Signal Separation with A Priori Knowledge Using Sparse Representation - Advances in Heuristic Signal Processing and Applications - page 311

The decomposition into dictionaries associated to short-time DFT and DCT is
conducive to the harmonic analysis of periodic signals. An overcomplete Fourier
or cosine dictionary can be constructed by making a finer sampling of frequencies.
Compared with DFT dictionaries, DCT dictionaries give decompositions in the real
space and have the advantage of adapting perfectly to the algorithms developed in
the context of sparse representation. The pure frequency analysis has some limita-
tions. Although it allows detecting the dominant frequencies of a signal, it does not
take into account its temporal characteristics. So DFT and DCT dictionaries are not
suitable for representing a nonstationary or temporally discontinuous signal.
Time-frequency analysis of a signal is an interesting approach when the frequen-
cies are varying. The transforms used for time-frequency analysis, such as wavelet
transforms, can model a nonstationary phenomenon well over a very short duration.
Therefore, dictionaries associated to time-frequency transforms are commonly con-
structed to represent nonstationary signals. Gabor dictionary is, for example, a kind
of time-frequency dictionary and its atoms are created by sampling the following
Gabor basis function: a s,τ,f (t)

s w( t − s )e 2 iπf(t − τ) , which can be described as a
window function w modulated by an oscillating frequency. τ represents the tem-
poral location of the window w , and s defines its width in the frequency domain.
There is no constraint for the choice of the window function. In general, we choose
the window functions so that they satisfy certain properties; one example is Ham-
ming window.
Unlike the prespecified dictionaries, adaptive dictionaries are constructed from a
set of given signals and based on learning methods, such as the dictionaries studied
in [ 1 ]. The advantage of these dictionaries is that they have the adaptability to repre-
sent any signal. However, it needs enough data sources for training the desired dic-
tionaries. In addition, the methods for constructing this kind of dictionaries usually
suffer from low computational efficiency. Choosing a dictionary method depends
on the application. It is therefore difficult to say which method is the best. Once the
dictionary is built, let us now present solutions to obtain the sparsest representation
of a given signal.

1

=

14.2.2 Pursuit Algorithms

In the past decade or so, several efficient pursuit algorithms have been proposed
to approximate the solution in Eq. ( 14.1 ). Greedy algorithms and algorithms based
on convex relaxations are two kinds of pursuit algorithms commonly used to find a
sparse representation.

14.2.2.1 Greedy Algorithms

For the problem of sparse representation, dictionaries can be supposed to be redun-
dant, which means signals can be well represented with only a small number of basis