Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 60

2.4.2.2.6 Wavelets in MATLAB In MATLAB wavelets analysis can be conve-
niently performed with the Wavelet Toolbox distributed by MathWorks. The col-
lection of functions delivered by this toolbox can be obtained by command help
wavelet .
Another MATLAB toolbox for wavelet analysis is WaveLab http://www-stat.
stanford.edu/ ˜ wavelab/Wavelab_850/index_wavelab850.html . WaveLab is
a collection of MATLAB functions that implement a variety of algorithms related
to wavelet analysis. The techniques made available are: orthogonal and biorthogonal
wavelet transforms, translation-invariant wavelets, interpolating wavelet transforms,
cosine packets, wavelet packets.

2.4.2.2.7 Matching pursuit—MP The atomic decompositions of multicompo-
nent signals have the desired property of explaining the signal in terms of time-
frequency localized structures. The two methods presented above: spectrogram and
scalogram are working well but they are restricted by the a priori set trade-off be-
tween the time and frequency resolution in different regions of the time-frequency
space. This trade-off does not follow the structures of the signal. In fact interpretation
of the spectrogram or scalogram requires understanding which aspects of the repre-
sentation are due to the signal and which are due to the properties of the methods.
A time-frequency signal representation that adjusts to the local signal properties
is possible in the framework of matching pursuit (MP) [Mallat and Zhang, 1993].
In its basic form MP is an iterative algorithm that in each step finds an element g γ n
(atom) from a set of functions D (dictionary) that best matches the current residue of
the decomposition R n x of signal x ; the null residue being the signal:
⎧
⎨

R 0 x

= x

R n x R n x R n + 1 x = , g γ n g γ n +

(2.109)

⎩

R n x g γ n = arg max g γ i ∈ D | , g γ i |

where: arg max g γ i ∈ D means the atom g γ i which gives the highest inner product with
the current residue: R n x . Note, that the second equation in (2.109) leads to orthogo-
nality of g γ n

and R n + 1 x ,so:

R n + 1 x

2 2

2

R n x R n x

= , g γ n + (2.110)

The signal can be expressed as:

n = 0 R n x , g γ n g γ n + R k + 1 x

k

x = (2.111)

It was proved [Davis, 1994] that the algorithm is convergent, i.e., lim k → ∞ R k s

2

= 0.

Thus in this limit we have:

n = 0 R n x , g γ n g γ n

∞

x = (2.112)