Next: Reception of a Signal
Up: The Statistical Theory of
Previous: A Receiver for a
Contents
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
20001119