Formulation of the problem of segregating two acoustic sources

In this paper, we define the problem of segregating two acoustic sources as ``the segregation of the mixed signal into original signal components, where the mixed signal is composed of two signals generated by any two acoustic sources.'' We formulate it as follows:

Firstly, we can observe only the signal f(t):

f(t)=f₁(t)+f₂(t), (1)

where f₁(t) is the desired signal and f₂(t) is a noise. The observed signal f(t) is decomposed into its frequency components by an auditory filterbank. Secondly, outputs of the k-th channel, which correspond to f₁(t) and f₂(t), are assumed to be

$$A_k(t)\sin(\omega_k t +\theta_{1k}(t))$$ (2)

and

$$B_k(t)\sin(\omega_k t + \theta_{2k}(t)),$$ (3)

respectively. Since the output of the k-th channel X_k(t) is represented by

$$X_k(t)=S_k(t)\sin(\omega_k t + \phi_k(t)),$$ (4)

where

$$S_k(t)=\sqrt{A_k^2(t)+2A_k(t)B_k(t)\cos\theta_k(t)+B_k^2(t)}$$ (5)

and

$$\phi_k(t)=\tan^{-1}\left(\frac{A_k(t)\sin\theta_{1k}(t)+B_k(t... ...t)}{A_k(t)\cos\theta_{1k}(t)+B_k(t)\cos\theta_{2k}(t)}\right),$$ (6)

then the amplitude envelopes of the two signals A_k(t) and B_k(t) can be determined by

$$A_k(t)=\frac{S_k(t)\sin(\theta_{2k}(t)-\phi_k(t))}{\sin\theta_k(t)}$$ (7)

and

$$B_k(t)=\frac{S_k(t)\sin(\phi_k(t)-\theta_{1k}(t))}{\sin\theta_k(t)},$$ (8)

respectively, where $\theta_k(t)=\theta_{2k}(t)-\theta_{1k}(t)$ and $\theta_k(t)\not= n\pi, n\in {\bf {Z}}$. Thus, if the four parameters, S_k(t), $\phi_k(t)$, $\theta_{1k}(t)$, and $\theta_{2k}(t)$ are calculated, A_k(t) and B_k(t) can be calculated by the above equations. Finally, f₁(t) and f₂(t) can be reconstructed by grouping constraints. $\hat{f}_1(t)$ and $\hat{f}_2(t)$ are reconstructed f₁(t) and f₂(t), respectively.

In this paper, we assume $\theta_{1k}(t)=0$ and $\theta_{k}(t)=\theta_{2k}(t)$. Additionally, we consider the problem of segregating two acoustic sources in which the localized f₁(t) is added to f₂(t).