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A. Milewski

Periodic Sequences with Optimal Properties for ChannelEstimation and Fast Start-up Equalization

The problems of fast channel estimation and fast start-up equalization in synchronous digital communication systems areconsidered from the viewpoint of the optimization of the training sequence to be trans mitted. Variou s types of periodicsequences having uniform discrete power spectra are studied. Some of the m are new and may be generated with data setscommonly used in phase modulation systems. A s a consequence of their power spectra being fla t, these sequences ensurema xi mu m protection against noise when initial equalizer settings are computed via channel estimate s and noniterativetechniques.IntroductionSequences (or codes) with good autocorrelation propertieshave been studied in comm unication s iterature for overtwenty years because of their applications to radar and thesynchronization of communicationssystems [1 , 2 ] . For acurrentan dmoregeneral eference on this ubject, seeAlltop [31. More recently, trainingequences with similarpropertieshave been studied nd used for ast tart-upequalization [4-71.

Telepho ne lines present large amo untsof linear amplitudeand phase distortion. Fast turnaround is an impo rtant ele-men t of mo dem perform ance since messages (especially onmultidrop lines) areoftenshortand esynchronization isrequired frequently. The use of training sequences permitsrapid equalization of the transmission channel withou t anyprior knowledge of signal distortions, provided th at the y arenot too extre me (e.g., they present no spectral nulls). Per-formingqualizationveryim eesynchronization isrequired has the advantagef simplifying t he overall controland maintena nce procedures. For example, n he case ofmultidrop lines, it avoids the necessity of stocking the equal-izer coefficients corresponding toeach econdary modemconnected to the ine, and hence it avoids the procedures orsettingan dmain tainin g hese coefficients in case of linechanges or variations.

All trainin g sequences sha re two properties:1. Theirautocor relation unction is small, except at he

origin.

2. Some limitation s imposed on the numericalvalues takenby the sequences.The first property is required to make these sequences asnearly as possible impulse-equivalent. Th e second isdue tothe requirement or peak amplitude limitationn all practicalimplementations.

The mai n distinction amon g th e m any ypes of sequencesis whether hey are periodic or not, since his affects thedefinition of the autocorrelation function and the spectralproperties. In this paperwe confine ourselves o the subje ctfperiodic sequences, becauseof their particular suitability forfast start-up equalization. More precisely, we discuss com-plex sequences having constant amplitude and zero autocor-relation (CAZAC sequences) and compare their perform-ance with the better known maxima l-length sequences.The channel modelLe t us consider a complex channel with additive white oisean d finite impulse response which is sample d at T intervals.The channel input a t t ime nT is a complex number u,, an dthe channel output , is

N2Y n =x kn-k +wn > (1 )k = N ,where r is a complex vector of length L =N , - N l + 1,defining the chann el impulse response, and w, is a complexrandom variable with the expectedalues

8 Copyright 1983 by Interna tional Business M achines Corp oration. Cop ying in printed form for private use is permitted without payment ofroyalty provided that (1 ) each reproduction is done without alteration and 2) the Journal reference and IBM copyright notice are included onthe first pag e. The title and ab stract, but no other portions, of this paper m ay be copiedr distributed royalty free without further permissionycomputer-based and other information-service system s. Permission to republish any other portion of this paper must be obtained from the426 Editor.

A . M IL E WSKI IBM J. RES. DEVELOP. VOL. 27 NO. 5 SEPTEMBER 1983

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E(w,,z,,,) =a2 if n =m,=0 if n f m ,nd

E(w,,) =0 for all n.This channelmodel can beconsidered as an idealizati on f aquadrature-amplitude-modulated (QAM) sampled channel.

The spe ctrum f this idealized channel is given byN2x kezrJkT,

a Four ier series of L coefficients, whe ref s the frequency andT is the sampling interval. Since the spectrumf a sampledchannel is a periodic functio n o ff, it can be approx imated asclosely as is needed by taking L lar ge enough.

k = N ,

Training sequences for fast equalizationFast start-up digital equalization requires rapid, accurate,and dependable estimationf channel characteristics (repre-sented by r"), and also rapid and hardware-implementablecalculation of the equalize r coefficients from the data fur-nished by the estimation.

There re two main dvantageso using trainingsequence for the purpose of estimating the channel charac-teristics. T he transm itted symbol s known to the receiver(thusdetection rrors re l iminated), nd he rainingsequence can bechosen to have certain desirable properties.Among these are that the estimation method is hardware-implementable and tha t therocess is insensitive t o noise.

An important property f a training sequence s the lengthof its period. In general, the onger period leads to the betterchannel estimate. However, it is impo rtant o note hat ifthere is no noise an d if the cha nne l response s of finitelength, then it ca n be completely estim ated using a train ingsequence period equal to this ength.

A well-known examp le of such a training seq uence is thesending of a single unit pulse every L-baud interval. The r,,are hen estim ated directly fr om he received signal . Thedisadvantage of this training sequences its low power and itsresultant sensitivity to noise.0 Calculating the channel response in the absence of noiseIn this section we assume thatwn=0 nd tha t the t ransmit -ted symbols u, come from a training sequence of period L ,such tha t L is equal to the length of the channel response;t h a t i s , r , , = O i f n < N , o r n > N , , a n d L = N , - N , + I .

The n the ith eceived signal can be writtenN2x i =x ui - , , r n . (2)

n=N,

IB M J. RES. DEVELOP. 1OL. 27 NO. 5 SELTEMBER 1983

If L successive signals are observed, we shall obtain Llinear equations in L unknowns and so the r , , a ren principleknown if and only if the matrix M =( u ~ - ~ )s nonsingular.Note that t he tra ining seque nce f the preceding subsectioncorresponds to M =I, (the unit matrix of order L). Notealso that if the training sequence is started at the instanti =0, then the first x , satisfying (2 ) is xN 2 . ince we requireL observations, complete knowledge of thechannelca nbe obtained ( N 2+L - 1)-baud intervals after the start ofthe training sequence, provided that it is continued duringN,-b aud intervals.

W e now consider th e problem of finding the r,, from the etof equations (2). Th e periodicity of th e training seque ncepermits a com putationally efficient solution, since the matrixM is a circulan t matrix and matri x multiplication can bereplaced by periodic convolution. Th us (2 ) can be rewrittenx = u * r , (3)where x is the vector consisting of L samples of the receivedsignal, etc.

Denoting the discr ete Four ier transfo rm (D FT)f x by X ,etc., we have X =U x R and so r = I D F T ( X I U ) =I D F T (Y X ) orr = v * x , (4)where v is the ID FT (inverse discrete Fourier transform) ofV =1/U. Since the components f U a re t heeigenvalues ofM, Vexists if an d only if M is nonsingular.

Estimating the channel response in the presence of noiseIf in (4)we replace x with y, where y =x +w a nd w is avector of noise samples, theni = v * y = v * x + v * w , ( 5 )where i is the estimate f r given by the linear estim ator .

Because of the assum ptions about the noise, we deducefrom (5) tha t E(?) =r, i.e., that the estimator v is unbiasedand tha t the mean square e rroror error variance) is

We now derive the condition for minimizing (6 ) over alltraining sequences of average unit power, i.e. all u, such tha tx ] u ; 1 2=L .

iW e have, by the DF T quivalent of Parseval 's formula,

and since y.=I / U; ,we have 4

A. M lL E W

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A. MILEWSKI

hence the rrorvariance esulting rom heuse of thesequence u is

Also,~ l V , 1 2=L x I u i 1 2=L 2 ,and it s easy to how that (7) is minimized when all the Vi 2are equa l ,1q.l=L i =0, 1, ..-,L - 1,that is, when the sequence u has zero autocorrelation. Theerror variance (7) then takes the valuer2.

Assuming that we haveno nformation oncerning R(other han hat t is nonzero), i t is clear hat he safesttraining sequence is such that the U, have constant ampli-tude,.or in other words tha t u has zero autocorrelation.

W e now evaluate the error due to periodic random data,i.e., a random vector of length L and average power = 1repeatedly transmitted. This approximates the transmissionof random data sim ilarly to theay in which c, approximatesthe true optimal coefficients cOpt,btained with a n infiniterandom sequence.

We sha l l use theotation

i= Oand denot e the data vector by d. W e first observe that theerror variance due to, is

It is interes ting ocompare his esultwith heerrorvariance obtained when a maxim al-length training s equencetaking thevalues +1 and - is employed. Su ch a sequen ce ischaracterized in the frequency domain by a D FT w h ic h ha su,=1,I u l ( 2 = ( u 2 ( 2 = . . . = ( u L - , ( 2 = L + 1 .Its error variance therefores

W e see tha t , for la rge L, a m aximal-length sequence is a t a3-dB disadvantage compared to a sequence with zero auto-correlation.

Writing c =c, +AC we have the error variance due to= I C ( 2 ~ ( r 2 + - E ( ( a c * r * d ) 2 ) ,= c 1 x (r2 + Ac * r 12, since d is assumed random, and= l c , + ~ c ( ~ ~ ( r ~ + ~ A C * r ~ ~ , O r~ ~ ~ , ~ ~ x ( r ~ + ) a c ) ~ x ~ ~ + ~ ~ c * r )since the correlation between c, and A C is small. Assumingthat the ignal-to-noise ratio is large, thisbecomes= ~ ~ , , ) ~ x a ~ + ( a c * r ( ~ .

I

L

Effect of th e training sequenceonfast equalizationW e now consider the effect of th e rainin gsequence onc a l c u l a t i nghequa l i z e roe f f i c i e n t s .Mue l l e rndSpaulding [5] have shown th at in the no-noise case, perfectequalization at a discrete number of frequency points can beachieved by the coefficients

U 1X RC, =I D F T -=I D F T - (8)

The ame method can be appliedo alculatinghecoefficients in the presence of noise, provided that a suitabletraining sequence s used. For if we calc ulate the oefficientsof the equalizerby the formulac =I D F T -UY (9)we must en sure that the denominatorY = U x R + W (10)does not contain any zero terms.

Now from (8 ) an d (9 ) it follows tha tAC=I D F T - --(k y)an d so

fo r W

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under certain assumptions. The most important members ofthis family are the polyphase sequences of lengths that areapowerof 2 because of their suitability for fast Fouriertranform processing.

This paper discusses the use of periodic training sequencesfor channel estimation and fast start -up equalization, both ofwhich are important in modem applications. A model of aquadrature-amplitude-modulated (QAM) channel is de-fined. It is then shown that in the presence of noise, errorvariance of the channel estimate depends on the trainingsequence used. It is also shown that, under certain assump-tions, the error variance is minimized by training sequencespossessing the zero-autocorrelation (ZAC) property. Analo-gous results are obtained for the error variance of equalizercoefficients calculated from a single period of the trainingsequence. The results obtained with ZAC sequences providea 3-dB advantage when compared with those obtained withmaximal-length sequences.

Other desirable properties of training sequences are fur-ther discussed. It is shown that a family of ZAC sequencescan be obtained from a maximal-length sequence by theaddition of a complex constant. One member of this family,which also has the CA property, is called the CAZACsequence. Previously ublished results about polyphasesequences, whichare also CAZAC, are summarized, and theconstruction of a family of mk+I-phase equences of lengthsmZk+ls given in he Appendix. The section ends with a set ofproperties permitting the generation of related CAZACsequences. The last of these, which is less well known, may berestated as follows: A sequence is CAZAC if and only if itsDFT is CAZAC.AppendixThe construction of rnk+l-phaseCAZAC sequences of lengthL =m2kfl k1 ) is shown below.

The existence of m-phase sequences of length m for m oddand 2m-phase sequences for m even is known [9].

Let u,, uI, ..., be such a sequence. We then form asequence of length L = n Z k + l by enumerating, row by row,the matrix zij =u ~ ( ~ ~ ~ ),wherei =0, 1, ...,M - M =mk+l,j =0, 1, ...,N - 1 N =mk,and Wis a primitive Mth root of unity.

This sequence is obviouslyCA and mk+-phase;we have toshow that it is ZAC.

We observe that the first column of the above matrix is430 simply the sequence u repeated N times. It is easy to show

that its DFT isu,, o,o, ...,0, VI,o,o, ..*,o, .*e, um-L,o,, ***,o,

M - 1 M - 1 M - 1where U is the DFT f u and the M- 1 indicates the numberof zeros in the row. Also, it is easily shown that the DFT fthe nth column is the above sequence rotated n places to theright. Hence the product of the DFTs of the columns is zero,and therefore the columns are orthogonal in pairs. It followsthat the autocorrelation coefficients A , of the sequenceobtained by enumerating this matrix row by row are zero forall I #0 mod N.

It remains to be shown hat A ,=0 also when l =0 modN ,except when I =0 mod MN.

Let I =cNwhere c # 0 mod M . Theni j

iJ

We now have two cases to consider:1. c # 0 mod m; then the first of the above two factors is2. c =0 mod m; then the second factor is zero, since Wis an

zero, since u is a ZAC sequence.Nt h root of unity, and W # 1 since c #0 mod M .

This completes the proof.References1 . R.C. Heimiller, Phase Shift Pulse Co des with Good PeriodicCorrelation Properties, IRE Trans. Info. Theory IT-6, 254-25 7 (October 1961).2. R. L. Frank and S. A. Zadoff, Phase Shift Pulse Codes withGood Periodic Correlation Properties, IRE Trans. Info. The-or y IT-S,381-382 (October 1962).3. W.0. lltop, Complex Sequences with Low Periodic Correla-tions, IEEE Trans. Info. Theory IT-26, 350-354 (May1980).4. R.W. Chang and E. Y.Ho, On Fast Start-u p Data Communi-cation Systems Using Pseudo-RandomTrainingSequences,BellS yst . Tech. J . 51,2013-2027 (November 1972).5. K.H. M ueller and D. A . Spaulding, Cyclic Equalization-ANew Rapidly Converging Equalization Technique for Synchro-nous Data Communication, Bell Syst. Tech. J . 54, 369-406(February 1975).6 . S . U . H. Qureshi, Fast Start-up Equalization with Periodic

Training Sequences, IEEE Trans. fnfo. Theory IT-23, 553-56 3 (September 1977).7 . D. . Godard,A 9600 Bit/s Modem for Multipoint Communi-cations Systems, National Telecomm unications ConferenceRecord, New Orleans, Dec. 1981, pp. B3.3.1-B3.3.5.8. R. Turyn, S equences with Sm all Correlation, Error Correct-ing Codes, H. B. Mann, Ed., John Wiley & Sons, Inc., NewYork, 1968, pp. 195-228.9. D. C . Chu, Polyphase Codes with Good Periodic CorrelationProperties, IEEE Trans. Info. Theory IT-18, 531-532 (July1972).

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10. R. L. Frank, Comments on Polyphase Codes with GoodPeriodic Correlation Properties, ZEEE Trans. Info.TheoryIT-19,244 (March 1973).11. A. Milewski,ProcM6 et Dispositifpour DCterminer lesValeursJnitiales des Coefficients dun Egaliseur Transv ersal Com -plexe, French Patent No . 7,540,417, Dec. 30, 1975. A.Milewski, Method and Apparatus for Determining the InitialValues of the Coefficientsof a Complex Transversal Equalizer,US . Pa ten t No. 4,089,061, May 9, 1978.Received O c t . 2I , 1982; revised Feb. 18, 1983

Andrzej Milewski IBM France. Centre dEtudes etRecherches, 06610 La Gaude , France. Mr. Milewski is an advisory

engineer working on signal processing algorithms with applica tion tomodems. He joined IBM in 1961 at Hursley, U.K., and has been atLa Gaude since 1962, except for a years assignment in the Sa n JoseGPD Laboratory in 1981/82. Mr . M ilewski eceived a B.S. nmathematics from the U niversity of the Witwatersrand, Johannes-burg, South Africa, in 1954 and an M.A. in mathematics from theUniversity of Cambridge, U.K., in 1956. Before he oined IBM, M r.Milewskiworked for ICL in the U.K. in operations research(1958-61). Since joining IBM, he has workedon programminglanguages and systems, line switching (network design and per -formance studies), signal processing with modem applications, mul-tiplexor performance, signal processing with voice applications, andrecently, signal processing with applications to magnetic recording.He has published two papers in the IEEE Transactionson Commu-nications and he has received two IBM invention achievementawards for his work.

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IBM J. RES. DEVELOP. VOL. 27 NO. SEFTEMBER 1983 A. MILEW