Page 342 - Fundamentals of Radar Signal Processing

filtering. For example, an FIR highpass filter could be designed using standard

digital filter design techniques such as the window method or the Parks-
McClellan algorithm. To be suitable as an MTI filter, the FIR filter frequency
response should have a zero at F = 0. In terms of the four recognized classes of
FIR digital filters (see Oppenheim and Schafer, 2010, Sec. 5.7), the MTI filter
can be either type I (even order with symmetric impulse response) or type IV
(odd order with antisymmetric impulse response). The transfer functions of type

IV filters always have a zero at z = 1 so that the frequency response is zero at f
= 0, ideal for an MTI filter. The two-pulse canceller (which has an order of 1)
is an example of a type IV filter. Type I filters do not necessarily have a zero at
f = 0, but can be made to have one by requiring that the sum of the impulse
response coefficients h[m] equal zero. The three-pulse canceller is an example
of a type I filter that has been designed to be suitable as an MTI filter.
Type II and III filters are unsuitable because they always have a zero at z =

–1, corresponding to a frequency response null at a normalized frequency of f =
0.5; this creates extra undesirable blind speeds (see Sec. 5.2.4). Alternatively,
infinite impulse response (IIR) highpass filters could be designed. Many
operational radar systems, however, use two- or three-pulse cancellers for the
primary MTI filtering due to their computational simplicity.

5.2.2 Vector Formulation of the Matched Filter

The N-pulse cancellers described previously can be remarkably effective and
have been widely used. Nonetheless, they are motivated by heuristic ideas. Can
a more effective pulse canceller be designed? Since the goal of MTI filtering is
to improve the signal-to-clutter ratio, it should be possible to apply the matched
filter concept of Chap. 4 to this problem. To do so for discrete-time signals, it is
convenient to first restate the matched filter using vector notation. This will also
aid in generalizing the matched filter somewhat.

Consider a complex signal column vector y = [y[m] y[m – 1] · · · y[m – N
T
+ 1]] and a filter weight vector h = [h [0] · · · h[N – 1]] . The superscript T
T
represents matrix transpose so that y and h are N-element column vectors. A
T
single output sample z of the filter is given by z = h y. The power in the output
sample is given by

(5.5)

where a superscript H represents the Hermitian (complex) transpose.
The matched filter is obtained by finding the filter coefficient vector h that
maximizes the SIR of the filtered data. Denote the desired target signal vector by
t and the interference vector by w, so that y = t + w. The interference is a

random process, but it is not assumed to be white or Gaussian, thus allowing
modeling of both noise and clutter. The filtered signal and interference are,