additional constraints

[logistic_guy]

here is the question

If $\displaystyle x[n] = \cos(\frac{\pi}{4}n + \phi_0)$ with $\displaystyle 0 \leq \phi_0 < 2\pi$ and $\displaystyle g[n] = x[n]\sum_{k=-\infty}^{\infty}\delta[n - 4k]$,
what additional constraints must be imposed on $\displaystyle \phi_0$ to ensure that $\displaystyle g[n]*\frac{\sin\frac{\pi}{4}n}{\frac{\pi}{4}n} = x[n]$?


my attemb
the question look complicated and i don't understand convolution
but i understand the idea of convolution, they want to recover the signal $\displaystyle x[n]$ from $\displaystyle g[n]$
if $\displaystyle \frac{\sin\frac{\pi}{4}n}{\frac{\pi}{4}n} =$ sinc$\displaystyle \frac{\pi}{4}n$, can i replace it to simplify the convolution further or it effect the solution?