simplified msk signal techniques

8/2/2019 Simplified MSK Signal Techniques

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433EEE TRANSAC TIONS ON COMMUNICATIONS, APRIL 1977

Concise PapersSimplified MSK Signaling Techniqu e

FRANK AMOROSO, SENIORMEMBER,IEEE, ANDJAMES A. KIVETT, MEMBER, IEEE

Abstruct-Modulation and demodulation in the minimum (fre-quency)-shift k eying (MSK) format are reduced to the form of coherentbiphase keying. Considerable simplification o f circuitry is thus effecte dwithout acrifice of performan ce. When theratio of the carrier fre-quency to the bit rate is high, then the technique described here doesnot require the precise relative phasing of any pair of oscillato rs in thetransmitter. Experimentalresultsdemonstrate he easibility of syn-thesizing the one filter that is unique to this signaling scheme. In thepresent instance, the filter s realized as a surface acoustic wave device.

I . EVOLUTION OF A SIMPLE MSK SIGNALSYNTHESIZER AND DEMODULATOR

Several authors [ 1 - [ 5 ] havenoted hat heminimum(frequency)-shift keying (MSK) format, with proper detection,offers hesamehighcommunicationsefficiencyascoherentbiphase (or quadriphase) shift keying. The practical promise ofrealizing this efficiency is evident in t he ex pression of th eMSKsignal as th e sum of wo antipodal pulse streams modulatingth e in-phase and quadrature channels of a single carrier. Thepresentpaper xtends hese esultsby educing he MSKsynthesizer to aimple iphasemodulatorcceptinghedata n erial orm.Thedemodulation, orrespondingly, isperformed with a biphase demodulator.The ssential lements of the ommunicati onsink reshown in Fig. 1. An arbitrary bit stream ak multiplies the localoscillator fl inabalancedmodulator.Thefilterh(t),as hefigure implies, is linear and time invariant. The output of theh(t) filter is th e MSK signal. At the receiver, a properly phasedfl entersn alanced emodulatorwhose utput passesthrough a zonal filter for removal of double frequency terms.The result is the recovered bit st ream ah . A matched filter maybe ntroduced at he receiver o gain he usual performanceimprovements in noise. Thiswill be discussed in Section111.

Theperationfhisynthesismethodependsncompliance with t he original definition of MSK given by Doelz[ 1 ] with additional constraint s recently stated by Sullivan 21 .During any keying interval of duration T, one of two frequen-cies, fl or f2 , s transmitted, where fl and f2 stand in a specialrelationship to the link bit rate 1/T. For some selected inte gerI 2

n + lf2 =- 2 T

n2 Tf1=- . (1 )

Paper approvedby the Editor fo r Communication Theory of theIEEE CommunicationsSociety for publicationwithout oralpresenta-tion. Manuscript receivedAugust 18 , 1976; revised December 17,1976.CA 92634.The authors are with he HughesAircraft Company,Fullerton,

In randomdata ransmission, he esultingspectrum will becentered on an apparent carrier ocated at f, = (211+ 1)/4T.The frequencies fl and f2 have been referred to [ 3 as markand space frequencies, respectively.Fig.2shows he MSK wave synthesi zedas hesum of astream of modulated pulses at the bit rate 1/T for the choicen = 2. Stric tly for graphical clari ty, he successive pulses areshown alternately in Fig. 2(a) and (b). Note t hat all pulses areidenticalexcept or sign and imedisplacement.There is aregular time overlap between the pulses in Fig. 2(a) and thosein Fig. 2(b), and the MSK wave of Fig. 2(c) is at all times thesum of two such overlapping pulses. Note that the frequencyturns out to be constant over each interval of duration T, asprescribed in t he MSK format.

A distinct, but equal ly egitimate MSK wave for IZ = 2 isgiven in Fig. 3. The transitions betweenfi and f2 take place atthe peaks of t he wave rather than at the zero crossings, whichwas the case in the more familiar form f Fig. 2. The enhancedwaveformcontinuity nFig. 3 is the esultof hedifferentformsofbasicpulsesshown nFig.3(a)and b).Here hepulseshavezeroslopeat hebitboundaries,apropertynotfound nFig.2.The signs and imeplacementsofpulses nFig. 3 are he same as n Fig. 2, and he resulting pattern offrequenc ies is the same.The basic data pulse for general IZ will be shown i n SectionI1 to be

for O G t G 2 T= 0, otherwise2)

where @ accounts or hedifferencebetween he ormsofMSK in Figs. 2 and 3.Thisexpression orp(t) sconsistentwith heprevailingconcept of MSK pulse structur e. The factor sin nt/2T is th eindividual pulse modulation envelope, shown as dashed lines inFigs. 2 and 3. The factor s in [ ( ( ~ I zt )/%T)nt+ @I representsthe apparent carrier at f,. To confirm that overlapping pulsesapp ea r t o derive from in-phase and quadrature sources off,, t is only necessary to note that

which means that from pulse to pulse the phase off, appearingin the pulse is, with respect o some reference phase, shiftedby some odd multiple of n/2 .ThesynthesizercircuitofFig.1operatesas ollows.Thedata wave entering he balance d modulato r is so phased hatevery zero crossing corresponds with some voltage peak of thelocal oscillator fi. In view of ( l ) , there are exactly 1 half-cycles


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43 4 IEEE TRANSACTIONS ON COMMUNICATIONS, A P R I L 1977

0 1 0 1 0 0 0 0BI T STREAM ak A OCAL OSCILLATOR fl

MATCHEDFILTER

PASS FILTE R

0 1 0 1 0 0 0 00 OCAL OSCILLATOR 11 BI T STREAM akFig. 1 . Transmissionand reception scheme.D A T A PULSE - (Q ) DATA PULSE - ( 0 )

I DATA PULSE - ( A ) 1 DATA PULSE - f b l

0 T Z T 3T 0 T 2 T 3 TFig. 2. Resolution into data pulses, n = 2 , @ = n/2. a) In-phase Fig. 3 . Resolution into data pulses, n = 2 , @ = 0. (a ) Quadraturepulses. (b) Quadrature pulses. ( c ) Transmitted total. pulses. (b) In-phaseulses. (c) Transmitted total.

of fi ineachkeying nterval T. Eachkeying nterval, herefore, 11 . A N A L Y S I S OF T H E SYNTHESIZER C I R C U I Tpresen ts a proper ly signed burs t of 11 half-cycles of fi at theinput of th e h ( t ) filter.Now he h ( t ) filter is so designed hat The impu1se response h ( t ) sit smpulseesponse is justonstantnvelopeurst of I I + 1 7 -half-cycles of f2, also of duration T. The response of h ( t ) ilterto he ypical nputburst is just he equired p ( t ) , and he h ( r ) = sin (27rf2 t ) = sinoverall output of the ( t ) ilter is then the requiredM SK signal.Hence, the remainder of the present paper will be concernedwithhease $J = 0. an dheypicalnputurstor 1 bit is

Thisarticularircuit will prod uce p ( t ) fo r @ = 0 only. = 0, otherwise (4)


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CONCISE PAPERS 435

( 5 )where he ak represent he data and 6' represents he relativephase of f 1 and the data transitions.

The respon se of h ( t ) o the typical input burst spans T :

r t

nt ( 2 n + 1)and

which agree with( 2 ) fo r 8 = 0, as mentioned.Now direct substitution will bear out that forn even,p o ( t ) = p , [ t f 2 m + 1 ) T l , for all integer m ( 1 l a )

and for n odd,p , ( t ) = - p e [ t f 2 m + 1) T], or all integer m . ( 1 l b )Thus, for n even, t is directly evident that all transmitted

pulses have the same for m except for sign (the data multipliera h ) and ime ocation.Fo r IZ odd,note hata sign reversalfrom pulse o pulse is indu ced by he fact hat here are anod dnumber of half-cycles of the requency fi perkeyinginterval. The input pulse g k ( t ) is, in effect, inverted on succes-sive keying intervals.

Th equadrature arrierproperty is seen n he elationbetween the factors

( h + l ) T sin ( 2 n f l x + 8 ) sin 27rf2(t - x) dx,fo r ( k + l ) T < < k + 2)T

0, otherwise. ( 6 )Appropriate manipulations will reduce p e ( t ) to the form

sin [7 t ]( 2 n + 1 )

( 7 )Substitutions from (1)will reduce ( 7 ) to

71 ( 2 n + 1) np @ , ( t ) =- COS- - sin ___ -: [ 2 T [ 2 T + 8 ]1

2n + 1 2Tfor all odd values ofk. and to

T 1+ 1 co s [2 1 sin[y2 rr + 1) n t / T ] }(2n + 1 )

for all even valuesof k.(8a) and (8b) reduce toObserve that when and only when 0 = n/2, he expressions

an d

cos [?2 n + 1) in p , ( t ) and p, ( t ) , respectively.Duringanykeying nterval, here s an overlapbetweenp , ( t ) an d p , ( t ) . The constant envelope property of MS K and

the emergenceof instantaneous f l an d f 2 follow from summingp , ( t ) and p , ( t ) :

It is interesting to explore the consequences of allowing 8to depart from n/2 in the expressions (8a) and (8b). The sump e , ( t ) an d p e e ( t ) will not have a constant envelope, as Fig. 4illustrates for n = 2 and 8 = O..The terms-__ sin (5 - 0 ) * co s(7)n + 1 7rt

2 n + 1an d

1

in (sa) and (8b), respectively, become undesirable for 0 f n / 2


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436 IEEE TRANSACTIONS ON COMMU NICATIONS, APRIL 1977A -0 ---A - T 2TT

A T0 - I C

-A T 2T 3T

A , /-DESIRED

-A ' T 2T 31

DESIREDA

0

-A

A -UNDESIRED

0 .--..L- I II-A T 2TT

,--DESIREDA /

0

-A

A - - NDESIRED A .0 - II 0 .

-A T 2T 3T -A 1 T 2T 3T

A - TRANSMITTEDOTAL A0 - 0

Fig. 4; Effect of allowing e = 0 when n = 2 andA = -AT / 7~ ( ( 2n 2 ) / ( 2 ni-)) . Fig. 5. Effect of allowing e = 0 when rz = 16 an d A =T/ n ( (2n + 2 ) / ( 2 n+ 1)).

because they conflict with the desired envelope. However, forlarge n , the factor - / ( 2 n + 1) causes the undesired terms tovanish, eaving the resulting MSK synthesis ndepedent of 0 .Fig. 5 illustrates his for n = 16 by showing he desired andundesirederms or 6 = 0. Thedeparture rom nvelopeconstancy is considerably reduced.

111. ANALYSIS OF THE DEMODULATORDemodulationof he ype hown nFig. 1 has onven-tionallybeenpredicatedon he ss umption hat hedata

bandwidth is small compared with the band center frequency.That assumpti on will be maintained for he present purposesof analysis; equivalently, Iz % 1. Now the phase of the receivedwavelreckoned with respect o f, is called a( t) .According to th eMSK format, a( t ) s linear with time over each keying interval,traversing just *7~/2with respect to an apparent carrier at f,.The + sign indicates hat f2 was ransmitted and he - signindicates that fl was transmitted. Now the phase at the mixerout put will be a( t )- 2 m t , where s = fl - f, = -1/4T.Low-pass filtering produces a baseband wave equal o cos

The waveform relationships in the demodulator are shownin Fig. 6, where all phase plots are referenced to the apparentcarrier phase 2rrfct. Here a data sequence is trac ed thro ugh thestepsofmodulation nddemodulation.Reading romhebottom, Fig. 6(a) shows first the data stream. Fig. 6(b) and thesolid line in Fig. 6(c) s how the instantaneous frequency1 andinstantaneousphase of the rans mitte d signal. The eceivermatched ilter is assumed bsent or he imebeing.Thedashed ine nFig.6(c) s he nstantaneousphaseof hereceiver local oscillator . It "slips" 90" per bit with respect to aphase axis which represents an apparent carrier at f, at zerophase.Th eneed or hecorrectstartingphase of the ocaloscillator is typical of any coherentPSK system.Fig. 6(d) shows the instantane ous phase difference betweethe ocal oscillator fl and he receiv ed signal. This diffe renceeitheradvances 180" over a bit nterval~or remains constantover that interval. The cosine of the difference phase is, then,the ecovereddata,Fig.6(e).This ecovery sverifiedbycomparison with Fig. 6(a).It is interesting to analyze by tracing individual pulses that

transmitter. otherwise.


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C O N C I S E P A P E R S 437

0 1 0 1 0 0 0 0 1 0 1 1 1 0 0RECOVERED BIT STREAM

I

' I PHASE ATIXERUTPUTI II

-180 -PHASE TRANSMITTED

Fig. 6 . Phase relationships. (a) Bit stream. (b) Frequency transmitted. (c) Phase transmitted. (d) Phase at mixer output.(e) Recovered bit stream.

the demodulator has the demonstrated effect. This is done byrepresenting hemodulation-demodulation ormat gain nterms of pulses, as in Fig. 7.

The half cosinepulses nFig.7(d)and e) epresent heconventional n-phase and quadrature pulse rains familiar oMSK. These pulse trains are called I ( t ) and Q ( t ) , respectively.With fl used, the new in-phase and quadrat ure outputs, calledI l ( t ) and Q l ( t ) , are given in terms of t he c onventional( t )andQ ( t ) y a known [ 6 ] ransformation:

I l ( t )= I ( t ) cos (27rst + YO ) -tQ ( t ) in (27r3t + a o )Q l ( t )= Q


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438 I E E E TRANSACTIONS ON COMMUNICATIONS, APRIL 1977RECOVERED BI T STREAM

I 0 1 0 1 0 0 0 0 1 0 1 1 1 0 0I ( t ) COS 2nSl I Q(t) SI N Pnrt

I I l t ) IN-PHASEULSES

I Q(t1 QUADRATUREULSES

2 FREQUENCY(b ) I l l 1 I 1

1 - IBIT STREAM

(a )

0 1 0 1 0 0 0 0 1 0 1 1 1 0 0Fig. 7 . Pulse synthesis, recovered bit stream. (a) Bit stream. (b) Frequency. (c ) Phase. (d) Quadrature pulses. (e) In-phase pulses. (f) cos 2nst and sin 27rst. (g ) Z( t ) cos 27rst and Q(t)sin 27rst. (h) Recovered bit stream.

R ( t )= - p ( t - x ) p ( - x ) dxT -=

where

p(x) = cos-71X2 T - T < x < T= 0, otherwise (14)

andwhere he actor1/Tpreceding he ntegralnormalizesthe peak ofR ( t ) .

Routine manipulations reduceR ( t ) o the form

= 0 , otherwise(15)

as llustrated in Fig. 8(a). Demodulation with f l will convertR ( t ) into R l ( t ) = R ( t ) co s 2ns t , plotted nFig.8(b).Th edemodulatoroutputpulse tream will thenhave he ormC a k R l ( t - k T ) , asplotted nFig. 9. Note hat he freedomfrom intersymbol interference inherent in R ( t ) s preserved inR l ( t ) because R l ( t ) haszerocrossingsat headjacentbitsampling imes t = rtT. The peak valueofunity is alsopre-


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C O N C I S E PAPERS 439

-2.00 -1.50 -1.00 -0.50 0.00 0.50.00.50 2.06 2.50TIME IN B I T INTERVALS

(b)Fig. 8. Effect of matched filtering o n pulses. (a ) Matched filtered pulse envelope. (b) Pulse contribution to recoveredbit stream.

. ~0.00 2.00 4.00 6.00 8.00 10.00 12.00

TIME IN BIT INTERVALSFig. 9 . Matched filtered recovered bit stream.

2.00

1.oo

0.00

-1.00 . V I \ I0 1 0 1 0 0 0 0 1 0 1 1

-2.000.00 2.00 4.00 6.00 8.00 10.00 12.00

TIME IN BIT INTERVALSFig. 9 . Matched filtered recovered bit stream.

served from R ( t ) to R l ( t ) . These roperties onfirmheearlier statements that the MSK modulatio n and demodula tionsystem presented here has the same communications efficiency2as conventional coherent biphase.

I V . EXPERIMENTAL RESULTSSeveral transceivers were built incorporating the modulator

and demodulator approac h shown in Fig. 1. These used an f,of 8 2 . 5 MHz as nntermediate requency nd eyinginterval T of 0.1 ,us. Both the pulse shaping filters h ( t )and thedatapulsematched ilterweresurfaceacousticwavedevicesconstructed on lithium niobate substrates. The local oscillatorfi was at 80 MHz for both he ransmitter and he receiver,thus producing 17 = 16.The pulse shaping filter used a wide-band input transducer,and it had an output transducer with an impulse response ofapproximately17half-cycles t 85 MHz. The imedomainresponse of this filter to a 0.1 ,us pulse at 80 MHz is shown in

Communications efficiency is measured by error rate performance.

Fig.10(a).Thepulseshape is very close toacosineshapedenvelope, and the instantaneous frequency within the pulse isprecisely 82.5 MHz. Fig.10(b)shows he requencydomainresponse of the same filter, with a repetitive pulse input. Whendriven with a pseudorandom data bit stream, the filter producesa earlyonstant utputnvelope,ndheirstpectralsidebands are approximately 2 3 dB down from the main lobe,just as they should be for an ideal MSK signal. Because of th erelativelyhortransducerengths,ooticeableilterdegradationwas bserved verhe peratingemperaturerange from o t o SO'C.As a result of additional filtering included in the transmittertoeducenterferenceo djacent hannel sers,otalamplitude modulation of about five percent was ntroduced.No attemptwasmade ovary hephaseof he witchingtransients.However, t is suspected hat he elatively argenumber of half-cycleslarge I ? ) andheelativelylowswitching (about 5 ns) would have made it difficult to demon-strate any significant relationship between switching phase andamplitude modulation.


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- (b) . (b)Fig. 10 . Transmitted signal characteristics. (a) Datapulseenvelope. Fig. 1 1 . Characteristicsof the matched filter.(a) Impulse response ofScale: 0.05 ps/div. (b) Transmitted spectrum. Scales: horizontal, the matched filter. Scale: 0.05 ps/div. (b) Frequency responseof the5 MHz/div; vertical, 10 dB/div. (Center frequency of the display s matched filter. Scales: horizontal, 5 MHz/div; vertical, 1 0 dB/div80 MHz.) (Center frequency of the display is 82.5 MHz.)The receiver pulse matching filter used an input transducer

with 1 6 half-cycles at 80 MHz and an output transducer with17 half-cycles at 85 MHz. The time domain impulse responseand the s wept CW spectral response of this filter are shown inFig. 1 1 . Again, the main lobe and the first sidebands are veryclose t o predictions. The higher order sidebands are relativelyunimportant both because of heir small energy content andbecause the other receive filtering suppresses them anyway.The transceivers sampled the low-pass filtered outputs andperformed further digital processing on th e signals, which werehighly redundant. Because of the complications due to specialfiltering ddedoheransceivers nd ueohe igitalprocessing, no meaningful bit error rate data are available.

V . CONCLUSIONS

bit rate is high, the phasing of the transmitter local oscillato rwith respect to the data keyin gs not critical.

Th eattendantconcept hatal lpulseson hechannelbeidentical n form was anticipated n a previous work [ 7 ] oncoherent quadrature modulation.

ACKNOWLEDGMENTThe authors are indebted to F. L. Morse for the analyticalderivations presented herein and for his careful reading of the

manuscript.

REFERENCES[ 11 M. L. Doelz and E. T. Heald, Minimum-shift data communicatiosvstem. U.S. Patent 2 9 7 7 4 1 7 .Mar. 28. 1 9 6 1 .

The MSK transmission chemepresentedherein equires [2 ] W . A . Sullivan, High-capacitymicrowave system for digital datafewer components (mixers, filters, and digital logic elements) transmission, IEEE Truns. Commun. (Concise Papers),vol. COM20, part I, PP. 4 6 6 -4 7 0 , June 1 9 7 2 .than previously proposed schemes. When he ratio of carrier o [ 3 ] R.. DeBuda; Coherent demodulation of frequency-shift keyin


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CONCISE P AP ERS 441with low deviation ratio, IEEE Trans. Commun. (ConcisePapers), vol. COM-20, part I, pp. 429-435, June 1972.[4] H. R. Mathwich, J . F. Balcewicz,and M. Hecht, The effect oftandem band and amplitude limiting of the Eb/No performanceof minimum (frequency) shift keying (MSK), IEEE Trans. Com-mun., vol. COM-22, pp. 1525-1540, Oct. 1974.[SI F. Amoroso, Pulseand spectrum manipulation in the minimum(frequency) shift keying (MSK) format, IEEE Trans. Commun.(Corresp.), vol. COM-24, pp. 381-384, Mar. 1976.[6] -, Note on coherent quadrature modulation, IEEE Trans.Commun. Technol., vol. COM-17, pp. 581-583, Oct. 1969.[7 ] -, Bandwidth efficient modulation for voiceband data trans-mission, Telecommun., vol. 2, pp. 19-24, Apr. 1968.

Linear Minimum Mean-Square Error EstimatorsApplied to Channel EqualizationA . CANTON1 AND P. BUTLER

Abstruct-This concise paper investigates various aspects of the ap-plication of unbiased linear minimum mean-square error (ULMMSE)estimators to the equalization of channels used for digital data trans-mission.One application is to the equalization of the channel to a responsefree from intersymbol interference. Previous results regarding this appli-cation are clarified andextended.The other application is to the equalization of the channel to afinite short memory response. This stems from .an interest in reducingthe complexity of the Viterbi algorithm. Various new results on theoptimum ULMMSE equalizer and its performance are presented.In both applications, the optimum ULMMSE equalizer is a stable,

secondproblem,somemeans of deconvolving he output ofthe equalizer is still required. The Viterbi algorithm could beused to carry out estimation of he symbol sequence n hislatter situation.Theapplicationof appeddelayequalizers of both finiteand infinite length to the second problem has previously beenconsidered by Qureshi and Newhall [7] , Fal cone r and Magee[4], and Cantoni and Kwong [SI, [61.

For the general equalization problem, the optimization ofboth the equalizer and best linear combination of symbols isconsidered. The coefficients of the best linear combination ofsymbols orrespondo initememory esirablempulseresponse (DIR) of the equalized channel. The solution of thisjointoptimizationproblem is presente d,and esultson heperformance of theequalizer as a unction of equalizationdelay and DIR memory length are derived.The optimum ULMMSE equalizer is shown to be a stable,realizable , finite dimensional recursive filter. The optimal DIRis shown o be he solut ion of a n eigenvector problem. Thislatter esults imilar to he esult or appeddelay ineequalizers as reported n [ 4 ] and SI.Th eadvantageof heequalizerdeveloped n hisconcisepaper s hat t is finitedimensional and its performance is shown to be a monotonicfunction of both equalization delay and DIR memory length.This esultdoesnotnecessarilyhold or inite appeddelayline equalizers in the same situation. It is not valid to make acomparison with an infinite apped delay ine as it does nothaveealizablempulseesponse.t is shown, owever ,that heperformanceof he ULMMSE equal izerapproachesuniformly the performance of the unrealizable infinite tappeddelay ine.A close relat ionbetween he woDIRs n helimiting case is also established by invoking some recent results[ 131 onhe roper ties of eigenvectors of persymmetri c

realizable, finite dimensionalecursive filter. matrices.I. INTRODUCTION 11. CHANNEL MODEL

of the ULMMSE equalizerProblemconsidered n heabove- where = ( h o , h l , ...,hN - - l ) is the channel impulse responsementioned eferences.Rather hanattempt ominimize heMSE of the qualize r utput omparedwith ome ingle vector nd { n h } is a equence of independentdenticallydelayed nputsymbol(no ntersymbol nterferenceequaliza- distributedGaussianrandomvariableswithdistribution N(0,tion), the problem considered is to minimize the MSE of the u2) . The input sequence {uk} is assumed to be an uncorrelatedequalizeroutputcomparedwitha inearcombinati on of a sequence of m-ary symbols with zero mean and variance 2 .finitenumber of inputsymbols.The ormerproblem, sof It will be convenient to work with the state space represen-course, a special case of t he la tter. In the first problem, symbol tation of the channel, defined byby symbol detection with a slicer is usually used, while in the x ( k + 1 ) = @x ( k )+ G U h + l

Paper pprovedby the Editor for Communication Theory of the. y ( k ) = ~ x ( k )n ( k )IEEE Communications Society for publication after presentation atthe International Conference on Information Sciences nd Systems, wherePatras, Greece, August 1976. Manuscript received April 12, 1976; re-vised November 15, 1976. This work was supported by the AustralianResearch Grants Committee and the Radio Research Board.The authors are with the Department of Electrical Engineering, Uni-versityfewcastle, New South Wales, Australia. G = [ l , O ; - , O I TX ( k ) 6 [uk , uk-1. .., uk-N+1] Ta

simplified msk signal techniques

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