Performance of Digital Communications Systems in an Arbitrary Fading Rate and Jamming Environments

The purpose of this theoretical investigation is to study the effects of fading rate on the performance of binary digital communications systems. The performance of three systems is evaluated from the aspect of probability of error per information bit when the desired and undesired signals are subject to statistically independent Rayleigh fading. Diversity reception is not included in the system comparison. The results, which apply only for a nonselective fading environment, of this investigation are the following: The effect of fading rate on communication system performance was evaluated. The information bit error probability function was explicitly related to the signal envelope and phase fluctuations. A Rayleigh-type fading with arbitrary fading rate was assumed for the signal in order to get an understanding of the system performance in a fast fading environment. Three basic types of communication systems were investigated. The first system, called the quadrature detection system, employs large-bandwidth signal duration product (WT) with quadrature detection. The next two systems employ a WT product approximately equal to one. These last two systems were the noncoherent frequency-shift keyed (FSK) and the differentially coherent phaseshift keyed (DPSK), also known as the phase comparison-type system. (Section I.) The comparison of three systems illustrating the effect of Rayleigh signal fading on the mean error probability
(\bar{Pe})
showed that the phase comparison system was most critical to fast fading while the noncoherent FSK was somewhat better than the quadrature detection system. The figures show that, for a fading signal bandwidth information bit duration (BT) product of one half, the phase comparison system rapidly degraded to a
\bar{P}e
of one half. The other two systems showed a slower rate of degradation as the BT product increased. The quadrature detection system and the FSK receiver with incoherent detection are not identical but are equivalent for slow fading. The performance of these two systems as a function of BT is definitely different. Also, the methods of improving the performance of both systems in a fast fading environment are completely different. The graphs of error probability covering ranges of 10
²
are of practical interest because they illustrate the comparison of the different systems as a function of fading rate and are necessary for an experimental evaluation of the systems. The exact expressions for error probability are given for evaluating any of the three system performances for ranges not covered in the graphs. (Section II.) In the error probability analysis, the basic function which is considered is the conditional probability of error per information bit averaged over the additive noise for a given desired and undesired signal envelope and phase fluctuations. This conditional probability of error will vary from one information bit to the next and, therefore, it must be considered to be a random process. Since this conditional error probability is a random process, it is important to determine its distribution function. It is shown, for the quadrature detection system, that there is a rapid degradation of the quartiles as the BT product increases, whereas the spread of the distribution decreases. An attempt was made to do a similar analysis on the other systems but the analytical problems were more difficult and time did not permit a solution. (Section III.) Received December 19, 1962. RCA Defense Electronic Products Division, Moorestown, N.J. An increase in transmitted signal power always results in a reduction in errors in the quadrature detection and FSK (filters sufficiently separated so there is no intersymbol interference) systems,but in the presence of fading there is a limit on the degree of error reduction which can be attained in the phase comparison system. In the presence of fading these theoretical results show there exists some signal level above which the error rate for the FSK system is less than that for the phase comparison-type system. These conclusions have been verified by field tests. (Section IV.) The performance of the phase comparison systems depends upon the phase stability of the propagation medium. Since these phase stabilities have not been investigated in any great detail, the complete capabilities of the phase comparison systems cannot be made for the various media. (Section V.) The shape of the spectral density of signal fading can have a significant effect on error probability performance. The single-tuned fading spectrum gave the worst error probability performance, the Gaussian spectrum was in the middle and the ideal band-pass spectrum had the least severe effect. (Section VI.) Energy detection of noncoherent FSK signals shows a very significant improvement in system performance over envelope detection in a fast fading environment. In addition, energy detection could result in better performance in a fast fading environment compared to a slow fading environment. This is completely reversed from the envelope detection system. (Section VII.) In quadrature detection using orthogonal coded-type waveforms, it is necessary to have maximum correlation over the entire waveform for minimum error probability. Fast fading will destroy this correlation and thereby increase the error probability. Improved performance may be obtained by subdividing the information bit so that the correlation of the subdivided waveform is high. The correlator outputs of this subdivided information bit are squared and added. The resulting error probability will show a significant improvement in performance in a fast fading environment compared to the nonsubdivided information bit technique. (Section VIII.)