# Nyquist sampling theorem / Nyquist - Shannon theorem evaluation over M-PSK

I am trying to simulate an M-PSK Tx-Rx System on Simulink and analyse what the effect of sampling rate reduction would have on it. More, particular I am trying to prove what Nyquist sampling theorem states that the sampling frequency should be twice the maximum frequency component.

The communication chain is as follows:

Random integer generator -> M-psk mapping (modulation) -> RRC (interpolation) -> up-conversion (carrier) -> sampling (sampling rate reduction) -> down-conversion ->Interpolation (if necessary)-> RRC (decimation) -> M-psk De-mapping ->BER

The figure bellow confirms the above arrangement.

So far I have proven that by reducing the sampling rate I simply achieve to “mix down” the signal (all the BW that sits around the carrier) within some bandwidth limits defined by the updated (reduced) sampling frequency. That stops at some point (rendering the signal unrecoverable) where the sampling rate becomes to small; probably confirming Nyquist-Shannon theorem that the sampling rate suffices to be 2 times the BW.

So, no matter how high the carrier frequency is, the signal will always return in between some frequency limits defined by the sampling frequency (-fs/2 to fs/2) following a cyclic manner (modulo). The new (aliased) frequency component will sit apart from the sampling frequency following : f_alias = abs(Fs-Fc).

For example, if Fc is exactly 2 times higher than Fs then the spectrum will return to zero (eg Fs= 8e6 Fc = 16e6 @ around 0), if is 2MHz more then it will sit 2Mhz further (eg Fs = 8e6 Fc =10e6 @ around +2Mhz), if is exactly 4 times higher then again around zero etc (eg Fs= 8e6 Fc = 32e6 @ around 0).

The above simple experiment contradicts the Nyquist theorem that states that the sampling frequency should be at least 2 times greater that the maximum frequency component.And my question is how can I evaluate that theory given a similar communication system arrangement?

If all that above is correct then the selection of and ADC's rate would merely be based on the BW of a signal to be sampled rather than the signal's modulating frequency (carrier)?

• For bandpass signals, the Nyquist Sampling Theorem is Not twice the highest frequency component but rather twice the signal bandwidth. – Dan Boschen Dec 3 '19 at 13:36
• To complement @DanBoschen's comment: the Nyquist sampling theorem states sufficient, but not necessary, conditions to recover a continuous-time signal from its samples. – MBaz Dec 3 '19 at 14:01
• @MBaz thank you for your reply,could you please expound on that a little bit?I was naively believing that the fs should be at least 2 times the highest component?Do you mean that "at least" some times is not enough? – Rizias Dec 3 '19 at 14:12
• @DanBoschen thanks for that. Missing the bigger picture and fundamentals some times... – Rizias Dec 3 '19 at 14:13
• @Rizias What I'm saying is that twice is always enough (it is a "sufficient" condition), but sometimes you can sample at less than that (it is not a "necessary" condition). – MBaz Dec 3 '19 at 16:40