Where can I find an authoritative (peer-reviewed or textbook) reference to sampling-induced beating?

Question

I presume we are all here well aware about foldback aliasing when sampling signals above the Nyquist frequency; i.e. half the sampling rate.

By contrast, the phenomenon of beating occurs when sampling a signal with a frequency slightly below the Nyquist frequency. You can see and hear some beats on this educational site.

Here is an example of a 60Hz signal sampled at 128Hz resulting in a 4Hz beat tone with a 8Hz beat. The beat tone frequency is calculated as follows: $f_{t}=\frac{f_{s}}{2}-f_{m}=\frac{128}{2}-60=4\,Hz$. The beat has a frequency twice this tone frequency.

In preparation of a paper, I am looking for a peer-reviewed or textbook reference about beating. I browsed through almost 20 DSP textbooks and found nothing.

On the web, I only found a Proceedings of the 2003 American Society for Engineering Education Annual Conference & Exposition session paper by Kostic which included the nice explanatory figure shown below. I am looking for something more authoritative though; peer-reviewed or textbook.

Answer 1

I think you are mixing two things that are actually not related. "Beating" happens if you add two sine waves that are close in frequency. What you describe is sampling sine wave close to the Nyquist Frequeny. If you plot the samples, it looks like there is beating going on, but that's not actually the case. All information is properly preserved and if you were to properly reconstruct the analog signal you would see a perfectly good sine wave with no beats at all.

The somewhat tricky issue here is "proper" reconstruction. Since your signal is very close to the Nyquist frequency you need a very steep anti-aliasing filter.

Here is a code example of how to reconstruct the sine wave with no beats. Any residual beating is simply an artifact of an insufficient anti-aliasing or interpolation filter.

%% create a 60 Hz sine wave sampled at 128 Hz
fs = 128;
n = 256;
x = sin(2*pi*(0:n-1)'*60/fs);
figure(1); clf;
plot(x); 
% this looks like beating but it isn't

% let's upsampe to 32kHz so we can actually listen to it. We need a very
% steep anti-aliasing filter, so we go with a 64 tab Kaiser Window and a
% beta of 15
y = resample(x,250,1,64,15);
% plot a few periods from the steady state portion
figure(2); clf;
plot(y(10000:20000,:));
% looks like a perfectly good sine wave
sound(.9*y, 32000);
% sounds like one too.

Answer 2

Beating just below the Nyquist frequency occurs when an attempt is made to reconstruct the time continuous signal without the use of sinc interpolation.

This sinc reconstruction method and, in fact, requirement for the Nyquist–Shannon sampling theorem to hold true, is the Whittaker–Shannon interpolation formula.