# 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? – jojek Jun 25 '16 at 18:09
• Sorry, I wish I could, but I'm taking this from an old PDF. – robertneville777 Jun 25 '16 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 $h[n]$. – robert bristow-johnson Jun 25 '16 at 18:18
• @robertneville777: It is extremely easy. Here is some tutorial. – jojek Jun 25 '16 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. – Matt L. Jun 25 '16 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).