Inter-channel Phase Difference¶
Inter-channel phase difference (IPD) is a useful feature in the estimation of the direction-of-arrival (DOA) of a sound source.
While the actual form of the IPD varies from implementation to implementation, it should always reflect the phase differences between the audio channels.
Phase Difference between audio channels¶
Given
- \(X_m \left( f,t \right) \in \mathbb{C}\): The short-time fourier transform (STFT) of the audio signal of the \(m\)-th channel at frequency \(f\) at time frame \(t\)
- \(m_r\): The integer index of the reference audio channel
The phase difference between channel \(m\) and \(m_r\) is then
\[
\delta_{m} \left( f,t \right) = \angle X_m \left( f,t \right) - \angle X_{m_r} \left(f,t \right)
\]
Possible forms of IPD¶
Form 1¶
One may define IPD as
\[
F_m \left(f,t \right) = \log \left( {X_{m_r}\left(f,t\right)} \over X_m\left(f,t\right) \right)
\]
such that the phase difference can be derived from
\[
\delta_{m} \left( f,t \right) = - \mathfrak{I} \left( F_m \left(f,t\right) \right)
\]
where \(\mathfrak{I}\left( \cdot \right)\) is the imaginary part operator
See https://ieeexplore.ieee.org/document/6882340
Form 2¶
Another way to define IPD is
\[
F_m \left(f,t \right) = X_m \left( f,t \right) \cdot X_{m_r}^* \left( f,t \right)
\]
such that the phase difference can be derived from
\[
\delta_{m} \left( f,t \right) = \angle F_m \left(f,t \right)
\]