Page 231 - Fundamentals of Radar Signal Processing

(4.12)

where the second step follows from Parseval’s relation. Using Eq. (4.12) in Eq.
(4.11) gives

(4.13)

Equation (4.13) states the remarkable result that the maximum achievable SNR
depends only on the energy of the waveform and not on other details such as its
modulation. Two waveforms having the same energy will produce the same

maximum SNR, provided each is processed through its own matched filter.
Although it is the ratio of the peak signal component power to the noise
power, the SNR of Eq. (4.13) is called the energy SNR because the peak signal
power at the matched filter output equals the energy of the transmitted signal. To
see this, note that the peak signal component at the matched filter output is given

by Eq. (4.10) with t = T M

(4.14)

Also, the duration of the signal component of the matched filter output is exactly
2τ seconds, since it is the convolution of the τ-second pulse with the τ-second
matched filter impulse response.

The previous results can be generalized to develop a filter that maximizes
output signal-to-interference ratio (SIR) when the interference power spectrum
is not white. In radar, this is useful for example in cases where the dominant
interference is clutter, which generally has a colored power spectrum. The
result can be expressed as a two-stage filtering operation. The first stage is a
whitening filter that converts the interference power spectrum to a flat spectrum

(and also modifies the signal spectrum in the process); the second stage is then a
conventional matched filter as described earlier, but designed for the now-
modified signal spectrum. Details are given by Kay (1998).

4.2.2 Matched Filter for the Simple Pulse
To illustrate the previous ideas, consider a simple pulse of duration τ:

(4.15)