In digital audio,
a simple square wave has some interesting properties. On the face of
it, you can actually represent the square wave digitally with no error
whatsoever. The quantization error, such as it is, becomes merely
a scaling factor. If you adjust the input gain, you can get it so that
the quantization error is zero for every single recorded value.

Imagine that - a waveform captured digitally with no loss whatsoever.
Even though it is of no practical value (who wants to listen to recorded
square waves?) you would imagine that it would be useful for test and
measurement purposes if nothing else. And people do imagine that. In
fact, it is not unusual to see reproductions of square waves, captured
on an oscilloscope, as part of a suite of measurements on a DAC. But in
digital audio things are seldom quite what they may logically appear to
be. Lets take a closer look at square waves and see what we come up
with.

A true and perfect square wave can be analyzed
mathematically and broken down into what we call its Fourier components.
These are nothing more scary than simple sine waves, each one having
its own frequency, phase, and amplitude. The Fourier components (sine
waves) which make up a square wave comprise the fundamental frequency
plus each of its odd harmonics. The fundamental frequency is simply the
frequency of the square wave itself. The odd harmonics are just
frequencies which are odd multiples of the fundamental frequency (3x,
5x, 7x, 9x, 11x, etc). The amplitudes of each Fourier component are
inversely proportional to the frequency of the harmonic. We’ll ignore
phase.

If that sounds complicated, it really isn’t. Start with
a sine wave, at the frequency of the square wave which I will refer to
as ‘F’. Now add to it a sine wave of its third harmonic, which is three
times the fundamental frequency, ‘3F’. This sine wave should have an
amplitude 1/3 of the amplitude of the fundamental sine wave. OK? Next
we add a sine wave of the fifth harmonic, 5F, with an amplitude 1/5.
Then the seventh harmonic (7F) with an amplitude 1/7, the ninth harmonic
with amplitude 1/9, and so on, ad infinitum. The more of these
harmonics you add into the mix, the closer the result approaches a
perfect square wave. A truly perfect square wave comprises an infinite
number of these Fourier components. Here is a link to an animated illustration which might help make this clearer.

Some interesting things happen as we add more and more Fourier
components to build up our square wave. The ripple on the waveform gets
smaller in magnitude, and its frequency gets higher. The ‘spikes’ at
the start and end of the flat portions of the waveform decay more
rapidly. These things can all be seen clearly on the web page I
mentioned. However, one surprising thing is that the ‘spikes’ don’t
ever disappear, no matter how many Fourier components you add in. In
fact, their magnitude doesn’t even get smaller. As you add Fourier
components, the 'spikes' converge to about 9% of the amplitude of the
square wave, something referred to as the ‘Gibbs Phenomenon’. The spike
itself ends up approximating a rapidly decaying oscillation at the highest frequency
Fourier component present.

So now lets go back to the idea at
the start of this post, that we can easily encode a square wave with
absolute precision in a digital audio stream. If this were actually
true, it would imply that we are encoding Fourier components comprising
an infinite bandwidth of frequencies. But Nyquist-Shannon sampling
theory tells us that any frequencies that we attempt to encode which are
higher than one half of the sampling rate actually end up encoding
‘alias’ frequencies, which are things that are seriously to be avoided.
The square wave that we thought we had encoded perfectly, and with
absolute precision, turns out to contain a whole bunch of unwanted
garbage! And that, to many people, is bizarrely counter-intuitive.

We’ve gone over this in previous posts, but the bottom line is that any
waveform that you wish to capture digitally - square waves included -
need to be first passed through a low-pass filter called an
‘anti-aliasing’ filter, which blocks all frequencies above one half of
the sampling rate. For the purpose of this post, lets assume that the
anti-aliasing filter is ‘perfect’ and does nothing except remove all
frequencies above half the sampling frequency. What this filter does to
our square wave is, in effect, to strip off the vast majority of the
square wave’s Fourier components, leaving only a few behind.

Lets look at what this means in the context of Red Book (44.1kHz sample
rate) audio. If the square wave is, for example a 1kHz square wave,
then its first ten Fourier components (the 3rd through 21st harmonics)
are left intact. However, if the square wave is a 10kHz square wave,
not one single odd harmonic component makes it through the filter! This
means that if you attempt to record a 10kHz square wave, all you could
actually ever capture would be a pure 10kHz sine wave.

What does
this mean for DAC reviews where you are presented with a screenshot
illustrating the attempted reproduction of a square wave? The bottom
line here is that for such a screenshot to make any sense at all, you
need to know exactly what a perfectly reproduced square wave ought to
look like. For example, it ought to have leading and trailing edge
‘spikes’ which approximate to half cycles of an approximate 20kHz sin
wave, and with an amplitude which is about 9% higher than the amplitude
of the square wave itself. It needs to have ripple. I could be a lot
more specific about those things, but the fine details end up being
governed by issues such as the frequency of the square wave, and - far
more complex - the characteristics of the anti-aliasing filter used to
encode the original signal. Without this information a square-wave
screenshot is of limited value. If nothing else, it can provide a level
of confidence that the person conducting the test knows what they are
talking about. Not to mention avoiding allowing the reader to infer or
imply erroneous conclusions of their own.