IEEE - Msk And Offset Qpsk Modulation

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-24, NO. 8 , AUGUST 1976

IV. CONCLUSIONS The paper has summarized some results on the properties of the eigenvectors and eigenvalues of persymmetric matrices. Persymmetricmatricesappearquiteoften in communication and informationtheory. Recently the eigenvectors of symmetric matrices have occurred as solutions to a number of importantproblems in datacommunication. Thepropertiesof the eigenvectorspresentedcan be used to characterizeand simplify the solution to these probiems. This has been demonstrated in the paper for some specific examples but it is felt that there may be many other applications.

REFERENCES P. Butler and A. Cantoni, “Eigenvalues and eigenvectors of symmetric centrosymmetric matrices,” Linear Algebra and its Applications, vol. 13, pp. 275-288, Mar. 1976. A. Cantoni and P. Butler, “Eigenvalues and eigenvectors of symmetric centrosymmetric matrices and applications,” Univ. of Newcastle, N.S.W., Australia, TechRep.EE7403, ISBN 0-7259-0143-8, Apr. 1974. S. A. Fredricsson, “Optimum transmitting filter in digital PAM systems with a Viterbi detector,” IEEE Trans. Inform. Theory, VOI.IT-20, pp. 479-489, July 1974. F. R. Magee, Jr., and J. G. Proakis, “An estimate of an upper boundon errorprobability on channels having finite-duration pulse response,” IEEE Trans. Inform.Theory, vol. IT-19, pp. 699-702, Sept. 1973. R. W. Chang, f‘A new equalizer structure for fast start up digital communication,” Bell S p t . Tech. J., vol. 50, pp. 1969-2014, July 1971. A. Cantoni, “A new adaptive receiver structure for PAM signals,” in Proc. 7th Hawaii Int. Conf on System Sciences, pp. 144-146, Jan. 1974. D. D. Falconer and F. R. Magee, Jr., “Adaptive channel memory truncation for maximumlikelihoodsequence estimation,” Bell Syst. Tech. J., vol. 52, pp. 1541-1562, Nov. 1973. U. Grenander and G. Szego, Toeplitz Forms and Their Applications. Berkeley, CA: Univ. of California Press, 1958.

[9]R.

M. Gray, “On theasymptotic eigenvalue distribution of ToeDlitz matrices,” ZEEE Trans. Inform.Theory, vol. IT-18, pp. 725-730, Nov. 1972. [ l o ] R. R. Anderson and G. J. Foschini, “The minimum distance for MLSE digital data systems of limited complexity,” ZEEE Trans. Inform.Theory, vol. IT-21,pp.544-551,Sept. 1975.

* AntonioCantoni (”74) was born in Soliera, Italy, onOctober30, 1946. He received the B.E. degree with first class honors in 1968 and the Ph.D. degree in 1972 both from the University of Western Australia, Nedlands, Australia. He was, a Lecturer in Computer Science at the Australian National University from 1972 to 1973. Presently, he is with the Department :.., ., , ’, of Electrical Engineering at the University of Newcastle, New South Wales, Australia. Heis interested in digital communicationand computer systems. He has also been a consultant on systems program development and industrial electronics. _ 1

,

* Paul Butler (S’72-M’75)was born in Sydney, Australia, on April 7,1941. He received the B.Sc. degree fromSydney University, Sydney, in 1962 and the M.Sc. degree from the City University, London, England in 1972. From 1965 to 1972 he was a Lecturer, then a Senior Lecturer in Mathematics at Trent Park College of Education, Barnet, Herts., England. From 1972 to 1975 he was engaged in research for the Ph.D. degree in theDepartment of Electrical Engineering, University of Newcastle, New South Wales, AuIstralia. He is currently working on problems of graph enumeration in the Department ofMathematics, University of Newcastle.

MSK and Offset QPSK Modulation STEVEN A. GRONEMEYER, MEMBER,IEEE, AND ALAN L. McBRIDE, MEMBER,

Absfruct-Minimum shift keying (MSK) and offset keyed quadrature phase shift keying (OK-QPSK) modulation techniquesare often proposed for use on nonlinear, severely bind-limited communication channels because both techniques retain‘ low sidelobe levels on such chanPaper approved by the Editor for Communication Theory of the IEEE Communications Society for publication after presentation at the National Telecommunications Conference, New Orleans, LA, December 1975. Manuscript received September 20,1975; revised February9, 1976. Theauthorsarewiththe Collins Radio Group, Rockwell lnternational Corporation, Dallas, TX 75207.

EEE

nels, while allowing efficient detection performance. A more detailed performancecomparison of the two techniques on suchchannels is, therefore, of interest. In this paper a Markov process representation is developed which is applicable to either the MSK’or OK-QPSKwaveform.Thisrepresentation is employed to illustrate the similarity between themodulation processes and to obtaintheautocorrelations of thetwo waveforms. This Markov and powerspectraldensities process representation may be similarly employed with other modulation waveforms of the same class. The autocorrelations and power’ spectraldensities of MSK and offset QPSK provideinitial insight to expected performance on band-limited channels.

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AUGUST COMMUNICATIONS, ON IEEE TRANSACTIONS

The resul+s o f , a digitalcomputer,simulation are presented. The simulltion,compares ?e bit errorrates (BER’S) of MSK and .offset QPSK on nonlinear, band-limited double-hop links such as encduntered in satellitecommunications. The simulationresults are presented as Eb/No degradation with respect to ideal detection,versus channei noise bandwidth. The.eiror probability was usbd a perfoniiance,metric, and equd idjacent channel interference as a constraint. For the c h h nels simulated, hSK ,is found to providesuperiorperformance when 1.1 times the binary d a b the channel noise bandwidth exceeds about, rate. For narrower bandwidths, offset QPSK providessuperiorperformance.

I. INTRODUCTION

B

OTH minimum shift keying (MSK) [ l ] and offset keyed quadraturephase. shiftkeying (OK-QPSK) modulation have been.considered for use onband-limited, nonlinear channels. as an alternative to conventional (nonoffset) QPSK for several reasons. If either an MSK or offset QPSK waveform is band-limited and then hard-limited, the degree of regenerationofthe filtered.sidelobes isless than is the case for conventional QPSK 121. Both techniques achieve the matched filter coherent detection bit error rate (BER) performance of antipodal PSK on linear, infinite bandwidth, white Gaussian noise (WGN), perfect reference channels. Either technique has carrier reference recovery features providing an advantage with respect to conventional QPSK [3] -[5] . A feature of MSK which is often useful is that it can be noncoherently detected by ,adiscriminator [6] ,whereas QPSK systems require either a fully coherentor differentially coherentdetection system. This noncoherent detection propertyo f MSK permits inexpensive demodulation when the received signal-tokoiseratio is adequate,yet allows forcoherentdetectionwith efficiency identical tocoherent QPSK , inlimitedsignal-tolnoise ratio situations. Investigations ofthecoherentdetectionperformance of MSK and conventional QPSK on band-limited or combined band-limited and nonlinear channels have previously been reported [ 7 ] , [8]. Thispaperdevelops the theory and connectionsbetween MSK andoffset QPSK andpresentsa performance comparison between the two techniques. MSK can be viewed as either a special case of continuous phase frequency shift keying (CPFSK), or a special case of offset QPSK with sinusoidal symbol weighting. In this paper offset QPSK, referenced withoutfurtherqualification, will refer to “square-pulse”offset QPSK. special, eases of offset QPSK will be qualified by the appropriate symbol weighting used, such as the half-cycle sinusoidal Weighting of MSK. The relationship between the ,two views of MSK is deveioped by constructing a first-order Markov process representation which has a transition probability matrix which is common to both MSK and offset QPSK. The Markov representation is then used to derive theautocorrelation.and power spectraldensity properties of MSK and offset QPSK which provide insight to the BER performance of the two modulationtechniques. The results of adigital computer simulation ofthetwo techniques over nonlinear, band-limitedchannelsare presented. A definitive comparison of the two techniqueson such channels is difficult, since the details of channel nonlinearities and passband characteristicsstrongly influencesystem performance. The approachused in comparing the two techniques

1976

is to simulate the BER performance of each over identical channels, with post-simulation adjustment t o account for ah adjacent channelinterferenceconstraint. While the receiver implementations are notoptimum over the channels considered,the simulationresultsprovide relative performance measures of the MSK and offset QPSK modulation techniques as a function of system parameters. The results of tlie simulation show thatboth MSK and offset QPSK requirea certain critical bandwidthand,for nariower channels, performance rapidly degrades due to intersymbol interference and pulse distortion. For the type of filter we’d in the simulation, the BER performance of MSK is found to be superior to that of offset OPSK only when the channel bahdwidth exceeds about 1 . 1 times the binary data rate: This relationship holds when bothmodulationtechniques are subjected to tlie same channel filtering, or when a constant adjacent channel interference constraint is applied in a multiple carrier situation. 11: CHARACTERIZATION OF MSK When viewed as CPFSK, the MSK waveform can be expressed as 191

where-w, is the carrier, or center, radian frequency, uk = is bipolar data being transmitted at a rate R = 1/T, and xk is a phase constant which is valid over the kth binary datainterval kT < t < (k + 1)T. Fig. l(a) illustrates the FSK nature of the MSK waveform, with aradian frequency w, +.n/2T being transmitted for uk = 1 and radian frequency d,- n/2T being transmitted for. uk = -1. The tone spacing in MSK is one-half that employed in conventionalorthogonal FSK modulation, giving rise to the name “minimum” shift keying. During each T second data interval, the value of xk is a constant determined by the requirement that the phase of the waveform be continuous at the bit transition instants t = k k Applying this requirement totheargument of (i) results in the recursive phase constraint

Fig. l(b) illustrates the continuous pHase, constant amplitude MSK waveform. For coherent detection, a reference value of x k , say xo, can be set to zero without loss of generality. This assumption will be used in all that follows, with the result, using ( 2 ) , that xi, = 0 or n,modulo 27r. Define e ( t ) as

e ( t ) is a piecewise-linear phase function of the MSK waveform

81 1

GRONEMEYER AND McBRIDE: MSK AND OFFSET QPSK MODULATION

(A)TONE SPACING IN MSK. DATARATE IS 1 BIT PERT SECONDS. CARRIERFREQUENCY IS w c = 2 n f, RAD/SEC

( 6 ) CONTINUOUS PHASE NATURE OF MSK

f IC1 EXAMPLE OF EXCESSPHASE S i l l FOR A PARTICULARDATASEOUENCE Yk

-112"

k

Oil'

ID1 LATTICE OF POSSIBLE EXCESS PHASEPATHS

xk

'k

0 /

/

2

~1

r

1 1

-2" -2"

1

-2"

Fig. 1. Characteristics of MSK. (a) MSK tone spacing. (b) Continuous phase MSK waveform. (c)Example ofexcess phase function. (d) MSK excess phase trellis.

in excess of the carrier term's linearly increasing phase. Using the recursive phase constraint, O(t) is plotted in Fig. l(c) for a particular data sequence u k . The phase constant x k is the phase axis intercept and nuk/2T is the slope of the linear phase function over each T second interval. The phase function in Fig. l(c) is a particular path along the phase trellis of possible paths shown in Fig. l(d). Fig. l(d) illustrates that over each T second interval, the phase of the MSK waveform is advanced or retarded precisely 90" with respect to carrier phase, depending uponwhetherthedataforthat interval is +1 or -1, respectively. Using trigonometric identities and the property that xk = 0 , n modulo 2n, the MSK waveform representation of (1) can be rewritten as

weighting, and cos ( x k ) is the data-dependent term. Similarly, thequadrature-phasechannel,or Q channel, is identified as uk cos ( X k ) S ( t ) sin (o,t),where sin (act) is the quadrature carrier term, S(t) is the sinusoidal symbol weighting, and u k cos ( x k ) is thedata-dependent term.Since the data, U k , can change every T seconds,it mightappear thatthedata terms cos ( x k ) and u k cos ( x k ) in (4)can also change every T seconds.To the contrary, it is shown in Appendix A that as a result of the continuous phase constraint the term cos ( x k ) can only change value at the zero crossings of C(t) and the term uk cos ( x k ) can only change value at the zero crossings of S(t). Thus,thesymbol weightingin eitherthe I or Q channel is a half-cycle sinusoidal pulse of duration 2T seconds and alternating sign. The I and Q channel pulses are skewed T seconds with respect t o one another. Finally, the data are conveyed at a rate of one bit per 2T secondsineach the I and Q channels by weighting the I and Q channel pulses by cos ( x k ) and u k cos ( x k ) , respectively. Recall that for coherent detection, x k = 0 , n modulo 2r, so that both cos ( x k ) and u k cos ( x k )take on only the values + I . In the discussion above, the MSK waveform was initially viewed as a CPFSK waveform. As a result,thequadrature signaling waveform, derived as (4), containeddataterms in which the binary data u k appeared encoded as cos ( x k ) and u k cos ( x k ) and in addition, the symbol weighting pulses in either the I- or Q-channel alternated in sign. However, for bitto-bitindependentdata u k , the signs of successive I- or Q-channel pulses are also random from one 2T second pulse interval to the next. Thus, when viewed as a quadrature signaling waveform, (4) can be rewritten with a more straightforward data "encoding" as u2k-lC[t

f

-

-u2k-$[t

2kT] COS (act) -

(2k - 2)T] sin ( q t ) ,

(2k- 1)T.

(1 1)

Thus,detecting MSK as twoorthogonalbinary channels provides a 3-dB advantage over detection of orthogonal FSK. All ofthe foregoing discussion has centeredon ideal channels having infinite bandwidth, additive WGN, and perfectly recovered carrierandtimingreferences. Section IV will consider theperformanceof MSK andoffset QPSK on band-limited, nonlinear channels. As a preliminary discussion tothistopic,it is useful t o consider the powerspectral densities of the MSK and offset QPSK moduiation processes. The powerspectral density of e'ach process is theFourier transform of the corresponding autocorrelation function, and are given by the following expressions:

where

f

PC

frequencyoffsetfrom carrier power in modulated waveform.

normalized to thebinarydata rate R = 1/T. The MSK spectrum falls off at a rate proportional to u/R)-4 for large values of f/R. In contrast, the offset QPSK spectrum falls off at a rate proportional to only u/R)-2. The main lobe of the MSK spectrum, however, is wider than that of the offsetQPSK spectrum, the first nulls falling at f/R = 0.75 and f / R = 0.5, respectively. A measure of the compactness of a modulation waveform's spectrum is provided bythefractionalout-of-bandpower, Pob , defined as

Equation (14)hasbeenevaluated forthe MSK and offset QPSK power spectral density expressions (12) and (13) as a function of bandwidth B normalized to the binary data rate, and the results plotted in Fig. 7 . Figs. 6 and 7 suggest that for system bandwidths exceeding about 1.5 R, MSK should providelower error rate .performance than offset QPSK for equal transmitter power. However, as system bandwidth is decreased from 1.5 R t o 1 .OR, a point should be reached at which the performance of offset QPSK will besuperiortothat of MSK. As systembandwidth is increased, the performance of the two systems converges, to the infinite bandwidth case (10). The precise boundaries of the regions of superior performance for each technique are difficult t o determine inpractical situations, since the detailed

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IEEE TRANSACTIONS ON COMMUNICATIONS, AUGUST 1976

intersymbolinterferenceeffects,intermodulationdistortion, etc. One approach to determining the error-rate performance of a modulation technique on a complex channel is through the use of digital computer simulation techniques.

IV.COMPUTER SIMULATION &RESULTS FOR SATELLITE LINK

A computer simulation was performed to examine the BER performance o f MSK and offset QPSK onband-limited, nonlinear channels,such as thoseencountered in satellite communications systems. Fig. 8 is ablock diagram of the simulated system. The simulation is carried out at baseband to minimize computer run time. The elements in the model correspond t o the elements in a satellite communication link as follows. Filter number 1 simulates band-limiting elements at thetransmittingearthstation. Noise addedatthe satellite receive system is represented by WGN source number 1. The bandpasshard-limiterwhichsimulates the satellite output power adplifier obeys the input-output relationship

Downlink, noise which is added at the receiving earth station is represented by WGN source number 2. A second filter is used to simulate band-limiting elements in the receive system. The matched filter is matched to the transmitter pulse shape. Both MSK and offset QPSK were simulated with this system model, simply by switching between rectangular and half-cycle sinusoidshapesin the “pulse shaping”and “matchedfilter”

system elements. Power is normalized prior to the points at which theuplink (WGN source 1) anddownlink (WCN source 2) noise is added to facilitate setting the desired up and downlink values of Eb/No (energyper bit to additive noise spectraldensity). This normalization also resuitsin equal uplink transmitted power at the output of filter number 1 for eithermodulationtechnique whichsimulatesfixed EIRP earth stations. In running the simulation, the noise samples of WGN source 2 are not actually added to the simulationsample stream. Rather, an estimation technique similar tothat employed by Lebowitz and Palmer [8] is employed. As long as the systemadditive noise is dominatedbythe down-link component,thistechnique provides simulation run time savings relative to an error counting BER estimation technique. In the simulations reportedhere,thetwosystemfilters werechosen to beidentical. The c h h n e l bandwidth B is defined as the noise bandwidth of the two identical filters in cascade. The simulatedfilters were 7 pole, 0.1-dBripple Chebyshevlow-passequivalentfilters. A highfilter Q-factor was chosen so that a high degree of filter symmetry was achieved. This resulted in tlie primary mechanism of crosstalk between the I and Q channels ljeing the bandpass hard-limiter, via (15). Since many practicalsatellitelinksare limitedby downlink thermal noise, the uplink E b / N o was maintained at a high levei forthe results reported here. Underthe above restrictions, several MSK and offset QPSK system simulation runswere performed for varying values of downlink Eb/No and channel bandwidth B . The resulting data were used to derive the curves shown in Fig. 9. Each curve applies to either MSK or offset QPSK, as labeled. The curves show the variation I

817

GRONEMEYER AND McBRIDE: MSK AND OFFSET QPSK MODULATION 12k-lIT 6 t C i2k + 1lT PN SEQUENCE "2k-1 PULSEGEN (I-CHANNEL)

PN SEOUENCE PULSEGEN (Q-CHANNEL)

"2k

1

= 'l

=''

1

NOWGN SOURCE

PULSE SHAPING

PULSE a

2kT C t C (2k+2lT

j

= 6 i SAMPLER (2k+l)T

RE (.)

t =

MATCHED FILTER

I-CHANNEL

4

BER ESTIMATOR

I

SAMPLER i2k+2lT

IM i.)

t =

Q-CHANNEL+

WGN SOURCENO 2

Fig. 8. Simulationblock diagram.

I

0.5

1.0

1.5

2.0

BT - CHANNEL BANDWIDTH NORMALIZED TO BINARY DATARATE(HZIBITISECI

Fig. 9. MSK andoffset QPSK Eb/No performancedegradation with respect to ideal antipodal PSK as a function of cascaded filter noise bandwidth with errorprobability as a parameter.

ofperformancedegradation as a functionofchannelbandwidth for the indicated constant value of bit-error probability. Performance degradation is defined as the ratio of the total link Eb/No required onthe simulatedlink t o achieve the desired BER to the Eb/N, required on an infinite bandwidth, WGN channel t o achieve the same BER. The ideal link performance is given by (10). Timing in the simulated system is nearideal. Several samples ofthematchedfiter near the expected optimum sample time are inspected t o determine the optimum sample for use in the BER estimator.The sample time is not, however, bit-by-bit adaptive, but is constant for the entire simulated bit sequence. As conjectured in Section 111, the performance curves show a crossover point, such that for channel bandwidths exceeding approximately 1.1 R, where R is the binary data rate, the performance of MSK is superior to that of offset QPSK. Conversely, for B < 1.1 R, offset QPSK has anadvantage over MSK. The reason for the performance crossover is basically thenarrowermainlobeof the spectral density of offsetQPSK as compared to MSK. Both

techniques begin to suffer severe degradations as their main spectral lobesencounterthechannelband edge. Since the main lobe of MSKis wider than that of offset QPSK, MSK begins to degrade sooner than offset-QPSK as channel bandwidth is decreased. The above discussion is based upon a parametric variation of filter bandwidth only. Other system elements can influence the effect of this parametric variation. For example, Poza and Berger [ 141 briefly touch on a simulated comparison of offset QPSKl and MSK on a channel model containing an AM/PM systemelementratherthan hard-limiter. a Their results indicate that the Eb/No advantage of MSK over offset QPSK onsuch a channelextendsto slightly narrower channel bandwidths than the 1.1 R crossover bandwidth reported in the present paper. In satellite communicationsystems,and indeed in most radio systems, adjacent channel interferenceis a critical design factor.Thequestion arises as to whether MSK, since its unfiltered spectrum falls off faster than that of offset QPSK, requires less filtering than offset QPSK for practical values of interchannel spacing and adjacent channel interference. If so, the performance crossover points of Fig. 9 must be re-examined. Define a carrier t o adjacent channel interference ratio as

C/A =% *

w )I ~ ( f14)d f / I w

m

I H(f) l2 I HCf+ Af) l2 df

(1 6)

where C ( f ) is the power spectral density of the MSK or offset QPSK waveform [(12) and (13)] , and H(f)is the common transfer function of the twosystem filters. The factor I H ( f ) l4 is the magnitude-squared transferfunctionoftwoidentical filters in cascade, with a commoncenterfrequencyand cascaded noise bandwidth B. The factor 1 H ( f ) l2 IH(f + Af3 l2 represents a cascade of the same two filters with the exception that one of the filters is now offsetfromtheotherbythe channel spacing variable Af Equation (1 6) was calculated for a range offilterbandwidths and channel spacings, and the 'While referred to as QPSK in [ 141,it was pointed out during the Presentation of the paper at ICC '75 that the modulation technique actually considered was offset QPSK.

818

IEEE TRANSACTIONS ON COMMUNICATIONS, AUGUST 1976

results plotted as curves of channel spacing normalized to data rate versus channel bandwidth normalized to data range with C/A as, a constant parameter. Two such curves are shown in Fig. 10 for MSK andfor offset QPSK. These curve.s canbe used to adjust the performance degradation curves of Fig. 9 to conform to a constantadjacent channel interference criterion. As anexample2 of the use of Figs. 9 and 10, considerTable 111. The table presents acomparison of BER degradations for MSK and offset QPSK as a function of allowable channel spacing with BER and C/A specified to by and 25 dB,respectively. The classifications ofthe various bandwidth restrictions are somewhat arbitrary, and are meant to apply t o the 7-pole Chebyshev filters used in the simulation. For either conventional or offset QPSK, narrower bandwidths can be used without significant degradation if delay equalization is used [ 151 . It remains for future work to determine the allowable bandwidth for MSK when equalization is employed. Someinitial work in this area is reported in [16] . For the -I simulation reported here, Table I11 indicates that offset QPSK 0.5 1.O 1.5 is most advantageous ii~ bandwidth limited systems, while ET - C H A N N E L B A N D W I D T H N O R M A L I Z E D TO B I N A R Y DATA RATE (HZ/BIT/SECl MSK is superior inpower-limitedsystems withadequate Fig. 10. Permissiblechannelspacing as afunction of channelbandbandwidth for thewider main spectral lobe of MSK. width for constantcarrier to adjacentchannelinterference. (R = 1/T = binary data rate.)

v. CONCLUSIONS MSK and offset QPSK. The main thrust of the simulationwas MSK and offset QPSK have been compared as modulation to compare the performance of MSK and offset QPSK based techniques and their performance on band-limited, nonlinear on a BER criterion. The performance of either system could channels has been determined by means of digital simulation. be improved by using a noise whitening matched fiter, transThe generation of both can be represented as two orthogonal, versal equalizer,or phase equalization.Incertain satellite antipodal binary systems with the symbol timing in the two applications it may be necessary to include an additional filter channels offset by one-half of a symbol duration. MSK uses in the satellite subsequent tothebandpasshard-limiter.In half-cycle sinusoid pulse shapes while offset QPSK uses these cases MSK enjoys an advantage since it would suffer less rectangular pulse shapes. Because of the sinusoidalpulse satellite output power loss due to the filteringprocess. The shaping in MSK, it was shown that MSK canbe viewed as advantagecouldbesignificant in apower-limitedsatellite continuous phase FSK with one-half thetoneseparation of system.The results ofthe analysis andsimulation indicate conventional orthogonal symbol FSK, or coherent, sinethe advantages of one modulation technique over another are weighted, symbol offset, QPSK where, in the latter view, the not always clear cut, and the nature of the application and detection efficiency of antipodal PSK on an ideal channel is communication channel characteristics are critical to choosing achieved. Incoherentdetectionof MSK, e.g., discriminator a technique for implementation. detection, could provide a low-cost flexibility feature in some systems. A Markov process representation of the two modulation processes was developed and used to derive APPENDIX A autocorrelation and power spectral density properties for the We show the following relationships among the data terms processes. Computer simulation of MSK and offset QPSK on channels of (4): such as those encountered in satellite communication systems resulted in the definition of ranges of channel bandwidth for which one or the other technique was superior with respect to signal-to-noise ratio required to achieve a given BER. .For the Chebyshev fiters used, MSK was found to have a performance advantagewhen the allowed channelbandwidth exceeded about 1.1 timesthe binary datarate. Atlower bandwidths, offset QPSK was superior. The effect ofimposing ai^ adjacent channel interference constraintwas illustrated. For the example system simulated, the added constraint did not strongly influence the crossover point in the choice between Suggested by an anonymousreviewer.

819

GRONEMEYER AND McBRIDE: MSK AND OFFSET QPSK MODULATION TABLE I11 PERFORMANCE DEGRADATION AS A FUNCTION OF NORMALIZEDCHANNEL SPACING FOR lop4 BER AND 25 dB CIA

Bandwidth Required Performance for 25 dB C / A

Degradation (dB) at lop4

Channel Frequency Spacing

Classification of Bandwidth Restriction

MSK

O-QPSK

MSK

0-QPSK

1.4 R 1.2 R 1.1 R 1.0 R 0.8 R

Very Slight Slight Moderate Tightb Very Tight

1.45 Ra 1.15 R 1.05 R 0.93 R 0.82 R

1.30R 1.15 R 1.10 R 1.02 R 0.90 R

0.15 0.50 0.75 1.30 2.40

0.50 0.65 0.75 0.85 1.10

a Note that the filter bandwidth of B = 1.45 R for each of two adjacent channels with a frequency separation of 1.4 R between band centers results in some overlap in passbands for adjacent channels. Tight filtering for sharp-cutoff case without delay equalization. With equalization, a bandwidth of 0.625 R and comparable spacing is allowable for offset QPSK.

The proof of the above relationships is based on the recursive phase relationship (2), the assumption that x k = 0, n modulo 2 n , and trigonometricidentities.First,substitute (2) into cos x k and expand by trigonometric identities cos x k = cos

[Xk-i

+ (Uk-1

= COS ( X k - 1 )

-sin

-Uk)(nk/2)]

COS [ ( U h - - l

(xk-1)

sin

[(Uk-l

-Uk)(Tk/2)] -u k ) ( n k / 2 ) ] .

We note the following:

1) sin ( x k - 1 ) = 0 for x k - 1 = 0 , n modulo 277 2)

3 ) sin 4,

- U h = - 2 , 0,+2 for u k =

t(k-1

cos

[(Uk-l

- u k ) ( n k / 2 ) ] = 0 for k any integer

f(Uk-1

-Uk)(nk/2)1

=I

1 for k even or for k odd and u k = U k - 1

-1 for k odd and U k # U k - 1 .

Therefore, COS ( X k )

= COS ( X k - 1 ) for k even Or for k odd and U k = U k - 1

COS ( X * )

= -COS

(Xk-1)

for k odd and U k # U k - 1 .

Equivalently, replacing k evey by 2k and k odd by 2k we have

+ 1,

cos ( x g k ) = cos ( X 2 k - 1 )

cos ( x 2 k + l ) = cos ( x 2 k + l - 1 ) = cos ( X Z k ) for U 2 k + l = U Z k cos ( x 2 k + l ) =-cos

(xzk)

for U 2 k + l

=-UZk.

Thisestablished ( A I ) and (A3). The relationships (A2) and (A4) follow in a similar manner.

REFERENCES [ l ] M. L. Doelz and E. H. Heald, “Minimum-shift data communication system,” U.S. Patent No. 2,977,417, Mar. 28,1961 (assigned to Collins Radio Company). [2] S. A. Rhodes,“Effects ofhardlimiting on bandlimitedtransmissions with conventional andoffset QPSK modulation,” in Proc. Nat. Telecommunications Con$, 1972, pp. 20F/1-7.

[3] M. K. SimonandJ. G . Smith, “Offset quadrature communicationswith decision feedbacksynchronization,” IEEE Trans. Commun., vol. COM-22, pp. 1576-1584, Oct. 1974. [4] R. DeBuda,“Coherent demodulation of frequency shiftkeying with lowdeviationratio,” IEEE Trans. Commun., vol. COM20, pp. 429-435, June 1912. [ 5 ] S. A. Rhodes,“Effect of noisy phase reference on coherent detection of offset QPSK signals,” IEEE Trans. Commun., V O ~ .COM-22, pp. 1046-1055, Aug. 1974. [6] W. R. BennettandJ. Salz, “Binary data transmission by FM over arealchannel,” Bell Syst. Tech. J., vol. 42,pp. 23872427, Sept. 1963. [7] H. R. Mathwich, J. F. Balcewicz, and M. Hecht, “The effect of tandem band and amplitude limiting on the Eb/N,-, performance of minimum(frequency)shiftkeying (MSK),” IEEE Trans. Commun., vol. COM-22, pp. 1525-1540, Oct. 1974. [ 8 ] S. Lebowitz and L. Palmer, “Simulation results for intersymbol interference loss; QPSK transmission through several filter types,” in Proc.Nat.Telecommunications Conf., 1973, pp. 32D/1-7. [9] G. D. Forney, “The Viterbialgorithm,” Proc. IEEE, vol. 61, pp. 268-218, Mar. 1973. [ l o ] S. Stein and J. J. Jones,Modern Communication Principles with Application to Digital Signaling. New York: McGraw-Hill, 1961, ch. 3. [ l l ] M. Hechtand A. Guida, “Delay modulation,” Proc. IEEE (Lett.), vol. 57, pp. 1314-1316, July 1969. I [12] R.C. Titsworthand. C. R. Welch, “Power spectra of signals modulated by random and pseudorandom sequences,” Jet Propulsion Laboratory, Tech. Rep. 32-140, Oct. 10, 1961. [13]J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. New York: Wiley, 1965. [14] H.B. Pozaand H. L. Berger, “Performancecharacterization of advanced wideband data links,” in Proc. IEEE Int. Con$ Communications, 1975, pp. 4/23-27. [15] F. Assal, “Approach to near-optimum a transmitter-receiver fiiter designator data transmission pulse-shaping networks,” Comsat Tech. Rev., vol. 3, pp. 301-322, Fall 1973. [16] V. Z. Viskanta,“Effectoffiltering on MSK signals,” in Proc. Nat. Telecommunications Con$, 1915, pp. 3011-5.

* Steven A. Gronemeyer (S’68-M’72) was born in Milwaukee, WI, on February 28, 1947. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Wisconsin, Madison, in 1968, 1970, and 1973, respectively. From 1969 to 1973 he was ateaching and researchassistant in the Department of Electrical Engineering at the University of Wisconsin. In 1973 he joined Collins Radio Group of Rockwell International Corporation, Dallas,

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-24, NO. 8 , AUGUST 1916

TX, where he has worked on tropospheric scatter andsatellite communication systems. His present assignment in the GovernmentTelecommunications Division is the developmentof packet switching radio systems.. His primaryareas ofinterestarecommunicationtheory, informationtheoryand coding. During 1975 he has been a Visiting Industrial Professor atSouthern MethodistUniversity, Dallas, TX, where he has taught graduate courses in communication theory. He is the 1975-1976 Program Chairman of the Dallas Chapter of the IEEE Communications Society. Dr. Gronemeyer is a member of Eta Kappa Nu, TauBeta Pi, and Phi Kappa Phi.

Alan L. McBride (S’59-M’60) was born in Stroud, OK, on January 6 , 1933. He received the B.S.E.E. and M.S.E.E. degrees from the Universityof Oklahoma, Norman, in 1958 and 1959, respectively, and the

Ph.D. degree from Southern Methodist University, Dallas, TX, in 1968. He fiist joined Bell Laboratories as a member of the Technical Staff, where he was assigned to the Nike-Zeus Antimissile Project, and also completed their Communication Development Training Program. InJune1962hejoined Texas Instruments, Inc., Dallas, where he did work in logic design, spectrum analysis, and synchronization techniquesforantijamcommunication systems. In 1974 he joined Collins Radio Group of Rockwell International Corporation, Dallas, TX, where he is now Manager of the Advanced SystemDevelopment,Space/ Satellite CommunicationsDepartment.He also holds a Visiting Industrial Professor position at Southern Methodist University Institute of Technology, where he teaches graduate courses in communication theory. Dr. McBride received the 1970IEEE Barry Carlton Award.

Second-Order Statistical Momentsof a Surface Scatter Channel with Multiple Wave Direction and Dispersion

Absstrocl-Underwater acoustic communications channels involving surface reflection usually exhibit extensive time and frequency spreading. The phenomenon was studied in a recent paper [ 5 ] and the scattering function of the channel was obtained for large Rayleigh parameters usingaFresnel-correctedgeometric opticsmodeland asimpleonedimensional surface correlation function. It was found that while extensive timespreading and ,Dopplerwereindeed predictedby this so model, the Doppler shift and time delaywerestronglycorrelated that the scattering function had the form of a very narrow parabolic ridge. The problem considered in this paper is the extent to which this phenomenon persists when more realistic correlation-function models, dispersion, and small Rayleigh parameters are considered. It is found that the general form of the scattering function is not very much affected by fairly gross changes in the assumed surface statistics, and that substantial correlation between time delay and Dopplercan be expected as long as the surface is not completelyisotropic. The effect isalso observed in amodified form witha small Rayleigh-parametermodel, Paperapproved by the Editor for Communication Theory of the IEEE Communications Society for publication without oral presentation. Manuscript received December 7, 1975; revised March 2, 1976. Thisresearch was supportedbytheDepartment of the Navy, under Contract N00014-75-C-0298, issued by the Office of Naval Research under Contract Authority NR 083-322. However, the content does not necessarily reflect the position or the policy of the Department of the Navy or the Government, and no official endorsement should be inferred. The United States Government has at least a royalty-free, nonexclusive, and irrevocable license throughoutthe world for Government purposes t o publish,translate, reproduce, deliver, perform, dispose of, and to authorize others so t o do, all or any portion of this work. F. B. Tuteurand H. Tung are with the Department of Electrical Engineering andApplied Science, Yale University, New Haven, CT 06520. J. F. McDonald is with the Department of Electrical and Systems Engineering, Rensselaer Polytechnic Institute, Troy, NY i2181.

but since this model also features a strong coherent component it may be of less practical significance.

I. INTRODUCTION

I

N underwater acoustic communications, channels involving a reflection of the acoustic signal from the air-water surface are frequently encountered. The random motion of the rough wind-drivensurfaceresults in severe timeandfrequency spreading of the incident signal. This phenomenon has been extensivelystudied;agoodsummary and list ofpublished papers is contained in Fortuin [ I ] . In the present paper this problem is attackedbyassumingthatthesurface-reflection channel can be modeled as a random, time-varying linear filter operatingontheincident signal. Thetimevaryingtransfer functionofthisfilter,obtainedbyapplyingtheFresnel approximation to aKirchhoffintegral,hasrecentlybeen obtained [ 2 ] in the form1

,. -m

The sign conventions used in the exponents are those used in textbooks onphysical optics.

consistent with