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Suppose that the detector output, , contains either noise alone, , or both a signal and noise, . The maximum likelihood receiver returns the quantity

 (13.1.287)

where is the probability of obtaining the output given that there is a signal present and  is the probability of obtaining the output given that there is no signal present. The likelihood ratio  can be viewed as the factor which relates the a priori probability of a signal being present with the a posteriori probability of a signal being present given the detector output:
 (13.1.288)

In general, there is no universal way of deciding on the a priori probabilities and , so one is limited to the construction of the likelihood ratio . However, as grows larger, the probability of a signal increases, so we can use it to test our hypotheses as follows:
• If then decide that there is a signal present.
• If then decide that there is no signal present.
Here, is some threshold. Lacking any a priori information about whether there is a signal present, the threshold  should be determined by setting a desired probability for a false alarm and/or false dismissal.

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