Signal Separation with A Priori Knowledge Using Sparse Representation - Advances in Heuristic Signal Processing and Applications - page 309

We focus here on the single-channel signal separation problem in which a single
observed signal is available and it is composed by a linear combination of sev-
eral source signals and a noise signal. For decomposing the observed signal into its
subcomponents, two kinds of methods are proposed in the literature. In one kind of
methods, these subcomponents can be modeled as some mathematical functions and
then the signal separation problem can be converted to a parameter estimation prob-
lem. Some linear or nonlinear parameter estimation algorithms can be used to esti-
mate these subcomponents. In the area of magnetic resonance spectroscopy (MRS)
data analysis, this kind of methods is commonly used to divide an observed spectrum
into several resonances associated respectively to different metabolites [ 10 , 23 ]. In
the other kind of methods, it is assumed that source signals are disjoint in the time
domain, frequency domain or time-frequency domain, and then source separation
can be achieved in one specified domain. For example, a frequency domain filtering
method can separate sources which are in different frequency bands. However, the
assumption that source signals are disjoint in certain domains is rather restrictive.
The sources are usually non-disjoint in all these domains and even time-frequency
filtering methods cannot separate them.
Recently, some researchers propose sparse representation-based methods which
relax the disjoint condition by allowing the sources to be non-disjoint in the time-
frequency domain, such as the methods proposed in [ 4 - 6 , 12 , 21 ]. In these methods,
it is assumed that the sources can be sparsely represented over different dictionaries.
Then they are finally recovered by estimating the sparse representation of the ob-
served signal over these dictionaries. The construction of proper dictionaries, which
is one of the keys for this kind of methods, is usually based on some a priori knowl-
edge about the characters of source signals. For example, the method of morpho-
logical component analysis proposed in [ 21 ] separates textures from the piecewise
smooth components by estimating the sparse representations of mixed images with
respect to a wavelet dictionary for cartoon source images and Gabor dictionary for
textures.
In this chapter, we introduce in detail how to achieve a single-channel signal sep-
aration based on a priori knowledge and using sparse representations. The chapter
is organized as follows. We first introduce the basic scheme of this kind of signal
separation methods. The two key steps are then presented: dictionary constructions
and pursuit algorithms employed to find sparse representations. Finally, applications
to separate MRS data are presented.

14.2 Signal Separation Using Sparse Representation

The research of sparse representation has attracted more and more interest in signal
processing domain in recent years. It is widely used for denoising [ 4 ], signal sepa-
ration [ 5 , 21 ], direction-of-arrival estimation (DOA) [ 11 ], and so on. A basic signal
representation model can be described as

y = Dw (14.1)