# Sampling & Aliasing

## 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 \(f = 100 \ \mathrm{Hz}\):

f_m = 100

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 \(f_m = 100 \ \mathrm{Hz}\):

### Interactive Example

Pitch (Hz):

Output Gain:

Frequency Domain: