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I have been trying to use the logic that both X and Y should have same Z transform, but according to the definition, Y is not anti causal.

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Hint: Question says $y[n]$ has length $10$, but the alternate DFT coefficients of $y[n]$ i.e. $Y(e^{j\omega})|_{\omega = 2\pi \frac{k}{5}}$, matches with $X(e^{j\omega})$ evaluated at those $\omega$ exactly.

This should draw your attention towards upsampling of 5-point DFT $X(e^{j\omega})|_{\omega = 2\pi \frac{k}{5}}$ or equivalently periodization of a length $5$ segment of $x[n]$.

Like upsampling of time-domain sequence by $N$, by inserting $(N-1)$ zeros between samples, shrinks the spectrum in frequency domain and brings $N-1$ more copies of spectrum inside $[-\pi, \pi]$, similarly, upsampling in frequency domain by inserting $(N-1)$ zeros between DFT samples will create more copies of time-domain sequence.

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  • $\begingroup$ Thanks! I was only thinking in terms of 5 point DFT. Upsampling the same makes much sense! $\endgroup$ – Ruhi Jul 17 at 11:26
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Hint: Solve for the Z-transform of $x(n)$ which is $X(z)$. Then from this solve from $X(e^{j\omega})$ with $\omega = 2\pi k/5$ (the missing j is certainly a typo) and the k's given.

If you are unable to solve the Z-transform, try to use the geometric series directly with the equation for the z-transform and this should help:

$$\sum_{n=0}^\infty r^n = \frac{1}{1-r} \space \space r<1$$

This should get you past the clueless point.

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