Another application in which matched filters and correlation functions play an important...

Another application in which matched filters and correlation functions play an important role is radar systems. The underlying principle of radar is that an electromagnetic pulse transmitted at a target will be reflected by the target and will subsequently return to the sender with a delay proportional to the distance to the target. Ideally, the received signal will simply be a shifted and possibly scaled version of the original transmitted signal.

Let p(t) be the original pulse that is sent out. Show that

That is, Use this equation. to deduce thail if the waveform that comes back to the sender is

where a is a positive constant, then

(Hint: Use Schwartz's inequality.)

Thus, the way in which simple radar ranging systems work is based on using a matched filter for the transmitted waveform p(t) and noting the time at which the output of this system reaches its maximum value.