Steered-response power (SRP) is a family of acoustic source localization algorithms that can be interpreted as a beamforming-based approach that searches for the candidate position or direction that maximizes the output of a steered delay-and-sum beamformer. Steered-response power with phase transform (SRP-PHAT) is a variant using a "phase transform" to make it more robust in adverse acoustic environments.
Algorithm
Steered-response power Consider a system of M {\displaystyle M} microphones, where each microphone is denoted by a subindex m ∈ { 1 , … , M } {\displaystyle m\in \{1,\dots ,M\}} . The discrete-time output signal from a microphone is s m ( n ) {\displaystyle s_{m}(n)} . The (unweighted) steered-response power (SRP) at a spatial point x = [ x , y , z ] T {\displaystyle \mathbf {x} =[x,y,z]^{\mathsf {T}}} can be expressed as
P 0 ( x ) ≜ ∑ n ∈ Z | ∑ m = 1 M s m ( n − τ m ( x ) ) | 2 , {\displaystyle P_{0}(\mathbf {x} )\triangleq \sum _{n\in \mathbb {Z} }\left|\sum _{m=1}^{M}s_{m}{\big (}n-\tau _{m}(\mathbf {x} ){\big )}\right|^{2},}
where Z {\displaystyle \mathbb {Z} } denotes the set of integer numbers, and τ m ( x ) {\displaystyle \tau _{m}(\mathbf {x} )} would be the time-lag due to the propagation from a source located at x {\displaystyle \mathbf {x} } to the m {\displaystyle m} -th microphone. The (weighted) SRP can be rewritten as
P ( x ) = 1 2 π ∑ m 1 = 1 M ∑ m 2 = 1 M ∫ − π π Φ m 1 , m 2 ( e j ω ) S m 1 ( e j ω ) S m 2 ∗ ( e j ω ) e j ω τ m 1 , m 2 ( x ) d ω {\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 }
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