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### A Receiver for a Known Signal

Consider the case in which the signal has an exactly known form, . In general, signals can occur with a variety of amplitudes; we let be the known signal with some fiducial normalization and we write the actual signal (if present) as where is the amplitude of the signal. Further, we assume that the noise samples are drawn from a stationary Normal distribution, though there may be correlations amongst the noise events (colored noise). The noise correlations may be expressed in terms of the one-sided noise power spectrum, , where is the Fourier transform of the noise , and  denotes complex conjugation. The probability of obtaining an instant of noise, , is  , where the inner product is given by

 (13.1.289)

The likelihood ratio is the ratio of the probabilities (since given the signal is present) and  :
 (13.1.290)

where and . Since the likelihood ratio is a monotonically increasing function of , and since the output  appears only in the construction of , we can set a threshold on the value of  rather than on .

It is straightforward to compute the false alarm and true detection probabilities for any choice of threshold . Under the assumption that there is no signal present, the false alarm probability is the probability that :

 (13.1.291)

Notice that we have used the absolute value of  since the actual signal may have either positive or negative amplitude. Similarly, the true detection probability (the converse of the false dismissal probability) is the probability that when a signal is present:
 (13.1.292)

Here, the complementary error function is defined by . Using these equations, the threshold can be set for any desired false alarm probability, and then the probability of a true detection can be computed.

Next: A Receiver for a Up: The Statistical Theory of Previous: Maximum Likelihood Receiver   Contents
Bruce Allen 2000-11-19