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$\{x(n)\}$ is a real-valued periodic sequence with a period N. $x(n)$ and $X(k)$ form $\mathrm{N}$-point Discrete Fourier Transform (DFT) pairs. The DFT $Y(k)$ of the sequence $y(n)=\frac{1}{N} \sum_{r=0}^{N-1} x(r) x(n+r)$ is

1. $|X(k)|^{2}$
2. $\frac{1}{N} \sum_{r=0}^{N-1} X(r) X^{*}(k+r)$
3. $\frac{1}{N} \sum_{r=0}^{N-1} X(r) X(k+r)$
4. $0$