Single channel (univariate) signal - Practical Biomedical Signal Analysis Using MATLAB - page 61

FIGURE 2.20: Illustration of the iterative MP decomposition. a) Signal to be de-
composed (black) and the residue after six iterations (dotted line), b) from top to
bottom the six consecutive selected atoms, c) the sum of the six atoms.

In this way signal x is represented as a weighted sum of atoms (waveforms) from the
dictionary D . The iterative procedure is illustrated in Figure 2.20. Taking into account
(2.110) we can see that the representation conserves the energy of the signal:

n = 0 R n x , g γ n

∞

2

2

x = (2.113)

The MP decomposition can be considered as an extension of the atomic decompo-
sitions offered by STFT or CWT. The main advantage of the MP paradigm is the
relaxation of constraint between the frequency band and frequency resolution. The
MP algorithm performs decomposition in an extremely redundant set of functions,
which results in a very flexible parametrization of the signal structures.
In principal the dictionary D can be any set of functions. In practical implemen-
tations (e.g., http://eeg.pl/mp ) the dictionary contains a base of Dirac deltas, a
base of sinusoids and a set of Gabor functions:

2

(

t − u

)

e − π g γ ( t )= K ( γ ) sin ( 2π f ( t − u )+ φ ) (2.114)

σ

with K are the pa-

rameters of functions in the dictionary ( u -time translation, f -frequency, σ - time

( γ ) normalization factor such that || g γ || = 1, and γ = { u , f , σ , φ }