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### A Receiver for a Signal of Unknown Phase

Suppose that the signal is known up to an arbitrary phase, where and  are known waveforms and is the unknown phase. For simplicity, suppose further that and  , i.e., the waveforms are orthogonal. Again assume that the noise is a stationary (but colored) normal process. Then the likelihood ratio, averaged over a uniform distribution of the unknown phase, is

 (13.1.293)

where with and . Here, is the modified Bessel function of the first kind of order zero. Since this function is monotonic, a threshold level can be set on the value of the statistic  rather than .

When there is no signal present, the statistic  assumes the Rayleigh distribution

 (13.1.294)

and, in the presence of a signal, the statistic assumes a Rice distribution
 (13.1.295)

Thus, the false alarm probability for a threshold of is
 (13.1.296)

and the true detection probability is
 (13.1.297)

The -function, which is defined by the integral
 (13.1.298)

has the properties , , and the asymptotic forms
 (13.1.299)

For any desired value of the false alarm probability, we can calculate the threshold  . Then the probability of true detection can be evaluated for that threshold using the -function.

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