Page 543 - Fundamentals of Radar Signal Processing

with N = 20 and a threshold scale factor of α = 10. Compute the average
OS
false alarm probability P for the order statistic k = 15, 16, …, 20.
FA
24. For the OS-CFAR in the previous problem with k = 15, how many outliers
(e.g., other targets) could be tolerated in the lead and lag windows without
serious degradation of the detection and false alarm performance?

_____________
1
In some detection problems, a third hypothesis is allowed: “don’t know.” Most radar systems, however,
force a choice between “target present” and “target absent” on each detection test.
2
A fourth probability can be defined, that of choosing H and thus declaring a target not present when in
0
fact the test sample is due to interference only. This probability, equal to 1 – P , is not normally of direct
FA
interest.
3 The exception occurs if points are added to or subtracted from for which p (y|H ), p (yH ), or both are
1
Y
1
0
y
zero. In that case the corresponding probability is unchanged.
4
Some subtleties that can arise if the PDFs are noncontinuous are being ignored. See Johnson and Dudgeon
(1993) for additional detail.
5
A monotone decreasing operation would simply invert the sense of the threshold test.
6
All of the following development is fairly easily generalized for the case when m is negative or is of
unknown sign. It will be seen later that radar detection generally involves working with the magnitude of the
signal, thus it is sufficient to work with a positive value of m.
7 ™
The definitions of Eqs. (6.18) and (6.19) are the same as those used in MATLAB .
–1
–1
8 The erf (·) function will often be used here, even when erfc (·) gives a slightly more compact
–1
®
–1
expression because of the wider availability of erf (·) functions than erfc (·) in MATLAB and similar
computational software packages.
9 Here T refers to receiver temperature, not the detection threshold. Which meaning of T is intended should
be clear from context throughout this chapter.
10
An alternative approach called the generalized likelihood ratio test (GLRT), in which the unknown
parameter(s) are replaced by their maximum likelihood estimates, is discussed in many detection theory
texts (e.g., Kay, 1998).
11 For instance, the earlier discussion of unknown signal energy corresponds to choosing A = 1/E.
12 The MATLAB® function gammainc is consistent with this definition.
13
I N–1 (x) is the modified Bessel function of the first kind and order N – 1, not to be confused with the
incomplete gamma function I(u, M).
14
Values of N > 100 are outside the range for which Albersheim’s equation claims good accuracy.
Nonetheless, the general trend of the proportionality of G to does tend to hold for larger N and also
nc
for fluctuating target models.
15 Unless otherwise stated, this same arrangement of lead, lag, and guard cells is used in all examples in this
chapter.
16
The base of the logarithm affects the specific offset needed to set the threshold but is otherwise
unimportant. In Eq. (6.156) it is assumed that the log data are on a decibel scale.
17
This terminology is unfortunate because all CFARs are adaptive in that they estimate the detection
threshold from the measured data. “Adaptive CFARs” are those for which one or more of the parameters
of the estimation algorithm itself are varied depending on data characteristics.