On the statistical detection problem for multiple signals

The problem of detecting signals in noise is reviewed for the multiple input model, where each of the inputs can contain one of many possible signals. The detection procedure for this model becomes, in general, the testing of multiple hypotheses. Two detection criteria are examined for choosing among multiple hypotheses and it is found that, for both criteria, the decision is based on the likelihood functions for the various signals. Systems for computing likelihood ratios are examined in detail for the multiple input case. A multidimensional matched filter is considered and its relationship to the likelihood ratios is shown. Optimum signals are determined for the two-hypothesis problem.