Formulation of the problem of segregating two acoustic sources

First, only the mixed signal f(t), where f(t)=f₁(t)+f₂(t), can be observed in the proposed model. Here, f₁(t) is the desired signal and f₂(t) is a noise or the other signal. The observed signal f(t) is decomposed into its frequency components by an auditory-motivated filterbank (the number of channels is K). The output of the k-th channel X_k(t) is represented by

 

X_k(t) = X_1,k(t)+X_2,k(t) (1)

$$S_k(t)\exp(j\omega_k t + j\phi_k(t)),$$ = (2)

where X_1,k(t) and X_2,k(t) are components of f₁(t) and f₂(t) that have passed through the filterbank, respectively.

Second, the outputs of the k-th channel, which correspond to f₁(t) and f₂(t), are assumed to be

$$X_{1,k}(t)=A_k(t)\exp(j\omega_k t + j\theta_{1k}(t))$$ (3)

and

$$X_{2,k}(t)=B_k(t)\exp(j\omega_k t + j\theta_{2k}(t)).$$ (4)

Here, $\omega_k$ is the center frequency of the k-th channel (the auditory filter) and $\theta_{1k}(t)$ and $\theta_{2k}(t)$ are the instantaneous input phases of f₁(t) and f₂(t), respectively. Using this assumption, the instantaneous amplitude S_k(t) and the instantaneous output phase $\phi _k(t)$ are represented by

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

and

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

Therefore, the instantaneous amplitudes 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)

where $\theta_k(t)=\theta_{2k}(t)-\theta_{1k}(t)$ and $\theta_k(t)\not= n\pi, n\in{\bf {Z}}$. Focusing on the output value of the k-th channel at time t, the relationships between every instantaneous amplitude and every instantaneous phase are shown in Fig. 2.

Hence, since the instantaneous amplitude S_k(t) and the instantaneous output phase $\phi _k(t)$ are observable (see Sec. 3.1.1), and if the instantaneous input phases $\theta_{1k}(t)$ and $\theta_{2k}(t)$ are determined, then A_k(t) and B_k(t) can be determined by the above equations.

Finally, f₁(t) and f₂(t) can be reconstructed by using the grouping of the instantaneous amplitude and the instantaneous phase for all channels. Thus, $\hat{f}_1(t)$ and $\hat{f}_2(t)$ are the reconstructed f₁(t) and f₂(t), respectively.

However, in the above formulation, it is difficult to uniquely and simultaneously determine the instantaneous amplitudes (A_k(t) and B_k(t)) and the instantaneous phases ( $\theta_{1k}(t)$ and $\theta_{2k}(t)$) using S_k(t) and $\phi _k(t)$, because there are currently no equations for determining two such instantaneous phases and the segregation of two acoustic sources is an ill-inverse problem. Therefore, in this paper, we try solving the problem of segregating two acoustic sources by constraining the desired signal using the four regularities.