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:

DTFT

sinc

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

 

The transition band of the frequency response depends on the order of the filter.

To obtain an infinitely narrow transition band like the brick wall frequency response, how many coefficients would you need ?

 

Volume 18 of the MIT Radiation lab series has an appendix (appendix A, starting at page 721) that explains the application of the Paley-Wiener theorem to the question of relizability of filters. Download volume 18 here: https://www.jlab.org/info_resources/mitseries

Also, Wikipedia has a page about the theorem: https://en.wikipedia.org/wiki/Paley–Wiener_theorem

 

The transition band of the frequency response depends on the order of the filter.

To obtain an infinitely narrow transition band like the brick wall frequency response, how many coefficients would you need ?

An infinite amount of coefficients?

 

Correct.
It's also explained here

https://www.dsprelated.com/freebooks/sasp/Ideal_Lowpass_Filter.html