Consider a stable discrete time signal x n whose discrete time Fourier transform X(e j ) satisfies the equation X (e j ) X (e j() ) and has even symmetry, i e , x n x n (a) Show that X(e j ) is periodic with a period (b) Find the value of x 3 (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

Question: Consider a stable discrete-time signal x[n] whose discrete-time Fourier transform X(e j ) satisfies the equation X (e j ) = X (e j() )

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

X (e^jω) = X (e^j(ω–π))

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].

(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.

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