Blind equalization for short burst wireless communications

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 4, JULY 2000 1235

Blind Equalization for Short Burst WirelessCommunications

Byoung-Jo Kim, Member, IEEEand Donald C. Cox, Fellow, IEEE

Abstract—In this paper, we propose a Dual Mode BlindEqualizer based on Constant Modulus Algorithm (CMA). Theblind equalizer is devised for short burst transmission formatsused in many current wireless TDMA systems as well as futurewireless packet data systems. Blind equalization is useful forsuch short burst formats, since the overhead associated withtraining can be significant when only a small number of bitsare transmitted at a time. The proposed equalizer overcomesthe common problems associated with classic blind algorithms,i.e., slow convergence and ill-convergence, which are detrimentalto applying blind equalization to short burst formats. Thus, itcan eliminate the overhead associated with training sequences.Also, the blind equalizer is extended to a two branch diversitycombining blind equalizer. A new initialization for fractionallyspaced CMA equalizers is introduced. This greatly improves thesymbol timing recovery performance of fractionally spaced CMAequalizers with or without diversity, when applied to short bursts.Through simulations with quasi-static or time-varying frequencyselective wireless channels, the performance of the proposedequalizer is compared to selection diversity and conventionalequalizers with training sequences. The results indicate that itsperformance is far superior to that of selection diversity aloneand comparable to the performance of equalizers with shorttraining sequences. Thus, training overhead can be removed withno performance degradation for fast time-varying channels, andwith slight performance degradation for static channels.

Index Terms—Blind equalization, constant modulus algorithms,convergence Rayleigh channels, diversity methods, dual mode,equalizers, short burst, wireless communications.

I. INTRODUCTION

SINCE the pioneering work of Sato [7] and Godard [6],blind equalization has attracted a lot of attention from

communications engineers. The most significant advantage ofblind equalization over conventional adaptive equalization is itsself-recovering ability, as noted by Godard [6]. Self recoveringmeans that a blind equalizer does not require any trainingsequences for initial acquisition or subsequent adaptation. Thisproperty makes blind equalization very attractive to short burstformats used in many existing wireless communication appli-cations that use Time-Division Multiple Access (TDMA) suchas IS-136, GSM and future wireless packet data systems suchas EDGE. Short burst formats are used to reduce end-to-endtransmission delay, and also, to limit the time variation ofwireless channels over a burst. However, training overhead can

Manuscript received February 21, 1999; revised July 13, 1999. This work wassupported by Hughes Network Systems.

B.-J. Kim is with AT&T Labs, Research, Red Bank, NJ 07701 USA (e-mail:[email protected]).

D. C. Cox is with STAR Lab., Stanford University, Stanford, CA 94305 USA(e-mail: [email protected]).

Publisher Item Identifier S 0018-9545(00)04823-4.

be very significant for such short burst formats. This overheadranges up to 30% in many systems, e.g., IS-136, GSM, etc.The overhead of these systems can be recovered by employingblind equalization. Also, low tier standards such as PACS,DECT, and PHS, do not incorporate equalization [3], andconsequently, there are no training sequences in these systems.In cases where longer range or higher tolerance of delay spreadis needed, blind equalization can be used for these systemswithout changing physical link formats.

There are several classes of approaches to blind equaliza-tion. Classical approaches are based on nonconvex optimiza-tion. Examples are Constant Modulus Algorithm (CMA) [6],[16], [26],1 Sato algorithm [7], Stop-and-go algorithm [17], etc.Due to the nonconvexity of the cost functions of these algo-rithms, they suffer from slow convergence speed and ill-con-vergence. Another approach is based on higher order statistics(HOS) explicitly or implicitly [8], [15], [23]. Algorithms of thisclass usually optimize some convex functions of HOS to achieveequalization. These can be direct equalization, or channel esti-mation/equalization. However, due to the use of HOS, they re-quire a large number of samples. Hence, these algorithms arenot suitable for the application considered in this paper. Thenewest approach is based on the second order cyclic statistics(SOCS) present in oversampled digital communications signals[22], [24], [25]. These SOCS based methods have been shownto have limited practical use due to their restrictive uniquenessconditions and the algorithms’ sensitivity to them [32], [36]. Theclassical approach is the lowest in complexity, and reliable inpractice.

However, the performance of classical blind equalization al-gorithms when used for short burst communications over real-istic wireless channels has not been addressed.

In this paper, we propose a Dual Mode Blind Equalizer basedon CMA [6]. The equalizer is designed for short burst transmis-sion formats used in many wireless TDMA systems [3]. Thecontributions of this paper are summarized as follows.

• With short burst communication formats, each burst maynot be long enough for a blind algorithm to converge. Weexamine the effectiveness of reusing data from capturedbursts. We find that under some conditions, this yields asteady state performance similar to that of using very longsequences of data. Also, we examine the relationship be-tween the length of equalizers and the length of bursts inthe context of the effectiveness of data reuse.

• We show through simulations that even with a fraction-ally spaced structure, CMA suffers from ill-convergence

1Godard originally did not call his algorithm CMA in [6]. However, his algo-rithm is generally viewed later as the first form of CMA.

0018–9545/00$10.00 © 2000 IEEE

1236 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 49, NO. 4, JULY 2000

from symbol timing errors. we propose a new initializa-tion scheme for fractionally spaced CMA equalizers thatsignificantly improves the convergence property of frac-tionally spaced CMA equalizers [2].

• To further improve the steady state performance, we in-corporate a simple yet effective dual mode operation withCMA and decision-directed (DD) adaptation. Althoughthis is not new, we introduce a cluster detection methodand adaptive switching threshold that allow smooth tran-sition back and forth between CMA and DD depending onwhich mode reduces the error more effectively.

• We extend the dual mode blind equalizer to two antennadiversity combining.

Like most classic blind algorithms based on nonconvex op-timization, CMA suffers from slow convergence and ill-con-vergence [8], [9], [11], [18]. We have chosen a Fractionallyspaced CMA equalizer due to its low complexity, better conver-gence properties and robustness to channel disparity, noise [10]and symbol timing offset [12]. With short burst communicationformats, each burst may not be long enough for a blind algo-rithm to converge. The proposed equalizer solves this problemby reusing data from captured bursts. We show that this yieldsa steady state performance similar to that of using very long se-quences of data. As for ill-convergence, we propose a new ini-tialization scheme for fractionally spaced CMA equalizers thatsignificantly improves the symbol timing recovery performanceof fractionally spaced CMA equalizers [2] used for short burstsby avoiding ill-convergence from symbol timing errors. Also,we extend these results to two antenna diversity combining blindequalization, and examine its performance against time-varyingchannels. Also, the performance of conventional equalizers withshort training sequences is compared to that of the proposedequalizer.

First, we will discuss the channel model and the system usedin this paper, and describe the proposed equalizer in detail. Thereuse of captured data and the proposed initialization schemeare explained and extended to two branch diversity. The per-formance of the blind equalizer against static and time-varyingchannels is presented, and compared to those of other conven-tional methods such as selection diversity and equalization withtraining sequences.

II. SYSTEM AND WIRELESSCHANNEL MODEL

The time-varying wireless channel model used in this paper isbased on Jakes’ model [5]. We use 320 Rayleigh paths equallyspaced over ten symbol periods with an exponential averagedelay profile [see (2)] with different average delay spread valuesfor static channels in heavily build-up microcell environments,and 12 path GSM Hilly Terrain average profile [30] for time-varying channels in macrocell environments.

The baseband equivalent channel model used in this paper isrepresented in continuous time as follows:

(1)

wheretime-varying impulse response of baseband wire-less channel observed from time;average power delay profile. ;magnitude of each unresolvable wave;white random process uniform over [0, ];maximum Doppler frequency;arrival angle for each unresolvable wave, uniformover [0, ];number of waves for each delay ( in this paper.)

in part of the simulations in this paper is obtained froman exponential profile, described as

otherwise (2)

where is a average rms delay spread. In other cases,is ob-tained from the GSM HT profile which specifies average powerlevels at 12 multipath delays. The average rms delay spread ofHT profile is 9.11 s. Time variation is induced by various max-imum Doppler frequencies equivalent to vehicle speeds up to90 mi/h at 2 GHz carrier frequency. When static channels aresimulated, is set to 0. is uniformly distributed over [0, 2).

is set to 1, which yields a Rayleigh distribution for each delaypath. Since the ’s used in computer simulations are dis-crete in time, they are modeled as ,where ’s are the time indices for , is the numberof ’s, ’s are the values of at , and is the delta(impulse) function. Channels are assumed to be uncorrelatedfor different antennas and for different time slots assigned toa single user. This is a reasonable assumption considering thespacing of diversity antennas and the temporal separation ofbursts for a single user for most TDMA system [3], [4].

The system considered in this paper is a TDMA system witha short burst format. Short burst formats are used primarily tolimit end-to-end delay as well as to limit the impact of the timevariation of wireless channels on link quality. TDMA systemssuch as PACS, PHS, DECT, IS-136 GSM, or EDGE have about60–200 symbols in a burst and consequently the training over-head for these systems would be or is significant. For example,the training overhead of GSM is 18%. If training sequences wereadded to the formats that do not currently have training, such asPACS, PHS, and DECT, the overhead would range from 10 to30%. We have chosen the US low-tier wireless communicationstandard, Personal Access Communications System (PACS) [4]for our system model, since the uplink bursts have only 60 sym-bols and no training sequences. Thus, for any equalization to beused, it must be blind and must work with a limited number ofthe samples from a short burst. Note that the PACS downlink hassynchronization sequences that can be used for training, as weshow later. PACS uses 4QAM with a symbol period of 5.26

s. The transmitter and the receiver have square root raised co-sine filters with 50% excess bandwidth. Each frequency channelcontains eight time slots, and each time slot can carry 32 kbpsvoice traffic for a user. The system is designed operate over alimited range with no equalization. Thus, it cannot tolerate high

KIM AND COX: BLIND EQUALIZATION FOR SHORT BURST WIRELESS COMMUNICATIONS 1237

ISI (Inter-Symbol Interference) [27], or equivalently, a high nor-malized average delay spread2 value higher than 0.1 [27]. Also,to support higher mobility than walking speed, fast and frequenthand-off is necessary when the range of base stations is small asin low-tier PCS. Thus, longer range is also beneficial in termsof reducing the hand-off rate for a given user speed.

The system model is shown in Fig. 1 for two branch diversity.Note that we will consider a single branch case first, and twobranch diversity in the latter part of this chapter. In the figure,a single user’s signal,, goes through two channels: and ,which are sampled channels that include transmit/receive fil-ters.3 The outputs from the equalizers, and , are summedand fed to a blind algorithm that controls the two equalizers si-multaneously.

III. B LIND EQUALIZATION FOR SHORT BURST

COMMUNICATIONS

A. Reuse of Captured Data

The length of a burst to be equalized is very short in our ap-plication, especially considering the slow convergence speed ofmost classic blind equalizers. Mostly for its simplicity, we fo-cused on the CMA and its variations. Most of the variationsare focused on faster convergence speed and more convex costfunctions [8]. These techniques usually have better performanceoverall than the original CMA when there is a large enoughsupply of data. However, their improvements are marginal inour application with short bursts [1]. Also, when averaged overa large number of channels, our simulations of CMA and thevariations show that there is little (if any) improvement fromusing the CMA variations [1].

We have considered reusing captured data to converge CMAequalizers. Although this is not a new approach in practice, itis not well documented in the literature, especially for CMA.Here, we study the effectiveness of data reuse in terms of thelength of reused data bursts. The focus is whether useful con-vergence can be achieved with short lengths of data. When dis-crete time channels are modeled as FIR filters4 driven by a fi-nite alphabet input such as QAM, the channel output samplescan take only a finite number of possible values. If the length ofa random sequence burst is long enough to produce most or allof these potential channel output samples, the captured samplesfrom this burst should have enough information for adapting anequalizer. For blind equalization, the uncontrolled input5 shouldbe long enough to yield a sufficient number of channel samples.However, with no or little noise, once there are enough sym-bols in the input, more symbols do not improve convergence, asseen in Fig. 2. Since the performance of any equalizer, blind ornot, is also limited by residual ISI and noise, short bursts can

2The average rms (root mean square) delay spread normalized to the symbolperiod is called normalized average delay spread. The instantaneous ISI of achannel convolved with raised cosine pulses can often be significant even whenthe normalized average delay spread is as small as 0.1.

3For time varying channel simulations, these filters are treated separatelyfrom the channel.

4In most practical situations, a channel is considered to be a FIR filter whichis obtained by sampling and truncation.

5User data, rather than fixed training sequences.

be enough to achieve nearly the best possible performance for agiven channel/equalizer combination, even when there is signif-icant noise. We have verified that this is indeed the case throughextensive simulations with various channels conditions, noise,step sizes, and equalizer lengths. Here with Fig. 2, we presentan example of one of the scenarios to illustrate the relationshipbetween equalizer lengths and the length of data bursts.

Fig. 2 shows the mean square error (MSE) performance ofa convergedCMA equalizer that reuses captured data. Theaxis indicatesthe length of a data burstreused to converge theequalizer until the CMA cost function is no longer decreasing.The MSE is calculated from the CMA equalizer taps after theequalizer has fully converged by reusing captured data from asingle burst. The equalizer is converged by reusing received dataof the length indicated on the axis. For example, the MSEfor the value of 100 on the axis is obtained from reusing thesamples from a 100 symbol long burst until the equalizer isfully converged.6 In the example in Fig. 2, the channel is threesymbols long. Thus, with 4QAM, there are only pos-sible channel outputs. The MSE curve in the figure is nearly flatabove about 100 symbols. This indicates that a small numberof samples, when reused, can achieve almost complete conver-gence, which yields the same performance as equalizers con-verged with significantly longer sequences of samples. The per-formance of the ideal linear equalizer (LE) and that of the idealdecision feedback equalizer (DFE) for the simulated channel arealso shown in the figure for comparison. Note that the CMAequalizer used in this figure is based on a LE, and the MSE ap-proaches that of the ideal LE.7

The convergence speed of the data reuse is shown in Fig. 3which includes the learning curves for various burst lengths. The

axis is the total number of iterations, i.e., the total numberof symbols passed through the equalizer. Thus, the number ofreuses for a specific burst length can be obtained by dividingthe axis values by the length. The figure confirms that theconclusion from Fig. 2 also applies to the convergence speed asa function of burst length. The curves for burst lengths above 64symbols are virtually indistinguishable from one another. Thus,the reuse of a short burst results in a convergence speed and afinal MSE almost equivalent to those for a much longer burst.

It has been observed that CMA may diverge when there is asignificant single frequency component embedded in a receivedsignal [9], [11], as is the case when a pseudo-random sequencewith a short period is being transmitted. This can be seen asa “lock-up” behavior, since a single frequency sinusoid satis-fies the constant modulus property as well as distortion free4QAM signals do. This behavior of CMA is likely to occur ifthe single frequency component is strong and the length of theCMA equalizer is comparable to the period of the single fre-quency component. In this case, the equalizer has high enoughresolution to capture the frequency of the periodic component.If a CMA equalizer locks up to a single frequency, the resultingconverged equalizer is usually not capable of removing ISI.

6The total number of iterations until this convergence is achieved is almostconstant for the different lengths of bursts. For instance, 10 times reuse of a100 symbol long burst produces a similar MSE value as 5 times reuse of a 200symbol long burst. For more details, refer to [1].

7The SNR in Fig. 2 is 17 dB. This is the minimum required SNR for PACS.


Fig. 1. Two branch diversity combining equalizer system

Fig. 2. After converging the equalizers by reusing captured data from a singleburst, the mean square error performance of the equalizer is calculated for eachlength of reused bursts. TheX axis indicates the length of reused data bursts.SNR is 17 dB. The total number of updates until convergence is around 500updates once the length of bursts is larger than 32 symbols, and slightly largerfor shorter bursts, as shown in Fig. 3.

A similar “lock-up” behavior can arise from the reuse of cap-tured data, because of the periodicity of the reused data. InFig. 3, the learning curve for ten symbol long bursts shows betterconvergence performance than those of 32 and 64 symbol longbursts. This is due to the lock-up behavior of CMA, since the sixsymbol long CMA equalizer is long enough to lock-up to the pe-riod of the reuse of a ten symbol long burst. The equalizer haslocked-up to the period of ten symbols, and the resulting CMAcost is small. However, the poor MSE performance of 10 symbollong bursts in Fig. 2 clearly shows that the resulting equalizer isnot capable of removing ISI. This indicates a tradeoff betweenusing long equalizers for better ISI mitigation and using shortequalizers to avoid the lock-up behavior. We will discuss thislater with simulation results in Section III-F.

B. CMA/Decision-Directed Dual Mode Blind Equalizer for4QAM

Decision-Directed (DD) equalization8 can often open theeye of the received signal when a 4QAM signal is transmittedthrough low ISI channels [19]. In this case, DD is effectivebecause the DD cost function is very similar to a CMA costfunction when the constellation is 4QAM. Intuitively, 4QAMhas the lowest probability of error among all QAM given afixed ISI. Thus, DD may work well with 4QAM if ISI is low.

Dual mode equalization [20] refers to an equalizer thatswitches its update algorithm between two alternatives de-pending on operational conditions. When DD is used afterinitial convergence with a blind algorithm, it provides notonly channel tracking but also the faster removal of remainingISI left by the blind algorithm. Usually, a blind algorithmsuch as CMA can achieve complete convergence under someconditions. However, once ISI is significantly reduced, theremaining convergence is much slower than that of DD. DDcan remove the remaining ISI much faster [20] and also providea way to track the time variation of wireless channels. Also,DD results in lower steady state mean square error performance[20], since with few decision errors, the cost function of DDis closer to the mean square error cost function than that ofCMA [19]. Besides the improved convergence speed and bettersteady state performance, the dual mode operation with DDprovides another advantage when CMA is the initial algorithm.Frequency offset correction can be easily incorporated in theDD mode [6], [31], while CMA is largely insensitive to a smallfrequency offset [31]. This makes dual mode operation veryattractive.

Although the use of DD with CMA is not new, we propose inthe following sections a novel yet simple way to implement dualmode operation with DD and CMA that works well with the datareuse described earlier, eliminates the need to guess a switching

8The symbol decisions of an equalizer are used as the reference for its own up-dates. Once an equalizer is fully converged and produces only a small number ofdecision errors, its own symbol decisions are reliable enough to replace trainingsequences. This provides some tracking of time varying channels.


Fig. 3. Blind Equalization learning curves for various lengths of burst. From the top, 20 (dashed), 32 (dotted), and 10 (solid) symbol long bursts. Theremainingoverlapping curves are 64, 128, 256, 512, 1024, and 2500 symbol long bursts. SNR= 17 dB. Length of CMA equalizer= 6 symbols

threshold, and allows switching back and forth between CMAand DD depending on which mode provides better convergence.

C. Dual Mode Operation for Convergence

In this section, we discuss the proposed dual mode algorithmfor a single antenna in detail. The dual mode algorithm incor-porates the idea of data reuse discussed in Section III-A. Fortwo branch diversity, the only difference is in initialization asdescribed in Section IV.

The algorithm’s first mode is CMA. As the equalizer starts, itworks on an entire burst repeatedly until the average error valueover the entire burst becomes small enough to switch to DD. Thefollowing equations describe CMA for 4QAM of unity power:

(3)

(4)

(5)

equalizer tap coefficient vector (6)

equalizer input vector (7)

whereequalizer output;step size; fixed or adaptively varied, e./g./ NormalizedCMA;length of equalizer;complex conjugate transpose;transpose.

For time-varying channels, it is not desirable to always re-peat the adaptation from the beginning of captured data, since achannel can be significantly different at the beginning and theend of a burst. Thus, the proposed algorithm alternates its adap-tation (or filtering) direction between forward and backward intime. Thus, the direction ofth burst iteration is opposite in timeto the direction of th burst iteration. This also helps preventthe aforementioned “lock-up” behavior, as it effectively doublesthe period of data reuse.

After the CMA cost function is reduced below a threshold, theequalizer switches to DD operation. The threshold is based ontwo criteria: average CMA cost function over the current burst,and the reduction of the average CMA cost function comparedto the last run over the entire burst. If the average cost functionis below the current threshold (determined dynamically as de-scribed shortly), the operation is switched to DD. However, evenif the average cost function is not below the current threshold,the operation is also switched if the reduction of the average costfunction is less than 5 % since the repetition. This is to detect thecase where the dynamically set threshold is not being reacheddue to its quantization errors. The initial switching thresholdshould be set to give low error probability when decision-di-rected adaptation is performed. If the cost function in DD is


Fig. 4. Simplified block diagram of the algorithm.

too high after switching modes, switching back to CMA maybe necessary. However, using a simple threshold to determineswitching between CMA and DD may cause bouncing back andforth between the two modes, since the CMA cost function andthe DD cost function are very different except near full conver-gence. To avoid this problem and still provide a way to switchback to CMA, the threshold is adaptively changed after a hys-teresis period. To determine when to initiate the switching, thevalue of the average CMA cost function over a burst period isconstantly monitored and compared to a threshold value. Thevery different cost function of DD adaptation can produce largeerror values immediately after switching from CMA. As notedearlier, this can result in bouncing back and forth between CMAand DD modes. A hysteresis mechanism is introduced in thedual mode operation by forcing one algorithm to work throughseveral iteration cycles no matter how big the error value isimmediately after switching between the two operation modes.After the hysteresis period, if the average error value over a bursthas not decreased below the switching threshold, the CMA al-gorithm is again used with a lower switching threshold value(Fig. 4). Specific values of the initial switching threshold andhysteresis are considered in Section III-F.

For dual mode operations, the insensitivity of CMA to a con-stant phase error poses a problem when switching from CMA toDD. CMA cannot correct a rotation of the signal constellationinduced by wireless channels and/or a carrier phase offset [6].For example, when the constellation points equalized by CMAlie near the real or imaginary axis, switching directly to the DDmode results in many decision errors. Too many decision errorsare fatal to the operation of DD mode in general. However, inour application, only a few decision errors can be fatal as well,since the length of a burst is short. Estimating the rotation of thecoherent decision boundary for the initial operation of DD modeis crucial to a smooth transition from CMA to DD mode. Themethod used in the original PACS design to estimate this rota-tion is mainly focused on finding the average angle of rotationof an entire burst and performs well in low delay spread (smallISI) environments [4], [28]. When applied to the output of theCMA, however, its performance is significantly impaired whenISI is not reduced enough by the CMA adaptation. Also, sinceCMA is in essence trying to recover the constant amplitude of4QAM at the symbol rate, it sometimes forms more than fourclusters on the unit circle in the complex plane. This phenom-enon is likely to happen when ISI is large. Thus, in these cases,


DD adaptation can fail due to too many decision errors. A newmethod for estimating decision boundaries and detecting pos-sibly good clusters for DD has been designed and incorporatedinto the dual mode CMA studied in this paper. This method isbased on locating the most concentrated cluster of CMA equal-izer outputs on the complex plane. The symbols in this mostconcentrated cluster are used in the initial phase of DD adapta-tion to ensure that the most correct decisions are available.

First, the algorithm centers a phase angle window on one ofthe available symbols at the CMA equalizer output, and cal-culates the average deviation of every other symbol within thewindow from the center symbol. The deviation can be only thephase angle for 4QAM, as shown in (8). Euclidean distance canbe used for higher constellations9

(8)

where is the width of phase window. The width of thewindow ( ) in terms of phase is usually chosen smallerthan the width of the decision phase width for the constellation,e.g., for 4QAM, . This calculation continues withdifferent symbols until it finds the most concentrated cluster.The decision boundary rotation is estimated from the center ofthe most concentrated cluster and the equalizer taps are adjustedaccordingly

(9)

where is the center of the most concentrated cluster.The symbols in the most concentrated cluster are used first inthe DD mode. Therefore, the equalizer converges faster to thecorrect solution because fewer decision errors occur. After thisphase, all the other data are used for adaptation in DD mode.The effectiveness of the cluster detection is demonstrated laterin Section III-F.

A simplified block diagram of the entire algorithm is pro-vided in Fig. 4. The iteration continues until the next burst ar-rives for processing. Thus, the number of iterations possiblefor each burst would be dependent on the computing capabilityof receivers (of course, a tradeoff should be made between thenumber of iterations and computing cost.) If a CRC check hasnot indicated successful decoding by the end of the availablecomputation time, the corresponding burst is declared to haveerror(s). The burst is discarded for voice-oriented applications,and re-transmission is requested for data oriented applications.

D. Initialization for a Fractionally Spaced CMA Equalizerwith a Single Antenna

So far, we have described the dual mode reuse of captureddata to overcome the slow convergence problem of CMA. Inthis section, the ill-convergence problem is addressed throughthe initialization of fractionally spaced CMA equalizers.

Symbol rate equalizers need accurate symbol timing, es-pecially when raised cosine pulses with an excess bandwidth

9For 4QAM, the phase angel and the Euclidean distance result in similar per-formance. The phase angle is used for the simulations in the following sections.

Fig. 5. Symbol timing recovery range of FSE CMA. (Averaged over 1000randomly generated channels with the average normalized delay spread of 0.2.Average SNR= 17 dB. Exponential average delay spread profile.)

greater than zero are used [29]. Fractionally spaced equalizers(FSE) are robust to symbol timing errors. Consequently, accu-rate symbol timing recovery is not necessary with sufficientadaptation [12], [29]. However, the robustness of a short burstCMA FSE against symbol timing errors is not satisfactory dueto the characteristics of the conventional initialization of CMA.This significantly limits the performance of FSE CMA. Thissymbol timing problem stems mainly from local minima, eachmaybe associated with a symbol timing other than the optimalsymbol timing for a channel. An intelligent initialization canplace the nonzero initial tap near the optimal symbol timing sothat global convergence be achieved more rapidly.

The conventional initialization for symbol spaced CMAequalizers is to set one tap near the center to a nonzero con-stant,10 e.g., 1, and to set to 0 all other taps as suggested byGodard [6]. This initialization exhibits satisfactory convergencefor symbol spaced discrete time channels. The convergenceis global only for the symbol timing implied in a symbolspaced sampled channel. For fractionally spaced CMAequalizers, there are possible locations for this nonzerotap. If a fractionally spaced CMA equalizer could interpolatethe optimal symbol timing, as do equalizers with trainingsequences, the location of the nonzero tap with respect tosampling phase would be irrelevant. However, the nonzeroinitial tap of Godard’s initialization usually decays slowly. Thismakes it likely for the equalizer to converge slowly or to afalse solution, and result in poor performance for the scenarioconsidered in this paper, where convergence speed must be alsofast given the TDMA operation.

Our simulations indicate that if the nonzero initial tap iswithin less than a quarter symbol period around optimal symboltimings, the convergence is usually satisfactorily fast with datare-use. In Fig. 5, the burst error rate11 of FSE CMA is shown

10A value larger than the threshold constant suggested in [6]11The probability of a 60 symbol long burst having at least one bit error. Refer

to Section III-E.


for different symbol timing values of the initial nonzero tap.12

The burst error rates are obtained from 1000 channels randomlygenerated from an exponential delay spread profile [see (2)]with 0.2 average normalized delay spread. The change of theburst error rate curve indicates that there is a symbol timingrecovery range for FSE CMA where the convergence is quitefast. (Although, it is possible to achieve full convergence withthe nonzero initial tap outside of this range, we find that theconvergence speed is too slow to be useful for the applicationsconsidered in this paper.) The exact width and the location ofthe recovery range differ for each individual channel. Whenaveraged over many random channels, the width is slightly lessthan one half symbol period and the recovery range is centeredaround the optimal symbol timing, as shown in Fig. 5. Out ofthis recovery range, a FSE CMA is more likely to convergeto a local solution associated with the symbol timing nearthe initial nonzero tap or the convergence speed can be veryslow, thus resulting in high burst error rates. However, if theinitial nonzero tap can be placed within this recovery range, theconvergence is often satisfactory and fast and the remainingsymbol timing error is removed by FSE CMA. Thus, it issufficient to provide a rough estimate of the optimal symboltiming for initialization before beginning adaptation.

To be able to identify the sampling phase that is within thesymbol timing recovery range, at least four times oversamplingis necessary. Note thaton-the-flysampling phase adjustment isnot practical for the scenario considered in this paper. This ismainly due to the lack of training sequences, and also due to thedelay associated with symbol timing estimation and samplingphase adjustment. Thus, an entire burst must be oversampledwhen it arrives. With 4 times oversampling, at least one sam-pling phase or usually more than one sampling phase is boundto be within the symbol timing recovery range of FSE CMA,since the recovery range is approximately a quarter symbol pe-riod before and after the optimal symbol timing. The exact rangedepends on the specific channel. When the initial nonzero tap iswithin less than a quarter symbol period from the maximum eyeopening,13 the symbol timing recovery is satisfactory for mostchannels.

A rough measurement of eye opening among the samples ofoversampled signals can be done efficiently as described in thefollowing. This procedure is based on the observation that thehigher the ISI is at a sampling phase, the lower the average signalamplitude is at the specific sampling phase. Thus, the samplingphase with minimum ISI is found by maximizing the followingmeasured quantity over , where is the over-sampling factor for a spaced equalizer.

The procedure is to find thethat maximizes

(10)

where is the number of symbols in a burst , and isthe th sample of th symbol corrupted by ISI and noise.

12For details of the simulation parameters, refer to Section III-F13The eye may not be open due to ISI, but we can still consider it to be the

sampling point which yields the minimum ISI.

Fig. 6. Illustration of the proposed initialization

Fig. 6 illustrates (10) for . The figure shows the magni-tude of a short portion of a sampled baseband signal at the inputof the equalizer. The lower curve is the magnitude of the channelthat produces this example (Of course, unknown to the equal-izer). Four different sampling phases are marked with differentsymbols. In this example, the third sampling phase yields themaximum average amplitude, obtained according to (10). Thissymbol timing estimation is sufficient for most channels, sincethe symbol timing recovery range of FSE CMA is not too small.This method greatly improves the symbol timing recovery per-formance and increases the convergence speed, as shown in thefollowing sections. The dual mode equalizer is preceded by afilter with the transmit pulse shape. Thus, the proposed initial-ization is essentially uses the transmit pulse shaping filter as itsinitialization and adjusts the selection of delay based on mea-surements for symbol timing.

An intuitively more plausible approach can be devised by ob-serving that the higher the ISI is at a sampling phase, the morethe signal amplitude fluctuates at the specific sampling phase.This leads to locating the sampling phase that minimizes thevariance of the signal amplitude, . This can also beeasily computed from stored samples. Simulations show that theinitialization at the sampling phase based on the minimum am-plitude variance results in little improvement over the one basedon the maximum average amplitude, and sometimes even worseperformance is obtained at low SNR. This is perhaps becausethe signal amplitude variance is severely affected by noise atlow SNR, while the average amplitude smooths out zero-meanadded noise samples. With the above observation, the initializa-tion at the sampling phase with the maximum average signalamplitude is used in the simulation results presented in the fol-lowing sections.

E. Performance Criterion

Since ill-convergence yields an equalizer solution far awayfrom the correct solution for a channel, bit errors due to theill-convergence of blind algorithms are bursty. This is also true


for bit errors due to multi-path fading. We can treat all these er-rors together as burst errors, i.e., a burst error occurs if a singleburst has at least one bit in error. Thus, a burst error rate is morerepresentative of the practical situation than a bit error rate, un-less error correction and/or interleaving are implemented. Con-sequently, a burst error rate14 is used as the performance crite-rion for the scenarios considered in this paper.

It is likely that blind equalizers would have higher burst errorrates than equalizers with training. This higher error rate must beweighed against the additional overhead of training sequencesfor the conventional equalizers, when taking into account thedistribution of delay spread expected to be encountered by apotential system.

F. Performance of the Dual Mode Blind Equalizer for a SingleAntenna

For these simulations, start-up symbol timing for initializa-tion is estimated by finding the maximum average amplitudesampling point among the oversampled values of the receivedsignal, as described in Section III-D. The initial condition ofthe fractionally spaced blind equalizer is set to 1 at this esti-mated symbol timing near the center tap and 0 for all other taps.Also, the performance of equalizers with training sequences arecompared to the dual mode algorithm. Four times oversamplingis used for all simulations. For each normalized delay spreadvalue, 400 randomly generated static channels are used for allalgorithms, and 20 bursts are transmitted through each channel.

For comparison, the same simulations using the same chan-nels and the same data bursts are performed with training se-quences prepended to the bursts. The equalizers with trainingare based on linear transversal FIR filters to keep the complexityof both the blind equalizers and the trained equalizers similar ex-cept for some additive constants resulting from the algorithmicdifferences. The SNR, averaged over Rayleigh fading, is 17 dB.Instantaneous channels can have lower or higher SNR due tofading. For all the cases, the adaptation is done as described inSection III-D over available data from a burst of 60 symbols,until satisfactory convergence is achieved. However, there is amaximum number of repetitions set for all the equalizers, as in-dicated in Section III-D. Even if convergence is not achieved,adaptation stops after the maximum number of iterations, e.g.,when the next burst arrives. For the simulations, the equalizersdeclare a burst error if decoding is not successful after 50 datareuses (CMA and DD combined) over an entire burst. This valueis quite arbitrary, but the majority of successful burst decodingsis achieved within 20–30 reuses, and more than 50 reuses resultin little improvement.

The initial switching threshold for the dual mode operation is0.3 and is reduced by 20% each time the equalizer returns to theCMA mode if the DD mode fails to converge.15 The hysteresisperiod after switching is 5 runs over an entire burst. The width

14A burst error rate is often calledframe error rate(FER) for TDMA systems.15The initial value of 0.3 is arbitrarily chosen. If this value is too large for

a specific channel, subsequent reduction is made as the DD mode fails afterswitching.

Fig. 7. Comparison of the Dual Mode Blind Equalizer and the TrainedEqualizer with the same length of six symbols (T/4 fractionally spaced,Average SNR= 17 dB, Exponential average profile, Quasi-static channels).

of the phase window for the cluster detection algorithm is,as discussed in Section III-C.

For the equalizers with training sequences, a communicationburst is extended to include 10% or 20% more symbols to beused as training sequences. Since the length of a burst is 60 sym-bols in our scenario, 10% corresponds to six symbols (12 bits)and 20% corresponds to 12 symbols (24 bits) (note; IS-136 hasa 14 symbol training sequence in one 162 symbol long burst.).The RLS update algorithm is used for fast convergence over thetraining portion of received signals, and the LMS is used fordecision directed updates for the random data of the signal. Thesix symbol choice is the number of the PACS downlink synchro-nization symbols, which can be used as a training sequence fora possible equalizer in handsets.

The choice of the equalizer length of six symbols was madeafter observing the performance of equalizers with variouslengths. As mentioned earlier in Section III-A, long CMAequalizers could “lock-up” to the periodicity of the reuse ofcaptured data, while short ones have limited capability miti-gating ISI. The length of six symbols has little chance of thelock-up behavior16 as it is one-tenth of the burst length of 60,and is capable of mitigating significant ISI with its fractionallyspaced structure.

Fig. 7 shows the comparison between the blind equalizer andtrained equalizers of the same length (6 symbols long). Theperformance of the blind equalizer with the proposed initial-ization is comparable to the trained equalizer with six symbollong training sequences which is 10% overhead (66 symbol longburst compared to 60 symbol long burst). The trained equalizerwith 12 symbol training (20 % overhead) performs consider-ably better than the blind equalizer. Note that without trainingsequences as in the application that motivates this paper, thesetrained equalizers cannot be used.

16The authors have verified this by examining the burst error probability ofvarious lengths of bursts for each fixed length equalizer


Fig. 8. Comparison of the Dual Mode Blind Equalizer with and without the cluster detection algorithm (6 symbol long equalizers, T/4 fractionally spaced, averageSNR= 17 dB, exponential average profile, quasi-static channels).

Compared to the conventional initialization,17 the perfor-mance of the proposed equalizer initialization is far superior. Itis clear that the proposed initialization significantly improvesthe robustness of FSE CMA against symbol timing errors. If,as in many voice wireless communications systems, burst errorrates less than 3% are acceptable, the delay spread toleranceof the proposed equalizer is approximately twice that of noequalization (PACS receiver).

Note that the performance of the conventional initializationis actually worse than that of the PACS receiver (labeled as“NoEQ”) with no equalization where the normalized delayspread is small. While the conventional initialization suffersfrom symbol timing errors even though the ISI at the optimalsymbol timing is low, the PACS receiver uses 16 times over-sampling to locate the best symbol timing [4], thus yieldingbetter performance when delay spread is small.

Fig. 8 shows the performance of the blind equalizerwith or without the cluster detection algorithm described inSection III-C. The initialization is the same for both cases. Theblind equalizer without the cluster detection algorithm correctsonly the average constellation rotation using the method in[28] developed for PACS, and uses all the symbol outputs fromthe beginning of DD operation. However, the one with thecluster detection algorithm corrects the average constellationerror measured at the center of the most concentrated clusterof equalizer symbol outputs, and uses the symbols from thatcluster for initial DD operation as described in Section III-C.As shown in the figure, the performance is greatly improvedwhen the delay spread is relatively high, since it provides morereliable symbols for the initial DD operation. As noted earlier,a few decision errors can be fatal for our application, since

17The equalizer with the conventional initialization is the same as the pro-posed equalizer to overcome slow convergence speed, except, of course, for theinitialization.

short bursts are not long enough to recover from only a fewdecision errors.

IV. BLIND DIVERSITY COMBINING EQUALIZER

In this section, we extend the proposed initialization for theshort-burst single antenna CMA equalizer to the initializationfor multiple antennas, or multiple channels. For the single an-tenna case, we showed that the proposed initialization schemesignificantly improves the performance of CMA FSE. To extendthis to multiple antennas, we propose the following. Withoutloss of generality, we assume that there are two antennas.

Previously known initialization schemes for CMA with mul-tiple antennas are mostly variations of the following two ap-proaches. Based on a multichannel approach [33], one can ini-tialize only one branch. Alternatively, each branch can be equal-ized separately by CMA, and after convergence, the equalizeroutputs can be co-phased and combined. This involves the es-timation of the post-equalization channel phase and the qualityof each subchannel equalizer output. We have found that thisdoes not yield satisfactory performance for realistic channelsand the application considered here. The supporting results arepresented in Section III-D.

As with the single antenna case in Section III-D, the basicidea behind the proposed initialization is to place the initial con-ditions within the gravitational pull of the global solution forCMA. Of course, the global solution is unknown, but one canestablish a few properties that the global solution for a set ofdiversity channels must have. First of all, each diversity branchshould recover its respective symbol timing, as with the singleantenna case. Also, all branches should be co-phased to addup coherently. As done for maximum ratio combining diver-sity [3], the signals from all branches should be roughly scaledaccording to their power levels. An initial equalization setting


should encourage these conditions. With these conditions inmind, the proposed initialization for diversity combining CMAis described as follows. Without loss of generality, we assumethat there are two antennas.

Equation (3) for a single antenna can be easily extended tomultiple branches by applying the following substitutions basedon multi-channel approaches. For example with two channels,i.e., a two branch equalizer.

(11)

(12)

Here, and denote the regression vectors of receivedsamples from the first antenna and the second antenna, respec-tively. Also, and are the subchannel equalizers for thefirst antenna and the second antenna. The extension to more thantwo channels is straightforward.

For the maximum eye opening for each antenna found in (10),one can find their approximate average channel phase differenceand average amplitude ratio as follows. Note that the signalsfrom the two antennas are from the same source. By correlatingthe two, their relative arrival time difference can be found at theresolution of a symbol period. This is removed by offsetting thesamples between the two antennas. With the signals synchro-nized this way within a symbol period,18 the following measuredcomplex quantity reflects the average phase difference and theaverage magnitude ratio at the maximum eye openings

(13)

where and are obtained from (10).By initializing as follows

(14)

(15)

(16)

where the locations of the nonzero entries are determined byand . Thus, at the maximum eye opening sampling phase (nearthe center of the equalizer taps), the initial equalizer outputs areon averagein phase with each other and scaled proportional totheir power at the sampling phase found in (10). This providesa good initial tap setting near the gravitational pull of the globalsolution of CMA, as seen in the following section. After theinitialization phase, the operation of the equalizer is identical tothe single antenna case in Section III-F.

18If not synchronized within a symbol period, the equalizer might combinesignals from transmitted symbola for the first antenna and froma for thesecond antenna, respectively.

Fig. 9. Comparison between BEQ, selection diversity, and TEQ in delayspread(All equalizers are six symbol long, exponential profile, average SNR= 17 dB, random quasi-static channels, two branch diversity).

V. PERFORMANCESIMULATIONS WITH TWO BRANCH

DIVERSITY

As before in Section III-F, the simulations here are done withPACS uplink parameters. The following Monte-Carlo simula-tion results are obtained from more than 2000 independent ran-domly generated channel pairs (two branch diversity) for eachdata point. The other simulation parameters are the same as thesingle antenna case in Section III-C.

Fig. 9 shows the performance comparison between the pro-posed blind equalizer (BEQ), selection diversity (SD) [3] withno equalization, and an equalizer with six symbol long trainingsequences (TEQ). Two branch diversity is used. The channelsare static over a burst and independent from burst to burst. Theaverage delay spread profile is exponential and properly scaledto various normalized delay spread values. The figure showsalso the performance of the diversity BEQ with conventionalinitialization. The conventional initialization is to set only onetap near the center of or to a nonzero constant (usually1 for 4QAM). For comparison, Fig. 9 also contains a few datapoints from the single antenna results in the previous section(NoEQ; PACS receiver, BEQ No Diversity; the proposed BEQwith a single antenna).

The performance of the BEQ with the conventional initial-ization greatly suffers from symbol timing errors, especiallywhen the delay spread is low. This problem is almost nonex-istent for the proposed BEQ with the new initialization. Theperformance is greatly improved over the conventional initial-ization and it compares well with the TEQ. The proposed BEQincreases delay spread tolerance by the factor of 3 at 3% BERover the selection diversity. Note that the selection diversity canbe used without training sequences, while the TEQ cannot bydefinition.

The performances of the BEQ and the TEQ against timevarying channels are compared in Fig. 10. The GSM HTchannel profile is used for these simulations. The carrierfrequency is assumed to be 2 GHz. For the case in Fig. 10 with


Fig. 10. Comparison between BEQ and TEQ (6 symbol training is used forTEQ, HT profile, two branch diversity).

short training sequences of six symbols, the BEQ actually per-forms better than the TEQ at all average SNR19 and at all speeds.Recall that the TEQ has only a short training sequence, but theBEQalwayshas its blind cost function that provides an adapta-tion reference (although much poorer than training sequences.).Also, the finite tracking speed limits the performance of bothequalizers at high speeds.

After sufficient convergence from training, equalizers usuallyswitch to Decision-Directed (DD) mode [13]. The BEQ alsoswitches to DD mode after sufficient convergence with CMA.At this stage, both the equalizers rely on their own decisions totrack time variation. From this point on, the BEQ and the TEQare essentially the same. Thus, the tracking speed at this stageis similar for both, since an identical update algorithm is used.20

If the time variation of a channel exceeds the tracking speed ofa specific algorithm, errors are likely to occur. Thus, both theBEQ and the TEQ exhibit similar performance at high speeds.With proper modifications, faster converging algorithms such asRLS can be used for both the BEQ and the TEQ. However, theconvergence speed and tracking speed of an algorithm are notthe same in time varying environments as noted in [34], [35].21

Thus, using faster converging algorithms may not be more ben-eficial in terms of time varying channel tracking.

The performance of the BEQ is similar to that of conventionalequalizers with short training at most vehicle speeds, and issometimes better due to its blind adaptivity. The results indicatethat the blind equalizer increases useful delay spread toleranceconsiderably over receivers with no equalizer. With some addedcomplexity, a blind adaptive equalizer can be used without in-curring the overhead of training sequences. Where the addedcomplexity is tolerable, e.g., base stations or possibly in hand-sets in the future, the link quality can be significantly improvedwithout incurring this overhead.

19Average SNR over Rayleigh fading.20LMS is used in this chapter.21For a practical review on this subject, refer to [34]. In [35], Chun compares

various algorithms for convergence speed and time varying channel tracking inseverely fading environments.

VI. CONCLUSION

We have proposed a CMA based blind dual mode equal-izer and investigated its performance against various delayspreads and time-varying wireless channels through exten-sive simulations. The equalizer is devised for short burstcommunications and eliminates any overhead associated withtraining sequences.

The slow convergence speed of CMA is overcome by reusingcaptured data. It has been shown that a short burst of datacontains enough information for convergence. Also, ill-con-vergence caused by the small symbol timing recovery rangeof FSE CMA is minimized with the proposed initialization.The initialization scheme is extended to two branch diversitycombining with CMA. To provide a smooth transition betweenCMA and DD mode, a cluster detection method and adaptivethreshold are proposed and their effectiveness is verifiedthrough simulations.

With the new initialization schemes and the technique forreuse of captured data, the proposed blind equalizer can greatlyimprove delay spread tolerance compared to the previous CMAinitialization and selection diversity. This is achieved with notraining overhead. For high vehicle speeds or, equivalently, fasttime-varying channels, the performance of the proposed blindequalizer is comparable to conventional equalizers with shorttraining sequences. These techniques can be used to removetraining overhead or to augment systems with no training se-quences for equalization. These blind equalizers require addi-tions only to receivers.

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Byoung-Jo Kim (S’93–M’98) received the B.S. degree from Seoul NationalUniversity, Seoul, Korea, in 1993, and the M.S. and Ph.D. degrees from Stan-ford University, Stanford, CA, in 1995 and 1997, respectively, all in electricalengineering.

He had consulted for several wireless start-up companies in Silicon Valleyon wireless systems design and deployment issues before he joined BroadbandWireless Systems Research Department in AT&T Laboratories-Research, RedBank, NJ, in 1998 as a Senior Member of Technical Staff. He has worked ondigital control theories and systems, and signal processing techniques for wire-less communications, including radio propagation, equalization, diversity tech-niques, sequence estimation, and fixed wireless access. His current research in-terests are mobile communications, mobile-aware applications, mobile protocolinteractions, mobile networks/computing, and mobile network management, es-pecially wireless/wireline QoS, thin-client mobile computing, and “open” ap-proaches for cross-layer optimization of mobile networks from applications tophysical layers.

Donald C. Cox(S’58–M’61–SM’72–F’79) received the B.S. and M. S. degreesin electrical engineering from the University of Nebraska, Lincoln, in 1959 and1960, respectively, and the Ph.D. degree in electrical engineering from StanfordUniversity, Stanford, CA, in 1968. He received the Honorary Ph.D. degree ofscience from the University of Nebraska in 1983.

From 1960 to 1963, he did microwave communications system design atWright–Patterson AFB, OH. From 1963 to 1968, he was with Stanford Uni-versity doing tunnel diode amplifier design and research on microwave propa-gation in the troposphere. From 1968 to 1973, his research at Bell Laboratories,Holmdel, NJ, in mobile radio propagation and on high-capacity mobile radiosystems provided important input to early cellular mobile radio system devel-opment and is continuing to contribute to the evolution of digital cellular radio,wireless personal communications systems, and cordless telephones. From 1973to 1983, he was Supervisor of a group at Bell Laboratories that did innovativepropagation and system research for millimeter-wave satellite communications.In 1978, he pioneered radio system and propagation research for low-powerwireless personal communications systems. At Bell Laboratories in 1983, heorganized and became Head of the Radio and Satellite Systems Research De-partment that became a Division in Bell Communications Research (Bellcore)with the breakup of the Bell System in 1984. He was Division Manager of thatRadio Research Division until it again became a department in 1996. He con-tinued as Executive Director of the Radio Research Department where he led,and contributed to research on all aspects of low-power wireless personal com-munications, entitled Universal Digital Portable Communications (UDPC). Hewas instrumental in evolving the extensive research results into specificationsthat became the U.S. Standard for Wireless or Personal Access Communica-tions System (WACS or PACS). In September 1993, he became a Professor ofElectrical Engineering and Director of The Center for Telecommunications atStanford University, where he continues to pursue research on wireless mobileand personal portable communications. He was appointed Harald Trap Friis Pro-fessor of Engineering in 1994. He is the author and coauthor of many papers,conference presentations, including many invited and several keynote addressesand books. He holds 13 patents.

Dr. Cox is a member of Sigma Xi, Sigma Tau, Eta Kappa Nu, Phi Mu Epsilon,a member of the National Academy of Engineering, a member of CommissionsB, C, and F of USNC/URSI, and was a member of the URSI IntercommissionGroup on Time Domain Waveform Measurements (1982–1984). He is a Regis-tered Professional Engineer in the States of Ohio and Nebraska. He is a Fellow ofAAAS and the Radio Club of America. He was an Associate Editor of the IEEETRANSACTIONS ONANTENNAS AND PROPAGATION (1983–1986) and a memberof the Administrative Committee of the IEEE Antennas and Propagation Society(1986–1988). He was the corecipient of the 1983 International Marconi Prizein Electromagnetic Wave Propagaion (Italy) and was awarded the 1993 IEEEVehicular Technology Society Paper of the Year Award, the 1985 IEEE MorrisE. Leeds Award, the 1990 IEEE Communications Magazine Prize Paper Award,the 1991 Bellcore Fellow Award, the 1992 IEEE Communications Society L. G.Abraham Prize Paper Award, and the 1993 IEEE Alexander Graham Bell Medal“for pioneering and leadership in personal portable communications.”

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Blind equalization for short burst wireless communications