Answer Happy

= - - = 3. (40 pts) In this problem we will investigate whether a band-limited signal can be recovered from unevenly spaced samples at the Nyquist rate () ) (a) Let za(t) = Enez 0 (t Enez 5 (t - n - 1 - «). Find its Fourier transform Za(f). Find simplified expres- sions for the special cases a = 0 and a= 1/2. (b) Suppose that U(f) = 0 for If| > 1. Let ya(t) = u(t)za(t). For (< f <1, find a simple expression for Ya(f) in terms of U(f). For -1<f<0, find another simple expression. (C) For -1/2 <a < 1/2, find a function s(t) and its spectrum S(f) with the following properties: s(0) = 1, s(n) = 0 for n #0, s(n + { + a) = 0 for all integers n, and S(f) = 0 for [f|> 1. (d) Suppose that y is strictly baseband with bandwidth 1. For -1/2 < a < 1/2, find an expression for y(t) as an interpolation of samples at times ZU{n+*+a:n € Z}. Prove that the interpolation produces the original values of y.