# DTFT and Inverse DTFT Homework Problem

I'm trying to solve this signals homework problem:

So for part a, since multiplication in the time domain is convolution in the frequency domain, I just used a DTFT table, found the DTFT for $\left(\frac12\right)^n$ and $\cos(\pi n/2)$, convolved them, and solved for $H(\Omega)$. I got the same answer they have in part a.

Anyways...for part b, you would think you'd use the DTFT of $\cos(n \pi/2)$, multiply that by $H(\Omega)$ and take that whole result and Inverse DTFT it back to the time domain, then solve for $y[n]$.

However, I can't get the math to work, and I can't seem to follow their solution.

Can anyone show me mathematically or intuitively how they get that final $\frac43\cos(n\pi/2)$?

Thanks, Clint

• Hi, any chance of getti LaTeX instead of screenshoots and plain-text equations?
– jojeck
Jun 25, 2016 at 18:09
• Sorry, I wish I could, but I'm taking this from an old PDF. Jun 25, 2016 at 18:11
• then learn to do $\LaTeX$ here at dsp.se . also you should show us that you tried to compute the Z transform of $h[n]$. Jun 25, 2016 at 18:18
• @robertneville777: It is extremely easy. Here is some tutorial.
– jojeck
Jun 25, 2016 at 18:53
• @robertneville777: Check what I did to your text (click the edit button), so you can see how you can easily Latexify your questions. Doesn't take much time but makes it much more readable. Jun 25, 2016 at 18:58

$$x[n]=A\cos(\omega_0n)\Longrightarrow y[n]=A|H(\omega_0)|\cos(\omega_0n+\phi(\omega_0))\tag{1}$$
where $x[n]$ and $y[n]$ are the input and output signals, respectively, and $H(\omega)=|H(\omega)|e^{j\phi(\omega)}$ is the system's frequency response.
So if you got $H(\omega)$, you just need to evaluate it at the given input frequency $\omega_0=\pi/2$ and apply $(1)$. Splitting up the cosine into two complex exponentials, as suggested by the given solution, is not necessary (and not even easier I'd say).