TABLE OF CONTENTS Up to now, we have ignored the form of the signal term v(t) coming from the receiver, and have simply examined the implications of thresholding the combined signal plus noise. In practice, the shape of the signal coming from the receiver is largely under the control of the system designer, and its properties are chosen to meet such specified criteria as desirable time or range sidelobe levels, resolution, etc. The most fundamental criterion, against which all other criteria must be balanced, is the ability to detect targets, which, as we have seen, is a function of the SNR. Hence a natural priority is to design the receiver to maximize SNR. By Eq. (4.33), we know that the noise power ? m is dependent only on the gain of the receiver, not on the shape of its impulse response function. Hence, for fixed gain, the best SNR is obtained by maximizing the response to the signal term. This is achieved by an elegant, conceptually simple and readily implemented processing scheme, known as the correlation receiver, whose frequency-domain implementation is the matched filter.
The scheme relies on a fundamental result known as the Cauchy-Schwartz inequality. This states that, given two functions f(s) and g(s) of finite energy, then
with equality if and only if
for some constant c. Here the asterisk denotes complex conjugate, and we allow complex functions so that we can, if desired, use complex representations of real signals or...
