Multiple random fault sources adaptive blind separation in situation of time-varying source signals and system

2.1. Model of linear instantaneous mixing and time-varying mixed blind source separation

Without consideration of observation noise, the problem of linear instantaneous mixing and time-varying mixed blind source separation in situation of both sources and mixed matrix are time-varying can be expressed as follows [8-10]:

In Eq. (1), $\mathbf{X}\left(t\right)=\left[{\overrightarrow{x}}_1\right(t),{\overrightarrow{x}}_2(t),...,{\overrightarrow{x}}_m(t){]}^T$ stands for measured signals (or mixed signals), $m$ stands for the number of mixed signals, $t=\mathrm{1,2},\dots ,L$ is sampling time and $L$ is sampling length. $\mathbf{A}\left(t\right)\in {\mathbb{R}}^{m\times k}$ stands for the linear instantaneous mixed matrix and it changes with time $t$, $\mathbf{S}\left(t\right)=\left[{\overrightarrow{s}}_1\right(t),{\overrightarrow{s}}_2(t),...,{\overrightarrow{s}}_k(t){]}^T$ stands for the source signals.

With consideration of observation noise:

where $\mathbf{N}\left(t\right)$ stands for the noise. And the goal of time-varying blind source separation is to find the appropriate time-varying separation matrix $\mathbf{W}\left(t\right)\in {\mathbb{R}}^{k\times m}$ and let:

To make sure $\mathbf{Y}\left(t\right)=\left[{\overrightarrow{y}}_1\right(t),{\overrightarrow{y}}_2(t),...,{\overrightarrow{y}}_k(t\left)\right]$ is an effective estimation of source signals $\mathbf{S}\left(t\right)=\left[{\overrightarrow{s}}_1\right(t),{\overrightarrow{s}}_2(t),...,{\overrightarrow{s}}_k(t){]}^T$.

Fig. 1Model of linear instantaneous time-varying mixed blind source separation

Actually, the BSS is a multiple solution problem, since for an observed signal $\mathbf{X}\left(t\right)$, there may be an infinite group of mixing matrices $\mathbf{A}\left(t\right)$ when the source signals $\mathbf{S}\left(t\right)$ meet Eq. (2). So, to solve the problem valuably, we make two basic assumptions as below:

1) Mixing matrix $\mathbf{A}\left(t\right)$ is a column full rank matrix, presented as $rank\left(\mathbf{A}\right(t\left)\right)=k$.

2) The source signal vector is a stationary random process with zero mean value, and the components of vector are statistically independent with no more than one component of $\mathbf{S}\left(t\right)$ obeying Gaussian distribution.

In absence of prior knowledge, there are two indeterminate problems about BSS problem:

1) The amplitude value of output signals is uncertain.

2) The order of output signals is uncertain.

3) The number of independent source signal vectors is uncertain.

If the contribution of independent source signal vectors is not enough, it is hard to identify and separate them. So, without any prior knowledge, determination of the number of independent source signal vectors by BSS algorithm is difficult.

The information in a signal is mainly contained in the waveform, and then how to recover the waveforms of each unknown source signal from mixed signal is the principal concern in blind source separation problems. And the original permutation of each signal and its real amplitude are the second concern. Indeed, both original permutation and real amplitude of the source signals are unsolvable in the case that only the mixed signals are available but the mixing matrix and source signals are unknown.

2.2. Theoretical inference of RLS for time-varying mixed BSS

The EASE algorithm is one of the widely used blind source separation algorithms [3], which uses the mutual information as the objective function and the natural gradient algorithm to optimize the separation matrix. The basic form of the iterative matrix is as follows [11]:

Where $\eta$ stands for learning step, $F(\cdot )$ is cost function and $F(\cdot )$ can be expressed as follows:

where $y\left(t\right)=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(y\right)$ expressed as nonlinear function. Unlike the batch gradient iterations of the least squares method, RLS can use natural gradient iterations and on-line processing which make it able to track system changes in the real-time.

2.3. Theoretical inference of R-EASI for time-varying mixed BSS

R-EASI has a low accuracy at the initial stage of separation. With the increase of the number of recursive, the accuracy of the separation matrix and separation sources are improved. After entering the tracking phase, the separation matrix and separation sources are basically the same. Therefore, it is necessary to find a separation matrix with higher separation accuracy in the tracking phase.

Due to the change of the system, it is necessary to find a suitable reference separation matrix for each mixed matrix to keep the mixed signal. We consider selecting a reference point $t_0$ between the current observation sample $t$ and the $k$th observation sample, near the position of the current sample point. Let $t_0=⌈k+\gamma (t-k)⌉$, where $⌈•⌉$ indicates rounding up, $\gamma$ is a constant close to 1. In order to avoid the occurrence of large accidental errors in a separation matrix and make the separation effect worse, considering the selection of the reference point near the separation matrix as the basis of the separation matrix, as follows:

The $k$th to $t-1$ observation samples are recursively recalculated using the baseline separation matrix $W_0$:

And let $k=t$.