# Find response of discrete time LTI system given input and impulse response

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.

• 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. Commented May 8, 2021 at 10:46
• You're right, the lecturer for the course commented after the submission that the denominator n was supposed to be 4.
– Eiv
Commented Jun 10, 2021 at 20:31
• 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)? Commented Oct 5, 2021 at 15:04

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.