The antijam performance ofM-ary noncoherent FSK with fast frequency hopping (one or more hops per symbol) is studied. A summary of terminology, definitions, and an overview of the literature is given. A general analysis method based upon circularly symmetric signal theory is presented. It provides a means for exact analysis of performance forM-ary operation with any combination of signal tone distribution (frequency bins contiguously or noncontiguously spaced), jamming distribution, jamming form (partial-band noise or multitone), and receiver system noise. Some previous results are derived as subcases of the general theory. A simple robust diversity combining technique of transmitting a symbol onLhops followed by a majority vote decision is found to provide significant performance improvements. This improvement is comparable to that obtained by most error-correction coding methods. Illustrative calculations of performance were chosen to indicate trends rather than to be exhaustive. Performance is usually presented as probability of bit error versus fraction of the band jammed to emphasize worst-case operation.

This paper presents a general solution to the problem of calculating the error performance of binary NCFSK systems in the presence of multiple tone interference and system noise. The development uses the concepts of circularly symmetric signals and expresses the results in terms of a Bessel integral. This integral is then evaluated using a rapidly converging Fourier-Bessel series. The error in this approximation to the integral is controlled by two parameters which may be adjusted so that the desired accuracy is attained. It is determined that, in the useful probability-of-error range and when the total tone power is constrained, the error rate increases for an increase in the number of interfering tones. In addition, for equal amplitude tones, the performance is independent of the distribution of the tones within the channel. The technique used is also applicable to other detection problems where the threshold is a random variable.

Using the theory of spherical symmetric random vectors one can find an expression for the error probability of a wide variety of digital communications systems. These expressions, however, are in the form of Bessel integrals which are usually difficult to solve. In this paper we show how the Fourier-Bessel series can be used to solve the integrals numerically. The calculation error is found to depend on two series parameters which can be manipulated to make the error arbitrarily small. Two examples are used to show the utility of the technique. In the first the probability of error for a CPSK communications system operating in Gaussian noise and cochannel interference is found. In the second the error performance for a multilevel ASK communications system operating in the same corrupting environment is determined. The Fourier-Bessel series technique is a valuable practical tool for solving these and other signal detection problems.