Absrrucr-Expressions are found for the effect of an error in the delay of the preceding signal, which provides the reference phase for the decoding of the present signal in differential phase-shift-keying reception. The signal-to-noise ratio is allowed to be different for the two signals that are compared by the receiver's phase detector. The results are applicable to both binary 'and quaternary DPSK. In addition, an approximation is obtained for the error probability when the two Signal-to-noise ratios are equal to the same large value.. In the differentially coherent reception of binary phaseshift-keyed signals, the phase G1 of the present signal plus noise is compared with the phase $10 of the preceding signal plus noise. If the difference q51 -lies in the first or fourth quadrant, the decoded bit is taken to be a "mark," and, if G1 -$0 lies in the second or third quadrant, it is taken to be a "space." We shall suppose that a mark has been transmitted and that would be zero were it not for the effect of additive narrow-band Gaussian ,noise, which gives G1 the probability density function [ 1 , p. 601 P ( 6 cos 4 1 ) cos $1where rl is the signal-to-noise ratio and is the cumulative standard normal distribution function [ 2 ,p. 9311. In the absence of noise cp0, too, would ideally be zero, but an error in the delay of the preceding signal prior to its comparison with the present signal will cause the distribution of 40 to be centered on another value, say e. Thus, Go can be assumed to have the probability density functionwhere the ratio r , of signal power t o noise power in the preceding waveform may differ from that in the present waveform. In the case of quaternary DPSK, two reference phases must be used to decode the two bits carried by each signal, and so e is deliberately given two different values intended to be close to +n/4 for use in the two phase detectors.Paper approved by the Editor for Data Communication Systems of the IEEE Communications Society for publication without oral presentation.
GENERAL EXPRESSION FOR ERROR PROBABILITYTo facilitate the convolution of (1) with (2) and its integration over the second and third quadrants, which gives the error probability, we observe that (1) is a function of cos $1 and so we can express i.t in the form of a Fourier cosine series:* r l % m l F l ( ? h m ; m + l ; -r l ) c o s m @ l .
(3).Here (%m)! denotes r(%m + l), which for odd m is %m-(%m -1) ..-%*+, ,F1 is the confluent hypergeometric function [Z, ch. 131, and the coefficient of cos mG1 in this series has been found as l/n times J o 2 n p l ( @ l ) cos m@1 d g l , which is the mean value of cos m@l evaluated in [ 1 , p. 591 . Similarly, (2) can be expressed as a series in cos m(40 -e), and the two probability densities can be convolved by multiplying them together, replacing 41 by $0 + @, and integrating with respect to Go from 0 t o 2n. The result is the probability density function o f @ = @ ] -G o : * lFI('hm;m+ l;-rl)cosm($+e).(4) Integrating with respect t o I $ from %n t o 3n/2, we get the error p...