{a(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
$$y\left( n \right) = \frac{1}{N}\sum\limits_{r = 0}^{N - 1} {x\left( r \right)} x\left( {n + r} \right)$$ is

A) $${\left| {X\left( k \right)} \right|^2}$$ B) $$\frac{1}{2}\sum\limits_{r = 0}^{N - 1} {X\left( r \right)X'\left( {k + r} \right)} $$ C) $$\frac{1}{2}\sum\limits_{r = 0}^{N - 1} {X\left( r \right)X\left( {k + r} \right)} $$ D) 0

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Correct Answer: Option A

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{a(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 $$y\left( n \right) = \frac{1}{N}\sum\limits_{r = 0}^{N - 1} {x\left( r \right)} x\left( {n + r} \right)$$ is

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{a(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
$$y\left( n \right) = \frac{1}{N}\sum\limits_{r = 0}^{N - 1} {x\left( r \right)} x\left( {n + r} \right)$$ is