A Survey of Kurtosis Optimization Schemes for MISO Source Separation and Equalization - Advances in Heuristic Signal Processing and Applications - page 204

1. Compute the gradient direction d (k − 1 ) =∇ J κ ( w (k − 1 ) ) from Eq. ( 9.14 ).

2. Obtain the optimal step size μ opt as described above (Eqs. ( 9.31 )-
( 9.37 )).
3. Update: w (k − 1 ) μ opt d (k − 1 ) . w = +

4. Normalize: w (k) = w / w .

To quickly illustrate the benefits of RobustICA, we take up the simulation ex-
ample introduced at the end of Sect. 9.4.4.2 . Recall that the solid lines in Fig. 9.3
plot the kurtosis contrast as a function of the global filter angle Δθ for the ob-
served instantaneous orthogonal mixture of two sources. Despite the short obser-
vation window (just 50 samples), the kurtosis local maxima lie quite close to the
valid extraction solutions, thus yielding improved source estimates. Moreover, the
algorithm converges in just a single iteration whatever the initialization employed
(panels (a)-(b)).
Finally, let us point out that the optimal step-size cencept can also be used in the
context of monotonically convergent algorithms presented in the next section. The
basic idea is similar to RobustICA's, and the reader is referred to [ 13 ] for details.

9.4.6 Algorithms Based on Reference Signals

As justified in Sect. 9.1.2 , many contrast functions are based on higher-order cu-
mulants. A common example studied throughout this chapter is the kurtosis ( 9.8 )
giving rise to contrasts ( 9.9 )-( 9.10 ), which is a normalized marginal cumulant of
the extractor output. In this section, we introduce contrast functions based on cross-
cumulants . The advantage of these alternative contrasts is that they can be expressed
as quadratic functions (Sect. 9.4.6.1 ) and, as such, their optimization is highly fa-
cilitated. Indeed, we will see that they can be considered as a starting point for
developing monotonically convergent algorithms (Sect. 9.4.6.2 ).

9.4.6.1 Quadratic Contrast Functions

Crucial to the development of quadratic contrasts is the assumption that a reference
signal, denoted by z(n) , is available. The reference signal is defined as the output of
a MISO reference filter v (n) :

z(n) = v (n) x (n) = t (n) s (n)

with t (n) v (n) M (n) . At first sight, one could think of z(n) as a kind of prior

information about the source being targeted by the extraction procedure. In this

=