Cb. Papadias, Globally convergent blind source separation based on a multiuser kurtosis maximization criterion, IEEE SIGNAL, 48(12), 2000, pp. 3508-3519
We consider the problem of recovering blindly (i.e., without the use of tra
ining sequences) a number of independent and identically distributed source
(user) signals that are transmitted simultaneously through a linear instan
taneous mixing channel. The received signals are, hence, corrupted by inter
user interference (IUI), and we can model them as the outputs of a linear m
ultiple-input-multiple-output (MIMO) memoryless system. Assuming the transm
itted signals to be mutually independent, i.i.d., and to share the same non
-Gaussian distribution, a set of necessary and sufficient conditions for th
e perfect blind recovery (up to scalar phase ambiguities) of all the signal
s exists and involves the kurtosis as well as the covariance of the output
signals. We focus on a straightforward blind constrained criterion stemming
from these conditions. From this criterion, we derive an adaptive algorith
m for blind source separation, which we call the multiuser kurtosis (MUK) a
lgorithm. At each iteration, the algorithm combines a stochastic gradient u
pdate and a Gram-Schmidt orthogonalization procedure in order to satisfy th
e criterion's whiteness constraints. A performance analysis of its stationa
ry points reveals that the MUK algorithm is free of any stable undesired lo
cal stationary points for any number of sources; hence, it is globally conv
ergent to a setting that recovers them all.