Digital Signal Processing Reference
In-Depth Information
hierarchical and k-means clustering algorithms works better than using either
the hierarchical or k-means clustering algorithms individually.
The next step is to jointly minimize the cost function (“dual update”). In
this case the sources are modelled as a Laplacian and the noise is assumed to
be white Gaussian as mentioned before, so the resulting cost function - log
likelihood
L
is:
with the assumption that the noise covariance matrix is a unity matrix. Here
T
denotes the Hermitian transpose of a matrix and indices k
and
m
are not specifically mentioned to simplify the expression. This
expression is optimized by first finding
S
(k,m)
that minimizes
under the constraint that:
The second part of the procedure re-estimates
A
(k) so that the sources
will be more independent.
The easiest way to perform the first part of the “dual-update” method is to
recognize that there is a local minimum whenever there are
N - M
zeros in
the
S
(k,n)
vector. This can be shown by using a geometrical argument. First
we draw the shape formed by all points at a certain cost The resulting
shape is an
N
-dimensional cube with vertices located on the axes at a
distance away from the origin. Now the constraint has dimension
N - M.
So, when there is one more source than the sensor the constraint is a line. If
the line goes through the cube then the portion inside the cube is at a lower
cost and the portion outside is at a higher cost.
If the cube is shrunk until the constraint only touches the edge of the
cube, the point of intersection is the lowest cost. If the line is parallel to one
of the sides of the box, then there are an infinite number of solutions. This
case corresponds to
A
matrix having at least two identical column vectors.
The other case requires that the line intersect a vertex of the cube. Of course
this occurs when the line passes through a plane created by all combinations
of M axes or in other words there are N - M zeros in S(k,n), which yields a
finite number of points to check. The point with the lowest cost is the global
minimum. Inclusion of this geometric constrained based search not only has
speeded up our original “dual update” algorithm and also has generalized it
to handle more than two sensors.
