Page 477 - Fundamentals of Radar Signal Processing

(6.73)

(6.74)

Since z′ is the sum of N scaled random variables r , the PDF of z′ is the N-
n
fold convolution of the PDF given in Eq. (6.73) or (6.74). This is most easily
found using characteristic functions (CFs; see App. A). If C (q) is the CF
z
corresponding to a PDF p (z), the CF of the N-fold convolution of the PDFs is
z
the product of their individual characteristic functions, i.e. .
Under hypothesis H the CF of r can be readily shown to be
n
0

(6.75)

The characteristic function of z′ is therefore

(6.76)

The PDF of z′ is obtained by inverting its characteristic function using the
Fourier-like inverse CF transform, giving

(6.77)

Using Eq. (6.76) in Eq. (6.77) and referring to any good Fourier transform table
(with allowance for the reversed sign of the Fourier kernel in the definition of
the characteristic function), the Erlang density is obtained

(6.78)