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.
I don't quite understand their solution though, which leads me to believe there is a more intuitive way to think about this.
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 $\frac43\cos(n\pi/2)$?