Comparison of Modulation

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INVESTIGATION INTO PRSCONTINUOUS PHASE DIGITAL MODULATION SCHEMES

PRECODED, CONSTANT-ENVELOPE,

J. N. Golby and R. M. BraunUniversity of Cape Town, Private Bag, Rondebosch, 7700Department of Electrical and Electronic Engineering

Abstract - Constant envelope modulationsystems such as QPSK and M S K have highout-of -band signal levels. Partialrespon se signaling (PRS) has been usedin PAM systems to improve spectralef f ici en cy . Some systems such a sContinuous Phase Modulation ( C P M ) andCorrelative Phase Shift Keying (CORPSK)make use of PRS, and exhibit improvedperformance over these modem types. Abrief study of these schemes ispresented here, a s are some practicalresults f o r CORPSK.

I . INTRODUCTIONConstant-envelope modulation schemes arevery attractive for use in digital radiobecause of their high tolerance to th enon1 ine ar ope rat ion s convenient inhigh-power amplif iers and receiver IFstages. Traditionally, four-phasesystems such as QPSK and MSK have beenused because they represent a goodcompromise between necessary bandwidthand power efficiency (t he forme r becausestringent limitations on out-of-bandspectral components must often be met,and the lat ter because good errorperformance must be achieved with onlymod era te power levels). However, anyimprovement in out-of-band signal levelcan only be achieved by filtering of thesignal, and this introduces envelopefluctuations and therefore reduces theschemes immunity t o the ef fe cts ofnoni inear it es. In many insta nces,filtering of the output signal is simplynot possible because of th e high fi lt erQ-factors and signal powers involved.Any further nonlinear amplification ofthe signal simply causes spectralspreading.

the amplitude modulation (PAM) area,making use of Partial Response Signaling(PRS). The ir signa ls, however, haveflu ctu atin g envelopes. Thisinvestigation attempts to study theef fe ct of applying PRS t o frequency- andphase-modulated system s with the aim ofrealizing the same benefits in anglemodulation as are yielded in theamplitude domain.This study focuses on two schemes inparticular - Continuous Phase Modulation(CPM) [21, 131 and Correlative PhaseShift Keying (CORPSK) 111. TamedFrequency Modulation, o r TFM [41, ha sbeen shown t o be an example of bothsystems 111, [31. It has been shownthat these schemes are particular casesof pulse-shaped CPFSK 151.11. MODULATION DEFINITIONThe constant-envelope phase-varyingsinusoids studied here are of the formCos(@ + cP(t)), where the phasefunction, N t ) , follows an encodedpattern according t o some rule [ l l , [2l.Also, phase transitions are correlatedto ensure good spectral efficiency, aswell as variety in phase shif ts t oimprove power efficiency, and thereforesystem er ro r performance. Fig. 1 showsa block diagram of such a system.CPFSK systems employ some form ofbaseband pulse shaping filter followedby either a frequency modulator, or anintegrator followed by a phasemodulator, while PSK systems follow thepulse shaper with a phase modulatoronly. The pulse shaping cir cuit ry maybe either digital or analog, or acombina tion of both. Some

C

implementations, in particular, areideally suited to use in DSPenvironments [SI.

Clearly, there exists a need to developconstant-envelope modulation schemeswhose spectral efficiency is higher thanth at yielded by power-eff icient systemssuch as QPSK. Spect rally -ef f ici entmodems are currently being developed in

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111. P R S I N PA M S Y S T E M S with c = the i th b i t ,iFig. 2 i s a generalized scheme forimplementation of partial responsesignaling [61. Correlation isintroduced between transmitted symbolsby summing appropriately weighted,delayed replicas of the input datastream, thereby causing each transmittedsymbol to contain information from morethan one input digit. The result isthat intersymbol interference isintroduced, the effec t being t o shapethe system spectrum. In addition, theseschemes are more immune to timing errorsthan memoryless systems - the resul t i sthat data may be transmitted at theNyquist ra te. This can be easilyachieved by a transversal fi l tercascaded with a lowpass fi l ter, G ( o ) .The transversal fi l ter has a frequencyresponse F(w), where

n=O

Here T is the intersymbol spacing, andFID) is called the "system polynomial".Different choices of F(o) an d G(o) arepossible, each combination having itsown performance characteristics. IfG ( w ) is an ideal lowpass filter, thesystem is characterized entirely byF(w), or the system polynomial, F ( D ) .The syste m polynomial may be chosen t oset th e number of output levels and toprovide suitable spectral nulls.IV. CONTINUOUS PHASE MODULATION

(CPM)Aulin, Rydbeck and Sundberg [ZI , [31have studied in detail the behaviour ofCPM sys tems. Thes e sys tem s are definedas follows [31:

The phase function, $(t), ismP

t=-m -03

h = the modulation indexTb = the bit interval

g( t ) = the baseband pulse shapeAulin and Sundberg [21 have shown thatM-ary ful l respo nse CPM syst ems haveimproved spec tra l properties and yieldgain in Eb/N , compared to MSK. The useof multilevel data inputs as well assmoothing of th e phase tr aje cto ry attransition instants yield a fa r moreattractive trade-off between err orperformance and spectrum efficiency.Furt her improvements resu lt fr om usingbaseband pulses longer than one symbolinterval ie. partial response CPM. Usingthis approach, modulation schemes can befound that are both bandwidth- andpower-efficient [ 3 ] .D i g i t a l Phase Modulation (DPM) [51 i s asimpler approach to CPM that allowseasier recovery of carrier informationand is better suited to digitalimplementat on. This scheme does,however, create a slightly broaderspectral main lobe.V. CORRELATIVE PSK (CORPSK)The CORPSK signal [ I ] is of the formCos(W t + @(t) ) , where the signalbehaviour is defined by manipulation ofN t ) according t o a n input sequence{a } = {ao, al, ...a , . ; successive

phase shifts A@ occur at t = T , T thesymbol period, according to t hr eecriteria:

m mm s s

the phase is continuous, and A@ isgiven by mA@ = C .2Wnm m= @((m+l)T - N m T 1;

C is the information-carrying t e r mand n is the number of phasestates , n an integer.successive phase paths follow somecorrelation rule, so that each shiftis determined by the current inputlevel and that from at least oneprevious interval. Because of thistype of correlative coding, theremust be more possible phase shiftsthan input levels.

m

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(3) The phase path m u s t be smoothed.This ensures good spectralefficiency, since the derivative ofthe phase function is thencontinuous.

Many different forms of CORPSK exist,depending on the correlation rulewhereby C in criterion (1) is derived.The notat ion CORPSKfM-N, F(D)) i s used,where F(D) is a PRS system polynomial ofthe type described in Section 111. M i sth e number of input levels, and N i s thenumber of possible phase sh ift s. Forexample, CORPSK(4-7, l + D ) represents afour-level input system wit h sevenpossible output phase s hi ft s followingthe Duobinary rule [71.

m

The basic CORPSK modulator consists of aPRS encoder with precoder, followed by alowpass premodulation filter and a nangle modulator (see Fig. 3 1.This is equivalent to a baseband partialresponse system followed by anglemodulation. The premodulation fi lt erensures that the angle modulator followsa smoo th phase path. If frequencymodulation is used, this is equivalentt o a phase modulator whose input is theintegral of the par tia l response signal.This is because freque ncy modulation ha san inherent integrating effect.CORPSK generates a signal of the forms( t ) = Codw t + Q(t ) ) . (5)By trigonometric identities,s( t ) = COS(@ t )Cos(@(t))

C

- Sin(w t)S in(@ (t)) . (6)Clearly, CORPSK can be generated bymodulating the required values ofCos(O(t)) and Sin( @(t) ) onto thei rrespect ive quadrature c arrie rs andsumming them [ll, [51.The spectrum reported by Muilwijk [ I1for CORPSK(4-7, 1+D) is shown in Fig. 4,as is tha t of unfiltered QPSK (ie.constant-envelope) fo r ref ere nce , andfor Tamed Frequency Modulation, or TFM(CORPSK(2-5, (1+Dl2). Both schemes havenecessary bandwidths similar to QPSK,but fa r lower out-of-band signal levels.Fig. 5 shows the corresponding bit error

rate performance curves.All of these modulation schemes fallwithin the class of power-efficientmodems. CORPSK(4-7,1+D) yields approxi-m a t e l y 1 dB gain over the ideal DQPSKcase, and a fu rt he r 1 dB over TFM.The ideal CORPSK demodulator is acoherent correlation receiver.Symbol-by-symbol dete ction i s possible,but yields a reduced error performancein the presence of noise. A pract icalapproach is to demodulate coherentlyinto in-phase and quadra ture signalcomponents and then perform detectionand decoding. Coherent ca rr ie r recoveryis requ ired . Because CORPSK cr ea te s asignal constellation of nsymmetrically-located phase sta tes , thi scan be effected by processing of thereceived RF signal with a nonlinearityof order n, and extrac ting the resultingfrequency component, formed at n timesthe carrier frequency, with aphase-locked loop. Symbol timingrecovery may be performed either on thebaseband signals, or at the predetectionstage using the approach proposed forMSK by de Buda [81.V I . H A R D W A R E I M P L E M EN T A T I ON O F

CORPSKThe system CORPSK(4-7, 1+D) has a signalconstellation as shown in Fig. 6.Clearly, for any one of the fourstationary phase points there are sevenpossible phase paths of less than 271that can be traversed ie. +n/2, +n,+3n/2. Each start- and end-point has,therefore, two possible phase paths.Because of the unique coding propertiesof t h e Duobinary ru le [61, [71, only thepath length affects the value of thereceived and decoded signal.TRANSMITTERThis scheme was implemented in hardwareaccording to the approach shown in Fig.7, derived from (6 ) above.The phase tra ectory-shaping circuitrycomprises a binary to four-levelconverter, mod-4 precoder andtransve rsal f ilt er (to implement F(D)).These are followed by a digital lowpass

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fil ter . Because of th e duality between afrequency modulator and anintegrator/phase modulator, the phasetrajectory thus created has to be theintegral of the input PRS function.A factor- L oversampled lowpass fi lt erimpulse response is to be s toreddigitally and convolved in time with theseven-level PRS signal, which is incodeword form (L = 8 o r 16). Thisfi l ter impulse response and i ts tai lextend over more than one symbolinterval, so any output phase value is asum of t he cur ren t impulse response andthe tail of one or more previoussymbols, depending on th e trunca tionlength used. I t is this summed functionthat has to be integrated. I t i spermissible, therefore, to integrate theindividual phase symbols and then sumthem (the integral of a sum equals thesum of the two integrated terms).Consequently, the digital lowpass filteris, in practice, programmed with theintegral of the impulse response.The in-phase and quadrature symbolchannels are then created with sin andcos look-up ta bl es in EPROM. The sevalues are put into analog form andlowpass filtered before being modulatedonto their respective carriers andsummed. Fig. 8 shows a computergenerated eye diagram for one channel.An adv antage to using a n oversampledbaseband waveform is that the resultingspectrum, which has no sidelobes,repeats itself at multiples of th esampling frequency [51. Therefore, thecut-off points of the lowpass fi lt er sused to remove the repeated spectra canbe placed higher in frequency than thesignal band-edge - problems caused byfil ter transit ion region group delaysare therefore eradicated.Fig. 9 shows a computer-simulated signalspectrum, normalised in frequency.These results agree closely to those ofMuilwijk [l l .RECEIVERThe received signal is demodulated intoin-phase and quadrature components witha regenerated carrier, sampled anddecoded (see Fig. 10).

The carrier is regenerated by nonlinearprocessing of the bandpass signal with aprecision full-wave rectifier andphase-Iocked loop. Our computersimulations have also indicated thisapproa ch t o be effecti ve. Symbol timingrecovery is effected by locking ontosymbol rate components locatedsymmetrically about the fourth-ordercarrier component, and multiplying themt o give a symbol-rate clock.The received signal is sampled at thesymbol rate and successive samplescompared t o give the transmitt ed symbolvalue; th e decoder con verts thesesymbols into a binary data stream.V I I . CONCLUSIONSBecause of the ir immunity to nonlinearsignal operations, constant-envelopemodulation schemes are at t ract ive foruse in digital radio environments.Traditional systems such as QPSK and MSKhave high out-of-band signa l levels.Continuous Phase Modulation (CPMI andCorrelative Phase Shift Keying (CORPSK)are two modulation approaches that yieldimproved error and spectral performance.These systems have been briefly reviewedand some practical results for CORPSKhave been shown.REFERENCES[ l l

[21

131

D. Muilwi k, "Correlative PhaseS h i f t K e y i n g - A Class of ConstantEnvelope Modulation Techniques,"IEEE Trans. Commun., vol. COM-29,N o. 3, March 1981, pp. 226-236.T. Aulin and C.-E. Sundberg ,

"Continuous Phase Modulation - PartI : Full Response Signaling," IEEETrans. Commun., vol. COM-29, No. 3 ,March 1981, pp. 196-209.T. Aulin, N. Rydbeck and C.-E.

Sundberg, "Continuous PhaseModulation - Part 11: PartialResponse Signaling," IEEE Trans.Commun., vol. COM-29, No. 3, pp.210-225.

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I41

[SI

[61

[71

181

Data PULSEin - HAPER

L

F. de Jager and C.B. Dekker, " TamedFrequency Modulation - A novelmethod to achieve spectrum economyin digital transmission,'' IEEETrans. Commun., vol. COM-26, No. 5,May 1978, pp. 534-542.

ANGLE , SignalMOD. - ou t

T. Maseng, "Digitally PhaseModulated ( D PM ) Signals," Trans. Fig.3. Basic CORPSK modulator structure-Commun., vol . COM-33, No. 9, Sept.1985, pp. 911-918.P. Kabal and S . Pasupathy,

"Partial -Response Signaling," IEEETrans. Commun., vol. COM-23, No. 9,Sept. 1975, pp. 921-934.A. Lender, "The Duobinary Techniquef o r high-speed data transmission,"IEEE Trans. Commun. Electron., vol.82, May 1963, pp. 214-218.R. de Buda, "Coherent Demodulationof FSK with Low Deviation Ratio,"IEEE Trans. Commun., vol. COM-20,No. 3, June 1972, pp. 429-435.

Fig. 1. Typical modulator structure

Fig.2. Generalized PRS sys temFig.5. Bit er ro r performance of

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--3lT12

Fig.6. CORPSK(4-7,1+D) signalconstellation

cos wct

Data4 PHASESHAPER

* 1

0

-1

Fig.8. Eye diagram of one channel.

Cf-fc)/f6

Fig. 9. CORPSK(4-7, 1+D) signal spectr um.Sin yt

Fig.7. Quad ra tu re CORPSK modulator

a

Fig.10. Demodulator structure

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Comparison of Modulation