partial-response signaling kabal and pasupathy

8/13/2019 Partial-Response Signaling Kabal and Pasupathy

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IEEE TRANSACTIONS ON COMMUNICATIONS,VOL. COY-23, O. 9, SEPTEMBER 1975 921

Partial-Response Signaling

Absfrucf-This paper presents a unified study of partial-responsesignaling (PRS) systemsand extends previous work on he compari-son of PRSschemes. A PRSsystem model is introducedwhichenables the nvestigation of PRS schemes from the viewpoint ofspectralproperties such as bandwidth, nulls, and ontinuity ofderivatives. Several desirable properties of PRS systems and theirrelation to system functions are indicated and a number of usefulschemes, ome of them notpreviously analyzed, are presented.These systems are then compared using as figures of merit speedtolerance,minimum ye width,and signal-to-noise atio SNR)degradation over idealbinary transmission.A new definition of speedtolerance, which takes into account multilevel outputs and the effectof sampling time, is introduced and used in the alculation of speed-tolerance figures. t is shown that eyewidth, a performance measurethat has not been usedreviously in comparing PRS systems, an becalculated analytically in many cases. Exact values as well as boundson the SNR degradation for the systems under considerat ion arepresented. The effect of precoding on system performance is alsoanalyzed.

P I. INTRODUCTIONULSE-AMPLITUDE modulationPARI) is oftenused to convey digital nformation. The usual con-straint on permissible PARI signal waveforms is that theyshould not cause intersymbol nterference. Signal designbased on this criterion can sometimes lead to a completeintolerance of timing errors or to incompatibilities withsome channel characteristics. Some of these disadvantagescanbe removed withpartial-response signaling (P RS )(also known as correlative level coding) wherein the con-stra int on waveforms is relaxed so as to allow a controlledamount of intersymbolnterference. PRS designs arebasedon the premise that since the intersymbol nter-ference is known, its effect can be removed.

One of the merits of P RS is tha t the controlled inter-symbol nterferencecanbe used to shape the ystemspectrum, for instance to place nulls in he frequencyresponse. Also, this spectral shaping can make the systemless sensitive to timing errors. This allows practical PRSsystems to transmit t th eNyquist rate,a feat not ossiblewith ordinary PAM. In addition, a PR S spectrum mightbe chosen to complement a nonideal characteristic inorder t o reduce the residual undesired intersymbol inter-ference. The partial-response coding formatas thefurther advantage thatviolations in the code can be used

Paper approved by the Associate Edit.orfor Data CommunicationSystems of the IEEE Communications Society for publicationwithout oral presentation. Manuscript received December 30, 1974;revised May 16,1975. This workwas upported n partby heNational Research Council of Canada.The authors are with the Department of Electrical Engineering,Universit,yof Toronto, Toronto, Ont., Canada.

t i .

to monitor error performance or even to correct errors.On the negative side, PRS systems using symbol bysymbol detection possess reduced noise marginsdue tothe fact that thesuperposition of signal waveforms causesthe number of output levels to be larger than the numberof input levels.

Lender [l] first introduced duobinary PRS as a data-transmissionmethod. IG-etzmer [a] categorized thecharacteristics of several PRS schemes and comparedthem on the basis of speed tolerance and signal-to-noiseratioSNR) degradation. PRS techniques ave beenapplied to various modulation schemes such as FM [3]and SSB [4] as well as o baseband ystems.Severalauthors [5]-[7] have nvestigatederrordetection andcontrol for PRS. Lender introduced precoding for duo-binary [l]. Gerrish and Howson [SI discussed precodingfor multilevel inputs. Tomlinson [9] and Harashima andMiyakawa [lo] have ntroduced a generalized form ofPRS precoding. Kobayashi [ll], in addition to comparingPRS with other coding techniques, has provided a goodbibliography. Maximum-likelihood decoding for PRS hasalso been an active area of research [la], [13]. The specialproblems associated with adaptive equalization for PRShave also been examined [14], [15].

This paper, partly tutorial in nature, presents a unifiedstudy of PRS and extendsprevious work on the comparisonof PRS schemes. Firs t a cohesive framework for studyingIRS systems is introduced. To this end, a general schemewhich separates PR S waveform generation into two pa rtsis described in Section 11.This allows the investigation ofPR S spectra from the viewpoint of useful properties suchas nulls and continuity of derivatives.

In Section I11 a number of candidate PR S systems areselected by choosing ones with desirable spectral proper-ties and a small number of output levels. ThePRSschemes under consideration include several whose per-formances have not been assessed previously. The processof decoding and precoding and the effect of error propaga-tion are then iscussed in Section IV. The est of the paperis devoted to a comparison of P RS systems using threecriteria. The sensitivity to timing errors is measured bytwoparameters, speed tolerance and he minimum eyewidth, while the effect of an increase in the number ofoutput levels is measured by the SNR degradation overideal inaryransmission. Speed-tolerance figures arecalculated using a new definition of speed tolerance whichtakes into account multilevel systems and samplinghase.The eye-width criterion has not been used previously forcomparing PRS systems. A procedure is given here which


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922 IEEE RANSACTIONS ON COMMUNICATIONS, SEPTEMBER 1975greatly simplifies the calculation of eye widths. Com-parativeSNRdegradation figures are also presented.These include exact values for systems without precoding,figures not previously reported.

11. AGENERALIZEDPRSSYSTEMThe basebandmodel of a synchronous da ta communica-

tions system is shown in Fig. 1. The model is applicable tovariousmodulationschemes.The overall transfer func-tion H ( w ) encompasses the ransmitter filter (or signalgenerator characteristic), the equivalentaseband channel,and the receiving filter (which may include an equalizer) .

First the discussion will concentrate on an ideal, noise-less system(ideal n that here isno distortiondue tochannel imperfections or sampler offsets). Such a systemcan be characterized by he samples of the desired im-pulse esponse h ( t ) . Let N be he smallest number ofcontiguoussamples thatspan all he nonzero samples.Then if { fn), n = 0,1,2,...,N - 1 are hese N samplevalues, the PR S system polynomial is given by

N-1F ( D ) = C f n D n 1)n=O

where D s the delay operator. For a given input symbolsequence { x n the output sequence { y n is given by

Y ( D ) = X ( D ) F ( D ) ( 2 )where

W 03X ( D ) = xnDn and Y ( D ) = ynDn.The { x n will be assumed o be independentm-ary symbolstaking on the equallyikely values { - m- 1) - m- ) ,

Though most f the desirable properties of a PR S systemcan be stated in termsof the impulse responseh( t ) manyare best illuminated by frequency domain considerations.Fig. 2 showsamethod of generating hePRSsystemfunction H ( w ) which gives an insight into requencydomain properties.

The system consists of a tapped delay line with coeffi-cients { f n } in cascade with a ilter with frequency responseG w ) . The ransversal filter has he periodic frequencyresponse (period 27r/T)

n=O n=O

- . , m - 3 ) , ( m- 1 ) 1.

( W ) = F ( D ) D--exp -jwT)N -1

= fn exp (-jwnT) 3)where Tis the symbol spacing. It can be easily shown tha th ( t ) has he samplevalues { fn } if and only if G(w )satisfies Nyquist 's first criterion,' th at is

n=O

of the impulse response be zero, i.e., g ( k T ) = 0, k 0, g 0) = 1 [16].Nyquist's first criterion requires th at all but one of the samples

NOISEI SYMBOL4 ETECTOR-Fig. 1. Model for a synchronous data communications system.

ADELAY-LINE

G(w) , = + { Y }SAMPLER

OUTPUTYH W)

Fig. 2. General partial-response systemmodel.

m G ( w - * ) = T.k = -m T

The PRS systemhasbeenseparated nto woparts:% ( w ) forces the desired samplevalues but is periodic,while G w ) preserves the sample values but may be usedto bandlimit the resulting system function. For a givensystemolynomial F ( D ) , different choices of G w )satisfying Nyquist'sirst criterion resultn ifferentsystem functionsH ( w ) ,all of which have identical sampledresponses. The advantage of this separation will be moreapparent when we study the various desirable propertiesof H ( w ) and see how some of these properties (such asspectral nulls) are influenced by the choice of the samplevalues ( fn ] and by the choice of G ( w ) . The separation isof course an artifice-the actual implementat ion may beorganized quite differently. While Fig. 2naturally suggestsa digital partial-response encoder (i?e., ( w ) ) , it may beadvantageous in some cases to implement the total systemfunction H ( w ) in an analog fashion.111. CHOOSING THE PRS SYSTEM POLYNOMIALA . System. Bandwidth

In order to maximize the data rate in the available bandwidth,manyPRSsystemsare designed to occupy theminimum bandwidth which supports transmission withoutundesired ntersymbol interference, i.e., H ( w ) = 0 forw I > T / T . Such systems also avoid the aliasing whichoccurs when theoutput of a nonminimum bandwidthsystem is sampled at thesymbol rate. This aliasing can inextreme cases causedips in he middle of the Nyquistequivalent channel which are difficult t o equalize.

For minimum bandwidth systems (see 4 ) ) ,I T , I w I 5 ?r/T0, elsewhere.G ( w ) = 5)


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KABAL AND PASUPATHY: PARTIAL-RESPONSESIGNALING

Th e corresponding system mpulse response is given by

sin - t - nT)PN -1 T

h( t ) = C f n (6)n=O P t - n T)T

Other choices for G ( w ) (occupying a larger bandwidth)are possible and , as we shall see, allow the use of systempolynomials (suchas 1 - D) which areunsuitable orminimum bandwidth systems.B. Spectral Null at w = P / T

It is well known that if H ( w ) and ts first K - 1derivatives re ontinuous and heKth derivativesdiscontinuous, h ( t ) I decays asymptotically as 1/1 t IR 1[17]. Continuity of the function and its derivatives helpsto reduce the portion of the tot alsignal energy in the ailsof h,( t ) and hence the undesirable intersymbol interferencein nonideal systems.Forminimumbandwidthsystems,5 ( w ) must have a zero at P I T (where G ( w ) has a discon-tinuity) for H ( w ) to be a continuous function. In generalthe conditions for the continuity of U (w )and its deriva-tives are summed up as follows.Proposition 1: The first K - 1derivatives of a minimumbandwidth H ( w ) are continuous if and only if F ( D ) has( 1 + D ) K as a factor.If F ( D ) has more than one zero a t D = -1, the roll-offnear w = P / T becomes less sharp and thus the design ofpractica.1 filters for the system becomes easier. On theotherhand, if F ( D ) has a largemultiplicity of 1 + Dfactors, the error performance t ends to be degraded dueto he increase in the number of ou tpu t levels. Thesensitivity to timing offsets also suffers because morecontrolled intersymbol interference terms are introduced.These affect the timing sensitivity more than the dist anttails which decay rapidly as more actors 1 + D areintroduced.

Forsystems that arenotrestricted to the minimumbandwidth,anull at P / T is sti ll useful-a pilot toneinserted a t this point can be used for clock recovery [lS].C . Spectral Null at w = 0

Reduced low-frequency components in he pectrumare desirable in ystems such as transformer coupledcircuits, dc powered cables, SSB modems, and arriersystems with carrier pilot tones. For a null at w = 0, itcan be seen from ( 3 ) that 1 - D mustbe a factor ofF ( D ) Multiple zeros at D = 1 cause a more gradual roll-off of the frequency components just above de which mayalso be desirable.

With combinations of just the two factors 1 + D and1 - D,most of th e common partial-response systemscan be developed. Table I shows a number of l'RS system

923

The class designations in the first column of Table I aredue to Kretzmer [a] Expressions for H ( w ) and h( )are given in Table 11. In th egeneral formula for h ( t ) ( 6 ) ,the sample fO occurs a t time zero; however a change oftime origin often simplifies the resulting expression. InTable 11, the center of the nonzero sampleshas beenchosen as the t ime origin, that is in (6), t = ( N - 1)T / 2is the new time origin. The expressions for the frequencyresponse in Table I1also assume this time origin.

The first en try in these tables, duobinary, also satisfiesNyquist's second criterion, i.e., th at th e pulsewidth shouldbe undistorted. The second system 1 - D is not practicalin the minimum bandwidth because of the resulting dis-continuity in the system function H ( w ) . Modified duo-binary, the next entry, has both a dc null and a null atw = r / T . The fourth entry, 1 + 2 0 + D2, as the sameresponse as a full raised cosine characteristic but sampleda t twice the usual rate . Also included in the tables aretwo pulse responses which have not been analyzed beforein the literature, namely 1 + D - D2 D 3 and 1 - D -D2 D3. These two systems have both dc nulls and nullsa t w = P / T .The last two entries in Table I suggest thatpolynomials andhe corresponding I H ( w ) I and h ( t ) _ _


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924 IEEE TRANSACTIONS ON COMMUNICATIONS, SEPTEMBER 1975TABLE I1

SYSTEMSCHARACTERISTICS OF MINIMUM ANDWIDTHARTI.4L-RESPONSESYSTEM POLYNOMIAL FREQUENCY RESPONSEF(D1 IMPULSE RESPONSE h ( t ) NO.OFH w ) for I w S TIT OUTPUl

I t D 2T co s T 4T cos(nt/ T)T2 - 4 t 2~ 2m-1

1 - D j 2T s i n T 8T t cos(nt /T)4 t Z - T2 2n-1

1 - D2 j2T s i n UT 2T2 sin(nt /T) .n t? . 2 zm- 1

1 + 2D + D2 4T cos2 ZT3 sin( nt /T )n t T? . 2 4m-3

1 + D - D2 - D 3 j4T co s s i n wT _ _ 4T3tos(nt1T) 4m-3(4t2-gT2)4tZ-T2)

- D - D2 D 3 -4T sini n w T 16T2 cos(nt /T) (4t2-3TZ)TI (4t2-9T2)(4tZ-T2) 4- 3

1 - 2D2 + D4 - 4Tin2 UT __T3 sin (n tIT 1 4m- 3ITt t 2 . 72 + D - D 2 T + T cos wT + j 3 Tin wT $ i n ( n t / T ) ( E ) 4m- 3

t2-T2

2 - D 2 - D4 -T + T cos 2wT + j 3 Tin 2wT < i n ( n t / T ) ( Z T - S f ) 4m- 3t2-4T2any appropriate polynomial in D may be used to modifythe basic polynomials 1 D.

If the system bandwidth is allowed to increase beyonds / T , he Nyquist filter ( w ) need not havediscontinuities.In this case, it is permissible for S(W) o be nonzero a tw = s/T . Fig. 3 shows an example of anonminimumbandwidthPRSspectrum fora pulse esponse 1 - Dand a raised cosine Nyquist filter. For this example it istheNyquist filter G (w ) which controls the spectralproperties at t he high end, while the system polynomialF ( D ) controls the low-end response.D. The Number of Output Levels

A PRS systemwith M nonzero pulse amples willhave mM output levels for m-ary input unless there arespecial relationships etween the sample alues. Thenumber of ou tput levels L lies in the range

M ( m - l ) + l < L < m M . ( 7 )with the minimum value being obtained when th e pulsesamples have the same magnitude. The number of outputlevels for a practical PRS system is limited both by thecomplexity of implementation and the inevitable distor-tionspresentn eal ystems. In additionhere is atendency for the error performance a t a given SNR todegrade with a large numberf output levels. It can easilybe shown that if the system polynomial has 1 D asa fac tor, some of the output levels coalesce. Thus thesefactors are also desirable from the viewpoint of reducingthe number of outpu t levels.If we restrict t he number of output levels to be lessthan 5 for binary inputs (multilevel inputs will result inconsiderably more output levels), the task of searchingforcandidate ystemsapproachesmanageability. If inaddition we require that 1 + D and/or 1 - D be a factorof F ( D ) and that there e no nullsor severe ripples in the

w

Fig. 3. Partial-response ystem using a raised cosine filter. (a)Raised cosine filter (50 percent excess bandwidth). (b) Partial-response spectrum for 1 - D pulse response.middle of t he passband, an investigation of t he possiblesystems shows that all the suitable ones havealreadybeen listed inTables I and 11. For binary nput hesesystems have either 3 or 5 output levels.

IV. DECODING AND PROBABILITY OF ERRORA . DecodingThe output of an ideal noiseless system is given by

N 1Y n = f G n + C fixn-i. (8)

i=l

The receiver can recover the data x, by subtracting outthe effect of previous input symbols. In practice thereceiver makes estimates of the data and then uses theseto cancel the tails of the pulse response. Thedetectorstructure (Fig.4) s the same as that f a decision feedbackequalizer [19]; indeed the tasks of decoding and equaliza-tion for deviations from the desired pulse response couldbe combined.

When additive noise is present, the input to the slicer(Fig. 4) s given by

where {, is a noise sample. r , is hen quantized to thenearest allowable data level to give the next data estimate2,. If we define a decision erroras e, = x, - 2,, thenfrom (S) and (9),

(10)We see from (10) that past decision errors can adverselyaffectsubsequent decisions. A techniqueforpreventingthiserrorpropagationphenomenon,namely precoding,will be discussed later.B. Probability of Error (No Precoding)

The robabili ty of (symbol) error for the systemshown in Fig. 4s given by2log, m times the bit error rate.For Gray coded input, the symbol error rat e is approximately


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KABAL AND PASUPATHY: PARTIAL-RE SPONSE SIGNALING 925

6 ) andDATAOUTSLICER DELAY-LINEU

U

uFig. 4. Partial-responsedecoder (no precoding). ( 1 5 )

N-1P[I l n+ fie,-; I > o ] ( 1 1 ) For a larger number f states, an analyti colution becomesi=l unwieldy andumericalpproach becomes morep-

where for simplicity fo is assumed to be positive. Thisexpression is difficult to evaluate n ts full generality.However, with uitable ssumptions, some results anbe obtained.A lower bound on P, can be found by ignoring errorpropagation, .e., etting e, = 0. Then if the noise isGaussian, the lower bound is

For whiteGaussian noise Duttweiler,Mazo,ndMesserschnlitt [ 2 0 ] have given a simple upper bound on P,for binary inputs. Thisbound, extended to consider m-aryinput symbols, is

mm - 1 P,L(mN-l - 1 ) + 1

Equation ( 1 4 ) shows th at error propagation increases theerror probability by a t most a factor w P 1 . For small m.and N this effect is modest.If the noise is white, the exact probability of error canbe obtained for small m and N by modeling the system asa Markov chain [ a l l with states where 1 is the numberof distinct values the error e, can assume 1 = 2 m - 1 ) .Analytically for m = 2 and N = 3,

pealing.Increases in the probabilityof error due to the ropaga-

tion of errors (i.e., P,/PeL) for the various PRS systemswere calculated and appear in Table111.The values haveessentially reached their asymptotic values a t P e L = lo+.It can be seen tha t the effect of error propagation canincrease drasticallywith the number of input levels insome cases.C . EquivalentSystems

The probability of error remains unchanged for certaintransformations of system polynomials (assuming un-correlated noise samples). Kobayashi 's proof of the dualitybetween 1 + D and 1 - D [ l a ] can be extended to showtha t the systems F ( D ) and F ( - ) have identical errorrates.

The systems F ( D ) and F ( D h ) also have quivalentperformances since the latte r s a Ic-fold interleaved versionof t he former. Then the system polynomial may for thepurposes of erroranalysisbe reduced in order, if thegreatest common divisor of th e exponents of D is greaterthan unity. Thisrder reduction tightens he upper bound( 1 4 ) andgreatly reduces the number of states in theMarkov chain model.

The PRS systems under consideration fall into the fol-lowing four groups, withall responses withinagrouphaving the same error performance. (These equivalencesdo not carry over to timing sensitivity.)

1 ) 1 + D,1 - D , l - D 22 ) 1 + 2 0 + D2,1 - 20 + D 43) 2 + D - D2,2 - D 2 - D 4

whereA, = [ l - tQ(f+2 f l + 2 f 2 ) ] [ (fa - fl + 2 f 2 ) ] 4 ) 1 + D - 2 - D3,l - D - D 2 + D 3 . 1 6 )

U1- Q

U D. PrecodingQ ( - 2fl - 2f2) & (fa + 2fl - 2 f 2 ) Precodings used to alleviate the error-propagationU U problems of the previous decoder. The precoder eliminates


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926 IEEE TRANSACTIONS ON COMMUNICATIONS, SEPTEMBER 1975TABLE I11INCREASEN PROBABILITYF ERRORUE TO ERROR PROPAGATION

SYSTEM PeL = 10-2

m=2 4 81 + D ,1 - D , 1 .9 3 . 8 7.11 - D21 + 2D + D 2 ,1 - 2D2 + D 4

3 . 7 11 28- .-.~

2 + D - D 2 ,2 - D2 - D4 1 . 9 3 . 7 7 . 1

1 + D - D 2 - D3,1 - D - D 2 + D 3 4 .5 161

=====IeL = IO-^m=2 8

2 . 0 8 .0. 0

4 . 0 433

2 . 0 8.0.0

5 . 0 961

the effect of previous symbols at the source where theyare known precisely. Consider a new data sequence (w,}defined by3,4

1 N -1wn = - x, - C fiW,-i]. (17)f n lHowever wn can take on more than m values. By inter-preting w, modulo m, the redundancy in the values of w,is eliminated. The coefficients ( f n ) must then be integervalued. For convenience let them be normalized so thattheyhave a reatest common divisor of uni ty.Theprecoded symbols (w,} are the m-ary values which satisfy

WRfO = xn- fiw,-i(mod m) . (18)A solution exists and is unique if and only if fo and mare relatively prime [22]. Equation (18) can be modifiedslightly if j n , - - , l-1 are zero modulo m and f i is relativelyprime to m,

N -1

i=l

N-1Wn-ljl = X,, - fiwn-i (mod) . (19)

i=l+lThis form is useful for PRS systems for which (18) is notapplicable, e.g., 2 + D - D2 with binary input.

Equations (18) or (19) canbe rearranged t o becomeN-1

x, = jiw,-i (mod m ) (20)This indicates th at in the absence of noise the originaldata can be recovered by interpreting the received signalmodulo m, a memoryless operation. A real (noisy) systemwould in addition employ a slicer to resolve the channelout put to the nearest allowable level.

id

The precoder formulation follows Gerrish and Howson [8].The alphabet (O,l,***,m - 1 ) will be used in hissubsectionmetic.to simplify t.he subse quen t equations which involve modulo arith-

A precoder also acts t o some extent as a scrambler. Thesource data may contain long runs or have some periodicsubsequences. The precoder tends o breakheseup,especially in systems with the more complicated systempolynomials.

The statistics of the precoded sequence {w,) are neededto evaluate system performance. It can be shown tha t ifthe nputdata x,} areequally likely andstatisticallyindependent, the precoded sequence will also have equi-probablendependent symbols [23]. In other words,precoding replaces one data sequence by another with thesame statis tics. The equally likely nature of the precodedda ta will be used in the calculation of the probability oferror ; heir independence will be needed for computingthe SNR degradation.

The restrictions imposed on 19)mean thatnot allPRS systemsan be precoded (though precoding ispossible in mostcircumstances for the systemsunderconsideration here). A generalized precoding scheme [SI,[lo] can be formulated for any PRS system, specificallythe coefficients { f,,) need not be integers. The .precoderhas a form similar to (18) but with a modulo operationdefined on real numbers. The precoded digits are not ingeneral integers and span a range 0 to m (cf., 0 to m - 1for ordinary precoding). The ordinary precoder is imple-mented easily withdigital logic, while the generalizedform needs much more complicated circuitry. Any refer-ences to precoding in he sequel will meanordinaryprecoding although some of the resultscarry over togeneralized precoding. A major obstacle to th e analysisof systems using generalized precoding is that the tatisticsof the precoded sequence are difficult to obtain.E. Probability of Error with Precoding

In order to keep the error rate expressions simple, itwill be assumed tha t the output levels are evenly spaced.The output levels of a PR S system are not equiprobableif a,ny of the levels coal e~ ce .~owever, the outer levelsalwayshaveprobabilities of l/mM.Theprobability oferror of a precoded system is then approximately

where u2 is the noise variance a t th e decoder and d is thedecision distance (half the level separation, equal t o unityfor all of the specific examples considered here) . Sincetheoutputdataare aken modulo m from the sliceroutput,errors which carry he slicer out put m levelsaway may still be correct modulo m. The probability oferror 21)s an approximation in hat his effect isignored.F. Error Detection and Optimal Decoding

The output of a P RS system has a built-in redundancysince the number of output levels is arger than he

1/4; for M = 3, L = 5, the probabilities ar e 1/8, 1/4, /4, 1/4, andFor M = 2, L = 3, the output s have probabilities 1/4, 1/2, and1/8; and for M = 4, L = 5, he probabilities are 1/16, 1/4, 3/8,1/4, and 1/16.


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AND PASUPATHY: PARTIAL-RESPONSE SIGNALING 927of input levels. This redundancy can be used to

errors and hence monitor performance [7].Maximum-likelihood sequence estimation is a decoding

e which makes fulluse of th e correlations betweenlevels at different sampling times [la], [13]. Much

PRS is thus recovered. This improved performance is,however, a t th e expense of a substantially increased com-plexity and a decoding delay.

V. SPEEDTOLERANCEOne of the reasons for using PRS is that it can allow

signaling at the Nyquist rate. Ordinary PAM signaling isnot practical a t this ate since the resultingsystem isintolerant of timingperturbations. Some measuresarethen needed to gauge the timing sensitivity of a particularsystem or to compareonesystemwithanother. Speedtolerance and minimum eye width are two such figuresof merit. While speed tolerance measures the sensitivityof a system to changes in the signaling rate, eye widthis a measure of sensitivity t o changes in he samplerphase. (It is assumed that when therate changes, thesynchronous nature of t he system allows the receiver tosample the outpu t at thehanged rate.)A . Speed Tolerance and Peak Eye Closure

Speed-tolerance figures for some PR S systems appearin the literature [a]. They seem to have been based ona definition of speed tolerance as t.he increase in transmis-sion rat e atwhich the peak eye closure just becomes unity[24, p. 901.

Lucky, Salz, and Weldon [24] offer two definitions forthree-level systems, the zero level, and outer level peakeye closures. These definitions depend on specific choicesfor the decision thresholds. Corresponding definitions formore than three output evels have not been given and itis not clear from the literature what criterion has beenused in such cases to determine the speed-tolerance figures.

The basic problem with he above definitions is th ata unity value for the peak eye closure merely indicatesth at th e distortionhas exceeded an arbitrary thresholdand not whether the eye pat tern is open or closed. Also,when the transmission rate is no longer the nominal value,the problem of finding the best sampling phase mus talso be considered in calculating speed tolerance.B. Speed Tolerance and Zero Eye O pening

In order to avoid th e ambiguities associated with theprevious definitions of speed tolerance, we will considerthe conditions under which a noiseless system Yails.The corresponding ra te will then define the speed tolerance.

Most PR S systems operate such that some of the moutput levels coalesce. When the signaling rate is largerthan thedesign value (with the transmission characteristicbeing kept fixed), the output evels generally split intomMlevels again. In addition, ntersymbol nterference romsamples tha t are nominally zero (at the nominal trans-mission rate and nominal sampling phase) will cause the

LEVELS ATCENTRE OF%{% (lhol+lh11)/2Ihol+IhllSAMPLING TIME

THETHE EYE

LEVELS AT THECENTRE OF EYEI h o l + I h l I

Fig. 5. Received levels for a three-level partial-response system.(a) Undistorted reception. (b) Dis torted reception.

levels to spreadoutaround hese mM levels6 (Fig. 5 ) .If the decision thresholdsare placed midway in thegaps between the levels and the sampler phase is kept a tthe point where the eye opening is the largest, a noiselesssystem will not make errors until the evels overlap in oneof the eyes (i.e., the eye closes). In general, the thresholdpositions as well as the sampler phase will have to beconstantly readjusted as he signaling rate is varied, inorder to keep the thresholds in themiddle of the gaps andthe sampling instant at the oint of maximum eye height.Above the rate atwhich one of the eyes closes, some datasequences will cause errors even in the absence of noise.When hishappens, wewill deem the system to havefailed. For other threshold locations, the system will failat even lower transmission rates in the absence of noise.Thus it is clear tha t this rate is an upper bound on per-missible transmissionate, the ctual thresholds willdicta te how close to this maximum rate the speed mayapproach.

We shall define speed tolerance as the increase in trans-mission rate at which the smallest eye opening is zero (theeye opening being the maximum height of an eye). The

symmetrical about zero. In addition, the probability distribution of6 For equiprobable bipolar d ata , the probability of the output isthe unwanted ntersymbol inteference is symmetric about. each ofthe mM levels individually.7 It should be noted th at thi s placement of the thresholds doesnot minimize the average probability of error n the presence ofGausslan nolse.


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928 IEEE TRANSACTIONS ON COMMUNICATIONS, SEPTEMBER 1975distortion from the nominally zero pulse samples at timeto + nT, is given by

m5 s , -h ( to + iT,) 5 ( m- 1 ) I h(t0 + T , ) I+m 2=- m

i Q l I N Z ) i $ f r N Z llD,, (22)

where T, is the sampling interval (not necessarily equalto th e ominal symbol pacing T ) o represents the samplerphase ( I to I 5 T,/2) (INZ) s the set of subscripts i forwhich h ( i T ) 0, and D, is the peak (or worst case)distortion from the nominally zero pulse samples.

For a three-level system (only possible with binary in-puts), the eye opening is given by (Fig. 5)

p = I h o I + I h l l - IhoI - h J l -2D,, (23)where hoand hl are samples of t he impulse response cor-responding to he twononzero pulse coefficients. Thecondition p = 0 can be rewritten as

DP( I ho I + hl I - / ho I - hl I l ) /2 = 1. (24)

This is analogous to the peak eye closure criteria [24].C . Inner and Outer Eye Openings

For systems with more than three output levels, speedtolerance is derived from the signaling rate which causesat least one of t he eyes to be closed.

Consider a five-level PRS system with hree nonzeropulse coefficients. Let the corresponding impulse responsesamples be ho, hl, and hz ordered such that I ho 2 I hl I 2I hz I The positive received levels due to just thesesamples,for binary f l nput, are shown in Fig. 6.

The outer eye opening isPO' ( I h o I + l h l I + I h z l - D p )

- ( I h o l + I h l l - hzI +D,)= 2 I h2 I - 2Dp, (25)

while the inner eye opening ispr = I ho I - hl I + hz I - D,)

- ( I ho I - hl I - h2 + D,)2 I hz I - 2Dp, forIhoI2 I h l l + Ih21I 1hol - 2 I h l l - 2 D , , forIhoI 5 I h l l + l h z I .(26)

From (25) and (26) we see thatpr 5 2 I hp I 0 , = PO. (27)

Similarly it can be shown tha t for all five-level systems,the inner eye opening is always less than or equal to theouter opening. Then for binary input, the speed toleranceof the systemsunderconsideration s the increase intransmission ra te which causes the central eye to close.

; AT THEf OF THE EYE

Fig. 6. Received levels for a five-level partial-response system.

D. Conditions for Nonzero Speed ToleranceBefore calculating the speed tolerance for various PRS

systems on the basis of the zero eye opening criterion, theconditionsunder which the speed tolerance s nonzerowill be determined. From the definitions of eye openings,it is clear that speed tolerance is nonzero only if D, isbounded.For a minimum bandwidth PR S system the expressionfor h ( t ) ( 6 ) ,can be substituted into (22) to give

i $ ( r N Z )(28)

It can be shown that the series (28) defining the peakdistortion D, converges if and only if [23]

and/or- 2 ) to = k T and T, = IT, for k, integers. (30)

The condition (29) is quivalent to requiring th at1 + D be a factor of the system polynomial F ( D ) Thisrequirement is not surprising in light of t he discussion inSection I11 on how 1 + D affects the asymptotic decay ofthe impulse response. The second condition (30) is thatthe sampling instants should occur at th enominal points.These conditions lead to theconclusion th at for minimumbandwidth systems, if 1 + D is not a factor of the systempolynomial, t he speed tolerance is zero since any changein speed ( l / T , ) will cause the peakdistortion to beunbounded.E. Comparison of PRS S ys t ems o n the B a s i s of SpeedTolerance

Speed tolera,nce figures for PRS systems of interestwere calculated on a digital computer. The problem wasprogrammed as a wo-variableoptimization; the speedwas increased to close the eye while the sampler phase wasadjusted to maximize the eye opening. This procedure wasrepeated until an equilibrium was reached.

Table IV shows the speed-tolerance figures for thevarioussystemsunderconsideration for binary nputs.


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KABAL AND PASUPATH Y: PARTIAL-RESPONSE SIGNALING

TABLE IVSPEEDTOLERANCEND MINIMIJMYEWIDTH OR MINIMUMBANDWIDTHRS WITH B I N A R YNPUTS

92920

SYSTEM

I

SPEED TOLERANCE

4 2 . 5

2 6 . 6

7 . 4 3

1 5 . 5

7 . 4 2

1 4 . 1

4 . 8 1

2 . 9 6

MINIMUMEYE WIDTH

0 . 6 6 7 T

0 . 6 8 9 T

0 . 2 4 3 T

0 . 3 5 7 T

0 . 2 4 9 T

0 . 3 6 3 T

0 . 2 0 0 T

0 . 1 6 4 T

Comparison with igures previously published by Iiretzmer[218 shows significant differences in two cases, namely1 + 2D + D2 nd 2 + D - D2.One of the systems th at has ot been previouslyanalyzed, 1 + D - D2 D3, as nearly the same speedtolerance as the well known 1 - D 2 ystem. The two sys-tems are closely related, but 1 + D - 2 - 3 has anadditional null at th eNyquist frequency which makes thehigh-end roll-off more gradual.F . Eflect of Sampler Phase

The optimum sampler phase remained unchanged withrespect to th e center of t he impulse response for all thesystemswithymmetry (orntisymmetry)nheirimpulse responses. This is not unexpected since the eyepatterns for these ystemsare ymmetrical about hecenter of t he impulse response. Fig. 7shows how the'zeroeye opening criterion elates to he samplerphase andexcess speed for the two ystemswithunsymmetricalresponses. Theoptimum times were offset by -2.7 XlO+T and -2.9 X 1OP2Tfrom heir nominal points forthe systems 2 + D - D 2 nd 2 - D2 D4,espectively.The speed tolerances for these ystems are 7.4 and 3percent, respectively. For 2 + D - D2,he often-quotedspeed tolerance is 38 percent [a]. Fig. 7(a) shows clearlythat bo th pairs of eyes are completely closed at transmis-sion ates more tha n 8 percent over the nominal rate.G . N onmi n i mum Bandw i d t h Sys t ems

As noted in Section 111-C, PRS is possible with systemsnot having 1 + D as a factor of the system polynomial ifthe system bandwidth is allowed t o increase beyond theNyquist bandwidth. As an example, we will consider usingthe 1 - D response in conjunct'ion with a Nyquist filterhaving a gradual utoff. Theparticular filter chosenbelongs to th e class of raised cosine filters [24]

2 + D - D2, - D2,nd 1 - 2D2+ D4 re 43,40 8, 15, and8Kretzmer s speed-tolerance igures for 1 + D, + 2 0 + 028 percent, respectively.

10

0-2.7

-10

-20

RAT E)

-? (b)Fig. 7. Eye closureas a function of sampling time and tra.nsmissionrate. (a) 3 + D - D2.h) 2 - D 2 0 4 .

~ ( la ) / T 5 I w I 5 ~ ( la ) / T0, 4 1 + a ) / T 5 I W

(31)where a is the roll-off parameter, 0 5 a 5 1. Fig. 3illustr ates the spectrum of the 1 - D response used witha raised cosine filter ( Y = 1/2).Fig. 8shows how the speedtolerance of this system varies with a. For comparison,this graph also shows the curves for the binary 1 + Dand 1 - D 2 esponses. At CY = 1 / 2 the 1 - D system hasa speed tolerance of 20 percent. This is bett er than anyof th e minimumbandwidthsystems considered earlierwhich also have dc nulls. This improved speed tolerance isof course at theexpense of more bandwidth; however, thetradeoff may be useful. This example serves to illustratethat Nyquist filters other th an the minimum bandwidthone can be useful for generating PRS spectra.H . Speed Tolerance for a General PRS Systenz

The calculation of the eye opening which is used toarrive a t he speed-tolerance figure can be generalizedfor PRS systemsother than those considered indetailhere. For a general PRS system, the output due just to thenominally nonzero pulse samples is given by (cf. ( 2 2 )

u(to+ ? T8) z,-ih (to + i TB ) . ( 3 2 )i E z N Z 1

For a fixed value of to+ nT,, only M input digits con-tr ibu te to the sum. The worst-case excursions of each of


10/14

930 IEEE TRANSACTIONS ON COMMUNICATIONS, SEPTEMBER 1975100i LA40 BINARY 2

P IE 20 1-D2naul0 0.2 0.4 06 ' 0.8 ' I O 0

ROLL-OFF PARAMETER o(Fig. 8. Speed olerance as a unction of t8he oll-offparameter ofaaisedosine Nyquist filter. -I

the total f mjypossible values canbe calculated by adding+Dp to each of th e m M values. (Note thatD, varies witht o , see ( 2 2 ) ) As t o is varied and the M 2 which affect( 3 2 ) take on allombinations of 'values,he worst case to '512excursions traceuthe outline of the eye pattern. Fig. 9. Duobinary eye pattern.

-2

-T 0-The nominal output levels result when to = 0 andT , = T. For each of thesenominal levels there ~ i 1 1e nomid (see (29) and ( 3 0 ) . For the rest of the section,

corresponding M-tuples of input symbols { zn-i}, i E {Inz}.we shall that such is the case.Thegap corresponding to aparticular eye can eound The impulse response of a genera.lPRS (6) canby aki ng he difference between the minimum of thebe written asvalues of u(tn+ nT.) D, caused by-tuples which T N-l (- ) n j ncorrespond tohe nominal level above the eye andhe h( t ) = -sin (Pt /T) ~. ( 3 5 )maximum of the values caused by M-tuples which cor-resporld to the nominal level belom the eye. F~~ he The sine factor nserts he regularly spaced zero crossings.general case, several. eye openings mus t be examined, The second factor (i*e.j the sum) has at lnostsince the inner eye is not necessarily the smallest. oots and hence a t most N - 1 sign changes. Let T Landthe duobinary eye pattern for binary input sylnbols The peak distortion can be split into three terms with the( ~ i ~ .). The four 2-tuples { 1,1} , 1,- 1 , { - , 1 ) , and real roots of the second factor of ( 3 5 ) contained within{ - ,- } correspond to he . nominal ou tput levels the range Of the term,2, 0, 0, and -2, respectively. Consider the upper eye; the IL I 4lower eye boundary, due to the { 1, } or { - , 1} input DP ( m- 1) I h (to -I- i T ) + I h(to i T ) Ipair, is given by

P L=o t - nT

To help clarify the procedure, consider, for example, Tu be he smallest and largest eal oots,

i 00 i=IL+li Q ( r N 2 )max Ch,(to)- h(tn + T.),-h(to) + h(t0 + T.)]+ D p

( 3 3 ) + 5 h(t0 + i T ) I } (36)where h( t ) is the duobinary response given in Table 11. where I L s the greatest integer i satisfying to + T 5 TLThe upper boundary, in this case due only to the { 1 , l ) and i 5 - and Iu is the smallest nteger k satisfyinginput symbol pair, is given by to + k T 5 TU and k 2 N .

Expanding he first term of (36) using ( 3 5 ) and then(34) interchanging thesummationandmagnitude operations,

+I

h(to) + ( t o + T,) - D p .The eye opening may then be calculat,ed as the difference ILbetween (34) and ( 3 3 ) . I h(to + i T )

6 mVI. EYE WIDTH T I L N-1

To simplify the calculation of t he eye width,a new = -i- sin (li- to I/) i= OO n4 t~+ i - n)Tformula for peak distortion is first developed.A . Peak Distortion T I L N--l (- ) n j n= -sin ( T I to I/T) ___

li- i n=O to + TT o measure eye width,only he samplerphase to isvaried;he symbol spacing T, assumes the nominalvalue T . Thus, the peak distortion due to the nominallyzero pulse samples is given by (28) with T , = T . Minimum

(37)

bandm'dth systems have bounded D p and hence nonzero 9 If thereare no real roots, TL nd Tumaybe set to zero to beeye widths only if 1 f D is a factor of the system poly- consistentwith the sequel.


11/14

RABAL AND PASUPATHY:ARTIAIrRESPONSE 931The first part of the result (37) is zero since 1 + D isa factor of the system polynomial. Similar manipulationof the last term of (36) is possible. Then for minimumbandwidth systemswithboundedpeakdistortion, thepeak distortion can be written as a finite series,

The lower outline is generated bym .- ) , t o 2 0- m - I , t o < 0.5, = 5n-1 = (42)

The upper and lower outlines of t he smallest eye then are

Forall the specific PRS systems ntroduced previouslyT L and Tu lie in the interval [0, ( N - 1) T I giving IL =-1 and Iv = N in (3s).B. Eye Width

The eye opening as a function of t o can be calculated byadding D, o th e levels generated by the M nominallynonzero pulse samples. The width of the eyes can be cal-culated from the values of t o a t which the eyes jus t close.

As a performance measure for PRS systems, we mill usethe smallest of thewidths of the eyes. In he simplercases, the M-tuples of input which result in theboundariesof the smallest eye can be determined analytically.1 Theupper and lower edges of t he eye can be equated to givea polynomial in to from whose root's the eye width can bedetermined.

To illustrate he procedure, the duobinary 1 + D)eye width will be determined. From (35))

and

SubstitutingforD, from (39) and equating (43) and (44)we can solve for the two values of to at which the edgesof the eye meet. The eye width is then just the differencebetween them,

r

eye width = ~ L (45)4m - 5 'The results of Section V-C can be applied directly-

for binary input and the systems under consideration thecentral eye widthsare the smallest. The minimumeyewidths are given in Table IV.

The speed tolerance and eye width criteria have somecommon ground in that they measure the sensitivities totimingperturbations,albeitnot of the same type. Theresults show' tha t he systemswithnearlyequal speedtolerances have nearly equalye widths. The major changein ordering of t he PR S systems on the basis of these twomeasures occurs with the interchange of 1 + D and1 + 3 0 + 2 , the system 1 + 2 0 + D2 has the best eye2 m - I ) sin ( P t o l / T )

D, = (39)From ( 3 2 ) the output from just the two main pulses is

width while 1 + D has the best speed tolerance.P 1 - ( ~ O / T ) ~ VII.SNRDEGRADATION

In this section, the SNR degradation over ideal binarytransmission is used to gauge the system performance

(ato/T) 1- (to/TI2) T withdditivehiteaussian noise present. We willagain concentrate on minimum bandwidth PRS.sin ( d o / T ) 2 (2, + 5 -1)+ 5 , - c,1-1 .)

(40)Fig. 9 shows these levels as afunction of t o for binaryvalued 5,'s. If to = 0, (40) akes on the value - x,--l.The term zn + zlL-l adds distortion when to 0. It canbe verified th at this distortion is worst for the inner eyesfor m-ary input symbols. Theupperboundary of the(upper) inner eye is generated by

- ( m ) ) t o 2 0xn = lm - 1, t o < 0

- ( m - l ) , t o 2 05,-1 = 41)

[m - 37 t o < 0.(1 - 0 2 ) in this manner. More recently work by Craig [26] has10 Smith [25] has found the eye width for modified duobinarycome to our attention.

A . Xystem ModelIn order to calculate the SNR degradation, we must

specify how the filtering is apportioned between the trans-mitter and the receiver. Two system models are shown inFig. 10. Thearrangement n Fig. 10(a) (which will becalled model 1 ) optimally distributes the shaping betweenthe transmitter and receiver for a perfect channel. Thephase response of the system function may be arbitrarilydistributed since it does not affect the system performancein the presence of noise. The noise at the decoder is cor-related at th e sample times meaning tha t the expressionsfor error probability which assume independence of noisesamples at th e detector cannot be applied for model 1.However the lower bound (12) still holds.

In Fig. 10(b) (model 2 ), the transmitter filter and thechannel determine the shape of the frequency response


12/14

932 IEEE TRANSACTIONS ON COMMUNICATIONS, SEPTEMBER 1975NOISE For model 1, (49) becomes

71 = [ @ ( P E / K ) jmH w ) d u l l . (50)t =nT 2ad -mI TRANSMITTERILTER RECEIVER For model 2, using Parseval's relationship and (6), the( 4

NOISE S N R caneritten asSAMPLER 2 N-l

1 Iwl


13/14

ABAL AND PASUPATHY: PARTIAL RESPONSE SIGNALING

TABLE VSNR DEGRADATIONOR BINARYNPUTS

933

SNR DEGRADATION (dB) at PE = IO-

Model 1 Model 2

* The three figures i n each entry represent the lower and upper bounds and the actual SNR degradation

class of systems s d = k. Hence theasymptoticSNRdegradation from (54) is

lim = u 2 (1 +i)For large k (and hence large number of output levels)thi s limit approaches he lower bound onSNR degradationfor m-ary input symbols. (The lower bound is achievedby ordinary PAM.) This class of systems demonstratesthat SNR degradation need not necessarily increase withan increase in the number of output levels.

The effect of error propagation on th e SN R egradation(as manifested in the value of K ) will also tend to besmall for this class as compared to other systems with thesamenumber of pulse coefficients (for k > 1 ) . This isdue t o the fact that the feedback coefficients are smallrelative to the main pulse sample on which the decisionis based. On the other hand, when precoding is appliedto systems in this' class, the decision distance decreasesto unity. The SNR degradation thenncreases with k ,

712PE-o 716

( 5 6 )

112 7129 b pE+o 71b2 lim - = aZ2(k2+ k ) . (57)

Two of the PRS systemsunder consideration here fallinto the category specified by ( 5 5 ) , 1+D ( k = 1) and2 + D - D ' ( k = 2 ) .C. Comparison on he Basis of S N R Degradation

Table V lists the SNR degradation in decibels for PRSsystemswithbinary nputsand at he representativeerror ate of loF5.ICretzmer [a] listsvalues of SNRdegradation or some of the systems considered hereusing model 1 and the asymptotic formula ( 5 3 ) . Qureshiand Newhall [27] use adiscretechannel model whichcorresponds to model 2 for SNR degradation calculationsfor the same systems Kretzmer considered.

Q

Table V lists a lower bound on the SNR degradation,corresponding to th e lower bound on the error rate ( 1 2 ) ,for model 1. For model 2, the SNR degradationorrespond-ing to the ounds on he probabilityof error, (12) and(1 4 ) ,as well as the exact SNR degradation from the Markovchain model are given. For precoded systems the degrada-tion values corresponding to he errorrate of (21) aregiven for both system models.

Previousresultshave included only lower boundsonthe SN R degradation for systems without precoding andthus it has not been possible to ascertain how seriouslythe error-propagation phenomenon affects the systemperformance. The results given here (forbinary nput)show that precoding does not decrease the SNR degrada-tion significantly and in factncreases it for some systems.(For 2 + D - D2 and 2 - D 2 - D4 a 6-dB penalty isincurred.)

The error-propagation phenomenon is more serious formultilevel input, though theworst-case sys teh inTable I1(nz = 8, P,/P,L = 96) requires only 1.6 dB more SNRwithout precoding than he same system precoded tomaintain an error rate of

VIII . CONCLUSIONSFive useful PRS systems have been studied previously[a]. Thisstudyhas analyzed some additional ones. Of

these systems 1 + D - D2 - D3deserves more attentionsince its performance rates above some of the previouslystudied PRS systems, for instanceKretzmer's class5 system [a]. The two systems which still stand out onthe basis of t he performance measures used here areduobinary 1 + D and modified duobinary 1 - D2.Theseare also the ones that have in the pasteen put topracticeboth because of their simplicity and their useful spectralshapes.

The exact figures for probability of error and henceSNR degradationhave been presented for one system


14/14

934 IEEE TRANSACTIONS ON COMMUNICATIONS, SEPTEMBER 1975model. This allows us o valuate he rue ffect ofprecoding. The results show that for reasonable errorrates (lop5or below for instance)) precoding does notdecrease the SNR degradation greatly and in some casesactually increases it. (This is due in part to the fact thaterrorpropagation affects the error ra tebyat mostamultiplicativefactor.) However, since precoding is rela-tively easy to implement, it serves a useful purpose formany PltS systems.I n real systems since the data are not truly random,selected input sequences could conceivably suffer severeerror propagation. Precoding does offer protection againstthis and in addition tends to cramble some repeated datapatterns, especially in the case of systems with the morecomplicated system polynomials.

Detectorssuchas hemaximum likelihood sequenceestimator were not emphasized here. Until theh igh costof implementing hese ypes of detectors decreases sub-stantially. PRS systems will continue to use decodingschemes as described here.Presently, PRS systemsstillten d o be applied where simplicity o f processing anddetecting s mportant. However, in he future, we canlook forward to achieving better performance for PRSwith more optimal decoding schemes.

ACKNOWLEDGMENTThe authors would like to tha nk the eviewers for their

helpful comments and suggestions.REFERENCES

[l ] A. Lender, Theduobin ary technique for high-speed da tatransmission, ZEEE Trans. Commun. Electron., vol. 82,pp.[2] E. R. Kretzmer, Generalization of a technique for binaryd a h communication, ZEEE Trans. Conamun. Technol. (Con-cise Pape rs), vol. COM-14, pp. 67-68, Feb . 1966.[3] T. L. Swartz, Performance analysis of a three-level modifiedduobinary digital F M microwave radio system, in Conf. Rec.,[4] F. K. Becker, E. R. Kretzmer,and J. R. Sheehan, AnewZEEE Znt. Conf . Communications,1974, pp. 5D-1-5D-4.signal format for efficient data transmission, Bell Syst. Tech.J . ,[5] J. W. Smith,Error control n duobinary systems by meansvol. 45, pp. 755-758, M a y J u n e 1966.of null zone detection, ZEEE Trans. Cornmun.Technol., vol.COM-16, pp. S25-830, Dec. 1968.161 J. F. Gnnn and J. A. Lombardi, Error detection for partial-

COM-17, pp. 734-737, Dec. 1969.response syst,ems, ZEEE Trans. Commun.echnol., vol.[7] H.KobayashiandD. T. Tang, On decoding of correlativelevel coding systemswithambiguity zone detection, ZEEETrans. Commun. Technol., vol. COM-19, pp. 467-477,Aug.1.971.[8 A. M. Gerrish and R . D. Howson, Multilevel partial-responsesignaling,n Conf. Rec., ZEEE Znt. Conf. Communications,1967, p. 186.[9] M. Tomlinson,New automatic equalizer employing modulonrithmetic, Electron. Lett., vol. 7, nos. 5/6, pp. 138-139, Mar.1971.[IO] H.Harashima nd H.. Miyakawa, Matched-transmissiontechnique for channelswith ntersymbol nterference, ZEEETrans. Commun., vol. COM-20, pp. 774-780, Aug. 1972.1111 H. Kohayash i, A survey of coding schemes for transmissionor recording of digital da ta , IEEE Trans. Commun. Technol.,

[12] -, Correlative level coding and maximum-likelihood de-vol. COM-i9, pp. 1087-1100, Dec. 1971.coding, ZEEE Trane. Inform. Theory,vol. IT-17, pp. 586-594,Sept. 1971.1131 G. D. Forney , Jr. , Maximum-likelihood sequence estimationof digital sequences in the presence of intersymbolnter-ference, IEEE Trans. Inform. Theory, vol. IT-18, pp. 363-378, May 1972.

214-218, Ma,y 1963.

[14] R. W . Chang, A new equalizer st,ructure for fast tart-updiEital communicxtion, Bell Syst. Tech. J . , vol. 50, pp. 1969-20714, July-Aug. 1971.[15] K. H. Mueller, A new, fast-converging mean-square algorithmfor adxntive enualizers withaartial-resaonse ienaline. Bell..

Syst . Tech. J . , vel. 51, pp. 143-153, Jan. i975.[16] 0. B. P. Rikke rt de Koe and P. van der Wurf, On some exten-sions of Nyquists t,elegraph transmission theory, Proc. ZEEE[17]W. R. Bennett, Introduction to Signal Transmission. New(Lett. ), vol. 57, pp. 701-702, -4pr. 1969.

[18] D. M.Baker,Analog/dlgltalhybrid radio, Can. Electron.York: RlcGra,w-Hill, 1970,,p: 16.

[19] D. A. George, R. R. Rowen, and J. R. Storey,An adaptiveEng., pp. 34-33, Feh. 1973.decision feedback equalizer, ZEEE Trans. Comnaun. Technol.,[20] D. L. Duttweiler, J. E. Mazo, an d D. G. Messerschmitt, Anvol. COM-18, pp. 281-293, Ju ne 1971.upper ound on t,he error probabil ity in decision-feedbackequalization, IEEE Trans. Inform. Theory, vol. IT-20,pp.490-497, July 1974.[21] W. Feller, A n Introduction to Probability Theory and Its Appli-cations, 2nd ed., vol. 1. New York: Wiley, 1957.[22] G. Rirkoff and S. MacLane, A Survey of Modern Algebra, 3rded. New York : Macmillan, 1965.[23] P. Kabal and S. Pasupathy, Partial-response signalling, Dep.Elec.Eng.,Univ. of Toronto, Toront,o, Ont.,Canada, Com-[24] R. W. Lucky, J. Salz, and E. J. Weldon, Jr., Principles of Dahmun. Tech. Rep. 75-1, Jan. 1976.[25 ] B. M.Smith, Some results for the eye patterns of class 4Communication. New York: McGraw-Hill, 1968.

parti al response da t,a signals, IE EE Trans. Commun. (ConcisePap ers), vol. COM-22, pp. 696-698, Ma y 1974.[26] J. W . Craig, Eyepatternsand iming error sensitivity npartial-response pulse transmissiou ystems,n Conf. Rec.,[27] S U. H. Qureshi and E. E. Newhall, An adaptive receiver forZEEE Znt. Conf. Communical.ions,1975.data transmission over time-dispersive channels, ZEEE Trans.Inform. Theory, vol. IT-19, pp. 448-457, July 1973.

.~

*Peter Kabal was born intockholm,Sweden, onNovember 3, 1945. He receivedthe B.A.Sc. and M.A.Sc. degrees in electricalengineering from the University of Toronto,Toronto,Ont.,Canad a, in 1968 and 1972,respectively. He is currently completing therequirements or thePh.D. degree at heUniversity of To ronto. He has held Nation alResearchCouncil of Can ada scholarshipswhile a t grad uate school.He worked for th e Hydro-Electric PowerCommission of Ontario from 1968 to 1969. He hasalso held a numberof summer positions including ones at the O ntar io Hy dro ResearchLaboratories, Toronto, and a t Bell Laboratories, M urray Hill, N.J.He has been a Teaching Assistant and a part-time Lecturer a t th eUniversity of Toronto . His esearch interes ts are n the field of digita lcommunication. *Subbarayan Pasupathy (73) was bornin Madras, ndia, on September 21, 1940.He received the B.E. degree in telecom-munications from th e University of Madras,Madras, India, in 1963, the M. Tech. degreeinlectricalngineeringrom thendianInstitute of Technology, Madras, ndia, in1966, and th e M. Phil. and Ph.D . degrees inengineering and applied science fromYaleUniversity, New Haven, Conn., in 1970and 1972, respectively.Dur ing 1965-1967, he was a ResearchScholar a t the ndianInstitute of Technology, and during 1968-1971 he was a TeachingAssistant at Yale University. Dur ing 1972-1973, he was a Post-doctoral Fellow at t he University of Toronto, Toronto, Ont., Can-ada, where he s now an Assistant Professor of Electrical Engineering.His urrent research interests include statistica l communicationtheory, array processing of signals, radar-sonar systems, and dat acommunication.

partial-response signaling kabal and pasupathy

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