The Effect of Phase Error on DPSK Error Probability

Abstract

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...

Symbol error rate bound of DPSK modulation system in directional wave propagation

“…causes a heavier tail of the phase difference pdf in (7), and therefore degrades the performance of DPSK demodulation in (10) and (11). Moreover, high SNR makes the performance curves approach the theoretical ones, which substantially supports our theoretical analysis.…”

“…=I 0 ðÞ and f m determines the influence of the fading rate in (7). Moreover, ¼ 0 represents the uniformly distributed AOA and leads to the results in the classic Jakes model [5], which is shown in Figure 1.…”

“…Accordingly, the authors devote themselves to the study of the evolution of the symbol error rate (SER) of DPSK modulation in directional mobile propagation channels. Exact expressions for the probabilities of the symbol error for DPSK over additive white Gaussian noise (AWGN) as well as fading environments are available in the literature [6][7][8][9][10]. For example, Blachman [7], Pauw [8] and Pawula [9] addressed the effects of phase error in an AWGN channel, and Proakis [10] developed a closed-form expression for the average SER of binary DPSK over slowly fading Rayleigh channels, which is based on the characteristic function method and does not take into account the influences of fading rate, i.e.…”

Cross-Phase Modulation Induced by OOK Channels on Higher-Rate DQPSK and Coherent QPSK Channels

“…In this section we will start from standard results on the BER evaluation for PSK modulated signals in the presence of both additive noise and a phase offset [14], and tailor them to find the BER of our upgraded optical (D)QPSK channels with XPM induced phase noise, similarly to the work in [15], [16]. In our optical system, the received optical field is the sum of a propagation-distorted signal component and amplified spontaneous emission (ASE) noise cumulated along the dispersion-managed line.…”

“…The extension to the case of both ASE polarizations is simple [16] and does not significantly change the numerical result Now, use (12), (11) into (10) to get a general expression of (13) where is the phase of the complex , and is similarly defined. Now, when is a circular Gaussian vector, with and zero-mean independent Gaussian RV's with common variance , one can prove that the angle has a Bennet PDF whose CF can be written explicitly as the following real quantity [14]: (14) where is the instantaneous SNR for the sample at time . Therefore, when the noises and are independent circular Gaussian RV's with identical variance, one can similarly define the SNR at time , and using (14) the MDPSK error probability formula (13) becomes (15) In coherent communications, the performance can be evaluated as for differential MPSK by considering that in this case the second beating field comes from the local oscillator and is thus not affected by ASE.…”