# Nyquist zero ISI criterion for rectanguar pulses in the frequency domain

Nyquist criterion for zero ISI states that in the frequency domain, the replicas of the frequency characteristic of combined:

TX pulse shaping filter - channel - RX pulse shaping filter

must add up to a constant value in order to ensure zero ISI.

How does it work in case of rectangular pulse signalling when the communication channel has an infinite bandwidth?

As I understand, the combined spectrum is then just a rectangular pulse spectrum (sinc function in the frequency domain) and I do not see how its replicas can add to obtain a constant value over the frequency. But in case of infinite channel BW, the pulses will not be smeared in the time domain, so ISI should not occur.

• It may be hard to believe, but it's true! Try adding a bunch of sincs in Matlab and see. I'll post some code tomorrow if nobody beats me to it or provides a better answer. – MBaz Oct 30 '19 at 23:30
• This answer over at math.SE should answer your question. Also, if you understand the proof of the Nyquist criterion, the fact that shifted sinc functions add up to a constant won't surprise you. – Matt L. Oct 31 '19 at 11:13
• @MattL. Even though I understood the proof of Nyquist's criterion at the intellectual level, I still found this surprising the first time I saw it, since it contradicted my (uneducated) intuition. – MBaz Oct 31 '19 at 15:00
• Interesting -- I was taught that it needed to be symmetrical around $1/T_s$ in the frequency domain, but they either left out (or I forgot) that you need to add everything up $\mod 1/T_s$. – TimWescott Oct 31 '19 at 16:35
• @MBaz: The result is indeed not very intuitive. But I probably got used to that my intuition lets me down once in a while. – Matt L. Nov 1 '19 at 7:57