Signal Types

Continuous vs Discrete

All physical signals we can observe in the real world - like sound - are time continuous. That means a value can be observed for any arbitrary point in time $t$:

$$x(t), ~ t \in \mathbb{R}$$

In digital signal processing, signals are time-discrete. For these signals, values only exist at the given sampling points $n$:

$$x[n], ~ n \in \mathbb{Z} $$

The conversion between continuous and time-discrete signals - sampling - will be treated later in the DSP module.

Stochastic Signals

Stochastic - or random - signals are those signals whose instantaneous value (the value of the signal at a given point in time) can not be predicted. They do not have a fundamental frequency or pitch. Such random signals can, however, be modeled through stochastic processes. This means a specific behavior and porperties can be expected - the frequency content and value range.

Some examples for stochastic signals:

• Noise (white, pink, brown)

• Breath/Noise in wind instruments

• Waterfall

White Noise Example:

Deterministic Signals

A deterministic signal is fully predictable and can thus be expressed in a mathematical function. For such signals the instantaneous value can be calculated for any point in time.

Typical deterministic signals are:

• periodic signals

• constant signals (aka DC)

Periodic Signals

Periodic signals are the most common type of deterministic signal. All periodic signals have a fundamental frequency - $f_0$ - the rate of repetition. The period $T$ of a periodic signal is the inverse of the frequency:

$$ T = \frac{1}{f}$$

Often, the angular frequency $\omega$ is used:

$$ \omega = 2 \pi f = \frac{2 \pi}{T}$$

For musical instruments and speech, the fundamental frequency is in the range of our hearing. It results in pitch, the perceptual concept related to fundamental frequency.

Some examples for periodic signals:

• sine wave

• square wave

• saw tooth

Sine Example:

Quasi-Periodic Signals

Music and speech consist of a mix of both deterministic and stochastic signal components. Even pitched instruments and the human voice are never fully predictable. However, in most cases it is perceptually sufficient to model musical signals as truly periodic.

Transients

In audio DSP, transients are short signal segments, characterized by a rapid change of signal properties. There is no exact definition, but in general they have a length between $ 1 - 100 \mathrm{ms} $. Most transients in speech and music signals are part of the attack - the beginning of a sound. They contain noise and rapidly changing sinusoidal components, like the struck of a guitar string, the hammer hitting a piano string or the early breath noise in wind and brass instruments. Transients are very important for recognizing instruments and defining their individual qualities.

Some examples for transients are:

• Handclap

• Snare drum

• Attack segments

• Noise bursts

Audio example for a transient sound - MFB-522 Clap:

Textures

Sound textures are created by a random sequence of similar events. Similar to their visual counterpart, they create a repeating surface.

Some examples for textures:

• Rain

• Fire crackling

• Vinyl crackling

• Frying bacon in a pan

Audio Example for a sound texture - grabbing chips:

Example - MFB-522 Clap:

References