540 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. COM-24, NO. 5 , MAY 1976

I Transmission Characteristics of an Wary Coherent PSK Signal Via a Cascade of N Bandpass Hard Limiters

Abstract-This paper provides a general analysis of the transmission characteristics of a coherent PSK signal, where an Mary coherent signal is transmitted via a repeater system composed of a cascade of N bandpass hard limiters. The probability density function of a composite phase of the signal and noise over the system is newly derived to obtain an error probability. In addition, the comparison with the repeater sys- tem composed of a cascade of N linear amplifiers is made to clarify the features of the transmission characteristics. I t is found that the proba- bility density function of the composite phase over the system is sym- metric about each input carrier-to-noise power ratio (CNR), and it reduces to some well-known results. I t is also shown that for a rea- sonably large input CNR in each stage, the presence of the bandpass hard limiter (BPHL) provides some improvement, but that for small input CNR it causes degradation in performance.

INTRODUCTION

T HE evaluation of the error probability for a coherent phase- shift-keyed (CPSK) system perturbed by additive noise

has received considerable attention in a great variety of papers which have appeared in the literature [ 11 -[3] . Considering these papers from the viewpoint of a transmission system, a repeater system1 between the transmitter and receiver is assumed to be a linear amplifier system. On the other hand, recently one can find the papers in which the input-output nonlinearity in the repeater is regarded as a bandpass hard limiter (BPHL) 141 -[6] . Those arise from the digital satellite communication system where the CPSK signal is transmitted via a satellite repeater, traveling-wave tube (TWT) with non-

Thus when we consider the transmission characteristics of the CPSK signal, taking into account the nonlinearity of the repeater represented by the bandpass hard limiter, it is desirable, from a more general viewpoint, to analyze the transmission characteristics of an M-ary coherent phase-shift- keyed (MCPSK) signal via a cascade of N BPHL's. However

Paper approved by thc Associate Editor for Far Eastern Contribu- tions of the IEEE Communications Society for publication after pre- sentation at the International conference on Communications, San Francisco, CA, June 16-18, 1975. Manuscript received May 2, 1975; rcvised September 30, 1975.

T. Mizuno was with the Department of Communication Engineer- ing, Osaka University, Osaka, Japan. He is now with the Research and Development Laboratory, Kokusai Denshin Denwa Company, Ltd., Tokyo, Japan.

N. Morinaga and T. Namekawa are with the Department of Com- munication Engineering, Osaka University, Osaka, Japan.

'This paper treats a nonregenerative repeater system such as the direct and heterodync systems.

'The TWT operating just at saturation gives a constant output power and no amplitude-dependent delay, and so this paper treats only the input-output amplitude nonlinearity.

the studies in this field have, to the authors' knowledge, been limited, so far, to the case of the transmission of the binary or M-ary CPSK signal through one repeater [4] -[6] .

This paper present! the general analysis of the transmission characteristics of the CPSK signal, where the MCPSK signal is transmitted via the cascade o f N BPHL's wherein noise is additively combined between the limiters. The general expres- sion for the probability density function (pdf) of a composite phase of a signal and noise over the transmission system is first derived, by the analytic method using N-1 averaging operations on the conditional pdf of the phase, and the result is applied in developing an expression for the error probability. In addition it is compared with the transmission character-. istics via a cascade of N linear amplifiers. The results of this paper bear strongly on the detection of the MCPSK signal in the digital satellite communication systems.

TRANSMISSION SYSTEM

The transmission system via the cascade ofN BPHL's consid- ered in this paper is shown in Fig. 1 , and Fig. 2 depicts the configuration of a BPHL.

The input x1 ( t ) to BPHL. 1 consists of the MCPSK signal s l ( t ) of angular frequency u0 and amplitude A I plus a sta- tionary narrow-band Gaussian noise nl( t ) with zero mean

where O,(t) is the modulating message signal and will have some value in the discrete set 27rJ/M, 0 < J < M - 1, corre- spondingly to the message. All M messages are assumed t o be equally likely. The input noises ni(t) (i = 1,2, -., N) are statis- tically independent of each other and of the signal si(t) (i = 1,2, -., N), and can be represented as

ni(t) = g i ( t ) cos wot - qi(t) sin mot,

i = 1 , 2 , -., N (3)

where &(t) and qi(t) are statistically independent low-fre- quency Gaussian random processes with zero mean and vari- ance ui2. Accordingly, in terms of envelope and phase repre- sentation, the input x l ( t ) is expressed as

MIZUNO et a[. : M-ARY COHERENT PSK SIGNAL 54 1

Fig. 1. Transmission system via a cascade of N bandpass hard limiters. The MCPSK signal passed through the cascade of^ B P H L ~ ~ is fed to an ideal phase discriminator. The detector examines the

mitted message. The detector operates with no timing error and with zero-width decision thresholds. The approach used to find the error probability is to develop the pdf of the phase

BbN)pp55 FILTER discriminator output and announces an estimate of the trans-

Fig. 2. Block diagram of a bandpass hard limiter.

where the ‘1 (t) and the phase $1 (t> of xl ( t ) are given of GN(t) , if the Nth-stage BPHL.N is assumed to be the limiter by at the receiver, and then to integrate the pdf over the error

r l 2 W = A 1 2 + $12(t) + 7)12(t) regions.

+ 2Al(El( t ) cos e ~ ( t ) + rll(t) sin e,(t)) (5) PROBABILITY DENSITY FUNCTION p($bN) OF

(6) As it is assumed that the phase eM( t ) of the MCPSK signal

s l ( t ) is equally likely to be any of the Mphases, we can set it to be zero phase without loss of generality, Some equivalent representations for the pdf P ( @ ~ ) are known in the literature. In this paper we consider the integral representation for P ( @ ~ ) ,

(7 ) V ~ Z . 171,

for the input xl(t), where A 2 is the amplitude of s 2 ( t ) . The exp (-P1 2>

signal is then transmitted to BPHL.2 and the noise n2( t ) is added to s2( t ) . Hence, the input xz(t) to BPHL.2 is

P(@l) = 2.rrol 2

r12 - 2Alrl cos @1 x2(t> = s2( t> + n2(t>. ( 8 ) f r1 exp (- 2012

Substituting (3) and (7) into (8) yields where p12 = A12/2u12 is the input carrier-to-noise power

ratio (CNR) in BPHL.1. @ ~ ~ ( t ) , the phase of x 2 ( t ) , can be re- (9) garded as the composite phase of the angle modulated wave

xz( t ) given by (9) passes through BPHL.2 to obtain the out- put S3(t)

Y

r22 - 2Azr2 cos (@2 - G1) rz exp

2 0 2 2

S 3 ( t ) = A3 cos (wot + @ 2 ( t ) ) . (12) for fixed G1, where pZ2 = A22/2022 is the input CNR in BPHL.2. ~ ( 6 ~ ) is, therefore, obtained by averaging (17)

s 3 ( t ) is again perturbed by the noise n3(t) and thus the input over @1 with the aid of (16) x 3 ( t ) to BPHL-3 becomes

xs(t) ~ 3 ( t ) + ~ ~ 3 ( t ) . (1 3) (1 8)

~ 3 ( t ) , furthermore, passes through BPHL.3 to obtain the out- Developing the integrand including the cosine term by using put s 4 ( t ) . In such a way, the output ~ ~ + ~ ( t ) of the Nth-stage the relation BPHL-N is given by

m

where A N + , is the amplitude of the output of BPHL-N, and where Z, ( a ) is the nth-order modified Bessel function, the the phase @N(t) is given by integral with respect to G1 is now evaluated as

-- (.; ) ( Y - P 1 m r - + I 1 ~ 1 - ; m + 1;-p12 m !

The remaining integral is attributed to Weber and Sonine, and its solution is given in terms of confluent hypergeometric function [8] . Hence, we obtain the following result:

where

(*)u average over u ; v - 1 gamma function; lFl(.; .; .) confluent hypergeometric function; f m Neumann factor.

Now P ( @ ~ ) , the pdf of the phase @ 3 ( t ) of x3( t ) , is also expressed by

for fixed G2( t ) as in (17) , where = A 3 2 / 2 ~ 3 2 is the input CNR in BPHL-3. Accordingly ~ ( 4 3 ) becomes

P ( @ 3 ) =I” P(43 I @21P(@2) dtJ2 (23) -n

as in (1 8). Expanding the integrand including the cosine term in (22) with the aid of (19), again using Weber and Sonine integral formula, the following result is obtained:

Thus the pdf p(@N) of @ ~ ( t ) , the output phase of BPHL-N, is finally represented as

by integrating p(@i/@i-1) N-1 times with respect to @+I,

where pi2 is the input CNR in BPHLSi. Note that in the definition of pi2 , A i 2 / 2 (i#l) contains both signal and the preceding BPHL input noise power. The newly derived result (25) reveals the general expression for the pdf of the output phase in the final stage BPHL, when the MCPSK signal is transmitted via the cascade of N BPHL’s.

Now some discussions are of interest in regard to (25). First, for the case where only one BPHL is included, putting N = 1 and utilizing the equality

in (25), we find that (25) becomes

The above is identical to that of Middleton’s result [9]. Equation (25), in addition, is symmetric about p i 2 , 1.e., ’ one need not consider p i 2 as the input CNR in the ith stage BPHL-i, but one may consider pi2 as the input CNR of BPHL in the arbitrary stage. Furthermore, when pi2 2=- 1 (i#k), using the asymptotic expansion

(25) reduces to (27) where p1 = p k and 6, = &. This is consistent with Davenport’s result [7] which shows that the cascade of N BPHL’s is equivalent to a single BPHL when the input CNR in each BPHL except one is large. The results of numerical evaluation for the pdf of the phase are shown in Figs. 3 and 4.

ERROR PROBABILITY

Assuming the Nth-stage BPHL-N to be the BPHL at the front end of the receiver, the error probability PN,M for the MCPSK signal which is passed through the cascade of N BPHL’s is given by usingp(@N) in (25), viz.,

MIZUNO et al. : M-ARY COHERENT PSK SIGNAL 543

Fig. 3. Probability density function of @N, p ( @ ~ ) for N = 3. . .

(29)

(29) is the general expression of the error probability for a single-sample detection model. On the other hand, for the case of a matched-filter detection model where the post- detection integration over one pulse duration is performed after the multiplication with a local reference, the error probability PE is given by

due to the majority detection, where ( y ) is the binomial coefficient, T is the duration per pulse, W is the bandwidth of an integrating filter, and TW is assumed to be odd. This paper treats PN,M on the understanding that PE can be deter- mined from (30). The numerical results of the error probabi- lity given by (29) are shown by the solid lines in Figs. 5 and 6.

COMPARISON WITH LINEAR SYSTEM

In this section the error probability (29) for the cascade

P O , 1

Fig. 4. Probability density function of @N, p ( @ ~ ) for N = 4.

the system which is composed of a cascade of N linear ampli- fiers (LA'S), under the condition of the same output power for both system^.^

The error probability PN,M' of the MCPSK signal for the linear system, in this case, is determined by

if the input CNR pN'2 at the receiver is given, where p ( @ ) is the. pdf in (27) with p1 = p ~ ' and G1 = @.

Now, in order to derive pNI2 as the function of pi2, (i = 1, 2, -, N) in the BPHL system, let us first obtain the input CNR pi '2 in LA-2. The signal power P2 and the noise power N2 in the output s2 ' ( t ) are given by

where it is assumed that

( [ ~ 2 ' ( t ) ] 2, = P2 + N2 = Az2/2 ( 3 3 )

due to the condition of the same mean power for s 2 ( f ) in (7) and s2'[f), and where

I.

P1 = p 2 m 2 (34)

because the output CNR (P2/N2) is equal to the input CNR

3s ; ( t ) and pi2 (i = 1,2, -., N ) denote the signal and the input CNR of N BPHL's is compared with that for a linear system, i.e., in LA.^.

544 IEEE TRANSACTIONS ON COMMUNICATIONS, MAY 1976

'"it-

p12 in the 'LA. The input noise power ([n2(t)] 2, in LA.2 can be expressed as the function of p22

([K?z(t)] 2, = CJz2 =L422/2p22. (35)

Therefore the input CNR p2I2 in LA92 can be obtained from (32) and (35) as

With the same procedure described above, the input CNR p 3 r 2 takes the same functional form as (36), viz.,

P2'2p32 P 3 1 2 = 1 + p 2 f 2 + p32 (3 7)

3; -dB

Fig. 6. Error probability of the quaternary CPSK signal for N = 3.

BPHL systems, the input CNR pNr2 of the Nth-stage LA*N is finally given by

N n P i 2

where

(39)

Substituting (36) into (37), we obtain the input CNR p312 of LA-3

P12P22P32

p312 = 1 + p12 + p22 + p32 + p12p22 + P 2 2 P 3 2 + P32P12'

(38)

Thus expressing the input CNR pi f2 in LA-I' in terms of the input CNR pi-1'2 in LA*I'-1 on the assumption that the output mean power in each stage is same for both LA and . . "

Consequently, the error probability P n r , ~ ' for the case of the cascade of N LA'S can be obtained by substituting (27), (39), and (40) into (31). The results of numerical evaluation of the error probability for the LA system are shown by the dotted lines in Figs. 5 and 6.

Although the error probability is plotted as a function of p32 (it may be regarded as the input CNR at the receiver in this case) in Figs. 5 and 6, one obtains the same results for p12 or p22 as the abscissa due to the symmetric characteris- tics about pi in (25). It is shown that the BPHL system yields better performance than the LA system when the input CNR in each stage is comparatively large, because the in-phase component of noise is eliminated by the limiter. It is also shown that the improvement of the error probability due to the limiter is more significant in the quarternary CPSK corn- Dared with the binary CPSK. For 02+ =, the results for the

MIZUNO et a[. : MARY COHERENT PSK SIGNAL 545

binary CPSK are in agreement with ~~i~ and Blachman [ 5 ] [9] D. Middleton, An Introduction to Statistical Communication

where the transmission system with two BPHL’s is discussed. N. Morinaga, T. Mizuno, and T. Namekawa, -Transmission When the input CNR is small, the LA system gives a slight characteristics of an M-ary coherent PSK signal via a cascade of improvement on the error probability over that in the BPHL N bandpass hard limiters-application to digital satellite com-

system, i.e., this is because of the small-signal suppression munication systems,” in Proc. IEEE Int. Con$ Communications, pp. 35-26,35-30, June 1975.

effect in the limiter. Thus it is interesting to note that the error probability characteristic is significantly correlated to the improvement effect of CNR in the limiter.* *

Theory. New York: McCraw-Hill, 1960.

CONCLUSION

This paper gives the general analysis of the transmission characteristics when the MCPSK signal is transmitted via the repeater system composed of the cascade of N BPHL‘s. Assum- ing that the Gaussian noise is added to the input of each BPHL, the pdf of the composite phase of the signal and noise over the system has been newly derived to obtain the error probability. The comparison with the repeater system com- posed of the cascade of N LA’S is also made to show more clearly the features of the transmission characteristics. The error probability of the BPHL system is found to be smaller than that of the LA system for large vdues of CNR’s. An application of the results in this paper to the digital satellite communication system including more than one satellite repeater is discussed in some detail by the authors [ 101 .

ACKNOWLEDGMENT

The authors wish to thank Dr . T. Satoh and Y. Hirata , both with the Research and Development Laboratory, Kokusai Denshin Denwa Company, Ltd., for providing many useful suggestions.

REFERENCES C. R. Cahn, “Performance of digital phase-modulation communi- cation systems,” IRE Trans. Communications Systems, vol. CS-7, pp. 3-6, May 1959. C. W. Helstrom, “The resolution of signals in white, Gaussian noise,”Pfoc. IRE, vol.43, pp. 1111-1118, Sept. 1955. V. K. Prabhu, “Error-rate considerations for digital phase-

vol. COM-17, pp. 33-42, Feb. 1969. modulation systems,” ZEEE Trans. Communication Technology,

I. Jacobs, “The effects of video clipping on the performance of an active satellite PSK communication system,” IEEE Trans. Commuhication Technology, vol. COM-13, pp. 195-201, June 1965. P. C. Jain and N. M. Blachman, “Detection of a PSK signal trans- mitted through a hard-limited channel,” ZEEE Trans. Informa- tion Theory, vol. IT-19, pp. 623-630, Sept. 1973. R. G. Lyons, “The effect of bandpass nonlinearity on signal detectability,” IEEE Trans. Communications, vol. COM-21. pp. 51-60, Jan. 1973. W. B. Davenport and W. L. Root, Random Signals and Noise. New York: McGraw-Hill, 1958. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. New York: Springer, 1966.

Toshio Mizuno (S’76) was born in Osaka, Ja- pan, on August 11, 1948. He received the B.E. degree in communication engineering from Osaka Electrical Communication University, Osaka, Japan, in 1971, and the M.E. and Dr.E. degrees in communication engineering from Osaka University, Osaka, Japan, in 1973 and 1976, respectively.

From April 1971 to March 1972 he did research at the Research and Development Laboratory, Kokusai Denshin Denwa Com-

pany Ltd., Tokyo, Japan, in connection with a joint research program. In April 1976, he joined the Research and Development Laboratory, Kokusai Denshin Denwa Company, Ltd., where he has since been en- gaged in the research of the satellite communication systems. He has been working in the field of communication theory and its application.

* Norihiko Morinaga (S’64-M’68) was born in Hyogo, Japan, on June 6, 1939. He received the B.E. degree in electrical engineering from Shizuoka University, Harnamatsu, Japan, in 1963, and the M.E. and Dr. E. degrees in com- munication engineering from Osaka University, Osaka, Japan, in 1965 and 1968, respectively.

He is currently an Assistant Professor in the Department of Communication Engineering, Osaka University, working in the areas of detection theory, satellite communication,

Dr. Morinaga is a member of the Institute of Electronics and Com- and laser communication.

munication Engineers of Japan.

* Toshihiko Namekawa (”72) was born in Hyogo, Japan, on October 28, 1922. He re- ceived the B.E., M.E., and Dr.E. degrees in communication engineering from Osaka Uni- versity, Osaka, Japan, in 1942, 1950, and 1962, respectively.

He was an Associate Professor in the Depart- ment of Electrical Engineering, Kobe Univer- sity, Kobe, Japan, from 1955 to 1960. In 1961 he was appointed Associate Professor in the Department of Communication Engineer-

ing at Osaka University, Osaka, Japan, where he is now a Professor on the Faculty. He has worked on noise theory, electronic circuits, com- munication theory, and CATV systems.

Dr. Namekawa is the Vice-president of the Institute of Television Eneineers of Jaoan. a member of the Institute of Electronics and Com- -

A BPHL gives 3 4 6 improvement of CNR for large input CNR, and munication Engineers of Japan, the Institute of Electrical Engineers ~L 1 ~

1-dB degradation for small input CNR. of Japan, and the Institute of Television Engineers of Japan.