cpfsk coherent detection

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vol. COM-19, pp. 113-119, Apr. 1971.communicationystems, I E E E T r a n s .Commun. Tcchnol.,[I61 A . J. Viterbi, Phase-locked loop dynamics in the presence ofnoise by Fokker-Planck techniques, P ro c. I E E E , vol. 51, pp.[17] R. W. Lucky, J. Salz, and E. J. Weldon, Principles ojData1737-1753, Dec. 1963.[IS] D. D. Falconer and R. 1. Gitlin, Bounds on error-patternCommunica t ion . New York: McGraw-Hill, 1968.probabilities for digitalommunlcationsystems, I E E ETrans . Commun. , vol. COM-20, pp. 132-139, Apr. 1972.[I91 W. Feller, An Introduction to ProbabilityTheoryand I t s A p -plications, vol. 2. New York: Wiley, 1966.[20] 11. Matyas and P. J. McLane, Data-aided track ing loops for

Proc. 1973 In t . Com munications Conj., pp. 33-8-33-13.channels with phase jitterand intersymbol nterference, in*Robert Matyas (S70-M73) was born inMontreal, P. Q., Canada, on December 10,1949. He received the B.Eng.degree in electri-cal engineering from McGill University,Montreal, in 1971 and he M.Sc. degree inelectrical engineering from Queens Uni-versity,Kingston, Ont.,Canada, n 1973.Concurrentwith his graduate tudies a tQueens University, he was a Teaching andResearch Assistant wit h the Depart ment ofElectricalEngineering.His area of researchas a graduate student centered on t,he effect of phase error on digitalcommunications. Since joining the Eart h Sta tion ngineering Group,

Telesat Canada, Ottawa, Ont., Canada, in973, he has been involvedin the implementation of a time division multiple access (TDMA)link in the Canadian domestic satellite system.*Peter J. McLane (S68-M69) was born nVancouver, B. C., Canada, on July 6, 1941.He received the B.A.Sc. degree from the Uni-versity of Briti sh Columbia, Vancouver, in1965, the M.S.E. degree from th e Universityof Pennsylvania,Philadelphia, n 1966, andthePh.D . degree from the University ofToronto, Toronto, Ont.,Canada,n 1969,all in electrical engineering. At the Universityof Pennsylvania he held a Ford FoundationFellowship and a t the University of Torontohe held a National 12esesrch Council of Canada Scholarship.From 1966 to 1 )67 he was a Junior Research Officer with theNational Research Council of Canada, Ottawa, Ont., Canada. Heheld summer positions with this organization in 1965 and 1966 andwith the Defence R.esearch Board of Canada in 1964. Since 1969 hehas been a faculty member with the Department of Electrical En-gineering, Queens University,Kingston,Ont., Canada, where heis currently an Associate Professor. His research in teres ts are nsignal processing for ommunications and adar nd n vehiclecontrol in transportat ion systems. He has served as a consultant on

research problems with the Canadian Department of Communica-tions and he Canadian Institut e of Guided Ground Transport a tQueens University.

Coherent and Noncoherent Detection of CPFSKWILLIAM 1. OSBORNE, MEMBER, IEEE, AND iVTICHAEL B. LUNTZ

Abstract-Continuous phaserequencyhiftkeyingCPFSK)is potentially an attractive modulation scheme for use on channelswhose performance is limited by thermal noise. In this paper resultsfor the performance available withCPFSK are given for coherentdetection and noncoherentdetection withrbitrary modulationindices and arbitrary observation intervals.

This work serves two purposes. First, it provides interesting, newresults for the noncoherent detection of CPFSK which indicate thatthe performance of such a system an be better than the erformanceof coherent PSK. Secondly, t provides a complete analysis of theperformance of CPFSK at highS R s well as low SNR and therebyunifies and extends the results previously available.

I INTRODUCTIONN SEVERAL recent papers the performanceain avail-ablebymultiplebitdetection of continuousphasefrequencyshift keying ( CPFSI I) signals has been dis-cussed. Pelchat et al. have discussed the distance proper-tiesand, hence,high SNR performance of coherentlydetected CPFSII waveforms for two and three bit obser-Theory of the IEEE Communications Society for publication with-Paper approved by he Associate Editor for Communicationout omi presentaiion. Manuscripc recewed Gctober 4 iSZ.The authors are with Radiation, Inc., Melbourne, Fla. 32901.

vation ntervals [l]. In addition, hispaper discussesoptimumcoherentdemodulationwith infinite observa-tion interval. De:Buda [a] has discussed the performanceof coherent CPFSIC with a modulation index of 0.5 andgiven a self-synchronizing receiver structure for this case.lcorney [SI has discussed the use of the Viterbi algorithmfor detection of coherent CPFSK and, in particular, themodulation index0.5 case studied by DeBudas examined.PelchatandAdams [4] have discussed the minimumprobability of bit error noncoherent receiver for the three-bit observation interval and they have shownhat the owSNR performance an e stimated y the averagematched filter concept. In this paper receiver structureswhichminimize theprobability of biterror for bothcoherent and noncoherent detectionor arbitrary observa-tion intervals are presented. The performance of both thecoherent and he noncoherent demodulators s boundedemploying the average matched filter concept at low SNRand employing the union bound a t high SNR. This com-bination of bounds forms a performance bound which is agood estimate of t he performanceavailable with hesereoeivers a t all SNRs.The papers organized in hree major ections. These are


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1024 IEEE TRANSACTIONS ON COMMUNICATIONS, AUGUST 1974

coherentdetection,noncoherentdetection,anda um-mary. In th e irst two sections the receivers are presentedfollowed by low and high SNR bounds. In the final sectionthe results of t he first two sections are discussed. In addi-tion, the realizability of t he various demodulators is dis-cussed briefly.COHERENT DETECTION O F CPFSK

The detection problem o be ddressed in this paper on-sists of observing n bits of a CPFSK waveform and pro-ducing an opt imum decision on one bit . In the coherentcase, the decision is made on the first bit by observingthe waveform during this bit time and n - 1 additionalbit times. The data are assumed to be random =ti's andthe interference is additive white Gaussian noise.The CPFSK waveform during the first bit interval canbe expressed as

exp 6T( t ) s ( t , l , A ) t f A ) dA)= 3)

/ A exp ($lT( t ) s ( t , - 1 , A ) d t f A ) d A)where the integral S A dA is taken to mean th e n - 1 foldintegral L2 ,. .lna2 da3. * .dun.The density of A is given by, f A ) = f a2) f a3) * - f ( a n ) ,where f a i ) is the density function of the ith data bit , andthe data bits are ssumed to be independent. The densityfunction of the random data bits is given by,

f ( U i ) = +6 Ui - 1) + + 1). 4)Using (4) in 3 ) and carryingout he ntegration, helikelihood ratio becomes

=exp ( / r ( t ) s ( t , - - l ,A) clt +...+ exp ( / ~ ( t ) s ( t , - l , A , ) dtnT nTNo 0 fi-0 0

where al is the data, , is the phase of the RF carrier at thebeginning of the observation interval, and h, the modula-tion index, is he peak-to-peak frequency deviation dividedby the bit rate. Inccord with the continuity of phase, thewaveform during the ith bit timef the observation inter-val can be written as

U i T h t - i- ) T ) i-lT + C aj*h + e,j=1

i - l ) T 5 t 5 iT. 2)The objective is to design a receiver which observes nbit times of data and uses the fac t that the carrier phaseduring the ith bit time epends upon the data in the irstbit t ime to minimize the probability of bit error. For thecase of coherent detection to be treated in thi s section, 81is assumed known nd set toero with no oss of generality.

In thenext section the noncoherent case is treated wherein 8, is assumed to be a random variable uniformly dis-tributed between h ~ .Let he signal waveform during the observation n-terval e enoted y s t,ul,Ak) where A k representsa particular data sequence, .e., i t represents the n - 1tuple { a2,u3 . , a n ] , and he actual waveform sagaingiven by 2 ) . The detection problem is then to observes ( t , a l ,Ak ) in noise and produce an opt imum decision asto thepolarity of a] .The problem stated in this mannersthe compositehypothesisproblem reated n [5] andother texts. This solution is known to be the likelihoodratio est and for the CPFSB waveform the likelihoodratio, I can be expressed as

wherem = 2n-1.

The receiver s tructure defined by (5) is shown in blockdiagram in Fig. 1. The receiver correlates the receivedwaveform with each of the m possible transmitted signalsbeginningwith data 1, then forms the sum of exp c j )where c j is the correlation of the received waveform withthe j th ignal waveform beginningwith a data 1. A similaroperation of correlating and summing for the m possiblewaveformsbeginning with a data -1 isperformed andthe decision is based on the polarity of the difference inthe two sums.

PERFORMANCE O F THE COHERENTDEMODULATORThe performance of the optimum demodulator shown

in Fig. 1cannot be computed analytically. However, itsperformance can be bounded by two bounds. ne bound istigh t at high SNR and the other is tight a t low SNR.These bounds taken as a single bound are a reasonablygood performance bound at all values of SNR.U p p e r B o u n d on Performance-Low SNR

The receiver presented in the previous section computessums of random variables of the form

At low values of E b / N o he random variable X l k can beapproximated by


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OSBORNE AND LUNTZ: DETECTION OFCPFSK 1025-0- T - X P ( )7x1 11P 1 and for i = 1 E,

is given byT2E1 = (1 - Sinc 2h) ) . 2 2 )

Upper Bound on Per formance-High S N RThe equations presented above can be used to evaluate

a bound on the performance of CP FS K at low SNR s.These will be used in conjunction with the union boundwhich is tigh t at high SNR s to provide the compositebound. The probabi lity of error for the opt imum receiveris overbounded by

where xlz s output of the correlator matched to thesignals ( t : ,A 1 ) . Further,

where1 nTp ( Z , j ) = s ( t , - l , A l ) s ( t , l , A j ) dt.nEb

Thecorrelation coefficient p( Z , j ) canbeevaluatedbyusing ( 1 9 ) for the signal waveforms, integrating one bita t a time, and.summing the results over the observationinterval. Carrying out thisrocess, p 1 , j ) can be written as

where the a's are the data bits t , the b's are the data bitsAi, and where al = 1 and bl = - 1.Lower Bound on Performance

A lower bound on the performance of t he coherentCPFSK receiver can be obtained by supposing th at foreach ransmitted sequence the receiver needsonly todecidebetween that sequence and ts nearest neighbor.This receiver will perform at least as well as the receiverwhich does not know which of two sequences was trans-mitted but must compare with allossible sequences. Theperformance of this receiver is a lower bound to the per-formance of the optimumeceiver presented in he previoussection. This lower bound on the probability error in theCPFSK receiver can be written as,

Pr 1 2 5=1 Q (?$ ( 1 - * ( Z ) ) ) ) 27)where p * ( I = maximum of p ( Z , j ) over all j .Numerical Results-Coherent Case

In the revious section three bounds onhe performanceof a coherent CPFSK system with observation n'terval oflength of n were presented. The average matched filterbound is an upper bound on performance, which, by itsconstruction, should be an approximation to the true per-formance at low SNR. The union bounds an upper bound


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Eb/No IN dBFig. 4. Bounds on performance of CPFSK.

which is known to be tight at high SNR, and the boundgiven by (27) is a ower bound on performance at anySNR.In order to illustrate the usef these bounds o estimateperformance of coherent CPFSN systems all three wereevaluated and plotted for an observation interval of fivebits and modulation index, h, of 0.715. These results areplotted nFig. 4.FromFig. 4t canbe seen tha t thecomposite upper bound constructed by taking the smallerof the average matched filter bound and the union boundconverges to the ower bound at high SNR and, in fact,orerror rates less than heseareessentiallyequal.Thecomposite upper bound is within.5 dB of t he lower boundat all SNRs showing that the composite bound is a goodapproximation to he rue receiver performance a t allSNRs and is tigh t at high SNRs. Fig. 4llustrates thegoodness of the three boundsonly for one set of param-

eters, however, the authors use of these bounds in severalcases has shown similar results, i.e., the composite upperbound s a good pproximation to rue receiver per-formance at all SNRs. Further evidence of this is shownin the noncoherent section in the orm of a comparison ofcomputer simulation results with this bound.The modulation index of 0.715 was selected for evalua-tion because in [a] it was shown that themaximum valueof the minimum distance over all transmitted words for aCPFSK signal wasachieved by using thismodulationindex. In Fig. 5 theperformance of CPESII with hismodulation ndexversus the ength of the observationinterval is illustrated. The curves in Fig. 5are the com-positeupperbound esults for the variousobservationintervals. The resultsshow tha t li ttl e gain is available byusing observation intervals longer than three bi ts at anySNR. Again, this behavior is a characteristic of CPFSK


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OSBORNE AND LUNTZ: DETECTION O F CPFSK 1029

systems ndependent of modulation ndex, i.e., in othercases investigated th e gain achieved by using an intervalof more than three bits s very small. As has already beenpointedout n [a] but isagain llustrated nFig. 4,

phase must be taken and also in that the decision is per-formed on the middle rather than the irst bit. Performingfirst theexpectation overall transmitted sequences aswas done to obtain 5 ) , he likelihood ratio becomes,,- m

CPFSK with a modulation index of 0.715 does performbetter than coherentPSI


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1030 IEEE TRANSACTIONS O N C O M M U N I C A T I O N S ,AUGUST 1974

Eb/No IN dBFig. 5. Upper bounds on CPFSK performance.

anduadratureomponents of each of the possible trans- m mmitted signals. For each ossible signal the receiver forms c {1 + 1P} > c 1 + NO 2 Z-l? } 35)theootumquare of the inphasenduadrature com- i=l i=lponents and weights this root with an Io * ) nonlinearity. implies a 1was transmitted. Upon simplification thisThe sum of these numbers for all signals with a da ta one processor becomesin themiddle bit interval s compared with the sum for allsignals with a data - in the middle bit interval.Noncoherent Receiver Performance

1

m de c ide 1 mc XI? < c 2-li. 36)6 1 d e c i d e -1 i=l

N o closed form analytical solution for the performanceof the noncoherent eceiverexists.However, as is thecase or the coherent eceiver, the performance of thereceiver may be bounded. This bound, which is tight athigh and low SNR may be determined analytically. Thebounds on the performance of the noncoherent receiverare constructed n a manner similar to that sed to analyzethe coherent receiver in the previous section.Low SNR Bound

The low SNR approximation to the optimum receivermakes use of the fac t that for small arguments

It maybe shown th at he low SN R approximationprocessor described by (36) is mathematically equivalentto a pair of complex correlators. One correlator has as i tsreference the average of all transmi tted waveforms con-taining a data 1 in hecenterbit ntervalTheothercorrelator reference is he average of all waveforms with adata - in that bit interval. Thus, a test equivalent to36) is

where& ( X ) = 1 + x2/4. (34) ms ( t , l ) = exp [jmot + + l i t ) ]Making this approximation in 31), describing the opti-mum processor,yields the low signal-to-noiseprocessor and i=l


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O S B O R N EA N DL U N T Z :D E T E C T I O N OF C P F S K 103

z - l l

ms ( t , - l ) = C exp [ j u d + + - d t ) ] .The performance of this test maye computed by applyingthe results of Stein for the solution of t he general binarynoncoherent problem [SI.If z1 and zzare two complex Gaussian variables with

i=l

. M I = E z l )M , = E z2)

u2 = Var (21) = Var 22)and1P = > E C Z l - M l ) * ( Z 2 - M Z ) ]

s ~ t , - l , A ' 1 1 )' 2Fig. 6. Optimum noncoherentreceiver.

thenPr ( I zz l > I z1 2 = 4 [l - Q (b1/2,a1/2) Q ( ~ l / ~ , b l / ~ ) ]

(38)where

The minus of the sign is used with a and the plus isused with b. The function Q (z,y) is the MarcumQ functiondefined by


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1032 IEEE TRANSACTIONS ON C O M M U N I C A T I O N S , AUGUST 1974

3 t J )Q(z,y) =lrnxp (-y o z w ) wdw 40) = COS sh t - T) (cos sh)--l exp ( j sh )where lo is the modified Bessel function. For a given ~ t , - l ) Jinput signalaveform, cos (mot + 8 ( t ) ) , the received T 5 t 5 i + l ) T (42)signal is r t ) = cos (mot + e ( t ) ) + n t ) . The variablesz1 and Z2, are the resultf correlating r ( t ) with s t ,- 1) and where the + sign in the exponential implies s ( t ,1).

Upon reversing the time axis, it is found by symmetry(39) considerations that during the thbit nterval precedingtlhe middle bit the average waveform iss t , l ) . Hence, the variables required o evaluatebecome,

= / s t,-1) pd t ,2and

= - / ~ * ( t , - l ) s ( t , l )O dt.

s t , - l ) = S t , l ) = cos s h t c 0 s sh)i - ) T 5 t 5 iT. (43)

These equations may be used to compute u2 and p . Using(42) and (43) the integrals in (41) can be written as,

and= -

Upon performing the indicated integrations and simplify-ing, it is found t ha t

The complex correlator references s (1,- 1) and 3 t , l )may be found na manner similar to th at used in thecoherent case. It is assumed t ha t all possible signals havezero phase a t th e beginning of the middle bit and thattime t = 0 corresponds to th e beginning of the middle bitinterval. For continuous phase FSK with a modulationindex of h, during the middle bit the signal is exp sh t )where the plus sign is used for a data one and the minusused or a data - . During the nextbit nterval heaveragewaveform is half thesum of the twopossiblewaveforms, orS t , l ) = [exp , j sh t- T ) ) + exp ( - j s h ( t - T ) ) ]

-exp j s h ) T 5 t 5 2Tfor a data one in the middle interval ands(t,- l) = [exp j s h t- T ) ) + exp ( - j s h ( t - 7))

.exp - j sh ) T 5 t 5 2Tfor a - . In general, during the ith bit interval, after themiddle bit, the average waveforms are

and(1 + sinc 2h) 1 + exp - j2sh)) 1 - C O P (sh)1 - cos2 (sh)

+ exp ( - j h ) sinc h .The mean outputs, MI and M z , are dependent upon theinput signal. Let the input bit sequence be { b i ] with theindex ranging from -n to n. The middle bi t is, therefore,bo. Computation of M 1 and isperformed in hesamenlanner as before by computing the contribution due toeach bit interval. Thus for bo = 1

nill = xp (-jsh b - k ) cos (cos sh)i-1

i= 1 k = lnexp ( -jb-;sht) dt + 1 + exp ( - j sh bk)i-1

i=l k = l


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O S B O R N EA N DL U N T Z :D E T E C T I O N OF CPFSK 1033

andn i-1M1 = exp ( -jah b - k ) cos aht (cos nh) -l

i=l k=l I,n+ exp - j2nh) C exp ( -jah b k )i-1i=l k=1

I,cos nht (cos nh) exp ( -jbirht) dt. 47)In the above equations, the sum CiSyxi is defined to bezero. Evaluating the integrals of these equations yields

M 2 = A 1 + 1 + A 2and

M l = A l + exp ( -jab sinc h + exp ( -j2nh) 48)where

nA, = (cos ah) -l exp ( -jah C b k ji-1i=l k-1

(1 + sinc h )exp ( -jahb-i j )andAz = (cos ah) -l exp ( -jah C b k )n i-1

i=l k=l1 + sinc h )exp -jnhbij). (49)

When these equations are evaluated on a digital computer,a bound on the optimumreceiver at low SNR is obtained.This bound s equivalent to the average matched filterbound shown n Fig. 4 for the coherent receiver. In thenextsectionaunionbound will befoundwhich,whencombined with heaveragematched filter bound, willyield a composite bound similar to that shown in Fig. 5.

Thedemodulator using thestrategy of 52) could alsochoose the largest of all zki and then classify the largest ascorresponding to a data 1 or a data - . A decision erroris made if, given a one was transmitted, one of t,he zPliwas largest. Although an exact evaluation of the perform-ance of this detector is not possible, the union bound willgive a tightperformance estimate at reasonably high SNR.Suppose tha t a 2n + 1bit transmitted word is observedand hat he middlebit sadata 1. The ransmittedsequence, exclusive of the middle bit, is indicated by theindex k so th at an error s made if a t least one of the{ x-lj] is greater than l k . Then by the union boundPr (Error 1 Sequence k Transmitted) 5 Pr xWlj> Z l k ) .7n

j-15 3 )

The average probability of error maynow be computed byaveraging over all transmitted equences containing a onein the middle bit interval,Pr ( E ) = 5 Pr ( E I sequence k was transmitted)nl. h.=l

or1 m m

Pr ( E ) 5 C Pr (z-lj > z l k ) . (54)m k=l illIn (54) the computat ion of the bounding performanceof the detector described by 5 2 ) has been reduced to abinary error probabil ity problem for which the solution isknown [C]. For this situation the probabili ty of error is

Pr z- l j > X l k j ,= 1/2[1 - Qb12,a12) Q a12,b12j]5 5 )whcrc

High XNR BoundThe high SNR bound may be found by noting that orlarge arguments c lO 4* o z2) (50)

iwhere 22 is the largest of the set{xi}.With this approxima-tion, the optimum detector described by (3 1 ) becomes

Decide 1Io i o lZ) e, e -1 I o($ -lk) (51)where zlz is the largest of ( z l i ) and 2- l k is the largest of

~ - ~ i ] . ecause lo ismonotonic unction, (51) isequivalent to the testD e c i de 1

x11- lk. ( 52 )Decide -1

and S/2N is the SNR of z l k . The value of p is the correla- .tion between the transmitted waveforms corresponding tosequence j , with a data- in the middle bit interval, andsequence k , with a data n the bit interval, ands given by1 Zn+1-1

where { b k ] is hekthbit sequence, with bn l = I , and{ a k } is the jth bitsequence, with an+]= - .Numerical Results-Noncoherent Case

The equations presented in this sectionor the boundingperformance of the noncoherent receiver have beenevaluated on a digital computer for three bit and five bit


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Eb/NoFig. 7 . Performance of noncoherent CPFSK receiver.

observation intervals for FSK with h = 0.715. The resultsare plotted in Fig. 7. Also plotted in Fig. 7is the non-coherentdetectionperformance for binaryorthogonalsignals and the coherent detection performance for anti-podal signals. These wo curves represent the bes t per-formance possible with single bit demodulation. Demodu-lation by observing five bits is seen to outperform PSI


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OSBORNE AND LUNTZ: DETECTION OF C P F S K 1035

Eb/No IN d BFig. 8. Comparison of computed bound and simulation results.

simulation shows the differences between the bound andthe ac tual performance to be less than 1 dB at any SNRand much less for most SNRs. Similarly, in the coherentcase a lower bound was shown to differ from this compositeupperbound byabout 1.5 dB worstcase and, hence,demonstrated the goodness of the bound for this case.The equations presented serve to consolidate the per-formancecalculationsfor CPFSK systems n hat heyprovide a technique for computing performance which isapplicable for all SNRs, all modulation ndices and allobservation ntervals. The equationscontainall of thepreviouslypublished esults and, n addit ion, allow theinterestedeader to investigate the performance ofCPFSK systemswithparameters for which previousresults a re not available.The specific numerical results presented for a modulationindex of 0.715 employing coherent and noncoherent detec-

tion serve to answer some questions about CPFSK andtodemonstrate everalpoints. In [l] the question ofimproving the performance of CPFSK at low SNRs byemploying an observation longer than hreebits wasraised. This question is answered by the results in Fig. 5.There s mprovement nperformance a t low SNR byallowing longer observation ntervals, however, the im-provement beyond a three bit interval is minor. In fac t,for engineering purposes the three bit interval appears tobe he optimum ength forcoherent receivers since thegain beyond this length is minimal and the complexity ofthe receiver grows rapidly with the length of th e in t e r~a l .~The specific results presented for noncoherent detectionof CPFSK show a new and rather interesting result. or ahalf, there is no gain beyond observation intervals of one and twoa For modulation indices which are integers or integers plus abits, respectively.


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1036 IEEE TRANSACTIONS O N COMMUNICATIONS, AUGUST 1974

modulation ndex of 0.715 and a five bitobservationinterval noncoherent CPFSKcanoutperform coherentPSI

cpfsk coherent detection

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