Page 117 - Fundamentals of Radar Signal Processing

whether the probability density function has one or two free parameters. The

nonfluctuating, exponential, and all chi-square (once the order is stated) are all
one-parameter distributions. The one parameter in the form given earlier is the
mean RCS, . The non-central chi-square, Weibull, and log-normal are two-
parameter distributions, as is the chi-square with variable degree. The
2
parameters are and a for the non-central chi-square, and m for the variable-
order chi-square, B and C for the Weibull, and σ and s for the log-normal in the
m
forms given. For a one-parameter distribution, estimating the mean is sufficient
to characterize the complete PDF. For the two-parameter case estimates of two
parameters, usually the variance and either the mean or median, must be
computed to characterize the PDF. This distinction is important in the design of
automatic detection algorithms in Chap. 6.
Most radar analysis and measurement programs emphasize RCS

measurements, which are proportional to received power. Sometimes ζ, the
corresponding voltage, is of interest, particularly for use in simulations where
Eq. (2.50) is used explicitly to model the composite echo from a multiple
scatterer target. The probability density function for the voltage is then required
in order to properly model the probabilistic variations of the complex sum. The
PDF of |ζ| is easily derived from the PDF of σ using basic results of random

variables (Papoulis and Pillai, 2001). Because RCS is nonnegative, the
transformation 5

(2.54)

from RCS to voltage has only one real solution for σ, namely σ = ζ, . It then
2
follows that the PDF of ζ is given by

(2.55)

Equation (2.55) can be used to write the voltage PDFs by inspection from
Table 2.3. The results, given in Table 2.4, are expressed in terms of the
parameters of the corresponding RCS distribution from Table 2.3. Additional
information is given in App. A.