Steered-response power

Consider a system of ${\displaystyle M}$ microphones, where each microphone is denoted by a subindex ${\displaystyle m\in \{1,\dots ,M\}}$. The discrete-time output signal from a microphone is ${\displaystyle s_{m}(n)}$. The (unweighted) steered-response power (SRP) at a spatial point ${\displaystyle \mathbf {x} =[x,y,z]^{T}}$ can be expressed as

${\displaystyle P_{0}(\mathbf {x} )\triangleq \sum _{n\in \mathbb {Z} }\left\vert \sum _{m=1}^{M}s_{m}(n-\tau _{m}(\mathbf {x} ))\right\vert ^{2},}$

where ${\displaystyle \mathbb {Z} }$ denotes the set of integer numbers and ${\displaystyle \tau _{m}(\mathbf {x} )}$ would be the time-lag due to the propagation from a source located at ${\displaystyle \mathbf {x} }$ to the ${\displaystyle m}$-th microphone.

The (weighted) SRP can be rewritten as

${\displaystyle P(\mathbf {x} )={\frac {1}{2\pi }}\sum _{m_{1}=1}^{M}\sum _{m_{2}=1}^{M}\int _{-\pi }^{\pi }\Phi _{m_{1},m_{2}}(e^{j\omega })S_{m_{1}}(e^{j\omega })S_{m_{2}}^{*}(e^{j\omega })e^{j\omega \tau _{m_{1},m_{2}}(\mathbf {x} )}d\omega ,}$

where ${\displaystyle ()^{*}}$ denotes complex conjugation, ${\displaystyle S_{m}(e^{j\omega })}$ represents the discrete-time Fourier transform of ${\displaystyle s_{m}(n)}$ and ${\displaystyle \Phi _{m_{1},m_{2}}(e^{j\omega })}$ is a weighting function in the frequency domain (later discussed). The term ${\displaystyle \tau _{m_{1},m_{2}}(\mathbf {x} )}$ is the discrete time-difference of arrival (TDOA) of a signal emitted at position ${\displaystyle \mathbf {x} }$ to microphones ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$, given by

${\displaystyle \tau _{m_{1},m_{2}}(\mathbf {x} )\triangleq \left\lfloor f_{s}{\frac {\|\mathbf {x} -\mathbf {x} _{m_{1}}\|-\|\mathbf {x} -\mathbf {x} _{m_{2}}\|}{c}}\right\rceil ,}$

where ${\displaystyle f_{s}}$ is the sampling frequency of the system, ${\displaystyle c}$ is the sound propagation speed, ${\displaystyle \mathbf {x} _{m}}$ is the position of the ${\displaystyle m}$-th microphone, ${\displaystyle \|\cdot \|}$ is the 2-norm and ${\displaystyle \lfloor \cdot \rceil }$ denotes the rounding operator.

Generalized cross-correlation

The above SRP objective function can be expressed as a sum of generalized cross-correlations (GCCs) for the different microphone pairs at the time-lag corresponding to their TDOA

${\displaystyle P(\mathbf {x} )=\sum _{m_{1}=1}^{M}\sum _{m_{2}=1}^{M}R_{m_{1},m_{2}}(\tau _{m_{1},m_{2}}(\mathbf {x} )),}$

where the GCC for a microphone pair ${\displaystyle (m_{1},m_{2})}$ is defined as

${\displaystyle R_{m_{1},m_{2}}(\tau )\triangleq {\frac {1}{2\pi }}\int _{-\pi }^{\pi }\Phi _{m_{1},m_{2}}(e^{j\omega })S_{m_{1}}(e^{j\omega })S_{m_{2}}^{*}(e^{j\omega })e^{j\omega \tau }d\omega .}$

The phase transform (PHAT) is an effective GCC weighting for time delay estimation in reverberant environments, that forces the GCC to consider only the phase information of the involved signals:

${\displaystyle \Phi _{m_{1},m_{2}}(e^{j\omega })\triangleq {\frac {1}{\vert S_{m_{1}}(e^{j\omega })S_{m_{2}}^{*}(e^{j\omega })\vert }}.}$

Estimation of source location

The SRP-PHAT algorithm consists in a grid-search procedure that evaluates the objective function ${\displaystyle P(\mathbf {x} )}$ on a grid of candidate source locations ${\displaystyle {\mathcal {G}}}$ to estimate the spatial location of the sound source, ${\displaystyle {\textbf {x}}_{s}}$, as the point of the grid that provides the maximum SRP:

${\displaystyle {\hat {\mathbf {x} }}_{s}=\arg \max _{\mathbf {x} \in {\mathcal {G}}}P(\mathbf {x} ).}$

Accumulation limits

The accumulation limits can be calculated beforehand in an exact way by exploring the boundaries separating the regions corresponding to the points of the grid. Alternatively, they can be selected by considering the spatial gradient of the TDOA ${\displaystyle \nabla _{\tau _{m_{1},m_{2}}}(\mathbf {x} )=[\nabla _{x\tau _{m_{1},m_{2}}}(\mathbf {x} ),\nabla _{y\tau _{m_{1},m_{2}}}(\mathbf {x} ),\nabla _{z\tau _{m_{1},m_{2}}}(\mathbf {x} )]^{T}}$, where each component ${\displaystyle \gamma \in \left\{x,y,z\right\}}$ of the gradient is:

${\displaystyle \nabla _{\gamma \tau _{m_{1},m_{2}}}(\mathbf {x} )={\frac {1}{c}}\left({\frac {\gamma -\gamma _{m_{1}}}{\|\mathbf {x} -\mathbf {x} _{m_{1}}\|}}-{\frac {\gamma -\gamma _{m_{2}}}{\|\mathbf {x} -\mathbf {x} _{m_{2}}\|}}\right).}$

For a rectangular grid where neighboring points are separated a distance ${\displaystyle r}$, the lower and upper accumulation limits are given by:

${\displaystyle L_{m_{1},m_{2}}^{l}(\mathbf {x} )=\tau _{m_{1},m_{2}}(\mathbf {x} )-\|\nabla _{\tau _{m_{1},m_{2}}}(\mathbf {x} )\|\cdot d}$

${\displaystyle L_{m_{1},m_{2}}^{u}(\mathbf {x} )=\tau _{m_{1},m_{2}}(\mathbf {x} )+\|\nabla _{\tau _{m_{1},m_{2}}}(\mathbf {x} )\|\cdot d,}$

where ${\displaystyle d=(r/2)\min \left({\frac {1}{\vert \sin(\theta )\cos(\phi )\vert }},{\frac {1}{\vert \sin(\theta )\sin(\phi )\vert }},{\frac {1}{\vert \cos(\theta )\vert }}\right)}$ and the gradient direction angles are given by

${\displaystyle \theta =\cos ^{-1}\left({\frac {\nabla _{z\tau _{m_{1},m_{2}}}(\mathbf {x} )}{\|\nabla _{\tau _{m_{1},m_{2}}}(\mathbf {x} )\|}}\right),}$

${\displaystyle \phi =\arctan _{2}\left(\nabla _{y\tau _{m_{1},m_{2}}}(\mathbf {x} ),\nabla _{x\tau _{m_{1},m_{2}}}(\mathbf {x} )\right).}$