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1 Matched Filters Matched filters are commonly employed in practical applications of signals and systems to detect the presence of specific signal patterns in an observed signal corrupted by additive noise. If the signal pattern to be detected is p(t), the matched filter is an LTI system with impulse response h(t) = p(-t). If an observed signal x(t) contains one or more instances of p(t), these can be found by determining the time instants at which the matched filter output y(t) = x(t) * h(t) shows a peak (* denotes convolution). p(t) 1 -1 Figure A1-1 (a) ints] Consider p(t), shown in Fig. Al-1, which is the pattern to be detected. De- termine the impulse response of the matched filter, h(t), for this pattern, and sketch p(t) * h(t). Note when p(t) * h(t) is maximised. (b) ju puts The matched filter in (a) is not a causal system. To make it causal, one can shift h(t) to obtain h(t - to). Sketch w(t) = p(t) * h(t - to) for an appropriate to that makes the matched filter causal, and find t = tm at which w(t) is maximised. What is the relationship between tm and to