Page 348 - Fundamentals of Radar Signal Processing

given previously is easily extended to higher order MTI filters. As the order

increases, the corresponding N-pulse canceller becomes a poorer
approximation of the matched filter (Schleher, 2010).
It is interesting to consider the form of the optimum filter when the
dominant interference is noise rather than clutter, that is, . In this case the
optimum first-order MTI filter of Eq. (5.26) reduces to (ignoring overall scale

factors again)

(5.27)

Equation (5.27) states that in the presence of completely uncorrelated
interference and with no knowledge of the target velocity, the filter impulse
response reduces to a single impulse, h[m] = δ[m]. Since convolving any signal
with δ[m] just returns the same signal, the filter does nothing. In the clutter-
dominated case, the filter combined the two slow-time samples because, even
though constructive interference of the target could not be guaranteed, the high
correlation of the clutter did guarantee that the clutter signal would be
suppressed. On average the overall effect was beneficial. In the noise-

dominated case, there is still no guarantee that the target signal will be
reinforced, and in addition there is now no guarantee that the noise will be
suppressed. The filter therefore does not combine the two data samples at all.
The previous analysis assumes that the target Doppler shift is unknown and
therefore considers all target Doppler frequencies equally likely. It is easy to

modify the analysis to match the MTI filter to a specific Doppler shift or to the
case where the target Doppler extends only over a portion of the Doppler
spectrum. These alternative assumptions manifest themselves as alternate
models for the desired signal vector t. The second case is treated in Schleher
(2010) and in Prob. 8; here the case of a known Doppler shift for the target and
a two-pulse canceller is considered. The interference and signal models are

exactly the same as given earlier except that now the target Doppler shift in t is
not a random variable but a specific, fixed value. Therefore, it is not necessary
to take an expected value of t. The filter coefficient vector is

(5.28)

where again all overall constants have been dropped. While this result is easy