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

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.

• What is it that you're looking for? The "beating" phenomenon that you reference just comes from the fact that your sample rate isn't a multiple of the sinusoid's frequency. Therefore, the phase of each sample relative to the sinusoid function slips by a little bit over each period of the sinusoid. Over time, this phenomenon traces out the shape that you illustrated. I'm not sure what textbook-level discussion you're seeking; I fear this is probably less interesting than you expect. Jul 21, 2013 at 0:28
• You shouldn't hear any beating. If you do, your DAC's reconstruction isn't any good. Jul 21, 2013 at 4:14
• @chirlu The phenomenon was observed on the digital printer of a medical device. Jul 21, 2013 at 8:00
• @JasonR I am just looking for a mention of beat or beating in relation to sampling. In a typical DSP application beating might be less a problem than foldback aliasing. However, this does not take away from the fact that both phenomena are of the same order and origin. In comparison, the alias frequency is $f_{a}=2\frac{f_{s}}{2}-f_{m}$. Jul 21, 2013 at 8:09
• @JasonR With respect to the above comment, beat and alias tones can be considered different order intermodulation products of an equivalent switching sampling mixer with a signum local oscillator at half the sampling frequency; i.e. the Nyquist frequency. A reference other than the referenced patent would equally interest me. Jul 21, 2013 at 9:21

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.


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.