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For this question the guide says to use $Y(e^{j\omega}) = H(e^{j\omega}) X(e^{j\omega})$. I have been able to find the discrete time Fourier transform of the impulse function as $$H(e^{j\omega}) = \frac{1}{1-\frac12 e^{-j\omega}}$$ Finding the same for the impulse function is what I do not understand.

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    $\begingroup$ I'm pretty sure that the $n$ in the denominator of the definition of $x[n]$ is a typo or misinterpretation. I guess it's meant to be a $4$ or something similar. Whatever it is, the term in parentheses is very likely supposed to be a constant. $\endgroup$
    – Matt L.
    May 8, 2021 at 10:46
  • $\begingroup$ You're right, the lecturer for the course commented after the submission that the denominator n was supposed to be 4. $\endgroup$
    – Eiv
    Jun 10, 2021 at 20:31
  • $\begingroup$ 1: are you sure you're supposed to use the Fourier transform here? 2: What is the definition of $\delta[n]$ (i.e., the impulse)? What happens if you just plug that into the definition of the Fourier transform (or whatever tool you ought to be using)? $\endgroup$
    – TimWescott
    Oct 5, 2021 at 15:04

1 Answer 1


Note that the given definition of $x[n]$ is undefined for $n=0$. But even if you changed it by replacing the unit step $u[n]$ by the shifted step $u[n-1]$ to exclude the value $n=0$, then the problem would still be very hard to solve. So it's more than likely that the correct definition of $x[n]$ has no $n$ in the denominator, and the current form is just a consequence of a typo or of a misinterpretation.


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