Signal Types

Signal Types

Signals in music and audio DSP can be categorized based on their properties. The list of categories below is not exhaustive but contains the most frequent types of sinals.


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:

signal_properties_noise

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:

$$ T = \frac{1}{f_0}$$

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

$$ \omega = 2 \pi f_0 = \frac{2 \pi}{T}$$

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 Wave

For many applications in science and engineering, including DSP, the sine wave (or the cosine) is the most important basic function.

$$ x(t) = a \sin \left( 2 \pi f t \right) $$

signal_properties_sine

The sine wave is fully described by its frequency $f$ and amplitude $a$. The argument of the sine function is the (instantaneous) phase. In many applications, the relation between phase $\varphi$ and angular frequency $\omega$ is of importance. They have the same fundamental relationship as speed and accelleration.

Frequency-Phase Relationship

The frequency is the derivative of the phase:

$$ \omega(t) = \frac{d \varphi(t)}{ dt} $$

And the phase is the integral over the frequency:

$$ \varphi(t) = \int \omega(t) d t$$


Phase Wrapping:

In signal processing, the phase is often wrapped, when visualized. Since the sine wave is periodic within $2 \pi$, it only needs input values between $0$ and $2 \pi$ (or in the range of $-\pi ... \pi$, when shifted). Wrapping the phase, means keeping it within this range - when the phase exceeds $2 \pi$, it continues at $0$.

The following plot shows a sine wave with it's wrapped instantaneous phase:

properties_sine_phase