A method for obtaining the probability density function (PDF) and the cumulative density function (CDF) of sum-independent random variables is presented. The method is capable of determining the PDF and CDF of this sum for an input consisting of any combination of a signal tone, white Gaussian noise, and multiple interfering tones. It is based upon circularly symmetric function theory, Fourier-Bessel series, and Fourier series. To illustrate this method, applications are presented for a fast frequency-hopped noncoherent frequency-shift-keyed communications system. From the PDF and CDF of the received signal, performance values such as the error probability for demodulation, the probability of detection and false alarm for coarse-time synchronization and the mean and variance of timing-error estimates for fine-time synchronization are obtained.>

Five methods for estimating the time error in a frequency-hopped spread-spectrum (FHSS) system are analyzed and compared. These methods are based upon estimation of the second-order moments of the envelope-detected, match-filtered, received signal. The performance of these estimators is determined as a function of the Gaussian noise level and of the number of samples L used to form the estimate. The estimator mean and standard deviation are approximated by means of the central limit theorem. Solutions compare favorably with simulation results. Results for signal-to-noise ratios of 0, 5, and 10 dB and with L between 1 and 512 show that these estimators are superior to previously known methods.>

The performance of fast-frequency-hopped M-ary frequency-shift keying with a fixed hop rate is evaluated, utilizing the Chernoff union bound method. The performance criterion used is a throughput measure i.e., an information rate sustained by a system for a given bit error rate, normalized by the hop rate. Both uncoded and coded systems are considered. It is shown using the cutoff rate argument that coding can provide a few dB gain in throughput. This is confirmed by the performance evaluation of various convolutional and block codes. Both partial-band noise jamming and multitone jamming with one tone per M-ary band are considered. Jamming parameters are assumed to be the worst case against the coding channel. Determination of the optimum M is also addressed.>

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.

The technique that is described by Lanzl and Heitmann in "Direct coherent optical Fourier transform of curves,"1 is shown to be a form of area modulation. Simpler and equally useful techniques have already been described.