Sampling & Aliasing: Theory and Math

Sampling¶

Impulse Train¶

The representation of signals through equidistant, quantized values is the fundamental principle of digital sound processing and has an influence on several aspects of sound synthesis. Mathematically, a continuous signal $x(t)$ is sampled by a multiplication with an impulse train $\delta_T(t)$ (also referred to as Dirac comb) of infitite length:

$x[n] = x(t) \delta_T (t) = \sum\limits_{n=-\infty}^{\infty} x(n T) \delta (t-nT)$

This impulse train can be expressed as a Fourier series:

$\delta_T = \frac{1}{T} \left[1 +2 \cos(\omega_s t) 2 \cos(2 \omega_s t) + \cdots \right]$, with $\omega_s=\frac{2\pi}{T}$

$\delta_T = \frac{1}{T} + \sum \left( \frac{2}{T} \cos(n \omega_s t) \right)$

Fourier Transform of the Impulse Train¶

The Fourier transform of a time-domain impulse train is a frequency-domain impulse train:

$\mathfrak{F}(\delta_T) = \mathfrak{F}(\sum C_k e^{j k \omega_0 t})$

$\mathfrak{F}(\delta_T) = \sum\limits_{m = -\infty}^{\infty} \delta(T f)$

Fourier Transform of the Sampled Signal¶

The Fourier transform of the sampled signal is periodic:

$X[i] = \frac{1}{T} + \sum\limits_{n=-\infty}^{\infty} X(\omega -n \omega_s)$

Aliasing¶

Since the spectrum of a sampled signal is periodic, it must be band-limited, in order to avoid misinterpretations, known as aliasing. Since the spectrum is periodic with $\omega_s$, the maximum frequency which can be represented - the Nyquist frequency - is:

$f_N = \frac{f_s}{2}$

As soon as components of a digitally sampled signal exceed this boundary, aliases occur. The following example can be used to set the frequency of a sine wave beyond the Nyquist frequency, resulting in aliasing and ambiguity, visualized in the time domain. The static version of the following example shows the time-domain signal of a $900 \ \mathrm{Hz}$ sinusoid at a sampling rate $f_s = 1000 \ \mathrm{Hz}$:

Aliasing Frequency for a Sinusoid¶

For pure sinusoids, aliasing results in a sinusoid at the mirror- or folding frequency $f_m$:

$f_m = \Big| f - f_s \Big\lfloor \frac{f}{f_s} \Big\rfloor \Big|$

With $\lfloor x \rfloor$ as round to next integer.

At a sampling rate $f_s = 1000 \ \mathrm{Hz}$ and a Nyquist frequency $f_N = 500 \ \mathrm{Hz}$, a sinusoid with $f = 900 \ \mathrm{Hz}$ will be interpreted as one with $f = 100 \ \mathrm{Hz}$:

The following example can be used interactively as a Jupyter notebook, by changing the frequency of a sinusoid and listening to the aliased output. When sweeping the range up to $2500 \ \mathrm{Hz}$, the resulting output will increase and decrease in frequency. In the static version, a sinusoid of $f = 900 \ \mathrm{Hz}$ is used, resulting in an audible output at $f_m = 100 \ \mathrm{Hz}$:

Aliasing for Signals with Overtones¶

For signals with overtones, undersampling leads to inharmonic alisases and happen before the fundamental itself exceeds the Nyquist frequency. For a harmonic signal with a fundamental frequeny $f_0$, the alias frequencies for all $N$ harmonics can be calculated:

$f_m = \sum\limits_{n=1}^{N} \Big| n f_0 - f_s \Big\lfloor \frac{n f_0}{f_s} \Big\rfloor \Big|$

For certain fundamental frequencies, all aliases will be located at actual multiples of the fundamental, resulting in a correct synthesis despite aliasing. The following example uses a sampling rate $f_s = 16000 \ \mathrm{Hz}$, with an adjustable $f_0$ for the use as a Jupyter notebook. In the static HTML version, a square wave of $f_0 = 277 \ \mathrm{Hz}$ is used and the result with aliasing artefacts can be heard. The plot shows the additional content caused by the mirror frequencies:

Anti-Aliasing Filter¶

In analog-to-digital conversion, simple anti-aliasing filters can be used to band-limit the input and discard signal components above the Nyquist frequency. In case of digital synthesis, however, this principle can not be applied. When generating a square wave signal with an infinite number of harmonics, aliasing happens instantaneoulsy and can not be removed, afterwards.

The following example illustrates this, by using a 5th order Butterworth filter with a cutoff frequency of $f_c = 0.95 \frac{f_s}{2}$. Although the output signal is band-limited, the aliasing artifacts are still contained in the output signal.

Band Limited Square Wave¶

In order to avoid aliasing problems, most environments for audio signal processing and sound synthesis offer band-limited oscillators for specific waveforms. Users thus do not need to worry about this issue. The following example makes use of a Fourier series for generating a square wave signal with $f_0 = 177 \ \mathrm{Hz}$. Partials exceeding the Nyquist frequency can thus be excluded from synthesis: