Recall that the Fourier transform of and the Fourier transform of is...

Recall that the Fourier transform of and the Fourier transform of

is

(a) Determine the Fourier transform of

and roughly sketch versus Q.

(b) Now consider the exponential sequence

where is some arbitrary frequency in the range radians. Give the most general condition that must satisfy in order for x(n) to be periodic with period P (P is a positive integer).

(c) Let y(n) be the finite-duration sequence

where is a finite-duration rectangular sequence of length N and where x(n) is not necessarily periodic. Determine and roughly sketch for What effect does N have in

(d) Suppose that

P a positive integer

and

where I a positive integer. Determine and sketch the N -point DFT of y(n). Relate your answer to the characteristics of

(e) Is the frequency sampling for the DFT in part (d) adequate for obtaining a rough approximation of directly from the magnitude of the DFF sequence If not, explain briefly how the sampling can be increased so that it will be possible to obtain a rough sketch of from an appropriate sequence