Blind source separation of far field source is used to recover an unknown number of sources using observable mixture signals. Consider a noise free instantaneous mixture signals described below:

$\begin{bmatrix} x_1(w,t) \\ x_2(w,t) \\ \vdots \\ x_N(w,t) \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1M}\\ a_{21} & a_{22} & \cdots & a_{2M}\\ \vdots & \vdots & \vdots & \vdots\\a_{N1} & a_{N2} & \cdots & a_{NM} \end{bmatrix} \begin{bmatrix} s_1(w,t) \\ s_2(w,t) \\ \vdots \\ s_M(w,t) \end{bmatrix}$

where there are $M$ sources and $N$ observations. The signals are in time-frequency domain because speech signals have been shown to be sparser in time-frequency domain as opposed to time domain. Under the assumption that only one source is active per time-frequency bin, which implies the signal space is sparse, for each time-frequency bin, we have

$x_i(w,t) = a_{ik}s_k(w,t)$

For the 2 microphone case, this reduces to:

$\frac{x_1(w,t)}{x_2(w,t)} = \frac{a_{1k}}{a_{2k}}$

Using k-mean clustering, the ratios of $\frac{a_{1k}}{a_{2k}}$ are used to determine the M parameters up to a scaling term. In real situations, the signals are noisy and the ratios will not be exact. In such scenarios, an alternative approach is used. Consider the cross covariance matrix, denoted $R_{X(w,t)}$ given as

$R_{X(w,t)} = \begin{bmatrix} x_1(w,t) \\ x_2(w,t) \\ \vdots \\ x_N(w,t)\end{bmatrix} \begin{bmatrix} x_1(w,t) & x_2(w,t) & \cdots & x_N(w,t)\end{bmatrix} ^*$

where $*$ denotes the complex conjugate, then the SVD of $R_{X(w,t)}$ will give a single significant eigenvalue if there is only one active speaker per time frequency bin. $R_{X(w,t)}$ can be averaged to approach the expected value by using

$\mathbb{E}[R_{X(w,t)}] = \frac{1}{T} \sum\limits_{w} \begin{bmatrix} x_1(w,t) \\ x_2(w,t) \\ \vdots \\ x_N(w,t)\end{bmatrix} \begin{bmatrix} x_1(w,t) & x_2(w,t) & \cdots & x_N(w,t)\end{bmatrix} ^*$

where $T$ is the length of the STFT. Since there is only one active signal assumed, the eigenvalue corresponding to the principal eigenvalue is chosen as the estimate of the $k^{th}$ active signal, $a_{i,k} \forall i$. Clustering is subseqeuntly used to separate the desired signals.

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