# 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 $$X(e^{j\omega}) = \begin{cases} A - 1 &0\le | \omega| \le \frac{3}{2} \\ A & \frac{3}{2}\lt | \omega| \le \pi \end{cases}$$ It means no value of $$A$$ works. I don't know whether is my answer correct. Maybe I've neglected something.

• One methodological comment: $X(e^{j\omega})$ is either $0$ or $1$, for each $\omega$, not globally Jul 18, 2020 at 13:55
• @LaurentDuval So it means $A = 1$ is the answer? Jul 18, 2020 at 13:58

Systematically, you could just solve the problem by finding a solution $$A$$ that satisfies the following two equations:
$$(A-1)^2=A-1\\A^2=A$$
• That's right. It should hold for each $\omega$ not globally. As always I've made a silly mistake. Thanks. Jul 18, 2020 at 14:44