ieee - msk and offset qpsk modulation

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-24, NO. 8 , AUGUST 1976 809

IV. CONCLUSIONS [9] R. M. Gray, “On the asymptotic eigenvalue distribution of ToeDlitz matrices,” ZEEE Trans. Inform. Theory, vol. IT-18,

The paper has summarized some results on the properties pp. 725-730, Nov. 1972.

of the eigenvectors and eigenvalues of persymmetric matrices. [ lo ] R. R. Anderson and G. J. Foschini, “The minimum distance for

MLSE digital data systems of limited complexity,” ZEEE Trans. Persymmetric matrices appear quite often in communication Inform. Theory, vol. IT-21, pp. 544-551, Sept. 1975. and information theory. Recently the eigenvectors of symmetric matrices have occurred as solutions to a number of im- portant problems in data communication. The properties of

* the eigenvectors presented can be used to characterize and simplify the solution to these probiems. This has been demon- strated 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, Tech Rep. 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,

F. R. Magee, Jr., and J. G. Proakis, “An estimate of an upper bound on error probability 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 maximum likelihood sequence estimation,” Bell Syst. Tech. J., vol. 52, pp. 1541-1562, Nov. 1973. U. Grenander and G. Szego, Toeplitz Forms and Their Applica- tions. Berkeley, CA: Univ. of California Press, 1958.

VOI. IT-20, pp. 479-489, July 1974.

Antonio Cantoni (”74) was born in Soliera, Italy, on October 30, 1946. He received the B.E. degree with first class honors in 1968 and the Ph.D. degree in 1972 both from the Uni- versity of Western Australia, Nedlands, Aus- tralia.

He was, a Lecturer in Computer Science at the Australian National University from 1972 to 1973. Presently, he is with the Department

_ 1 :.., ., , , ’, of Electrical Engineering at the University of Newcastle, New South Wales, Australia. He is

interested in digital communication and computer systems. He has also been a consultant on systems program development and industrial electronics.

New South Wales, Au graph enumeration in Newcastle.

* Paul Butler (S’72-M’75) was born in Sydney, Australia, on April 7, 1941. He received the B.Sc. degree from Sydney 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 the Department of Electrical Engineering, University of Newcastle,

Istralia. He is currently working on problems of the Department of Mathematics, University of

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

Absfruct-Minimum shift keying (MSK) and offset keyed quadrature phase shift keying (OK-QPSK) modulation techniques are often pro- posed for use on nonlinear, severely bind-limited communication channels because both techniques retain‘ low sidelobe levels on such chan-

Paper 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

1976. 1975. Manuscript received September 20, 1975; revised February 9,

The authors are with the Collins Radio Group, Rockwell lnter- national Corporation, Dallas, TX 75207.

nels, while allowing efficient detection performance. A more detailed performance comparison of the two techniques on such channels is, therefore, of interest. In this paper a Markov process representation is developed which is applicable to either the MSK’or OK-QPSK waveform. This representation is employed to illustrate the similarity between the modulation processes and to obtain the autocorrelations and power spectral densities of the two waveforms. This Markov process representation may be similarly employed with other modulation waveforms of the same class. The autocorrelations and power’ spectral densities of MSK and offset QPSK provide initial insight to expected performance on band-limited channels.

810 IEEE TRANSACTIONS ON COMMUNICATIONS, AUGUST 1976

The resul+s o f , a digital computer, simulation are presented. The simulltion, compares ?e bit error rates (BER’S) of MSK and .offset QPSK on nonlinear, band-limited double-hop links such as encduntered in satellite communications. The simulation results 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 provide superior performance when the channel noise bandwidth exceeds about, 1.1 times the binary d a b rate. For narrower bandwidths, offset QPSK provides superior performance.

I. INTRODUCTION

B OTH minimum shift keying (MSK) [ l ] and offset keyed quadrature phase. shift keying (OK-QPSK) modulation

have been. considered for use on band-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 regenera- tion of the filtered .sidelobes is less 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 ,a discriminator [6] , whereas QPSK systems require either a fully coherent or differentially coherent detection system. This noncoherent detection property o f MSK permits inexpen- sive demodulation when the received signal-tokoise ratio is adequate, yet allows for coherent detection with efficiency identical to coherent QPSK , in limited signal-tolnoise ratio situations. Investigations of the coherent detection performance of MSK and conventional QPSK on band-limited or combined band-limited and nonlinear channels have previously been reported [7] , [8]. This paper develops the theory and connections between MSK and offset QPSK and presents a 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 without further qualification, 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 the autocorrelation. and power spectral density properties of MSK and offset QPSK which provide insight to the BER performance of the two modulation techniques.

The results of a digital computer simulation of the two techniques over nonlinear, band-limited channels are presented. A definitive comparison of the two techniques on such channels is difficult, since the details of channel nonlinearities and passband characteristics strongly influence system performance. The approach used in comparing the two techniques

is to simulate the BER performance of each over identical channels, with post-simulation adjustment t o account for ah adjacent channel interference constraint. While the receiver implementations are not optimum over the channels considered, the simulation results provide relative performance measures of the MSK and offset QPSK modulation techniques as a function of system parameters.

The results of tlie simulation show that both MSK and offset QPSK require a certain critical bandwidth and, 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 both modulation techniques 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 data interval kT < t < (k + 1)T. Fig. l(a) illustrates the FSK nature of the MSK waveform, with a radian 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 conventional orthogonal 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 to the argument 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 xk, 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

GRONEMEYER AND McBRIDE: MSK AND OFFSET QPSK MODULATION 81 1

(A) TONE SPACING IN MSK. DATA RATE IS 1 BIT PER T SECONDS. CARRIER FREQUENCY IS w c = 2n f, RAD/SEC

(6) CONTINUOUS PHASE NATURE OF MSK

IC1 EXAMPLE OF EXCESS PHASE S i l l FOR A PARTICULAR DATA SEOUENCE Yk

-112" k 'k xk

0

2 ~1 r 1 -2" 1 -2" 1 -2"

/ /

f Oil ' ID1 LATTICE OF POSSIBLE EXCESS PHASE PATHS

Fig. 1. Characteristics of MSK. (a) MSK tone spacing. (b) Continuous phase MSK waveform. (c) Example of excess 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, depend- ing upon whether the data for that 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

where

C(t) = cos (7Tt/2T)

S(t) = sin (ntf213.

This representation of the MSK waveform can be viewed as being composed of two quadrature data channels. The in-phase channel, or I channel, is identified as cos (xk)C(t) cos ( ~ , t ) , where cos (act) is the carrier, C(t) is the sinusoidal symbol

weighting, and cos ( x k ) is the data-dependent term. Similarly, the quadrature-phase channel, 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 the data-dependent term. Since the data, U k ,

can change every T seconds, it might appear that the data 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, the symbol weighting in either the 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 to one another. Finally, the data are conveyed at a rate of one bit per 2T seconds in each 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, the quadrature signaling waveform, derived as (4), contained data terms 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 bit- to-bit independent data 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 straight- forward data "encoding" as

f u 2 k - l C [ t - 2kT] COS (act)

- u 2 k - $ [ t - (2k - 2)T] sin ( q t ) ,

(2k- 1 )T<t<2kT YMSK(I) =' '

U ~ k - l C [ t - 2kTj COS (ac t )

- u 2 k S ( t - 2kT) sin (act),

\ 2kT < t < (2k + 1)T.

The "encoding" used in (5) is to demultiplex the data stream u k into odd and even data streams which are used to determine the symbol pulse signs of the I and Q channels during odd intervals, (2k - l )T < t < (2k + 1)T, and even intervals, 2kT < t < (2k + 2)T, respectively. This process is illustrated in Fig. 2.

111. COMPARISON AND MARKOV PROCESS REPRESENTATION OF MSK AND OFFSET QPSK

Equation ( 5 ) is one case pf a class of modulation waveforms which can be defined by replacing the symbol weighting shapes C(t) and S(t) by a variety of other pulse shapes. In this paper we give particular attention to (rectangular pulse) offset QPSK, defined as


BINARY DATA u1 "8 '7 u6 u5 u4 "3 u2

I 1 I 1

" 1 = 1 u 7 = 1 u 5 = 1 u 3 = - 1 I-CHANNEL DATA

I I I I I 1 ,

Q-CHANNEL DATA u 2 = 1 u 4 = 1 u 6 = - 1 u g = l

+1

WAVEFORM I-CHANNEL

WAVEFORM 0-CHANNEL

1

Fig. 2. Data and symbol timing relationships in MSK waveform.

(U2k-1Sqc(t - 2kt) COS ( a c t )

I - u 2 k - 2 s q s [ t - (2k - 2)T] sin ( a c t ) ,

(2k - l )T < t < 2kT (6)

uZk-1Sps(t - 2kT) cos (act)

- uakSss(t - 2kT) sin ( a c t ) ,

2kT < t < (2k + l )T

where

O<t<2T elsewhere

S,,(t) = { -T<t<T

elsewhere.

The only difference between the MSK waveform (5) and the offset QPSK waveform (6) is that the former employs a half- cycle sinusoidal symbol pulse and the latter employs a rectangular pulse.

Fig. 3 illustrates typical waveforms for MSK, offset QPSK, and conventional QPSK waveforms. The binary data rate in each case is assumed to be 1/T, so that the phase transitions in the MSK and offset QPSK waveforms occur every T seconds, but the conventional QPSK waveform has transitions only every 2T seconds. Typical phase transitions for each waveform are shown. The phase transitions shown for MSK are referenced to carrier phase, and accumulate linearly over T seconds. Offset and conventional QPSK have abrupt phase

transitions. In offset QPSK, the transitions can only be +90" or -90", while in conventional QPSK, 180" phase reversals are also possible. As will be shown later, the continuous phase nature of MSK is responsible for the more rapid spectral density fall-off for this waveform as compared to offset or conventional QPSK.

The class of quadrature modulation techniques employing offset, time-skewed symbol pulses in the quadrature channels, can be represented by a first-order Markov process. This process is completely described by the state transition probability matrix, initial state probabilities, and the waveforms associated with each state. Fig. 4 illustrates the state transition diagram of the Markov process of interest. The process remains in a given state for T seconds, during which time the corresponding state waveform is generated. Tables I and I1 show the state waveforms for MSK and offset QPSK, respectively. For convenience, low-pass equivalent notation [ l o ] is used, wherein the state waveform for state i is represented by

Sdt) = Sk(t) + jSj,(t) (7)

where sic(t) is the I-channel component and sis(t) is the Q-channel component and j is n. The actual signal is given by

Each T seconds the state of the process is modified according to the next data value uk, as illustrated in Fig. 4. In the figure transition arrows indicate allowable state transitions. Each arrow is labeled to indicate the data value, + 1 , which causes this transition, and the corresponding probability of occur- rance of this data value. It will be assumed in subsequent analysis that the data values are equally likely, as indicated in the transition diagram. The transition probability matrix P corresponding to the transition diagram is given by

r o o o o % o % o - 0 0 0 0 0 % 0 %

0 0 0 0 % 0 % 0

0 0 0 0 0 % 0

% % 0 0 0 0 0 0 P = j % 1 .

1 % % 0 0 0 0 0 0

0 0 % % 0 0 0 0

0 0 % % 0 0 0 0 ~

The (i, j)th element of P, denoted by p(il i) , is defined as the probability of transition from state i t o state j at the transition instant, given that the process is currently in state i.

The Markov representation described above can be used to compute the autocorrelation functions of MSK, offset QPSK, or any of the other modulation processes of the same class. Using techniques similar to those discussed in [ l 11 , [12] , it can be shown that the low-pass equivalent autocorrelation function is given by

GRONEMEYER AND McBRIDE: MSK AND OFFSET QPsK MODULATION 813

Fig. 3. Typical waveforms for MSK, offset QPSK, and conventional QPSK.

Fig. 4. Markov process state transition diagram.

R,(T) =Ry(7 ’ + mi“)

Sii*(x) complex conjugate of sj i (x)

+ pQ I i,m + 1)sji*(r’)) S d t )

(9) a waveform of state i number of states = 8

for P(i ) probability of state i on 0 S t G T = 1 /S pG/i, m) probability of state j on mT S t < (m + l ) T

r = r’ i- mT 2 0, m = integer 2 0, 0 < 7‘ < T. given state i on 0 S t S T

RY(7) = R, *(-T) for T < 0 (i. j)th element of the matrix Pm .

Evaluting the autocorrelation expression (9) for MSK and where offset QPSK results in the autocorrelation functions in Fig. 5 .


TABLE I MSK LOW-PASS EQUIVALENT STATE WAVEFORMS

si(f) = s i c ( t ) +is&) STATE

1

I

L -COS It-nT)]

5 1 C l N [ It-nT1 1

' l r -SIN I It-nT)]

K I -SIN [ It-"TI]

! - I SIN [-& It-"TI]

S I COS [ & It-nT)]

t 7 - I -COS [ j+ It-"TI]

& I COS [ $f It-"TI]

2 Sqclt-nTl -Sqrlt-nT)

3 -sqc It-nT1 SqSlt-nTl

-Sq,lt-nTI

t I I I I

6 1 SqSlt-nTl 1 -Sqclt-nT)

7

~

Sqclt-nT) -Sqs(t-nT1

6 -Sqclt-nTl -Sg,lt-nTl

Fig. 5 . Normalized autocorrelation functions for MSK and offset QPSK.

Note that both the autocorrelation functions are zero for time lags exceeding two bit times. If either MSK or offset QPSK is transmitted through an infinite bandwidth, linear, additive WGN channel, no crosstalk will be produced between the in-phase and quadrature-phase channels at matched filter sample times. Thus, both systems can be viewed as two orthogonal binary channels, where antipodal symbols of length 2T seconds are used in each channnel, and a half-symbol timing skew of T seconds exists between the orthogonal channels. For independent binary data bits, the signs of the symbols in each binary channel are independent from one 2T interval to the next, and the autocorrelation function would be expected to go to zero for lags exceeding 2T.

Fc r later comparison, the BER performance of MSK and offset QPSK on an ideal channel is of interest. An ideal channel is defined as a linear, infinite bandwidth channel, corrupted only by additive WGN. In addition, perfect carrier

and timing references are assumed available at the receiver. With these assumptions, and viewing both MSK and offset QPSK as orthogonal binary channels with antipodal signaling, the binary error probability is known to be [ 131

where

'X = <WQ E , = signal energy per bit

No/2= two-sided spectral density of WGN.

Note that the detection of each bit takes place after observing

I GRONEMEYER AND McBRIDE: MSK AND OFFSET QPSK MODULATION 815

0

0

0 N

- 1 e, -

! ? i 0 0 z .

0 W I

0

0 P- I

0

0

I m

one of two antipodal symbols in noise for 2T seconds. We observe that the optimum performance for coherently detected orthogonal FSK (tone spacing 1/T) is

Pe = e(->. (1 1)

Thus, detecting MSK as two orthogonal binary channels provides a 3-dB advantage over detection of orthogonal FSK.

All of the foregoing discussion has centered on ideal channels having infinite bandwidth, additive WGN, and perfectly recovered carrier and timing references. Section IV will consider the performance of MSK and offset QPSK on band-limited, nonlinear channels. As a preliminary discussion to this topic, it is useful to consider the power spectral densities of the MSK and offset QPSK moduiation processes. The power spectral density of e'ach process is the Fourier transform of the corresponding autocorrelation function, and are given by the following expressions:

where

f frequency offset from carrier PC power in modulated waveform.

normalized to the binary data 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 offset QPSK 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 by the fractional out-of-band power, Po b , defined as

Equation (14) has been evaluated for the 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 provide lower error rate .performance than offset QPSK for equal transmitter power. However, as system bandwidth is decreased from 1.5 R to 1 .OR, a point should be reached at which the performance of offset QPSK will be superior to that of MSK. As system bandwidth 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 to determine in practical situations, since the detailed


intersymbol interference effects, intermodulation distortion, 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 on band-limited, nonlinear channels, such as those encountered in satellite communications systems. Fig. 8 is a block diagram of the simulated system. The simulation is carried out at baseband to minimize computer run time. The elements in the model correspond to the elements in a satellite communication link as follows. Filter number 1 simulates band-limiting elements at the transmitting earth station. Noise added at the satellite receive system is represented by WGN source number 1. The bandpass hard-limiter which simulates 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 sinusoid shapes in the “pulse shaping” and “matched filter”

system elements. Power is normalized prior to the points at which the uplink (WGN source 1) and downlink (WCN source 2) noise is added to facilitate setting the desired up and downlink values of Eb/No (energy per bit to additive noise spectral density). This normalization also resuits in equal uplink transmitted power at the output of filter number 1 for either modulation technique which simulates fixed EIRP earth stations. In running the simulation, the noise samples of WGN source 2 are not actually added to the simulation sample stream. Rather, an estimation technique similar to that employed by Lebowitz and Palmer [8] is employed. As long as the system additive noise is dominated by the down-link component, this technique provides simulation run time savings relative to an error counting BER estimation technique.

In the simulations reported here, the two system filters were chosen to be identical. The chhne l bandwidth B is defined as the noise bandwidth of the two identical filters in cascade. The simulated filters were 7 pole, 0.1-dB ripple Chebyshev low-pass equivalent filters. A high filter 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 practical satellite links are limited by downlink thermal noise, the uplink Eb/No was maintained at a high levei for the results reported here. Under the above restrictions, several MSK and offset QPSK system simulation runs were 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

GRONEMEYER AND McBRIDE: MSK AND OFFSET QPSK MODULATION 817

12k-lIT 6 t C i2k + 1lT

1 WGN SOURCE NO 1 PN SEQUENCE "2k-1 = 'l PULSE GEN (I-CHANNEL)

PULSE SHAPING

PN SEOUENCE PULSE GEN (Q-CHANNEL)

PULSE "2k ='' a -

2kT C t C (2k+2lT j = 6 i

RE (.) t = (2k+l)T SAMPLER I-CHANNEL

4 MATCHED BER FILTER ESTIMATOR

I IM i.) t = i2k+2lT Q-CHANNEL+ SAMPLER

WGN SOURCE NO 2

Fig. 8. Simulation block diagram.

I

0.5 1.0 1.5 2.0

BT - CHANNEL BANDWIDTH NORMALIZED TO BINARY DATA RATE (HZIBITISECI

Fig. 9. MSK and offset QPSK Eb/No performance degradation with respect to ideal antipodal PSK as a function of cascaded filter noise bandwidth with error probability as a parameter.

of performance degradation as a function of channel bandwidth for the indicated constant value of bit-error probability.

Performance degradation is defined as the ratio of the total link Eb/No required on the simulated link to achieve the desired BER to the Eb/N, required on an infinite bandwidth, WGN channel to achieve the same BER. The ideal link performance is given by (10). Timing in the simulated system is near ideal. Several samples of the matched fiter near the expected optimum sample time are inspected to 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 an advantage over MSK. The reason for the performance crossover is basically the narrower main lobe of the spectral density of offset QPSK as compared to MSK. Both

techniques begin to suffer severe degradations as their main spectral lobes encounter the channel band edge. Since the main lobe of MSK is 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 system element rather than a hard-limiter. Their results indicate that the Eb/No advantage of MSK over offset QPSK on such a channel extends to slightly narrower channel bandwidths than the 1.1 R crossover bandwidth reported in the present paper.

In satellite communication systems, and indeed in most radio systems, adjacent channel interference is a critical design factor. The question 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 to adjacent channel interference ratio as

C/A =% w ) I ~ ( f ) 14 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 two system filters. The factor IH( f ) l4 is the magnitude-squared transfer function of two identical filters in cascade, with a common center frequency and 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 offset from the other by the channel spacing variable Af Equation (1 6) was calculated for a range of filter bandwidths 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

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 and for offset QPSK. These curve.s can be used to adjust the performance degradation curves of Fig. 9 to conform to a constant adjacent channel interference criterion. As an example2 of the use of Figs. 9 and 10, consider Table 111. The table presents a comparison 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 of the various bandwidth restrictions are somewhat arbitrary, and are meant to apply to 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. Some initial work in this area is reported in [16] . For the simulation reported here, Table I11 indicates that offset QPSK is most advantageous ii~ bandwidth limited systems, while MSK is superior in power-limited systems with adequate bandwidth for the wider main spectral lobe of MSK.

v. CONCLUSIONS

MSK and offset QPSK have been compared as modulation techniques and their performance on band-limited, nonlinear channels has been determined by means of digital simulation. The generation of both can be represented as two orthogonal, antipodal binary systems with the symbol timing in the two channels offset by one-half of a symbol duration. MSK uses half-cycle sinusoid pulse shapes while offset QPSK uses rectangular pulse shapes. Because of the sinusoidal pulse shaping in MSK, it was shown that MSK can be viewed as continuous phase FSK with one-half the tone separation of conventional orthogonal symbol FSK, or coherent, sine- weighted, symbol offset, QPSK where, in the latter view, the detection efficiency of antipodal PSK on an ideal channel is achieved. Incoherent detection of MSK, e.g., discriminator 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 autocorrelation and power spectral density properties for the processes.

Computer simulation of MSK and offset QPSK on channels 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 advantage when the allowed channel bandwidth exceeded about 1.1 times the binary data rate. At lower bandwidths, offset QPSK was superior. The effect of imposing ai^ adjacent channel interference constraint was illustrated. For the example system simulated, the added constraint did not strongly influence the crossover point in the choice between

Suggested by an anonymous reviewer.

IEEE TRANSACTIONS ON COMMUNICATIONS, AUGUST 1976

0.5 1 .O -I 1.5

ET - CHANNEL BANDWIDTH N O R M A L I Z E D TO B I N A R Y DATA RATE (HZ/BIT/SECl

Fig. 10. Permissible channel spacing as a function of channel bandwidth for constant carrier to adjacent channel interference. (R = 1/T = binary data rate.)

MSK and offset QPSK. The main thrust of the simulation was to compare the performance of MSK and offset QPSK based on a BER criterion. The performance of either system could be improved by using a noise whitening matched fiter, trans- versal equalizer, or phase equalization. In certain satellite applications it may be necessary to include an additional filter in the satellite subsequent to the bandpass hard-limiter. In these cases MSK enjoys an advantage since it would suffer less satellite output power loss due to the filtering process. The advantage could be significant in a power-limited satellite system. The results of the analysis and simulation indicate the advantages of one modulation technique over another are not always clear cut, and the nature of the application and communication channel characteristics are critical to choosing a technique for implementation.

APPENDIX A

We show the following relationships among the data terms of (4):

GRONEMEYER AND McBRIDE: MSK AND OFFSET QPSK MODULATION 819

TABLE I11 PERFORMANCE DEGRADATION AS A FUNCTION OF NORMALIZED CHANNEL SPACING

FOR lop4 BER AND 25 dB CIA

Required Bandwidth Performance Channel for 25 dB C/A Degradation (dB) at lop4

Spacing Bandwidth Restriction MSK O-QPSK MSK 0-QPSK Frequency Classification of

1.4 R Very Slight 1.45 Ra 1.30R 0.15 0.50 1.2 R Slight 1.15 R 1.15 R 0.50 0.65 1.1 R Moderate 1.05 R 1.10 R 0.75 0.75 1.0 R Tightb 0.93 R 1.02 R 1.30 0.85 0.8 R Very Tight 0.82 R 0.90 R 2.40 1.10

a Note that the filter bandwidth of B = 1.45 R for each of two adjacent channels with a frequency

Tight filtering for sharp-cutoff case without delay equalization. With equalization, a bandwidth separation of 1.4 R between band centers results in some overlap in passbands for adjacent channels.

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 trigonometric identities. First, substitute (2) into cos x k and expand by trigonometric identities

cos x k = cos [ X k - i + ( U k - 1 - U k ) ( n k / 2 ) ]

= COS ( X k - 1 ) COS [ ( U h - - l - U k ) ( T k / 2 ) ]

-sin ( x k - 1 ) sin [ ( U k - l - u k ) ( n k / 2 ) ] .

We note the following: 1) sin ( x k - 1 ) = 0 for x k - 1 = 0 , n modulo 277

2) t ( k - 1 - U h = - 2 , 0, +2 for u k =

3 ) sin [ ( U k - l - u k ) ( n k / 2 ) ] = 0 for k any integer

4, cos f ( U k - 1 - U k ) ( n k / 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 ( X k - 1 ) for k odd and U k # U k - 1 .

Equivalently, replacing k evey by 2k and k odd by 2k + 1, we have

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 ( x z k ) for U 2 k + l = - U Z k .

This established (AI) and (A3). The relationships (A2) and (A4) follow in a similar manner.

REFERENCES [ l ] M. L. Doelz and E. H. Heald, “Minimum-shift data communica-

tion system,” U.S. Patent No. 2,977,417, Mar. 28, 1961 (assigned to Collins Radio Company).

[2] S. A. Rhodes, “Effects of hardlimiting on bandlimited trans- missions with conventional and offset QPSK modulation,” in Proc. Nat. Telecommunications Con$, 1972, pp. 20F/1-7.

[3] M. K. Simon and J. G. Smith, “Offset quadrature communications with decision feedback synchronization,” IEEE Trans. Commun., vol. COM-22, pp. 1576-1584, Oct. 1974.

[4] R. DeBuda, “Coherent demodulation of frequency shift keying with low deviation ratio,” IEEE Trans. Commun., vol. COM- 20, pp. 429-435, June 1912.

[5] S. A. Rhodes, “Effect of noisy phase reference on coherent detection of offset QPSK signals,” IEEE Trans. Commun.,

[6] W. R. Bennett and J. Salz, “Binary data transmission by FM over a real channel,” Bell Syst. Tech. J., vol. 42, pp. 2387- 2427, 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) shift keying (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.

[9] G. D. Forney, “The Viterbi algorithm,” 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. Hecht and A. Guida, “Delay modulation,” Proc. IEEE (Lett.), vol. 57, pp. 1314-1316, July 1969.

I [12] R. C. Titsworth and. 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 Communica- tion Engineering. New York: Wiley, 1965.

[14] H. B. Poza and H. L. Berger, “Performance characterization of advanced wideband data links,” in Proc. IEEE Int. Con$ Com- munications, 1975, pp. 4/23-27.

[15] F. Assal, “Approach to a near-optimum transmitter-receiver fiiter designator data transmission pulse-shaping networks,” Comsat Tech. Rev., vol. 3, pp. 301-322, Fall 1973.

[16] V. Z. Viskanta, “Effect of filtering on MSK signals,” in Proc. Nat. Telecommunications Con$, 1915, pp. 3011-5.

V O ~ . COM-22, pp. 1046-1055, Aug. 1974.

32D/1-7.

* 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 a teaching and research assistant in the Department of Elec- trical Engineering at the University of Wiscon- sin. In 1973 he joined Collins Radio Group of Rockwell International Corporation, Dallas,

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

TX, where he has worked on tropospheric scatter and satellite communication systems. His present assignment in the Government Telecom- munications Division is the development of packet switching radio systems.. His primary areas of interest are communication theory, information theory and coding. During 1975 he has been a Visiting Industrial Professor at Southern Methodist University, 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, Tau Beta 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 Univer- sity of Oklahoma, Norman, in 1958 and 1959, respectively, and the

Ph.D. degree from Southern Methodist Univer- sity, 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. In June 1962 he joined Texas Instruments, Inc., Dallas, where he did work in logic design, spectrum analysis, and synchronization techniques for antijam communication systems. In 1974 he joined Collins

Radio Group of Rockwell International Corporation, Dallas, TX, where he is now Manager of the Advanced System Development, Space/ Satellite Communications Department. He also holds a Visiting In- dustrial Professor position at Southern Methodist University Institute of Technology, where he teaches graduate courses in communication theory.

Dr. McBride received the 1970 IEEE Barry Carlton Award.

Second-Order Statistical Moments of 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 using a Fresnel-corrected geometric optics model and a simple one- dimensional surface correlation function. It was found that while extensive time spreading and ,Doppler were indeed predicted by this model, the Doppler shift and time delay were strongly correlated so 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 af- fected by fairly gross changes in the assumed surface statistics, and that substantial correlation between time delay and Doppler can be expected as long as the surface is not completely isotropic. The effect is also ob- served in a modified form with a small Rayleigh-parameter model,

Paper approved 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. This research was supported by the Department 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 in- ferred. The United States Government has at least a royalty-free, non- exclusive, and irrevocable license throughout the world for Govern- ment purposes to publish, translate, reproduce, deliver, perform, dis- pose of, and to authorize others so t o do, all or any portion of this work.

F. B. Tuteur and H. Tung are with the Department of Electrical Engineering and Applied 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-driven surface results in severe time and frequency spreading of the incident signal. This phenomenon has been extensively studied; a good summary and list of published papers is contained in Fortuin [ I ] . In the present paper this problem is attacked by assuming that the surface-reflection channel can be modeled as a random, time-varying linear filter operating on the incident signal. The time varying transfer function of this filter, obtained by applying the Fresnel approximation to a Kirchhoff integral, has recently been obtained [ 2 ] in the form1

,. -m

The sign conventions used in the exponents are consistent with those used in textbooks on physical optics.

ieee - msk and offset qpsk modulation

Documents