Consider a stable discrete-time signal x[n] whose discrete-time Fourier transform X (ejω...

Consider a stable discrete-time signal x[n] whose discrete-time Fourier transform X (e^jω) satisfies the equation

and has even symmetry, i.e., x[n] = x[−n].

(a) Show that X (e^jω) is periodic with a period π.

(b) Find the value of x[3]. (Hint: Find values for all odd-indexed points.)

(c) Let y[n] be the decimated version of x[n], i.e., y[n] = x[2n]. Can you reconstruct x[n] from y[n] for all n. If yes, how? If no, justify your answer.