Digital Signal Processing Reference
In-Depth Information
where
L
corresponds to
log
(
P
())
.
The minimization of the negative log
likelihood function of then basically corresponds to minimizing
with respect to
s
since there is no prior information on
a.
Since the accuracy of estimated separated source signals
s
depends on the
accuracy of estimated
a
we think that by jointly optimizing the above log
likelihood function with respect to both
a
and
s
(as evidenced by simulation
results and as described in [9]) we can separate the sources signals from the
observations more efficiently. For this joint optimization, we developed a
“dual update” algorithm in [7] that is briefly described below.
Description of joint minimization algorithm - “dual update”:
For the
joint optimization problem, we consider a sparse domain i.e., the domain
where most of the coefficients that correspond to non-signals are small (near
zero). In other words, the sparse domain is a domain in which signals can be
efficiently represented. This has the advantage of reducing the complexity of
the problem (i.e., need to deal with sparse matrix compared to full matrix) of
separation of mixed signals. Examples of domains where signals can be
efficiently represented are Fourier and wavelet. Here we choose Fourier.
Note that in this chapter, when we refer to Fourier, we mean short-time
Fourier transform and we are not making a specific distinction between
Fourier and the short-time Fourier since it is a special case of Fourier i.e., the
windowed Fourier. When we compute the Fourier transform we use the fast
Fourier transform (FFT) technique. Next, we assume that (a) the source
signals are statistically independent to each other (which is not a strong
assumption since in practice source signals are statistically independent to
each other and researchers commonly make this assumption) and follow
Laplacian probability distribution function in the sparse domains (it has been
observed that the Fourier and wavelet coefficients do exhibit Laplacian
behavior [9]) and, (b) noise
v
is white Gaussian.
As mentioned above, we first transform the mixed signals in to the sparse
domain by applying the Fourier transform. We then apply the probabilistic
approach of BSS in the sparse domain. The observed mixed signals in the
transformed domain can be written as:
where W is the Fourier transform. This has the same form as the mixed
observed signals in the time domain (see Eq.(l)). Therefore, without loss of
generality, the problem of BSS in the signal domain and in the transformed
