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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
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Bruce Allen
20001119