Formulation of the problem of segregating two acoustic sources

In this paper, the problem of segregating two acoustic sources is defined as ``the segregation of original signal components from a noise-added signal, where mixed signal is composed of two signals generated by two acoustic sources.''

This problem is formulated as follows.

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

 

f(t)=f₁(t)+f₂(t), (1)

where f₁(t) and f₂(t) are the original acoustic signals. The observed signal is decomposed into its frequency components by an auditory filterbank as shown in Fig. . Secondly, outputs of the k-th channel, which correspond to f₁(t)and f₂(t), are assumed to be

$$f_1(t): A_k(t)\sin(\omega_k t + \theta_{1k}(t))$$ (2)

and

$$f_2(t): B_k(t)\sin(\omega_k t + \theta_{2k}(t)),$$ (3)

respectively. Here, $\omega_k$ is a center frequency of the auditory filter and $\theta_{1k}(t)$ and $\theta_{2k}(t)$ are input phases of f₁(t)and f₂(t), respectively. Since the output of the k-th channel X_k(t) is the sum of Eqs. () and (), then it is represented by

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

Here,

 

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

and

 

$${\phi_k(t)}$$

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

where $\theta_k(t)=\theta_{2k}(t)-\theta_{1k}(t)$ and $\theta_k(t)\not= n\pi, n\in{\bf {Z}}$. Since the amplitude envelope S_k(t) and the input phase are observable and if the input phases $\theta_{1k}(t)$ and $\theta_{2k}(t)$ are determined, the amplitude envelopes 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. Finally, for each auditory filter, determining of A_k(t) and B_k(t), f₁(t) and f₂(t) are reconstructed from Eqs. () and () synthesizing each frequency components. Here, $\hat{f}_1(t)$ and $\hat{f}_2(t)$ are the reconstructed f₁(t)and f₂(t), respectively.

In this paper, the analysis synthesis system (filterbank) as shown in Fig. is constructed using the wavelet transform. We assume that $\theta_{1k}(t)=0$, $\theta_k(t)=\theta_{2k}(t)$, and that f₁(t) is an amplitude modulation signal. We also assume that a center frequency of analytic filter matches a center frequency of f₁(t), and that f₂(t) is a bandpassed random noise which the center frequency is the same as f₁(t). We will also consider the problem of segregating two acoustic sources in which the localized f₁(t) is added to f₂(t).