A signal limited in bandwidth to | ω | < W can be recovered from nonunifonnly spaced...

A signal limited in bandwidth to | ω | < W can be recovered from nonunifonnly spaced samples as long as the average sample density is 2(W/2π) samples per second. This problem illustrates a particular example of nonuniform sampling. Assume that in Figure P7.37(a):

1. x(t) is band limited; X(j ω) = 0, | ω | > W.

2. p(t) is a nonuniformly spaced periodic pulse train, as shown in Figure P7.37(b).

3. (t) is a periodic waveform with period T = 2 π /W. Since f (t) multiplies an impulse train, only its values f(0) =a and f(?)=b at t= 0 and t = ?, respectively, are significant.

4. phase shifter, that is,

5. is an ideal lowpass; thatis,

Where K is a (possibly complex) constant.

(a) Find the Fourier transform of

(b) Specify the values of a, b, and K as functions of ? such that z(t) = x(t) for any band-limited x(t) and any ? such that 0 < ? < π/W.