Having a system like below, which (i think) is LTI, and $H(e^{j\omega}) = \begin{cases} 1 & |\omega|\leq \frac{\pi}{2} \\ 0 & |\omega| \gt \frac{\pi}{2} \end{cases} $

system block diagram

I've tried to find whole system's frequency response by shifting $H(e^{j\omega})$ with value of $\pi$ and summed it up with itself which then got $H_{total}(e^{j\omega}) = 1$ for all $\omega$ and if I'm correct with that I think the system is a wire which makes $y[n] = x[n]$

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    $\begingroup$ Don't think so. Any tone between pi/2 and pi will be blocked $\endgroup$ – Stanley Pawlukiewicz Jan 7 at 3:31
  • $\begingroup$ @StanleyPawlukiewicz would you provide any proof please? $\endgroup$ – no0ob Jan 7 at 4:02
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    $\begingroup$ Everything outside of the pass band of the first filter is blocked. Frequency translation of zero is zero. Is this LTI? $\endgroup$ – Stanley Pawlukiewicz Jan 7 at 4:23
  • $\begingroup$ I don't know if it is LTI or not, i assumed it is, because we are using Fourier transform here, if not i think we couldn't! $\endgroup$ – no0ob Jan 7 at 4:27
  • $\begingroup$ @StanleyPawlukiewicz If it's not LTI what should we do? Is there any calculable Impulse response for whole System? $\endgroup$ – no0ob Jan 7 at 4:32

The system is not just a wire, it is in fact time-varying, so it has no frequency response in the conventional sense. The sequence $w[n]$ is of course just a low-pass filtered version of the input signal. The filtered signal is band-limited to half the Nyquist frequency. Now just write the output $y[n]$ in the time domain as the sum of $w[n]$ and the modulated version of $w[n]$. You should be able to see that effectively every other (odd) sample of $w[n]$ is replaced by a zero. But there occurs no aliasing due to the low pass filter at the input.

  • $\begingroup$ If it is not LTI, there is no Frequency Response for this system? $\endgroup$ – no0ob Jan 7 at 11:36
  • $\begingroup$ @no0ob: No, only LTI systems are characterized by their frequency response and/or impulse response. $\endgroup$ – Matt L. Jan 7 at 11:36
  • $\begingroup$ I've got signals and systems course this semester and this was one of our final exam problems. the problem had many parts mostly talked about impulse response and freq response. i think i should talk to teacher about how this question is wrong. i wish there were more mathematical and equational stuff to have a better proof for him. $\endgroup$ – no0ob Jan 7 at 12:23

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