Consider the case in which the signal has an exactly known form, . In
general, signals can occur with a variety of amplitudes; we let be the
known signal with some fiducial normalization and we write the actual signal
(if present) as where is the amplitude of the signal. Further, we
assume that the noise samples are drawn from a stationary Normal distribution,
though there may be correlations amongst the noise events (colored noise).
The noise correlations may be expressed in terms of the one-sided noise
power spectrum,
,
where is the Fourier transform of the noise , and
denotes complex conjugation. The probability of obtaining an instant of
noise, , is
, where the inner
product
is given by

(13.1.289) |

(13.1.290) |

It is straightforward to compute the false alarm and true detection
probabilities for any choice of threshold .
Under the assumption
that there is no signal present, the false alarm probability is the
probability that :

(13.1.291) |

(13.1.292) |

Here, the complementary error function is defined by . Using these equations, the threshold can be set for any desired false alarm probability, and then the probability of a true detection can be computed.