Bandlimited square waves
Introduction
In the field of audio signal processing, square waves seem easy to generate but require some care to get a high quality result. As shown in the samples below, a naive rendering of a square wave will have noise and harmonic distortion, whereas a correct rendering sounds clean and undistorted.
Square waves are simple audio signals that are useful in practice. For example, early electronic devices could only generate square waves for melodic tones, because they are computationally easy to produce. This was the case for early personal computers, and firstgeneration game consoles like the Nintendo Entertainment System (NES) and Game Boy. Square waves were later superseded by more powerful technologies such as FM synthesis and samplebased synthesis.
In old electronic products, square waves were synthesized based on control parameter values (e.g. frequency, amplitude, duty cycle, offset) and the signals were sent directly to an analog audio output. But nowadays a lot of audio processing happens in samplebased formats. To play a square wave on a modern computer, the signal must be rendered as a sequence of samples in discrete time and sent to a sound adapter (which is designed only to accept samples, not synthesis parameters). Also, rendering a square wave to samples is convenient for playback (e.g. in any music player software) and for lossy compression (e.g. MP3, Vorbis).
The process of rendering a square wave to a sequence of samples has some pitfalls. The naive way is to simply flip back and forth between the values −1 and +1 at the appropriate sample positions. But a square wave has an infinite number of (nonzero) harmonics, extending to arbitrarily high frequencies. A samplebased waveform must be bandlimited to half the sample rate to prevent aliasing – for example, a wave file sampled at 48 kHz can only contain frequencies from 0 to 24 kHz, no higher than this. The naive rendering method suffers from not suppressing frequencies above the Nyquist limit, and from inexactness when the square wave transitions from −1 to +1 (or vice versa) at a point in time that is in between two sample points.
Wave samples
Here is an illustrative 440 Hz square wave, rendered in the simple naive way versus the correct bandlimited way, at different sampling rates.
Audio
Sample rate  Naive rendering  Bandlimited rendering 

8000 Hz  
11025 Hz  
16000 Hz  
22050 Hz  
24000 Hz  
32000 Hz  
44100 Hz  
48000 Hz 
In the naive square wave renderings, notice the buzzing noise (e.g. 24, 32 kHz), spurious lowfrequency tones (e.g. 8, 16, 24 kHz), choppedup tone sound (e.g. 8, 11, 22 kHz), and inconsistent timbre depending on sample rate (e.g. 22 vs. 24 kHz).
In the bandlimited square wave renderings, notice the pure and steady tone, free of any distortions. The sound becomes more shrill as the sample rate increases, but the basic quality stays the same.
As the sample rate increases, the naive renderings have less audible artifacts. This is because in the limit as the sample rate goes to infinity, the waveform of the naive rendering approaches the bandlimited rendering. To produce a bandlimited square wave, it is certainly possible to render a naive square wave at a very high sample rate, and downsample it properly using a bandlimited sinc filter. But in practice (as I argue on this page), it is easier to directly generate a square wave that is bandlimited by design.
Pictures
Visualization  Naive rendering  Bandlimited rendering 

Waveform  
Spectrogram 
In the waveform pictures, there isn’t a big difference between the naive and bandlimited renderings. The naive rendering purely uses the values −0.5 and +0.5 (ignore the interpolated curves), whereas the bandlimited rendering uses all sorts of values surrounding −0.5 and +0.5, and has noticeable ringing.
In the spectrogram pictures, the difference between the naive and bandlimited renderings is night and day. The naive rendering is full of noise between the real harmonics and extraneous frequencies surrounding the harmonics. The bandlimited rendering is clean (ignoring the endpoints due to sudden start/stop), containing only the true harmonics and zero other content.
These screenshots were taken in Syntrillium Cool Edit Pro 2.0 (which evolved into Adobe Audition). Note that CEP2’s builtin square wave generator uses the naive algorithm, with results being essentially the same as my naive implementation. Thus, even commercial software used in audio production can get this wrong.
Generator demo (JavaScript)
Wave type:  
Frequency:  Hz  
Duty cycle:  
Volume:  %  
Number of harmonics:  
This JavaScript application uses the Web Audio API, and is supported in most browsers except Microsoft Internet Explorer. The source code is available for viewing.
Pseudocode
 Naive synthesis algorithm

theta = sampleIndex × frequency / sampleRate.
sampleValue[sampleIndex] = if (theta mod 1 < dutyCycle) then 1 else 0.  Bandlimited synthesis algorithm

coefficients = (... finite length array ...).
theta = sampleIndex × frequency × 2π / sampleRate.
sampleValue[sampleIndex] =
coefficients[0] × cos(0 × theta) +
coefficients[1] × cos(1 × theta) +
coefficients[2] × cos(2 × theta) + ⋯ .This pseudocode omits how the coefficients are determined. They are discussed in the next section on Fourier analysis.
Fourier series analysis
To synthesize a bandlimited square wave, we need to know the amplitudes of each harmonic frequency, and only use harmonics that are below the Nyquist frequency limit (hence setting all higher harmonics to zero amplitude). The Fourier series method allows us to figure out these coefficients/amplitudes. To start, we illustrate a canonical square wave used in audio engineering, which alternates between the values −1 and +1, has a duty cycle of d (between 0.0 and 1.0), and has a period of P = 1/f:
But to make analysis easier, we shift and scale the waveform in time and amplitude into this form, so that it alternates between the values 0 and 1, and has a period of 2π:
Because we created an even function, only the cosine terms (a_{k}) of the Fourier series matter. The sine terms (b_{k}) are all zero. The zeroth harmonic, also known as the DC offset, is clearly \(a_0 = d\). As for the AC coefficients, for each \(k \in \mathbb{N}^+\):
\(\begin{align} a_k &= \frac{1}{π} \int_{d π}^{d π} (1)(\cos(kt)) \: dt \\ &= \frac{2}{π} \int_{0}^{d π} \cos(kt) \: dt \\ &= \frac{2}{π} \left[ \frac{\sin(kt)}{k} + C \right]_{t=0}^{d π} \\ &= \frac{2\sin(kd π)}{k π}. \end{align}\)
Therefore, our list of cosine coefficients \((a_0, a_1, a_2, a_3, \ldots)\) are \((d, \frac{2\sin(dπ)}{π}, \frac{2\sin(2dπ)}{2π}, \frac{2\sin(3dπ)}{3π}, \ldots)\). This is for a wave with duty cycle d that nominally goes between 0 and 1 (subject to overshoot and ringing).
For a wave that goes between −1 and +1, the \(a_0\) coefficient is affinetransformed, and all the remaining coefficients are doubled. Namely, \(a_0 = 2d  1\), \(a_1 = \frac{4\sin(dπ)}{π}\), \(a_2 = \frac{4\sin(2dπ)}{2π}\), et cetera.
Note that for the popular special case of a balanced square wave where d = 0.5, we have a_{0} = 1/2, a_{1} = 2/π, a_{2} = 0, a_{3} = −2/(3π), a_{4} = 0, a_{5} = 2/(5π), etc. (Note that all the even harmonics are zero.)
Calculation example
Suppose we want to generate a bandlimited square wave with the following set of parameters:
 Frequency = 440 Hz
 Duty cycle = 0.3 (30%)
 Low, high = −0.5, +0.5
 Sample rate = 48000 Hz
 \(a_0 = d  1/2 = 0.200.\) (0th harmonic, DC offset, 0 Hz)
 \(a_1 = 2\sin(dπ)/π \approx 0.515.\) (1st harmonic, fundamental, 440 Hz)
 \(a_2 = 2\sin(2dπ)/(2π) \approx 0.303.\) (2nd harmonic, 1st overtone, 880 Hz)
 \(a_3 = 2\sin(3dπ)/(3π) \approx 0.066.\) (3rd harmonic, 2nd overtone, 1320 Hz)
 (... 50 coefficients not shown ...)
 \(a_{53} = 2\sin(53dπ)/(53π) \approx 0.004.\) (53rd harmonic, 23320 Hz)
 \(a_{54} = 2\sin(54dπ)/(54π) \approx 0.007.\) (54th harmonic, 23760 Hz)
Thus our bandlimited square wave signal in the time domain is given by:
\(\begin{align} f(t) &= \displaystyle\sum_{k=0}^{54} a_k \cos((440\text{ Hz})(2π) kt) \\ &= a_0 + a_1 \cos(880 π t \text{ Hz}) + a_2 \cos(1760 π t \text{ Hz}) + \ldots + a_{54} (47520 π t \text{ Hz}). \end{align}\)
(Note that the value of t must have the unit of second (s) attached. When multiplied with hertz (Hz), these cancel out to 1.)
Source code
 Python: generatesquarewave.py
 Java: GenerateSquareWave.java
These command line programs take 4 or 5 parameters, and generate a square wave as an uncompressed WAV file. They can generate naive and bandlimited square waves, with the same phase alignment so that the audio signals can be compared/subtracted easily. Both the Python and Java versions take the same arguments and generate the same output (except for tiny differences in floatingpoint arithmetic).
More info
 Xiph.Org Video Presentations: Digital Show & Tell (talks about bandlimited square waves, along with other topics)