I have a few "conceptual" questions given to me in preparation for a signals and systems exam, and I can't seem to grasp this one.
Does there exist a linear time-invariant (LTI) system S such that:
$$x[n] = \frac{sin(\frac{\pi}{8}n)}{\pi n} \quad \underrightarrow{\quad S \quad} \quad y[n] = \delta[n] $$
I don't have much to go on here, although this is probably simpler than I'm getting right now. I know x[n] is a sinc function, and I know the output is the unit impulse function. I can't think of any reason that it wouldn't be possible. Am I missing anything?
Such a system cannot exist. The reason becomes obvious if you think about it in the frequency domain. The spectrum of the input signal has an ideal low pass characteristic, i.e. it is exactly zero above the cut-off frequency $\omega_c=\pi/8$. On the other hand, the spectrum of the output signal is constant over the whole frequency range. This means that the unknown LTI system would need to add frequencies that are not present in the input signal, which is impossible for an LTI system, because the spectrum of the output signal is given by the multiplication of the input spectrum and the system's frequency response:
$$Y(\omega)=X(\omega)H(\omega)$$
Consequently, for all frequencies $\omega$ for which $X(\omega)=0$, the output spectrum must also satisfy $Y(\omega)=0$.