Consider a signal x(n) with Fourier transform(a) Show that the signal x(n) can be recovered from its samples x(mD) if the sampling frequency ?x = 2?/D ? 2?m(fs = 1/D ? 2fm).(b) Show that x(n) can be reconstructed using the formula(c) Show that the bandlimited interpolation in part (b) can be thought as a two-step process: First, increasing the sampling rate by a factor of D by inserting (D ?? 1) zero samples between successive samples of the decimated signal xa(n) = x(nD) and second, filtering the resulting signal using an ideal lowpass filter with cutoff frequency ?c.

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x(n) X(w) -2 -10 1 2 .. (a) n= 0, ±2, ±4, ... n = +1, +3, ±5,... x(n). x,(n) 0.