Detection Statistics

Abstract

Up to this point, we have described the efficiency of an optical or infrared detector in terms of its output signal-to-noise ratio, without questioning the relationship between this measure and the operating characteristics of the system. In this section, we shall treat the properties of the detection process that are particularly important for a radar or communications system. These are the probability of detection p _d and the probability of error or false alarm p _fa. In essence, we shall determine the effect of the signal-to-noise ratio upon the expectation of detecting a radar return or communication signal in the presence of noise, subject to an always finite probability that the supposed signal is really caused by noise in the detection system. One of the critical properties that affects this system behavior is the statistics of the received signal; we shall first treat the expected behavior of backscattered returns from a radar target in terms of the spectral width of the transmitted signal and the target properties. In this case, we shall find two extremes, namely a constant return power for any return pulse or an exponential probability distribution (Rayleigh) of pulse powers, that is, a mean-square fluctuation in received power that is equal to the square of the mean. We shall not attempt to treat the expected fluctuations in a communications channel, because these are difficult to specify, “fading” being caused by a large number of different phenomena such as turbulence, multipath scattering, and receiver-pointing fluctuations. Because the constant and the exponential statistics represent two extremes of fluctuation statistics, their treatment effectively encompasses the range of expected behaviors in radar and communication systems.