{x(n)} is a real-valued periodic sequence with a period N. x(n) and X(k) form N-point Discrete Fourier Transform (DFT) pairs. The DFT Y(k) of the sequence y(n) = 1/N ∑r=0^N-1 x(r) x(n + r) is (a) |X(K)|^2 (b) 1/N ∑r=0^N-1 X(r) X^* (k + r) (c) 1/N ∑r=0^N-1 X(r) X (k + r) (d) 0

Question

[latex]\left \{ x(n) \right \}[/latex] is a real-valued periodic sequence with a period [latex] N.\, x(n)[/latex] and [latex]X(k)[/latex] form [latex]N[/latex]-point Discrete Fourier Transform (DFT) pairs. The DFT [latex]Y(k)[/latex] of the sequence

[latex]y(n) = \frac{1}{N} \sum_{r=0}^{N-1} x(r) x(n + r)[/latex] is

(a) [latex]|X(K)|^{2}[/latex]

(b) [latex]\frac{1}{N} \sum_{r=0}^{N-1} X(r) X^{*} (k + r)[/latex]

(c) [latex]\frac{1}{N} \sum_{r=0}^{N-1} X(r) X (k + r)[/latex]

(d) [latex]0[/latex]

Accepted Answer

DFT of [latex]y(n) = |x(k)|^{2}[/latex]

Hence, the correct option is (a).

Question : {x(n)} is a real-valued periodic sequence with a period N. x...