Digital Signal Processing Reference
In-Depth Information
9.4.4.1 Choice of the New Columns of
A
In the aforementioned algorithm, step 1 amounts to adding a column vector to the
current mixing matrix
A
. The most simple choice would amount to choosing this
vector at random. Wiser choices can also be made based on additional prior infor-
mation:
Decorrelation:
If the mixing matrix is assumed to be orthogonal, the new col-
umn vector can be chosen as being orthogonal to the subspace spanned by the
columns of
A
with rank (
A
)
=
n
s
−
1.
Known spectra:
If a library of spectra is known a priori, the new column can
be chosen from among the set of unused ones. The new spectrum can be cho-
sen based on its correlation with the residual. Let
A
denote a library of spectra
n
s
denote the set of spectra that have not been chosen
yet; then the
n
s
th new column of
A
is chosen such that
{
a
i
}
i
=
1
,...,
Card
(
A
)
, and let
A
l
]
.
N
1
a
i
a
i
(
Y
a
i
=
argmax
a
i
∈A
−
AS
) [
.,
(9.32)
2
ns
l
=
1
Any other prior information can be taken into account, which will guide the choice
of a new column vector of
A
.
9.4.4.2 The Noisy Case
In the noiseless case, step 2 of Algorithm 36 amounts to running the GMCA algo-
rithm to estimate
A
and
S
T
for a fixed
n
s
with a final threshold
=
α
λ
min
=
0. In the
noisy case,
in (
P
n
s
,σ
) can be closely related to the noise level. For instance, if the
noise
E
is additive Gaussian with
σ
2
3-4, as suggested
throughout the chapter. If the GMCA algorithm recovers the correct sources and
mixing matrix, this ensures that the residual mean squares are bounded by
E
=
σ
E
,
λ
=
τσ
E
with
τ
=
min
τ
2
σ
2
E
with
2
probability higher than 1
−
exp(
−
τ
/
2).
Illustrative Example.
In this experiment, 1-D channels are generated following the instantaneous linear
mixture model (9.2) with
N
s
sources, where
N
s
varies from 2 to 20. The number of
channels is
N
c
is chosen as
the Dirac basis, and the entries of
S
have been independently drawn from a Lapla-
cian pdf with unit scale parameter (i.e.,
p
=
64, each having
N
=
256 samples. The dictionary
1 in equation (9.25)). The
entries of the mixing matrix are independent and identically distributed
=
1 and
λ
=
∼N
(0
,
1).
The observations are not contaminated by noise.
This experiment will focus on comparing the classical principal components anal-
ysis (PCA), the popular subspace selection method, and the GMCA algorithm, as-
suming that
N
s
is unknown. In the absence of noise, only the
N
s
highest eigenvalues
provided by the PCA, which coincide with the Frobenius norm of the rank 1 matri-
ces (
a
i
s
i
)
i
=
1
···
,
N
s
, are nonzero. PCA therefore provides the true number of sources.
The GMCA-based selection procedure in Algorithm 36 has been applied to the
same data to estimate the number of sources
N
s
. Figure 9.1 depicts the mean num-
ber of sources estimated by GMCA. Each point has been averaged over 25 random
realizations of
S
and
A
. The estimation variance was zero, indicating that for each
of the 25 trials, GMCA provides exactly the true number of sources.
