Page 465 - Fundamentals of Radar Signal Processing

(6.50)

The expression Q (α, γ) is known as Marcum’s Q function . It arises frequently
M
in radar detection calculations. A closed form for this integral is not known.
Algorithms for evaluating Q (α, γ) are compared in Cantrell and Ojha (1987).
M
TM
TM
The Communications Toolbox and Signal Processing Toolbox optional
packages of MATLAB® includes a marcumq function to evaluate Q (α, γ);
M
another MATLAB® algorithm is given by Kay (1998).
By defining a change of variables the integral of Eq. (6.49) can be put into

the form of Eq. (6.50). Specifically, choose and .
Substituting into Eq. (6.49) and doing the integration gives

(6.51)

Finally, noting that is the signal-to-noise ratio χ and expressing the
threshold in terms of the false alarm probability using Eq. (6.48) gives

(6.52)

It is usually the case that the energy E in m or is not known. Fortunately,

Eq. (6.52) does not depend on E (or the noise power explicitly, but only on
their ratio χ, so that it is possible to generate the ROC without this information.
However, actually implementing the detector requires a specific value of the

threshold T′ as given in Eq. (6.48), and this does require knowledge of both E

and . One way to avoid this problem is to replace the matched filter
coefficients with a normalized coefficient vector . This choice simply
normalizes the gain of the matched filter to 1. The energy in this modified
sequence is , leading to a modified threshold

(6.53)

The modified matched filter gain and threshold result in no change to the ROC

so that Eq. (6.52) remains valid. Setting of the threshold still requires

knowledge of the noise power ; removal of this restriction is the subject of
Sec. 6.5. The handling of unknown amplitude parameters is discussed in