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Maximum Likelihood Receiver

Suppose that the detector output, $h$, contains either noise alone, $h=n$, or both a signal and noise, $h=s+n$. The maximum likelihood receiver returns the quantity

\begin{displaymath}
\Lambda = \frac{P(h\mid s)}{P(h\mid\neg s)}
\end{displaymath} (13.1.287)

where $P(h\mid s)$ is the probability of obtaining the output given that there is a signal present and  $P(h\mid\neg s)$ is the probability of obtaining the output given that there is no signal present. The likelihood ratio $\Lambda$ 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:
\begin{displaymath}
\frac{P(s\mid h)}{P(\neg s\mid h)} = \Lambda\,\frac{P(s)}{P(\neg s)}.
\end{displaymath} (13.1.288)

In general, there is no universal way of deciding on the a priori probabilities $P(s)$ and $P(\neg s)=1-P(s)$, so one is limited to the construction of the likelihood ratio $\Lambda$. However, as $\Lambda$ grows larger, the probability of a signal increases, so we can use it to test our hypotheses as follows: Here, $\Lambda_\ast$ is some threshold. Lacking any a priori information about whether there is a signal present, the threshold $\Lambda_\ast$ should be determined by setting a desired probability for a false alarm and/or false dismissal.


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