2
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ One methodological comment: $X(e^{j\omega})$ is either $0$ or $1$, for each $\omega$, not globally $\endgroup$ Jul 18, 2020 at 13:55
  • $\begingroup$ @LaurentDuval So it means $A = 1$ is the answer? $\endgroup$
    – S.H.W
    Jul 18, 2020 at 13:58

1 Answer 1

2
$\begingroup$

Systematically, you could just solve the problem by finding a solution $A$ that satisfies the following two equations:

$$(A-1)^2=A-1\\A^2=A$$

$\endgroup$
1
  • $\begingroup$ That's right. It should hold for each $\omega$ not globally. As always I've made a silly mistake. Thanks. $\endgroup$
    – S.H.W
    Jul 18, 2020 at 14:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.