# Cross Domain Equivalent to Nyquist Sampling Theorem?

In attempting to answer this question by @Oliver here: What characterizies 'causality' for a finite FFT? I have considered the minimum requirement to avoid time domain aliasing in the Discrete Fourier Transform, or more generally any application where the frequency domain is sampled. Similar to sampling in time at at least twice the highest frequency to represent the spectrum without the effects of aliasing, I suggest using a time duration that is at least twice the response time of the underlying continuous time signal to represent the continuous time domain signal (in the DFT) without the effects of time aliasing. Or when the time domain process is restricted to known causal processes, the time duration is at least as long as the response time.

This is the equivalent to Nyquist's Sampling Theorem in the frequency domain; ultimately "sampling in frequency" such that the duration of the time domain waveform is greater than twice its response time.

I understand that the same theory would apply, but given that Shannon in his paper provides the Nyquist theorem in the time domain specifically has made me curious if this property may go by other formally named theorems in other domains?

To illustrate this graphically consider the drawing by RBJ below except replace the frequency axis with the time axis.

Public Domain, https://commons.wikimedia.org/w/index.php?curid=1065579

• hay, i recognize that drawing. May 9, 2020 at 3:42
• @robert oh look at that! It says user "Rbj". Nice drawing Robert, hopefully I did the attribution properly! May 9, 2020 at 11:16
• Dan, if you read it, it says "No rights reserved". no need to attribute, you can take it and butcher the drawing, you can do anything with it you want. May 9, 2020 at 18:03