The antijam performance of M -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 for M -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 on L hops 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.

Antijam Performance of Fast Frequency-Hopped M-ary NCFSK--An Overview

A relatively complete analytical treatment of TCT (tone-calibrated transmission) error rate is presented. It includes the effects of Rice and Rayleigh fading. frequency offsets, and a selection of nonidealities in the receiver's pilot tone filter. Such nonidealities are shown to result in the reappearance of an error floor, which is quantified and shown to be insignificant except for extreme fading rates. Comparison with differentially detected PSK (phase-shift keying) shows that TCT provides a lower bit error rate in the noise-limited region and a much lower (by orders of magnitude) error floor in the random FM limited region. >

Performance of tone calibration with frequency offset and imperfect pilot filter (mobile radio)

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.

Error Performance of Binary NCFSK in the Presence of Multiple Tone Interference and System Noise

The authors derive closed-form expressions for the BER (bit error rate) performance of coherent M-ary quadrature amplitude modulated (QAM) system in the presence of an additive combination of Gaussian and highly impulsive noise by using a Fourier-Bessel series expansion method. The model used for the impulse noise, which closely models atmospheric and/or ignition noise, consists of a stream of Poisson impulses with areas distributed according to a bilateral Rayleigh probability distribution function. Although the methodology is general enough to accommodate any M-ary QAM scheme, they present analytical as well as some computer simulated results for 16-, 64-, and 256-QAM systems. An almost insignificant increase of the truncation error with the increase of the modulation levels verifies that the proposed method of analysis can be a useful tool in evaluating the performance of very large-alphabet M-ary QAM systems, e.g. M>or=256. >

Fourier-Bessel error performance analysis and evaluation of M-ary QAM schemes in an impulsive noise environment

The use of side information can provide increased coding gains in frequency-hop multiple-access communication systems. The internal generation of this side information by the receiver is investigated in the paper. Methods to determine which received signals to erase prior to decoding are derived from Bayesian decision theory. The performance of synchronous and asynchronous frequency-hop systems with Reed-Solomon coding is analyzed for channels with Rayleigh fading, multiple-access interference, and wideband Gaussian noise. The performance of the Bayesian technique is compared with that of receivers that make erasure decisions using the Viterbi ratio-threshold test. The Bayesian method is found to provide better performance. >

A decision-theoretic approach to the generation of side information in frequency-hop multiple-access communications

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.

D. George

Papers

The Use of the Fourier-Bessel Series in Calculating Error Probabilities for Digital Communication Systems