Page 456 - Fundamentals of Radar Signal Processing

model for any radar detection problem due to at least three major limitations.

First, coherent radar systems that produce complex-valued measurements are of
most interest. The approach must therefore be extended to the complex case.
Second, there are unknown parameters. The analysis so far has assumed that
such signal parameters as the noise variance and target amplitude are known,
when in fact these are not known a priori but must be estimated if needed. To
complicate matters further, some parameters are linked. Specifically in radar,

the (unknown) echo amplitude varies with the (unknown) echo arrival time
according to the appropriate version of the radar range equation. Thus, the LRT
must be generalized to develop a technique that can work when some signal
parameters are unknown. Finally, as seen in Chap. 2, there are a number of
established models for radar signal phenomenology that must be incorporated.
In particular, it is necessary to account for fluctuating targets, i.e., statistical
variations in the amplitude of the target components of the measured data when a

target is present. Furthermore, while the Gaussian PDF remains a good model
for noise, in many problems the dominant interference is clutter which may have
one of the distinctly non-Gaussian PDFs discussed in Chap. 2. The next
subsections begin addressing these shortcomings by extending the LRT to
coherent systems.

6.2.1 The Gaussian Case for Coherent Receivers

An appropriate model for noise at the output of a coherent receiver was
developed in Chap. 2. It was shown there that if the noise in the system prior to
quadrature signal generation is a zero mean, white Gaussian process with power
, the I and Q channels will each contain independent, identically
9
distributed zero-mean white Gaussian processes with power That is,
the noise power splits evenly but independently between the two channels. A

complex noise process for which the real and imaginary parts are i.i.d. is called
a circular random process. The expression for the joint PDF of N complex
samples of the circular Gaussian random process is

(6.26)

where m is the N × 1 vector mean of the N × 1 vector signal y = m + w, S is the
y
N × N covariance matrix of y

(6.27)

and H is the Hermitian (conjugate transpose) operator. In most cases the noise

samples are i.i.d. so that , which in turn means that .