We are given a real-valued signal $a(t)$. We want to find the filter having impulse response $h(t)$, that maximizes the ratio, $\frac{[a(t)*h(t)|_{t=0}]^2}{\int d\tau h^2(\tau)}$. Hint: use the Cauchy-Schwarz inequality, $[\int dt f(t)g(t)]^2 \leq [\int dt f^2(t)][\int dt g^2(t)]$, with equality iff $g(t) = c \cdot f(t)$, for some constant, $c$ a) Find the optimum filter (called the \"matched filter\") explicitly in terms of the signal; b) What is the frequency response of the filter?
