# Sampling - Higher order harmonics

I have been studying about the sampling theorem and it seems that even though we sample at a frequency with the nyquist criterion, the harmonics (due to sampling process) remain within the nyquist frequency range, as illustrated below:

If we sample a 5MHz sine wave with 20 MHz sampling frequency (nyquist frequency being 10 MHz), its fundamental and harmonic frequencies (up to 4th) lie at:

fin=5 MHz (fundamental)

1st Harmonics: (around fs) fs+fin=25 MHz fs-fin=15 MHz

2nd Harmonics: (outside nyquist) fs+(2*fin)=30 MHz fs-(2*fin)=10 MHz

3rd Harmonics: fs+(3*fin)=35 MHz fs-(3*fin)=5 MHz (Inside nyquist)

4th Harmonics: fs+(4*fin)=40 MHz fs-(4*fin)=0 (DC)

I understand that when doing a high level simulation in matlab, we do windowing or coherent sampling to overcome this harmonics falling into the nyquist range or signal band by choosing a prime number of cycles.

But practically, if we give an arbitrary frequency to an ADC, which usually has a sample/hold circuitry in the front, aren't these harmonics generated by it?

Or do we have the control of the harmonics' power so that we can adjust it to a minimum (depending on the application).

(This question also relates to my previous question - Higher order harmonics during sampling)

## 1 Answer

I don't quite understand why you feel "harmonics" are relevant to this discussion.

1. A 5 Mhz has sine wave has no harmonics. If the signal has harmonics, it is not a sine wave any more but a different signal (rectangular, triangle, etc.)
2. For ANY signal: you need to determine the highest frequency that's in the signal and then chose the Nyquist frequency to be higher. Otherwise you get either loss of information or aliasing
3. Most ADC do have an anti-aliasing low-pass filter build in: if you have harmonics higher than Nyquist, the filter will just chop them off.

Points 2 and 3 apply to ANY signal: noise, music, video, OFDM, and also harmonic signals like rectangle our saw-tooth waves. There really is nothing special about "harmonics".

• I agree points 2 and 3 in a practical scenario. In point 1 - practically, if we do a Sample/Hold circuitry, then (fin) could be distorted and its harmonics must be generated (although low in power) - Is that correct? (Because, we rate an RF-LNA through IIP3 or IIPn - thus measuring its sensitivity to harmonics - even though ideally the input signal is of single tone) – sundar Jan 15 '19 at 14:34
• An ADC comes with a specification on it's performance (THD, SNR, linearity, etc.). These captures all non-ideal behavior including sample & hold and a plethora of other properties. The ADC is guaranteed to capture a sine wave better than that. – Hilmar Jan 15 '19 at 17:36