Optimum detection of a randomly frequencymodulated carrier
Abstract
The logarithm of the likelihood ratio for detection of a frequencymodulated signal tone, received in the presence of white Gaussian noise, is derived and expressed in the form of an infinite Volterra series. The amplitude of the received signal tone and the received noise power density level are presumed known; the additive phase shift is uniformly distributed; and the frequency modulating process is Gaussian, with a spectrum such that no carrier tone remains in the frequency modulated signal spectrum. This is called a fullyrandom signal herein. When the time bandwidth product (of observation time and signal bandwidth) is large enough, and there is at least a moderate amount of frequency modulation, the optimum processor is wellapproximated by a filter followed by an energy detector; the filter passband is that of the spectrum of the received signal. The frequency modulation is termed moderate when the ratio of the RMS frequency deviation to the equivalent bandwidth of the frequency modulating process is of the order of 23. Numerous approximations have been necessary to facilitate evaluation of some of the multiple integrals; to what degree the sufficient conditions cited above can be relaxed, without violating the conclusions, is unknown.
 Publication:

NASA STI/Recon Technical Report N
 Pub Date:
 October 1979
 Bibcode:
 1979STIN...8021646N
 Keywords:

 Carrier Frequencies;
 Frequency Modulation;
 Signal Detection;
 Bandpass Filters;
 Variance (Statistics);
 Volterra Equations;
 White Noise;
 Communications and Radar