Steered-Response Power Phase Transform

Steered-Response Power Phase Transform (SRP-PHAT) is a popular algorithm for acoustic source localization, well known for its robust performance in adverse acoustic environments. The algorithm can be interpreted as a beamforming-based approach that searches for the candidate position that maximizes the output of a steered delay-and-sum beamformer.

Algorithm

Steered-Response Power

Consider a system of $M$ microphones, where each microphone is denoted by a subindex $m \in \$. The discrete-time output signal from a microphone is $s_m (n)$. The (unweighted) steered-response power (SRP) at a spatial point $\mathbf =$, y, z can be expressed as

$P_0(\mathbf) \triangleq \sum_ \left\vert \sum_^ s_m(n-\tau_m(\mathbf)) \right\vert^,$

where $\mathbb$ denotes the set of integer numbers and $\tau_m(\mathbf)$ would be the time-lag due to the propagation from a source located at $\mathbf$ to the $m$-th microphone.

The (weighted) SRP can be rewritten as

$P(\mathbf) = \frac \sum_^\sum_^ \int_^ \Phi_(e^)S_(e^) S_^(e^)e^d\omega,$

where $()^$ denotes complex conjugation, $S_(e^)$ represents the discrete-time Fourier transform of $s_m(n)$ and $\Phi_(e^)$ is a weighting function in the frequency domain (later discussed). The term $\tau_(\mathbf)$ is the discrete time-difference of arrival (TDOA) of a signal emitted at position $\mathbf$ to microphones $m_1$ and $m_2$, given by

$$