Digital Signal Processing Reference
In-Depth Information
sparse domain can be considered equivalent. Therefore, the general
probabilistic approach mentioned before applies in the transformed sparse
domain. However, to get the separated source signals back from the
transformed domain to the time domain, we apply the inverse Fourier
transform.
We start with the negative log likelihood function i.e., the cost function
in the sparse domain. With the assumption of Laplacianity of
source signals in the sparse domain the prior probability:
a unit vector. By applying the “Laplacianity” of signals, “Gaussianity” of
noise and no prior information on
a
,
it can be shown that:
where
is the noise covariance matrix.
For mathematical simplicity we assume that the noise covariance matrix is an
identity matrix. However, the proposed “dual update” approach works for
non-Gaussian noise with covariance greater than unity [7]. With unit
covariance assumption and re-writing the above equation in terms of
n =
1,2,···
K
we get:
where
&
are the column vectors of
x
&
s
.
From (4), it can be seen that the first term corresponds to L2 norm where as
the second term corresponds to L1 norm. Therefore, our “dual update”
approach corresponds to minimizing L2 and L1 norms simultaneously. Note
first, we consider the minimization of L2 norm that leads to the estimation of
unknown mixing matrix
a.
For this the above equation is differentiated with
respect to
a
and set to zero. By doing this we get:
