Application to biomedical signals - Practical Biomedical Signal Analysis Using MATLAB - page 146

way it was possible to find a filter extracting single ERP. However, the method re-
lied on average ERP in the identification of ARMA model, and as such was not free
from the assumptions pointed out in Sect. 4.1.7.1.1. A similar approach was pre-
sented in [von Spreckelsen and Bromm, 1988]. The method relies on the assumption
that the measured activity can be separated into its evoked and spontaneous parts.
A compound state-space model trying to incorporate the observable properties of
both parts is formulated on the basis of additivity of the two components. Within this
model, spontaneous activity is described as an autoregressive process, while the EP
is modeled by an impulse response of a parametrically described system. Based on
the state-space representation of the model, a Kalman filter for the observation of the
system's state can be designed which yields estimates for both activities.

4.1.7.2.3 Time-frequency parametric methods The discrete wavelet decompo-
sition (Sect. 2.4.2.2.3) is particularly suitable to parameterize the transients which are
time locked to the stimuli. The procedures aiming at extraction of single-trial ERP
based on discrete wavelet decomposition were proposed by [Bartnik et al., 1992].
In their approach the individual trials were first split into two parts: one which was
assumed to contain components related to the stimulus (ERP epoch) and the sec-
ond, which was assumed to contain only the spontaneous EEG activity (reference
epoch). Both datasets were decomposed by dyadic wavelet transform. The coeffi-
cients of the basis functions that comprise the detail functions were then analyzed by
regression and discriminate analysis to identify the coefficient best suited for distin-
guishing ERP from spontaneous EEG. Finally, the coefficients that most significantly
differentiated the ERP epochs from the reference epochs were selected and used for
reconstruction of the underlying single-trial ERPs. These reconstructed ERPs had
significantly improved signal to noise ratio. The authors verified the method on a
set of 40 auditory potentials. The correct classification to the group of ERP epochs
was obtained for 34 cases. It was demonstrated that by means of DWT and discrim-
inant analysis it is possible to construct a method capable of extracting components
strongly related to the ERP with minimal assumptions. Nothing has to be assumed
about the particular form of the components. A similar method based on wavelet
denoising, although less advanced in the selection of relevant wavelet coefficients,
was successfully applied to the study of the habituation effect of rat auditory evoked
potentials [Quian Quiroga and van Luijtelaar, 2002].
A further development of the idea of parametric decomposition application for
extracting single-trial ERPs, was proposed in the framework of multivariate match-
ing pursuit (Sect. 2.4.2.2.7 and 3.6.3). The flexible parametric approach offered by
MMP allows for explicit implementation of assumptions made about the ERPs. For
instance, one may assume that the ERP morphology is constant and that only the
amplitude varies across trials [Sieluzycki et al., 2009a]:

x i

( t )= a i s ( t )+ n i ( t ) (4.15)

where a i is the amplitude of ERP in trial i . This assumption is equivalent to imposing
constraints on the parametersγ

= { , , , }

u f σ φ (eq. 2.114) of the Gabor functions such