Determine constant $A$ such that $x[n] = x[n] \star x[n]$ Let $x[n] = A\delta[n] - \frac{\sin(\frac{3n}{2})}{\pi n}$. Determine constant $A$ such that for all $n$ $$x[n] = x[n] \star x[n] \tag{1}$$
I think it's not possible since $(1)$ leads to $$X(e^{j\omega}) = X(e^{j\omega})X(e^{j\omega})$$ And this means $X(e^{j\omega}) = 1$ or $X(e^{j\omega}) = 0$. Also $$X(e^{j\omega}) =  \begin{cases} A - 1   &amp;0\le | \omega| \le \frac{3}{2}   \\ A &amp;   \frac{3}{2}\lt | \omega| \le \pi \end{cases}$$
It means no value of $A$ works. I don't know whether is my answer correct. Maybe I've neglected something.
Systematically, you could just solve the problem by finding a solution $A$ that satisfies the following two equations:
$$(A-1)^2=A-1\\A^2=A$$