Matlab toolbox for separation of convolutive mixtures

  The deflation method

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The deflation method

We give a brief description of the deflation method implemented in the toolbox. Assume that the mixing model is the following one:

$$\forall n \qquad \mathbf{x}(n) = \sum_k \mathbf{H}(k)\mathbf{s}(n-k)$$

where:
- $\mathbf{x}(n)=(x_1(n),\ldots,x_Q(n))^{^\mathrm{T}}\!$ is the observation vector
- $\mathbf{s}(n)=(s_1(n),\ldots,s_N(n))^{^\mathrm{T}}\!$ is the source vector
- $\mathbf{H}(k),k\in\mathbb{Z}$ is the impulse response of the mixing filter with $N$ inputs and $Q$ outputs.

The contrast functions implemented in the toolbox allow one to extract one source from the mixture. After one source has been restored or if a filtered version of one source has been extracted, one can subtract its contribution in the observation signals. In so doing, the problem of separating $N$ sources from the mixture simplifies to the problem of separating $N-1$ sources. The so-called ``deflation'' method in source separation is based on this idea. More precisely, the algorithm (which has been implemented in Deflation.m) is the following one:

1. initialization: $p=1$; extraction of the first source $(y_1(n))_{n\in\mathbb{Z}}$ from the observations $(\mathbf{x}^{(1)}(n))_n = (\mathbf{x}(n))_{n}$.
2. for $p\in \{2,\ldots,N\}$,
   1. subtract the contribution of the source extracted at the $p-1$-th stage; this yields a signal $(\mathbf{x}^{(p)}(n))_{n}$ with $Q$ components but which corresponds to a reduced mixture of $N-p+1$ sources. (This is done in the program SubtractSource.m).
   2. extraction of a $p$-th source $(y_p(n))_{n}$ from the modified observations $(\mathbf{x}^{(p)}(n))_{n}$.

At stage 2a, in order to subtract from the observations the contribution of the $(p-1)$-th source, we compute:

$$\mathbf{x}^{(p)}(n) = \mathbf{x}^{(p-1)}(n) - \sum_{k} \mathbf{t}^{(p)}(k)\, y_{p-1}(n-k),\quad n \in \mathbb{Z}$$ (1)

where $(\mathbf{t}^{(p)}(k))_{k\in \mathbb{Z}}$ is the impulse response of a filter with $1$ entry and $Q$ outputs. Since the sources are mutually independent, this impulse response is obtained as the minizer of the following quantity:

$$\epsilon(\mathbf{t}^{(p)}) = \mathbb{E}\{\\ }{\vert\mathbf{x}^{(p)}(n)\vert^2\}.$$ (2)

In practice, the filters have finite impulse response and the above problem amounts to the least square solution of a linear system.
Note that deflation methods often lead to an accumulation of errors.

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