In this problem we develop the dual to the time-domain sampling theorem, whereby a time-...

In this problem we develop the dual to the time-domain sampling theorem, whereby a time-limited signal can be reconstructed from frequency-domain samples. To develop this result, consider the frequency-domain sampling operation in Figure P7.52

(a) Show

Where are the inverse Fourier transform of and P(jω), respectively

(b) Assuming that x(t) is time-limited so that show that x(t) can be obtained from through a “low-time windowing” operation. That is,

Where

(c) Show that x(t) is not recoverable from is not constrained to be zero for .