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196 IEEE TR ANSAC TIONS ON C OM M UNIC ATIONS, VOL. COM-29, NO. 3, M AR C H

Continuous Phase Modulation-Part I: Full ResponseSignaling

TOR A U L I N , MEMBER, IEEE, A N D CARL-ERIK w. S U N D B E R G , M E M B E R , IEEE

Abstract-The continuoushasemodulation (CPM) signalingschem e has gained nterest in recent years because of ts attractivespectral properties. Data symbol pulse shaping has previously beenstudied w ith regard to spectra, for binary data and modulation index0.5. In this paper these results have been extended to theM-ary case,where the p ulse shaping is over a one sym bol interval, the so-calledfullresponsesystems. Resultsar egiven ormodulation ndexes ofpractical interest, concerningot herformance and spectrum.Comparisons are madewith minimum shift keyingMSK) and systemshave been found which are significantly better n E b / N O for a largesignal-to-noise ratio SNR)withoutexpandedbandwidth.Schemeswith hesamebit error probability as MSK but withconsiderablysmaller bandwidth have also been found. Significant improvement inboth pow er and ba ndwid th are obtained by increasing the number oflevels M from 2 to 4.

I . INTRODUCTIONF R digital transmission over bandlimited channels, the de-mand for bandwidth efficient constant envelope signalingschemes w ith good reliability has increased in recent years. Asystem often used in practice is multilevel phase shift keying,M-aryPSK,whichhas hedrawback that,although, or Mequal 2 or 4, the receiver sensitivity is acceptable, thesignal istoo wide-bandbecauseofdiscontinuousphase.Thus,RF-fdteringhas obeperformedbefore ransmissioncausinganonconstant e n v e l o p e s i g n a l a n d a d e c r e a s e d r e c e i v e r s e n s i t i v -ity. The so-called minimum shift keying (MSK), or fast fre -quency shift keying (FFSK), binary signaling schemes openednew prospects since he error probability performance is thesame as coherent 2 - or 4-ary PSK but the spectrum s narrowerfor large frequencies. Choosing an M larger than 4 (e.g., M =8 or M = 16) in the MPSK system makes the main lobe of thespectrum narrower, but the sensitivity to noise is considerablyincreased.A general definition of continuous phase modulationCPM)systems isgiven in he next section. Assume that each datasymbol only affects the instantaneous frequency of the trans-mitted signal in one symb ol interval and that he phase is acontinuous unctionof ime. This defines hesubclass ullresponse CPM systemsconsidered n hispaper. nPart I1more general CPM schem es are considered. In some cases thephase is allowed to bediscontinuouswhilemaintaining hecoupling between the phase n successive symbol intervals.By increasing M , an interesting radeoff between symbol

Manuscrip t ece ived March 19 , 1980; ev ised September 19 , 1980.This work was supported by the Swedish Board of Technica l Develop-m e n t u n d e r G r a n t79-3594.Th e a u t h o rs a re wi t h t h e De p a r t me n tf Te l e c o mmu n i c a t i o n Th e o ry ,University of Lu n d , Fa c k , S-220 07 Lund, Sweden .

error probability at large signal-to-noise ratio (SNR) and strum is achieved. This trade off is studied for modulationdexes of practical interest and also for systems where thestantaneousrequen cy is not onst ant over eachymbinterval.The channel no ise is assumed to be additive,w h te Gauthroughout hepaper.Th esymbolerrorprobabili ty o ranoptim um detec tor at large SNR is calculated using the mmum Eu clidean distance between any two signals in th e sspace [3] . Theopt imumdetectoroperatescoherently,an ddue o he cont inuous phase , he detector must observe hereceived signal for more than one symb ol interval to m adecision about aspecific symbo l [3] .11. GENE RAL SYSTEM DESCRIPTIONFor CPM systems, the transmittedsignal is

~ ( t ,U)= - OS (2nf0t+ cp(t,a)+ cpo)IEwhere the idormation carryingphase is

cp(t,a)= 2nh I t 5 i g ( r - i T ) d r ; -

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AULIN A ND SUNDBERG: CONTINUOUS PHASE MODULATION-PART I 1

2 n h~- ~ foFig. 1. Schematicmodulator or CPM.pulses. A schematic modulator is shown in Fig. 1 . Note that aCPM signal always has a constant envelope.Defining the baseband phase response (phase pulse)

it is seen that the phase of the CPM signal is formed by

A causal CPM system is obtained if the freque ncy pulse g ( t )satisfies

I & ) S O ; t < O , t > L Tg( t )+O; O

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198 IEEE TRA N S A CTIO N S ON COMMUNICATIONS, V O L . COM-29, NO. 3, MARCH 1where Equation (19)an beritten

and t is not necessary for he detec tor o find the specificsequence G N which maximizes (13) a nd to choose Go as an es-timate of q,. nstead, the detector must find which group ofsequences 6 k , N , = + I , +3, *.-,+(M - 1) jointly maximizes(13), and take Zo as the group belonging. It is believed, how -ever, that for large SNR the two detectors have the same per-formance [7].Theprobabilityofanerroneou s decision canbeupperbounde d by using the union bound [3 ], [5 ], [ l o ]

where

and the summ ation is taken over all pairs of sequences definedby (15), with he restriction hat k # 1, k , 1 = k l , + 3 , .--,+(M- 1). D [(Yk N , (YI,N] is the Euclidean distance betweenthe signals i ( t , a k , N ) an d s ( t , ala) .Th e squared Euclidean distance can be written

Assuming 27rf0T S 1 , this can be wri t ten

= 2 E (N - lNTOS [ 2 ~ h ( f f i k- ffi)N - 1i= 0

- q(t -in dtI )The superscript denotes the value of the first symb ol in a se-quence of N symb ols, i.e.,

Thus, it is sufficient to consider the difference sequence

instead of the pair of sequences (Yk an dThe approximation

p e = r o - Q [ - ]mrnn, Nof (16) is good fo r large E/No. t is now assumed that Eis sufficiently large for hisapproximation to be valid. limitations of this assum ption is considered in detail in [ro s a positive constant, independen t of E/No , and D m iis the minimum of D ( a k , N , a13)with respect to the paisequences and a I3 with he estriction hat k # 1.quan tity can also be calculated using the difference sequeY N through

with the restriction that7 0 = 2 , 4 , 6 , ...,2(M - )yi = 0, +2, +4, ...,+2(M - ); i = 1 , 2 , .a, N - .

In Part 11, we w ill only deal with squared Euclidean distannormalized b y bit energy

n 2

Note that

where Eb is the bit energy. Thus, error probabili ty compasons for large SNR can be made directly n Eb/fVo betwsystems even if they have different M . Only values thatpowers of tw o (M = 2 , 4 , 8, -) will beconsidered. Areference poin t in the following, note hat d k i n = 2MSK, binary PSK (BPSK), andquaternary phase shiftking (QPSK). F or he case of full esponseCPFSK systcalculations of d k i n , N as been considered for both the b

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AULIN A ND SUND BERG : CONTIN UOUS PHASE MODULATION-PART I 19nary case [ 1 ] , [5] - [7] , [I31 and also for M = 4 an d M = 8to some extent [ l o ] , [ I l l , [ 1 3 ] .111. BOUNDS ON THE MINIMUM EUCLIDE AN DISTAN CE

An imp ortan t tool for the analysis of CPM systems is theso-called phase tree. This tree is form ed by all phase trajec-tories q(t, a) having a common start value zero at t = 0 . Theensemble is over the sequence Q! and Fig. 2 shows a part ofthe phase treefor abinary full response CPFSK system.Amore general case is shown n Fig. 3 where two phase treesfor a quatern ary CPM system having different freq uency base-band pulsesg(t) are shown.To calculate the minim um squared Euclidean distance foran observation length of N ymbols, all pairs of phase trajec-tories in the phase' tree over N ymbol intervals must be con-sidered. The phase trajectories must not coincide over the firstsymbol interval however. T he Euclidean distance is calculatedaccording to (21) or all these pairs, an d' th e minimum of theseEuclidean distances is the desired result. It is of great impor-tance to remember th at he phase must always be viewedmodulo 2n in conjun ction with distance calculations. A prac-tical metho d to do this is to for m a cylinder by folding thephase tree [16] [17]. Trajectories which seem to be far apartin the phase tree might actually be very close or even coincidewhen viewed modulo 2n.It is clear from ( 1 8 ) that, for a futed pair of phase trajec-tories, the Euclidean distance is a nondecreasing func tion ofthe observation length N . If just a few pairs of infinitely longsequences are chosen, an upper bound on the minimum Eu-clidean distanc e a t alj values of th e observation interval N sobtained . Good candidates for these infinitely long pairs arepairs that merge as soon as possible. Twophase rajectoriesmerge at a ce rtain tim e if they coincide all the. ime the reafter.These merges are called 'inevitable if the y occur inde pende ntlyof h . Thus, an upper bound on the minim um Euclidean dis-tance is obtained as a. function of he modulation ndex hfor all N .Applying this meth od to th echeme in Fig. 2;it is seen tha tif a pair of sequences is chosen as

1 a + 1 =+l , - l ,a2 ,a3; -a-1 = - 1 , + l ,a2 ,a3 , -28)the two phase trajectories coincide for all t > 2T. Thus, theupper bound on the normalized m inimum squared Euclideandistance isd ~ ~ ( h ) = 2 - - TO S [2nh(24(t)- q(t -Q)] dt (29)

where (21) was used. For thebina ry CPFSK system, i.e.,linear phase trajectories, the result is

/-

Fig. 3 . Phase rees for M = 4 CPM schemeswith wodifferent basband pulses g(t). Th e a, function [see (56) ] with a = 0.25, is p0 is shown by a solid line nd the HCS, half cycle sinusoid [see(SS)is shown by a dashed line.

It can be noted that instead of using the pair of sequenc(28), the single difference sequence 7= +2, -2,0,0, coube used together with (24) for calculation o f the normalizsquared Euclidean distance.

Turning to the quate rnary case, we can find pairs of phatrajectories merging at t = 2T . Fig. 3 shows tw o examples phase trees for this case. These merges occur at the poin ts lbeled A , B , C , D, nd E in the phase tree. Unlike the binacase, there is more than one merge point, and two differenpairs of phase trajectories can have the same merge poiThere are only thre e phase differences however, name ly, thohaving phase difference +2hn7+4hn,.and +6hn a t t = T .It is easily seen that an upper bound on the minimum Eclidean quareddistance for he M-ary case is obtainedbusing the difference sequences

and taking the minimum of the resulting Euclidean distance1.e.,2 TdB2(h)= log,? min { 2 - [ cos [2nh

I Q k G M - 1

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200 IEEE T R A N S A C T I O N S O N CO M M U N I CA T I O N S, VOL. COM-29, NO. , M A R C Hwhich, for theM-ary CPFSK system, specializes to

Fig. 4 shows the minimum distanced2 h) for binary CPFSKas a function of h and N . The upper bound dB2 is showndashed. Note the peculiar behavior a t h = 1 . This will be dis-cussed la ter.It canbenoticed hat or all full esponse CPM systemswith the property

(34)a merge can occur at t = T and thus thedifference sequences

yield the upper bounddB 2(h)= log20

T* min { 1- [ OS [2nh * 2q(t)] dt .1 4 4 M - 1 I

(36)This class of pulses is caged weak [20] ,, and is not consideredbecause distance properties are poor. Only positive pulses g(t)will be considered below. Furtherm ore, they are assumed tobe symmetric with

g(t) = g(T- ); 0 <

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AULIN AN D SUNDBERG: CONTINUOUSHASEODULATION-PART I 20correspondingEuclidean distance might be onhe pperbound dB2(h); for he third orderand higher it is strictlyabove. But from (39) t can be seen that the number, of first-order weak mod ulation indexes grows rapidly with M.For weak mo dulation indexes of the first order, it is suffi-cient to calculate the corresponding Euclidean distance overthe first symbol nterval. Since the calculation of the upperbound uses the two first symbol intervals, it can be concludedthat d2(h,) can be,and normally is, smaller than dB2(h,).Thus, for weak modu lation indexes of the first ord er, dLin(h,)might be smaller than dB2 h,) (for details, see [20] .Further-more, it is shown in detail [17] , 20] that only+,f the firstorder can influence the minimum distance calculation for fullresponse CPM systems.Tightness of the Upper Bound dB2(h)

A powerful property of the upper bound on the minimumEuclidean distance is that except for weak modula tionndexesof the first ord er, hemin imum Euclidean istancetselfequals this bound if the observation nterval is long enough[18] , 20] . Denotingheminimum normalizedquaredEuclidean istance for n N symbol observationntervald& n ,,(h), we have tha t

ifN 2 (h)h + h,: (firstrder). (42)

NB(h) is the num ber o f sym bol intervals required to reach theupper bound for thepecific modulation index h.If a pecific pair of phase trajectories never merge, theEuclideandistance will grow with out imit. This is true be-cause the Euclidean istance alculated over each symb olinterval is positive. Since the mi nim um distance was previouslyshown to be upperbou nded , he pair of phase trajectoriesgiving the minim um distance must eventually merge.For a modulation index near.a first-order weak modulationindex, the difference sequence (35) gives the smallest growingEuclidean distance with N :

Tmin { 1 - $ cos [2nhkyoq(t)]dt1 4 k 4 M - 1

+ (N - ) ( 1 -COS [nhkyo] (43)and the smallest growth rate per sym bol interval is

log, M min { 1 - os [nhkyo] .1G k 4 M - 1 (44)

For sufficiently large N in (43), he minimum distance for theconsidered h is given y dB2 (h)ince d2 (h)ill exceed dB2(h).

The tightness of the upper bound and the behavior of (4is illustrated in all the min imu m distance figures.Optimizationof he Upper Bound dB 2 (h )

For frequen cy pulses g(t) having the symm etry property(37) , the expression for the up per bound can be written

(4and since cos (.)2 - 1 , dB2(h) an never exceed 4 10g2Thus, at most, an improvem ent of 3 dB in E/No for a largSN R might be obtained for the binary ase compa red toMSKFurthermore, (45)can also be written

1G k 4 M - 1

- %,CO S [4nhkqo(t>- hk] t ) ] (46where

The binary case (M = 2 ) will be considered first. In this case,

dB2(h)= 2 1 -2 CO S nh[It can at once be observed that cosnh= 0 fo r h = 1/2,3/2,5/etc. Thus, he upper bound equals two, ndependen t of heshape of the frequency pulse g(t). Actua lly, this is true fo r apulses g(t) (except weak ones) [16] T h s is of particular iterest since much attention has been devoted to the case

To maximize dB2(h) for the binary case, the last term (48) mus t be minimized.Two differen t cases can be distinguished :

h = [ a , 121~ 1 4 1 , 1 5 1 .

Case I:O G h < $ w h e r e c o s n h 2 0 . (4Case11:3 S h < 1where cos nh

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202 IEEE T R A N S A C T I O N S ON C O M M U N I C A T I O N S , V O L .O M - 2 9 , NO. 3 , MARCHpulse, w hich maximized d B 2 @ )or case I , minimizes d B 2 @ )for case I1 and vice versa.To make the integral in (48) as small as possible, the argu-me nt inside the cosine must be as close to n as possible. Thisyields for the interval 0< h < 1

with the.resulting phase responses. . . .

Lan d

( 0 ; t = O

From he sign symm etry, o minimize d B 2 @ ) he phase re-sponses above have to be interchanged with each other or therespective cases.. ' The upperbound d ~ ' ( h ) or he wo phase responses isnow found as

. .

Thus, for any binary scheme,~~ h) s bounded by

11 - COS 2nh < < ~ ~ ( h )

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A U L l N A N D S U N D B E K G : C O N T I N U O U S P H A S E M O D U L A T I O N - P A K T I 20MSK, BPSK e tc .\

Fig. 5. Upper bound comparison for M = 2, 0 S h < 0.5. h e b o u n dfo r q l ( t ) is the upper bound on all upper bounds d s 2 ( h ) ,a n d t h e01 = 0 plateau function is the lower bound on all the upper boundsd&h) in the considered interval.

This pulse g(t) has o ne contin uou s derivative at the end pointst = 0 an d t = T. Since this frequency pulse is a raised cosinefun ctio n, it will be referred to as raised cosine (RC).When h =1/ 2, this scheme has previously been referred to as SFSK [12] .Fig. 5 shows the upper and lower bou nds on d B 2 @ )or allbinary schemes in the region 0 < h < 1/2, computed with(55). Fig. 5 also shows d ~ ~ ( h )or CPFSK, HCS (with form ulain [181), and a, functions with a = 0.25, 0= 0.5 an d a =0.25, p = 0 . The bound for the RC scheme is in betw een thatof QI= 0.25 ,p = 0 and HCS.The problem of finding frequency pulsesg(t) that optimized B 2 @ ) , iven h and M , is far more complicated in the generalM-ary case than for the binary case. This general problem hasnot been solved. The reason is that the upp er boun d is con-structed from the minimum of more than one function, and hvaries with fixed M , different unctions ake heminimumvalue. This is also true for futed h an d M when the frequencypulse g(t) is varied. For a nM -ary scheme with 0 N B ) d $ i , ( hequals dB2 (ho).For the quaternary case, the CPFSK scheme gives a mamum of d s 2 ( h o )= 4.232. In [17 ] it is shown tha t a schembased on the a,0 unction with a = 0 , = 0.17 for h = 0.gives a min imu m distance of 4.62 . This value came out ononexhaustive earch forquat ernary schemes yielding ladistance values. However, better schemes may exist.

IV . NUMERICAL RESULTS ON THE MINIMUMEUCLIDEAN DISTANCEOf course, also for M-ary schemes d B 2 @ ) 3) fo rbinary upper bound dB2 h), nd also how large N has to be made i

[201 . ferent values of N , th eumber of received signal interv

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20 4 IEEERANSACTIONS O N COMMUNICATIONS, VO L. COM-29, N O . 3, MARCH TABLE IOPTIMUM h-VALU ES AND CORRESPONDING NORM ALIZEDEUCLIDEAN DISTANCES FORM-ARY CPFSK SCHEMES

specific situation, o make he minimum Euclidean distanced$in(/z) equal to d B 2 @ ) .Plateau Functions

As an example of a plateau function, a binary scheme witha phase response very similar to that of QI= 0.05 an d /3 = 1/2wdl be chosen. The difference between the chosen phase func-tionand he a, /3 function is that hephasedoesnot varylinearly in the intervals 0 < t < aT an d ( 1 - a) T < t < T.Instead, hepha se varies like a raised cosine foranexactdefinition, see b-function s, b = 0.05 i n [16] ) .Fig. 6 shows d$ in(h ) for this binary system. The numberof observed bit intervalss N = 1 , 2 , --, . The upper bound onthe minim um Euclidean distance is also shownby adashedlinewhere heminimumEuclideandistance tilldoes notequal dB2 h).CPFSK and Mary PSK

Fig. 4 shows the well-know n minimum distance for binaryCPFSK fo r N = 1 , 2, 3, a n d 4 o b s e r v e dbi t n terval s . Alsoshown in Fig. 4 is th e upper bound dB2(h). t can be notedthat h = 1/2 corresponds to MSK and gives dkin(1/2) = 2 ,which is the same as antipodal signaling, e.g., BPSK. The re-quired observation interval for PSK is one bit interval, and f ordetectors making bit by bit decisions, PSK s optimum [3] .The required o bservation interval for MSK is two bit intervals,and the asymptotic performance in terms of error probabilityis the same as that for PSK. The optimu m modu lation indexfor CPFSK is h = 0.715 w hen the number of observed symbolintervals is 3. This gives theminimumEuclideandistanced$i,(O.715) = 2.43 and thus a gain of 0.85 dB in terms ofE,/No is obtained comp ared to MSK or PSK.The minimum n ormalized squared Euclidean distance ver-sus the modulation index h i s shown in Fig. 7 for the quater-naryCPFSK ystem see lso [l o] ). Note hat heupperbound dB 2(h ) shown by a dashed line where its not reached)is twice the minim um distance for a eceiver observation inter-val of he N = 1 ymbol.This is because he ectangularfrequency pulse g ( t ) has hesymmetryproperty 37).Th emaximum value of d B 2 h) s approximately reached forN = 8observed sy mbol intervals (com pare o Table I).It is clear from (39) that the first-order weak modu lationi n d e x e s i n t h e i n t e r v a l O < h < 2 a r e h , = 1 / 3 , 1 / 2 , 2 / 3 , 1 , 4 / 3 ,3/2 , 5/3, and 2 , and the effect of some of these early mergescan be clearly seen n Fig. 7. Note hat most of hese weak

3. 5

3.0

2 . 5

2. 0

1 . 5I v t m v I w v m D h.5 1 o

Fig. 6 . Normalized squared minimum distances &h) versus mlation ndex for a b-function with b = 0.5. This phase functivery similar to the a-p function with a = 0.5, fi = 1/ 2 (see [16[181).

indexes are catastrophic. The minimum Euclidean distancthese is no better than 2.It is interesting to compare the minimum distance for thquaternary CPFSK ystem to QPSK (phase esponse 4 1h = 1/4) As indicated n Fig. 7, theminimumsquareddistance for QPSK is dLi, = 2, and for the quaternary CPsystem it is slightlybelow this value for h = 1/4. This different relative performance level than that for M = 2. M = 2 , h = 1/2 all schemes have he minimum squared distance d & i n = 2, CPFSK and PSK included.The minimum distance for the octal M = 8) CPFSK syis given in Fig. 8 N = 1 , 2 , 3 an d in some interval for Nan d 5. The upper bound dB2 h),which, as usual, is showna dashed ine where t is not reached, is like the quaterncase reached with N = 2 observed symbol ntervals for ow

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AULIN A ND SUNDBER G: CONTINUOUS PHASE MODULATION -PART I 2

'VJFig. 7. Minimum normalized squared distance versus modulation n-dex forM = 4 CPFSK.

modu lation ndexes. Compared to hequaternary case, thenumber of firstarder weak (catastrophic) modulation indexeshas increased in th e interval of 0.3 5 .Distance properties of CPFSK schemes with larger values of

Mh ave been investigated in [17] , 20] .The maximum attain-able minimum distance value grows with M , but the number offirst-order weak (castastrophic) mod ulatio n inde x values alsogrows with M , as does the length of the observation intervalnecessary for reaching the upper bound d B 2 @ ) .However, fo rh 5 0.3, N = 2 is sufficient for all M . As an illustration to thebehavior discussed ab ove, Table I shows NB which is a lowerboundon he observationnterval for reaching theupperbound.

0 .5 1 oFig. 8. Minimum normalized squared Euclidean distance versus molation index forM = 8, CPFSK.

HCS, RC, and M a r y PSKThe HCS system yields a phase tree where the phase

jectories arealways raised cosine shaped over eachsymbinterval. Fig. 3 shows the quatern ary case.The upper bounddB2(h) or HCS,M= 2 is given by [18dB (h )= 2( 1 - os nh Jo (nh)) (

where .To(.) is th e Bessel f unctio n of the first kind and zerorder. The maximum value of d B 2 @ )s smaller than that binary CPFSK, but this maxim um value is still reached wN = 3 observed symbols (d&i, = 2.187 fo r h = 0.626, [18For h = 1/2 , dkin(h)equals that of MSK and PSK, as inbinary full response CPM systems. In the region of 0

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206 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-29,O. 3, MARCH 1

0 .5 1 .oFig. 9. Minimum norm alized squared distan ce versus modula tion n-de x h for M = 4, HC S system.

th e pulse shaping. The results for the oct al case (see Fig. 10)follow the same trend as for th e CPFSK system; i.e., the upp erbound is reached with N = 2 for low h-values and the numberof catastrophic modulation ndexes has increased compared tothe quaternary case.V. POWER SPECTRUM

The power spectral density for theull responseCPM schemesconsidered in this paper can be calculated with form ulas givenin [2]. The data symbols are assumed to be independent andidentically distribute d.'Fo r he case of full response CPFSKsystems, the spect rum can be expressed directly n terms ofelementary unctions [ 2 ] and or RC systems in ermsofBessel function s [121 .Spectra for systems with different values of M should becompared at the same bit rate. The bit rate ormalized variable

f.Tb is used wh ere'p

Hence, the power spectra R o o are plotted against the bit ratenormalized frequen cy eparation rom carrier.Plotsof the

0 .5 1 oFig. 10. Minimum normalized squared Euclideanistance vemodulation index h forM = 8, HCS system.commonly used fu nction [141, [15]

g o d n d fpOb(B>=which gives the fractio nal out of band power a t the one-sibandwidth B,will also be given.Fig. 11 shows th e power spectra (double-sided) for M-CPFSK with h = 1/M, M = 2 , 4, 8. The corresponding frt ional out of band power plots are shown inFig. 12 (for ovalues of h , see [17] , [2 0]) . I t i s well known that for f x ean d g(t), the spectrum widens for increasing h . For certaivalues discrete componentsoccu r. Fig. 11and hedistancefigures illustrate the fact that for a roughly fixed distancespectralmain lobe is decreasing wit h increasing M. Th e havior of he spec tra for large frequencies (i.e., the specttails) depends only on the number c of co ntinu ous derivatof he nstantaneous phase. It is shown n [8] hat he tdecrease withfas I f I-'(' + '1 . For CPFSK, c equals 0.Fig. 13 shows the spectra for the quaternary CPFSK, Han d RC schemes for h = 1/ 4. Note that for increased valuec the main lobe becomes larger. The main lobe widens intively due to thepresence of higher phase slopes over a port

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AULIN AND S U N D B E R G : CONTINUOUS PHASE MODULATION-PART I10.lolog R o ( f l [dB1

50

-1 0

-2 0

-3 0

-40

-50

-6 0

-7 0

-8C

1 3

Fig. 1 1. Normalized power spectral density in Sxi be ls fo r M-ary (M=2, 4, and 8) CPFSK with modulation inde xesH = 1 / 2 , 1 / 4 ,and 1/8,respectively.

1 o 1. 5

20

t-5 0 1

Fig. 12. Fractional out of band power in decibels for M-ary (M= , 4 ,an d 8) CPFSK with modulation nd exes h = 1/2, 1/4, 1/8, respec-tively.of the pulse for non-CPFSK schem es. The spectral ail of HCS d f in (close to 4) are reached with plateau functions witbehaves like f 6 , nd like f 8 fo r RC. Further spectra for M = 2 an d h close to 1. Fo r h = 1, the power spectrum cothese schemes are pl otte dn17] ,19] , [20] . tains spectral lines however.The spectra of schemes with plateau functions are investi-gated n [16], [18]. As might be expec ted, he rapid phase VI . DISCUSSION AND CONCLUSIONSchange in the beginning and the end of each symbol interval From the distance and spectrum results above and in [17gives wide spectra. The previously mentioned spectralai l [191 , [20] , t is evident that M -ary full response CPM schembehavior versus c is alsoapplicabe n this case. However, f have both bandwidth compaction properties and yield gain must be impractically large before this asympto tic behavior is Eb/No as compared to MSK. Schemes within this class of CPdom inating . Furthe rmore , t was concluded above that large systems can also be designed to give a large gain in Eb)No w

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208 IEEE TRANSACTIONS ON COMM UNICATIONS, V O L . COM-29, NO. 3 , MARCH

0 0.5 1t * A f . T1. 5 'TABLE I1BANDWIDTH/DISTANCE T R A D E O F F FOR SOMEM-ARYCPFSK SYSTEMS

-1 0PiI \h

I

b

Fig. 13 . Normalizedpowe r spectral densities in decibles for quatern-ary CPFSK with modulation index h = 1/4. The sc heme s are CPFSK(solid line ), HCS (dashed line), and R C (dash-dotted line).

the same bandwidth as MSK, or considerably smaller band-width at the expense of an increased E b / N o . This holds, forexam ple, for M-ary CPFSK . T he same also holds for systemsl ike HC S and RC, nd in t h e s e cases the t r a d e o f fb e t w e e nbandwidthand gain in E,/No at large SNR is even moreattractive.

In the binary case, plateau function s are a way to achieveconsiderable gains in ermsof E b / N o , whichunfortunatelygives poor spectra.In Table I1 comp arisons between various C PFSK schemesare. made, both concerning bandwidth and gain in erms ofE b / N o (dB) at large SNR. T he'referenc e system is MSK.Three different definitions of bandwidth will be used. Thenormalized b andw idth (double-sided) .is defined at 2BTb, fo rwhich 90, 99, or 99.9 percentof he otal signal power iswithin the frequency band f - o . I

8/3/2019 CPM part I

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A U LINN D I 20It is interesting t o note that for the schemes considered inthis p aper, a gain in terms of Eb/No is obtained without ex-panded bandwidth, compared to MSK. This is different fromthe case with a channel coded MSK system, where the spec-trum must be expanded by a factor of 1/R here R is the coderate [3 ] , [9] . For he CPM systems, no paritysymbolsaretransmitted, and the totalsignal energy is devoted to the infor-mation symbols.This paper explores the distance and bandw idth propertiesof full response CPM systems. In spite of the restriction thatthe schemes must be a full response typ e (i.e., the ns tanta -neous frequency only dependson one data symbol), we havefoun d considerable mprovem ents. However, larger improve-ments are obtainablewith partialesponseystems (theinstantaneousrequency epends nmorehan ne atasymbol). This class of system is considered in part 11. We haveintentionallyomitted all problems dealing with ransm itterand receiver considerations. These problems will be treated ina unified manner in Part 11.

r71

REFERENCESV . A .Kotelnikov, The heory of OptimumNoisemmunity.New York: Dover , 1960.R. R. Anderson and J . Salz, Spectra of digi tal FM, Bell S y s t .T e c h . J . , vol. 44. pp. 1165-1 189, July-Aug. 1965.J . M. Wozencraf t and 1.. M., Jacobs , Princip les of CommunicationEngineering . New York: Wiley . 965.R . W. Lucky , J . Salz. and E. J . W eldon .Jr . Principles of DataCommunica t ion. New York: McGra w-Hill, 968.M. G . Pelchat. R. C,. Davis,ndM. . untz.Coherentdemodulat ion of continuous phase binary FSK s ignals . i n P r o c .Inr. Telemetering Con f . , Wash ington, DC, 1971, pp. 181-190.R . deBuda, Coherent emodulat ion of frequency-shift eyingwith low deviat ion rat io , IEEE Tran s. Comm un. , vol . COM-20,pp. 429-436, June 1972.W .P .O s borne ndM. B. Luntz, Coherent nd oncoherentdetect ion of CPFSK. IEEE Tra ns. Comm un. , vol . COM-22, pp.1023-1036,Aug.1974.T. J . Baker , Asymp totic behaviour of digital FM spectra. IEEET ra ns .Co m m un. . vol.CO M -22 ,pp, 1585-1594. Oct . 1974.W .C .Lindsey ndM .K .S imon , Telecommunication ystemsEngineering . EnglewoodCliffs. NJ: Prentice-H all,1974.T. A. Schonhoff . Symbol error probabil i ties for M-ary coherentcontinuous phase f requency-shif t keying (CPFSK). in Proc. IEEEInt . Conf . Commun. Conf . Record . San Francisco, CA, 1975. pp.34.5-34.8.

T .A .Schonhoff , Bandw id thvsperformance cons iderat ions orCP F S K . in Proc. IEEE Nat ionalTelecommun.Conf .Record ,1975. pp. 38.1-38.5.F. Amoroso.Pulsean dspectrummanipu lation in theminimum(frequency) shif t keying (MSK) format, IEEE Trans. Commu nvol . COM -24, pp. 381-384, Mar .1976.T. A. Schonhoff , Symbol er ror probabil i ties , for M-ary CPFSKCoherent and noncoherent detect ion. IEEE Trans. Commun. .voCOM -24, pp. 64-52. June 1976.M .K .S imon , A general izat ion fheminimum-shif t-keyingIMSK)- typeignalingased upon ,inputat aymbo lulses hap ing , IEEE Trans.Co m m un. . vol .CO M -24 ,pp. 845-856A ug .1976 .M . Rabzel and S. Pasupath y. Spectra l shaping in minimum shifkeying IMSK) ype s ignals . IEEE Trans. Commu n. , vol. COM26.pp, 189-195, Jan.1978 .T . A u l inand C-E.Sundbern, BinaryCP F S K ype of signalinwithnput ataymbo l ulse shaping-Error probabilityndspectrum.e lecommunica t ionheory,echn .ep.R-99 ,Univ. Lund, Lund, Sweden, July 1978.pulse shaping-Minimum distance nd pectrum ,Telecommu-nicat ion Theory, Techn. Rep. TR-I 11 , Univ. Lund, Lund. SwedenAug. 978.-. Bounds on the erformance of binaryCPFSK type os ignal ing with nput data symbol pulse shaping, in Proc. IEEN ut .T e l e c o m m un.Co nf .Record , Birmingham,AL, 1978. pp6.5 .1-6.5 .5 .-, M-aryCPFSK ypeof ignaling with input data ymbolpulse shaping-Minimum distance and spectrum , in P io c . IEEI n t . Conf . Commun. Conf . Record . Boston. MA, 1979, pp. 42.3 . I42 .3 .6 .T . A u l i n ,CPM-A power and bandw idth efficient digital consenvelopemodulat ioncheme,Dr. echn. is ser tat ion, ele-communicat ion Theory. Univ. Lund. Lund. Sweden. Nov. 1979.T . A u l in , N . Rydbeck, nd C.-E. W . Sundberg, Continuousphase modulation-Part 11: Partial response signaling, this issue,pp. 210-225.

__ M-aryCPFSK ypeof signaling with inputdatasymbol

*Tor Aulin (S77-M80) . for a photograph and biography, see th is is sue.p . 195.

Carl-Erik W . Sundberg (S69-M 75), for a photograp h and biographsee his ssue, p . 195.