carrier offset recovery and estimate of theparameters ! (the timing offset) and W (the equalizer tapweight vector) are obtained by minimizing the mean dispersionof the sampler output and using the modi!cation in CMA costfunction [14]. Introducing new parameter, ! , to cost functionof Oh and Chin [9], the modi!ed objective function can bewritten as

Jk(! ,W ) = JR,k(! ,W ) + JI,k(! ,W ) (5)

where JR,k(! ,W ) and JI,k(! ,W ) are the cost functionsfor the real part &{y" (k)} = yR," (k) and imaginary part'{y" (k)} = yI," (k) of the equalizer output respectively, andare de!ned as

JR,k(! ,W ) = E{(|yR," (k)|2 ! &R)2} (6)

JI,k(! ,W ) = E{(|yI," (k)|2 ! &I)2}, (7)

where |·| is the modulus operator, E{·} represents the expectedor mean value and &R and &I are constants for the real&{ak} = aR,k and imaginary '{ak} = aI,k parts of theinput data sequence respectively. These constant factors are

proportional to the kurtosis of the input sequence and aredetermined as [8]

&R =E{|aR,k|4}E{|aR,k|2}

(8)

&I =E{|aI,k|4}E{|aI,k|2}

(9)

where input data sequence ak is assumed to be stationary andnon gaussian process and ak = aR,k + jaI,k, with i.i.d. realand imaginary parts [2].

(8) and (9) are determined subject to the constraint thatthe average gradient of the cost function (5) with respect tothe tap weight vector W is zero when the channel is perfectlyequalized. The objective cost function given by (5) is obtainedby using p = 2 (dispersion order) in the generalized Con-stant Modulus Godard’s cost function E[(|y" (k)|p ! &p)2] for&p = E{|ak|2p}/E{|ak|p} [8]. The value of p = 2 is chosenbecause the practical digital implementation with !nite lengtharithmetic (!xed point implementation) suffers from precisionor over"ow problems for large p [8]. Godard also demonstratedthat the cost function can be applied to non-constant modulussignals such as rectangular QAM constellation [8].Let W (k) = {w0, w1, . . . , wN"1}t be the adjustable tap

weights for N tap equalizer where superscript (·)t denotes thetranspose of a vector. If X(k) = {u" (k), u" (k!1), . . . , u" (k!N+1)}t are interpolated samples in the delay line of equalizerthen equalizer output is

y" (k) = Xt(k)W (k) (10)

The estimated equalizer weight vector W (k) is fed to theequalizer by Blind Equalization and Timing Adjustment blockwhich uses the steepest descent type method of stochasticgradient algorithm and optimizes the cost function (5) to adaptthe tap weight vector and timing offset [2]. In this case, themodi!ed steepest descent update equations are

W (k + 1) = W (k) ! µw (w Jk(! ,W ) (11)

!(k + 1) = !(k) ! µ" (" Jk(! ,W ), (12)

where µw and µ" are small positive step sizes which con-trol the convergence of parameter update equations and(wJk(! ,W ) and ("Jk(! ,W ) are the gradients of the costfunction (5) with respect to equalizer tap weights and timingoffset respectively. Solving these gradients and then substitut-ing in (11) and (12) results in

W (k + 1) = W (k) ! µwew(k)XH(k), (13)

!(k + 1) = !(k) ! µ" (|y" (k)|2 ! &)'|y" (k)|2

'!, (14)

where the superscript (·)H represents the conjugate transpose(Hermitian) operator and the error signal for weight updateew(k) = ew,R(k) + jew,I(k) is given as

ew,R(k) = (|yR," (k)|2 ! &R) yR," (k) (15)

ew,I(k) = (|yI," (k)|2 ! &I) yI," (k) (16)

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!2 !1 0 1 2!2

!1

0

1

2(a)

real

imag

inar

y

!2 !1 0 1 2!2

!1

0

1

2(b)

real

imag

inar

y0 500 1000 1500 2000 2500 3000

!40

!20

0

(c)

No. of iterations

MS

E (d

B)

0 1000 2000 3000 4000!0.1

0

0.1

0.2

0.3(d)

No. of iterationsE

stim

ated

tim

ing

offs

et

Fig. 2. Results for QPSK with multipath channel, fractional timing phaseoffset (!) = !0.4, frequency offset ("f/Fs) = 10!4 and SNR = 25 dB.(a) Channel output constellation. (b) Equalized output constellation. (c) MSEbetween equalizer output and transmitted symbols. (d) Timing offset recovery.

!2 !1 0 1 2!2

!1

0

1

2

real

imag

inar

y

(a)

!2 !1 0 1 2!2

!1

0

1

2

real

imag

inar

y

(b)

0 500 1000 1500 2000 2500 3000

!40

!20

0

No. of iterations

MS

E (d

B)

(c)

0 1000 2000 3000 4000!0.2

0

0.2

0.4

0.6

No. of iterations

Est

imat

ed ti

min

g of

fset

(d)

Fig. 3. Results for 16-QAM with multipath channel, fractional timing phaseoffset (!) = !0.4, frequency offset ("f/Fs) = 10!4 and SNR = 25 dB.(a) Channel output constellation. (b) Equalized output constellation. (c) MSEbetween equalizer output and transmitted symbols. (d) Timing offset recovery.

The derivative of |y" (k)|2 in (12) is computed using Euler’sapproximation by evaluating y" (k) at ! + ( where ( is verysmall increment and then using the following equation

'|y" (k)|2

'!) |y"+#(k)|2 ! |y" (k)|2

((17)

where ( is 0.0001 in our case.Intuitively, the algorithm tries to move the the real part of

the equalizer output to reside on the points of value +*&R or

!*&R on the real axis instead of a circle. Similarly, imaginary

part of the equalizer output is moved to lie on the pointsof value +*

&I or !*&I on imaginary axis. Since, the cost

function employs both modulus and phase of the equalizeroutput, channel induced phase offset "o and small carrierphase offset "(k) are corrected along with blind equalizationwithout the need of separate decision directed phase trackingloop [9]. Finally, the slicer makes a hard decision on equalizeroutput generating ak which are demodulated and decoded toproduce the output bits bk.

C. Algorithm

The whole blind setup can be summarized in the form ofthe following algorithm.

Proposed Algorithm

Initialization:Initialize equalizer tap weight vector

W with zeros and substitute 1 for thecentral tap. Initialize estimated timingoffset ! = 0 and vector X and Xd ofequalizer length with zeros. Let u be theresulting interpolated sample.Loop Processing:

1) Use (4) for linear interpolation onthe received symbols.IF 0 $ ! $ 0.5u = interpolation between current andthe next sample.

ELSE IF !0.5 $ ! $ 0u = interpolation between current andthe previous sampleEND

2) Update X by right shifting by onesample and replacing the first sampleby u.

3) Repeat steps 1 & 2 for Xd using ! + (in place of !.

4) Calculate y" = XtW and y"+# = XtdW.

5) Evaluate (17).6) Update W and ! using (13) and (14)

respectively.

The processing is done on every received symbol to handlesmall carrier offset and track the revolving constellation. Thecomplexity of the algorithm can be reduced in case of staticmultipath channels with no carrier offset by stopping theparameter updates, after the convergence has been achieved.

IV. SIMULATION RESULTS

The simulations are carried out in MATLAB. We haveshown the results for static frequency selective wireless chan-nel but they can be extended easily for slow fading channeldue to the symbol by symbol processing and adaptive natureof algorithm. We consider QPSK and 16-QAM modulations,which are widely-considered constellations. Simulation resultsare evaluated at 25 dB SNR under two different multipathchannels to validate the robustness of the algorithm. Theoversampling factor Q is set to 2 for implementing linearinterpolation. Root raised cosine !lters are used for transmitterpulse shaping and receiver matched !ltering with roll offfactor set to 0.25. The carrier frequency offset (#f/Fs)is set to 10"4 [9]. We have modeled the timing offset !produced by the sampler by !ltering the channel output withoffset exhibiting matched !lter samples. Mean Square Error(MSE) or average instantaneous squared error between theequalizer output and the reference transmitted symbol over60 realizations is calculated as the performance metric. The

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!3 !2 !1 0 1 2 3!3

!2

!1

0

1

2

3(a)

real

imag

inar

y

!2 !1 0 1 2!2

!1

0

1

2(b)

real

imag

inar

y0 500 1000 1500 2000 2500 3000

!40

!20

0

(c)

No. of iterations

MS

E (d

B)

0 1000 2000 3000 4000!0.4

!0.3

!0.2

!0.1

0

0.1(d)

No. of iterationsE

stim

ated

tim

ing

offs

et

Fig. 4. Results for QPSK with multipath channel, fractional timing phaseoffset (!) = 0.4, frequency offset ("f/Fs) = 10!4 and SNR = 25 dB.(a) Channel output constellation. (b) Equalized output constellation. (c) MSEbetween equalizer output and transmitted symbols. (d) Timing offset recovery.

!3 !2 !1 0 1 2 3!3

!2

!1

0

1

2

3(a)

real

imag

inar

y

!2 !1 0 1 2!2

!1

0

1

2(b)

real

imag

inar

y

0 500 1000 1500 2000 2500 3000

!40

!20

0

(c)

No. of iterations

MS

E (d

B)

0 1000 2000 3000 4000!0.6

!0.4

!0.2

0

0.2(d)

No. of iterations

Est

imat

ed T

imin

g of

fset

Fig. 5. Results for 16-QAM with multipath channel, fractional timing phaseoffset (!) = 0.4, frequency offset ("f/Fs) = 10!4 and SNR = 25 dB.(a) Channel output constellation. (b) Equalized output constellation. (c) MSEbetween equalizer output and transmitted symbols. (d) Timing offset recovery.

values of the step sizes are selected by hit and trial methodand are shown in Table I

TABLE IUPDATE STEP SIZES

Modulation Chan-1 Chan-2µ! µw µ! µw

QPSK 1e ! 2 5e ! 2 1e ! 2 5e ! 216-QAM 5e ! 3 5e ! 2 5e ! 3 5e ! 2

A. Results for Channel 1

The !rst channel (Chan-1) is taken from [4] and [15], whichde!ne the multipath channel impulse response as

C(z) = (0.4!0.6z!1+1.1z!2!0.5z!3+0.1z!4)ej$/4/1.41(18)

The results for Chan-1 are shown in Fig. 2 and 3 for QPSKan 16-QAM, respectively. The timing phase offset ! is set to!0.4 to introduce ISI. The equalizer is selected as a 7 tap!lter with central tap initialized to 1. The values for step sizesµ" and µw are shown in Table I.Fig. 2(a) and Fig. 3(a) shows the channel output. We can

see that amplitude distortion and ISI due to multipath channeland timing offset has deteriorated the transmitted symbols.The circular appearance of the constellation is due to carrierfrequency offset. Fig. 2(b) and Fig. 3(b) shows the equalizedoutput. MSE performance and estimated timing offset ! areshown in Fig. 2(c), 2(d) and Fig. 3(c), 3(d) respectively. !converges to +0.25 and +0.3 approx. for QPSK and 16-QAMrespectively to mitigate the introduced negative offset. Sincethere is also an ISI due to the multipath channel, the estimatedoffset converges to a different value instead of 0.4 to mitigatethe resultant channel dispersion. The MSE converges to thedesirable level of !50 dB for QPSK and !30 dB for 16-QAM, in just 500 iterations which is a great improvement onthe results shown by Zhang et al. [4], where the convergenceis achieved after 5000 iterations under same channel with nojoint timing recovery. This is accomplished due to the !nelytuned step sizes and updating equalizer taps and timing offsetjointly to minimize the mean dispersion and ISI.

B. Results for Channel 2

The second channel (Chan-2) is taken from [3], which isa multipath channel, de!ned in Signal Processing informationdatabase SPIB(#2) [16]. The results are shown in Fig. 4 and. 5for QPSK an 16-QAM modulations respectively. The timingphase offset ! is set to +0.4 to introduce ISI. The equalizeris selected as a 16 tap !lter with central tap initialized to 1.The values for step sizes µ" and µw are shown in Table I.Sub-!gures (a)-(d) demonstrate the same graphs as describedfor Chan-1. ! converges to !0.3 and !0.35 approximately.for QPSK and 16-QAM respectively to mitigate the resultantchannel dispersion. The MSE converges to an acceptable levelof !35 dB for QPSK and !30 dB for 16-QAM, in just650 iterations which is a considerable improvement on theresults shown by Ashmawy et al. [3], where the convergenceis achieved after 3000 iterations under same channel with nojoint timing recovery. This is accomplished due to the !nelytuned step sizes and updating equalizer taps and timing offsetjointly to minimize the mean dispersion and ISI.

V. CONCLUSION

We have accomplished blind timing recovery, equalizationand small carrier offset recovery jointly without any trainingsequence. We have modi!ed CMA and extended the objectivefunction to handle another parameter of timing phase offset.We have validated the robustness of algorithm by showingsimulation results under two different multipath channels.We have shown the fast convergence for MSE between theequalizer output and the expected value. Our simulation resultsshow approximately 5 to 10 times improvement in MSEperformance compared to the previous research papers, whileusing their channel and handling synchronization in addition.Moreover, we have summarized the algorithm for an easeof implementation. In future, the results can be extendedfor wireless block fading channel and handling larger carrierfrequency offsets by data reuse.

REFERENCES

[1] Z. Ding and Y. Li, Blind Equalization and Identi!cation, K. J. Ray, Ed.Signal Processing and Communication Series, 2001.

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[2] C. Y. Chi, C. C. Feng, C. H. Chen, and C. Y. Chen, Blind Equalizationand System Identi!cation. Springer, 2006.

[3] D. Ashmawy, K. Banovic, E. A. Raheem, M. Youssif, H. Manour, andM. Mohanna, “Joint MCMA and DD blind equalization algorithm withvariable-step size,” in IEEE International Conf. on Information Tech.,2009, pp. 174–177.

[4] Y. Zhang, J. Zhao, J. Li, and Y. Sun, “A fast convergent algorithm forjoint blind equalization and carrier recovery,” in Proc. ICSP, 2008, pp.1784–1787.

[5] Z. Jiang, M. Zhang, Z. Li, and S. Lu, “A newly high-speed MCMAalgorithm for QAM system,” in WiCOM, Oct. 2008, pp. 1–4.

[6] X. W. Zhang and W. Rao, “New concurrent blind equalization algorithmsuitable for higher order QAM signals,” in International Conf. on FuzzySystems and Knowledge Discovery, vol. 2, 2008, pp. 177–181.

[7] I. Chahed, J. Belzile, and A. B. Kouki, “Blind decision feedbackequalizer based on high order MCMA,” in IEEE Canadian Conf. onECE, vol. 4, 2004, pp. 2111–2114.

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[9] K. N. Oh and Y. O. Chin, “Modi!ed Constant Modulus Algorithm:Blind equalization and carrier phase recovery algorithm,” in IEEE Conf.on Commun. (ICC), vol. 1, 1995, pp. 498–502.

[10] C. I. Hwang and D. W. Lin, “Joint low-complexity blind equalization,carrier recovery, and timing recovery with application to cable modemtranmission,” IEEE Trans. Commun., vol. 1, pp. 120–128, 1999.

[11] N. K. Jablon, “Joint blind equalization, carrier recovery, and timingrecovery for high-order QAM signal constellations,” IEEE Trans. Com-mun., vol. 40, no. 6, pp. 1383–1397, Jun. 1992.

[12] G. Picchi and G. Prati, “Blind equalization and carrier recovery usinga “Stop-and-Go” decision directed algorithm,” IEEE Trans. Commun.,vol. COM-35, pp. 877–887, Sep 1987.

[13] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital CommunicationReceivers, Synchronization, Channel Estimation, and Signal Processing,J. G. Proakis, Ed. Wiley Series in Telecom. and Signal Processing,1998.

[14] W. Chung, W. A. Sethares, and C. R. Johnson, “Timing phase offsetreovery based on dispersion minimization,” IEEE Trans. Signal Process.,vol. 53, no. 3, pp. 1097–1109, March 2005.

[15] I. Santamaria, C. Pantaleon, L. Vielva, and J. C. Principle, “A fastalgorithm for adaptive blind equalization using order-a renyi’s entropy,”in IEEE Conf. on Acoustics, Speech and Signal Processing, vol. 13,no. 2, 2002, pp. 2657–2660.

[16] Signal Processing Information Base (SPIB). [Online]. Available:http://spib.rice.edu

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