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can someone explain why the ideal frequency response of a low pass filter cant be implemented using the inverse discrete time fourier transform.

i understand the concept of why it cant be implemented because a digital filter is infinite and non-causal, but i can't explain it using the equation. can someone help?

looking at this equation for an ideal low pass filter

enter image description here

how can i explain using this that the ideal frequency response cannot be implemented in practise?

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  • $\begingroup$ Can you please clarify your question? You seem to already know the answer (the ideal filer has infinite-length response and is non-causal). How is that unclear? $\endgroup$ – MBaz Nov 11 '18 at 20:05
  • $\begingroup$ how do i prove it in an equation. look at the equation of the ideal low pass filter. how can i refer to this equation to prove that this is not possible in practise? $\endgroup$ – nothing9099 Nov 11 '18 at 20:06
  • $\begingroup$ Prove that there is no $L$ such that $h[n] = 0$ for all $|n|>L$. Does that help? $\endgroup$ – MBaz Nov 11 '18 at 20:09
  • $\begingroup$ Sorry can u explain a little more please. What is L? $\endgroup$ – nothing9099 Nov 11 '18 at 20:19
  • $\begingroup$ Don't vandalize your own questions. $\endgroup$ – Marcus Müller Nov 17 '18 at 16:03
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Since the impulse response of an ideal low pass filter extends to plus and minus infinity, the corresponding filter is not causal, i.e., in order to implement it (exactly) you would need to know the complete input signal (including its future) to compute the current output value. Furthermore, the sinc function only decays as $1/n$, which makes the filter unstable. In sum, the ideal low pass filter is non-causal and unstable, and for that reason it can't be implemented. Of course, we can approximately implement it by making it causal and stable in such a way that its desirable characteristics are not changed too much for the given application. That's the topic of the field filter design. An little overview for the discrete-time case can be found here.

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