Blind source separation (BSS) is a problem that arises when we desire to separate a number of sources from a mixture measurement. The system is illustrated in Figure 1 below.

m data-srcset= 2 far field source impinging 2 microphones” width=”316″ height=”245″>

Figure 1: m source signals impinging 2 microphones

The accuracy of a BSS algorithm is not measured only with the accuracy of the mixing matrix estimate but also on the accuracy of the extracted signals from the original unmixed signals. Each source estimate of the signal s_n(t) can be characterized as \hat{s}_n(t) = \alpha_n s_n(t) + \gamma_n where \alpha_n is a scaling factor and \gamma_n is the estimation noise. It is easy to see that even in the case of a perfect separation — \alpha_n =0 \forall n –, conventional definition of relative distortion of

D_n = \frac{|\hat{s}_n(t) - s_n(t)|^2}{|\hat{s}_n(t)|^2} = \left|1-\frac{1}{\alpha_n}\right|^2 =0 \iff \alpha_n = 1

does not present a smooth criteria for distortion quantification. Any other value of \alpha_n will render the distortion non zero. An alternate definition of

D_n = \frac{\left|\mathbb{E}\left[\frac{s_n(t)}{\hat{s}_n(t)}\right]\hat{s}_n(t) - s_n(t)\right|^2}{|s_n(t)|^2} \approx \frac{\left|\frac{\gamma_n}{\alpha_n} \right|^2}{|s_n(t)|^2} =0 \iff \gamma_n = 1

can be used to make the quantification smoother. Thus, the scaling does not matter, if the noise term goes to zero, the distortion goes to zero. Another performance measure that is used is the total relative distortion which is given as

D_{Tn} = \frac{|\hat{s}_n(t)|^2}{|s_n(t) \times \hat{s}_n(t)|^2} -1

The total distortion metric has the added advantage that when the estimate is orthogonal to the desired source, the denominator causes the distortion to approach infinity. So called metrics such as source to interference ratio (SIR) and source to artifact ration (SAR) are all based off the total distortion. Here the noise term \gamma_n is separated into noise, artifact due to non linear zeros, and interference from other sources.

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