A Survey of Kurtosis Optimization Schemes for MISO Source Separation and Equalization - Advances in Heuristic Signal Processing and Applications - page 186

9.2.1 Convolutive Mixtures

We consider an observed Q -dimensional ( Q ∈ N , Q ≥ 2) discrete-time signal x (n)

following the linear model

x (n) M (n) s (n) = M (p) s (n − p). (9.2)

p ∈Z

Hence, the observed data x (n) are the output of the convolutive MIMO channel rep-
resented by the ( Q

×

N ) matrix impulse response M (n) excited by the unknown

N -dimensional ( N Q ) source input s (n) .The (i,j) th entry of

the matrix M (n) , denoted m ij (n) , represents the scalar channel transforming source
s j (n) before adding its contribution to mixture y i (n) . The above model is assumed
noise-free. The objective of BSS is to restore the sources by exploiting the obser-
vations alone, without any knowledge of the MIMO channel M (n) and the sources
s (n) . Clearly, some further assumptions are necessary to prevent this problem from
being ill-posed.
A first type of assumptions concerns the convolutive mixing system M (n) .We
assume that it admits a left inverse or MIMO separating filter W (n) such that

∈ N , N ≥ 2, N ≤

y (n) W (n) x (n) = W (p) x (n − p) (9.3)

p ∈Z

recovers all sources in the separator output y (n) . Since the source ordering and spec-
tral profiles cannot be identified by using criteria based on statistical independence
only, the separation is considered successful whenever the global system

G (n) W (n) M (n) = W (p) M (n − p) (9.4)

p ∈Z

is of the form

G (n) = D (n) P (9.5)

where D (n) is an invertible diagonal convolutive MIMO system modeling the
source spectral ambiguity, and P a permutation matrix modeling the source order-
ing ambiguity. Remark that these ambiguities are inherent to blind processing and
are acceptable in most applications. When the sources are independent and iden-
tically distributed (i.i.d.), the scalar filtering represented by the diagonal entries
of D (n) reduces to a simple delay n i ∈ Z with a possible scale factor d ii ∈ C ,

and so d ii (n) d ii δ n − n i , where δ n denotes Dirac's discrete delta function; see,

e.g., [ 16 , 17 , 22 , 59 ] for more details.
Another set of assumptions concerns the source signals:

=

A1. For all i , the source sequence s i (n) is stationary and zero-mean. In addition,
the fourth-order cumulants, C { s i } , exist and are assumed to be nonzero.