Mathematics in Independent Component Analysis

Chapter 20. Signal Processing 86(3):603-623, 2006 269
adjusted with the aid of an incorporated graphical user interface [18]. In the
present work, the thresholds were set to 10% for all the signals so that we may
compare all obtained results easily. The algorithm looks for zero-crossings on
the signal and defines a waveform as the samples of the considered signal comprised
between two consecutive zero-crossings. If the maximum (respectively,
minimum in the case of a negative waveform) is above (respectively, below)
the threshold, that waveform is kept in the signal given as output. Otherwise,
the waveform is substituted by a zero-voltage signal on that period.
It is quite important to use such a filter, because although BSS algorithms
are generally able to separate the noise as a component, they are usually
unable to separate more source signals than available channels. Generally in
our experiments there were more MUs than available channels even at low
levels of contraction, but only few of them had amplitude above the noise
level. In order to eliminate the activity of some MUs we delete the MUAPTs
belonging to distant MUs, whose MUAPs are less powerful. This is enhanced
by PCA preprocessing and dimension reduction as discussed later in section
3.2.
2.2 Blind source separation
Linear blind source separation (BSS) describes the task of blindly recovering
A and S in the equation
X = AS + N (1)
where X consists of m signals with N observations each, put together into an
(m × N)−matrix. A is a full-rank (m × n)−matrix, and we typically assume
that m ≥ n (complete and under-complete case). Moreover N models additive
noise, which is commonly assumed to be white Gaussian. Depending on the
assumptions on A and S we get different models. Our goal is to estimate
the mixing matrix A. Then the sources can be recovered by applying the
pseudoinverse A + of A to the mixtures X (which is optimal in the maximumlikelihood
sense when Gaussian noise is assumed).
In the case of s-EMG data, the original signals are the MUAPTs generated by
the motor units active during a sustained contraction. In this setting, A quantifies
how each source contributes to each observation. Of course additional
requirements will have to be applied to the model to guarantee a satisfactorily
small space of solutions, and depending on the assumptions on A and S we
will obtain different models.
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