Accoustic Source Characterisation

Let us consider a microphone array comprising $n$ microphones at known locations (see figure above). These microphones register the sound that is emitted by a number of sources with unknown locations. The characterization of these acoustic sources requires the estimation of their location and strength.

The propagation of sound from a source position $x$ to a receiver at position $y$ can be modelled by a Green’s function. In the following we assume that the source is always a monopole. In that case the sound pressure amplitude at the receiver position for a given discrete frequency $\omega$ is defined by

\begin{equation} p \left( r, \omega \right) = q_{0} \frac{1}{r} \exp\left( -\imath \omega \frac{r}{c_{0}} \right) \end{equation}

with $\imath$ being the complex unit, $q_{0}$ being the source strength, and $r$ denoting the distance between the source and the receiver. Finally, the constant $c_{0}$ denotes the speed of sound. The signals from any given source are evaluated at a known reference point $y_{0}$ at distance $r_{0}$ from its source:

\begin{equation} p_{0} \left( r_{0}, \omega \right) = q_{0} \frac{1}{r_{0}} \exp\left( -\imath \omega \frac{r_{0}}{c_{0}} \right) \end{equation}

Introducing the reference point into our model formulation helps us to eliminate the source strength and leads to the following description

\begin{equation} p\left( r, \omega \right) = a p_{0} \left( r_{0}, \omega \right) \end{equation}

with

\begin{equation} a := \frac{r_{0}}{r} \exp\left( \imath \omega \frac{r_{0}-r}{c_{0}} \right) \end{equation}

The latter equation yields the sound pressure amplitude at a receiver position depending on the sound pressure induced at a reference location by the source.

The estimation approach that we follow here additionally assumes that the real locations of the sources are restricted to $m$ possible source locations. Since a superposition principle holds in our model, we can account for multiple sources by adding all contributions. Thus, the sound pressure at a microphone $j$ is given by $\sum_i a_{ij} x_i$ where the sum is taken over all possible source locations $i$. The coefficient $a_{ij}$ is defined in accordance with:

\begin{equation} a_{ij} := \frac{r_{0,j}}{r_{ij}} \exp\left(\imath \omega \frac{r_{0,j}-r_{ij}}{c_{0}} \right) \quad i = 1, \ldots, m,\ j = 1, \ldots, n \end{equation}

with $r_{ij}$ denoting the distance between sender $i$ and receiver $j$.

In a typical setting the number $m$ of possible source locations is much larger than the number $n$ of microphones, but the number of actual sources is less than $n$. Therefore, most of the $x_{i}$ in the sum are zero.

Gathering all possible coefficients $a_{ij}$ in a matrix $A$, and using a vector $c \in \mathbb{C}^{n}$ to hold all microphone sound pressures yields

\begin{equation} c=Ax, \end{equation}

where $x \in \mathbb{C}^m$ is the sparse vector of source strengths. Using the Hermitian cross-spectral matrix $C := E[cc^\top]$ of microphone sound pressures, we can reformulate the previous equation as

\begin{equation} C=A\underbrace{E\left[xx^\top\right]}_{=: X} A^\top \end{equation}

where the operator $E$ denotes the expected value and where $X \in \mathbb{C}^{m,m}$ is the cross spectral matrix of source levels. This matrix is sparse, Hermitian, and in case of uncorrelated source signals also diagonal. By estimating $X$, the task of characterizing the sources is solved.

This task has been analysed and published in Optimization and Engineering and presented at an optimization workshop in Münster, Germany. In our approach we used sparsity favouring convex optimization models and solved these with the split Bregman algorithm. In addition, we used clustering techniques to refine the estimated source locations in presence of noisy measurements.

Our publication was awarded the Howard Rosenbrock Prize 2018.

Source Code

The source code for the accoustic source characterization is here.