# Acoustic source localization

Acoustic source localization[1] is the task of locating a sound source given measurements of the sound field. The sound field can be described using physical quantities like sound pressure and particle velocity. By measuring these properties it is (indirectly) possible to obtain a source direction.

## Overview

Traditionally sound pressure is measured using microphones. Microphones have a polar pattern describing their sensitivity as function of the direction of the incident sound. Many microphones have an omni-directional polar pattern which means their sensitivity is independent of the direction of the incident sound. Microphones with other polar patterns exist that are more sensitive in a certain direction. This however is still no solution for the sound localization problem as one tries to determine either an exact direction, or a point of origin. Besides considering microphones that measure sound pressure, it is also possible to use a particle velocity probe to measure the acoustic particle velocity directly. The particle velocity is another quantity related to acoustic waves however, unlike sound pressure, particle velocity is a vector. By measuring particle velocity one obtains a source direction directly. Other more complicated methods using multiple sensors is also possible. Many of these methods use the time difference of arrival (TDOA) technique.

Some have termed acoustic source localization an "inverse problem" in that the measured sound field is translated to the position of the sound source.

## Methods

Different methods for obtaining either source direction or source location are possible.

### Particle velocity or intensity vector

The simplest but still a relatively new method is to measure the acoustic particle velocity using a particle velocity probe. The particle velocity is a vector and thus also contains directional information.

### Time difference of arrival

The traditional method to obtain the source direction is using the time difference of arrival (TDOA) method. This method can be used with pressure microphones as well as with particle velocity probes.

With a sensor array (for instance a microphone array) consisting of at least two probes it is possible to obtain the source direction using the cross-correlation function between each probes' signal. The cross-correlation function between two microphones is defined as

$R_{x_1,x_2} (\tau) = \sum_{n=-\infty}^{\infty} x_1(n)\ x_2(n+\tau)$

which defines the level of correlation between the outputs of two sensors $x_1$ and $x_2$. In general, a higher level of correlation means that the argument $\tau$ is relatively close to the actual time-difference-of-arrival. For two sensors next to each other the TDOA is given by

$\tau_{\mathrm{true}} = \frac{d_{\mathrm{spacing}}}{c}$

where $c$ is the speed of sound in the medium surrounding the sensors and the source.

A well-known example of TDOA is the interaural time difference. The interaural time difference is the difference in arrival time of a sound between two ears. The interaural time difference is given by

$\Delta t = \frac{x \sin{\theta}}{c}$

where

$\Delta t$ is the time difference in seconds
$x$ is the distance between the two sensors (ears) in meters
$\theta$ is the angle between the baseline of the sensors (ears) and the incident sound, in degrees

### Triangulation

Main article: Triangulation

In trigonometry and geometry, triangulation is the process of determining the location of a point by measuring angles to it from known points at either end of a fixed baseline, rather than measuring distances to the point directly (trilateration). The point can then be fixed as the third point of a triangle with one known side and two known angles.

For acoustic localization this means that if the source direction is measured at at least two locations in space, it is possible to triangulate its location.