A new vision on the averaging technique for the estimation of non-stationary Brainstem Auditory-Evoked Potentials: Application of a metaheuristic method
Introduction
The classical averaging technique has been widely used in signal processing applications for the estimation of stationary signals corrupted by a very heavy noise. This popular method became very appropriate for solving some typical biomedical problems, generally for extracting evoked potentials (EP) and especially for Brainstem Auditory-Evoked Potentials (BAEPs). Unfortunately, with increasing experiences, it has been quickly noticed that, in some cases, the averaging method absolutely failed. The reasons are well known: basically, it is because the averaging technique is based on some principles and performs well only if some important conditions are satisfied. Among these conditions, the averaging technique considers that the signal to be estimated is repeated many times without any modification. This first condition requires that the signal to be estimated should be stationary. The second condition (i.e. the hypothesis) concerns the noise which corrupts the desired signal. In fact, this noise is considered as a zero-mean stationary process. In the practical cases, it is clear that such a hypothesis is not always satisfied. If we consider a typical biomedical engineering application, namely the estimation of BAEP, it is obvious that in this case extracting the desired signal from the ongoing electroencephalogram (EEG) requires hundreds of responses. The Experimentation has shown that the signal to noise ratio can be as low as () in some cases. So, many papers in the literature have been devoted to solve this problem, particularly by taking into account the EEG behaviour. Many variants of the classical averaging have been proposed in the literature. For example, one can mention the weighted averaging, that consists in multiplying each response by an optimal coefficient [1], latency-dependent averaging [2], a posteriori filtering [3], [4], [5] and deconvolution methods [6], [7], [8]. However, such techniques are quite limited since the main advantage consists in reducing the number of responses required for extracting the desired signal. Unfortunately, like the classical averaging method, these approaches were not sensitive to BAEP variations and suitable methods were those attempting to estimate BAEPs from a single trial, using adaptive filters [9], [10], [11], [12], [13], [14]. These techniques are very efficient but they can behave poorly in case of low SNR or in the case where the noise is extremely correlated with the desired signal. The denoising process in such situations becomes very complicated, if not impossible.
The idea that we would like to highlight in this paper is the fact that since the averaging technique is not efficient if one or more conditions are not satisfied, one can modify the process in order for the conditions to be satisfied. In other words, in the case where the signal to be estimated varies from one response to another one according to a specific known model, we can show that application of an inverse model, optimally fitted, allows to estimate the averaged signal as if it was extracted from a stationary process. This technique will be described in the following three sections. In Section 2, we present the basic idea behind the technique which can be used to solve some typical problems. In Section 3, an example related to BAEPs is considered. In Section 4, some special cases are discussed. Finally, a conclusion is given in the last section.
Section snippets
Averaging technique
Let be the sampled signal recorded in response to the th acquisition, wherewhere N represents the number of samples for a given signal, is the desired signal to be estimated, corresponding to the ith acquisition.
The averaged signal, denoted by can be expressed by the following equation:where M is the number of the averaged responses.
If the noise is considered as a zero-mean
An example of application to real signals
In this section, we try to apply the simulated annealing method, described above, to solve a special case of the problem presented in Eq. (5). We consider here that each response is randomly delayed from one response to another one [15]. This case can be observed in some pathological recorded signals, especially for endocochlear patients. Each filter corresponding to a specific response could be considered as a linear one having a non-stationary phase and producing a random temporal delay.
Special alternative cases
In this section, two different cases are discussed. The first one concerns the particular situation where some of the BAEPs are missing, and the second case consists in including a priori information to improve the estimation process.
When some BAEP responses are missing
If we assume that during the recording process, K BAEP responses are missing (which may occur for a given abnormal case), the energy corresponding to the averaged signal will be exactly the same as if one uses only (M–K) BAEPs
Conclusion
This paper shows that the classical averaging can still be used to solve some complex problems in relation to estimating BAEPs. The complexity is due to the non-stationarity of BAEPs and the presence of significant non-stationary noise. When the noise is recorded from an EEG, in practice, the energy of the averaged signal is low when a large number of accumulations are used. Using this property the study has been concentrated only on the problem of the non-stationarity of BAEPs. It has been
Amine Nait-Ali was born in 1972 in Oran (Algeria); he received his B.Sc degree in Electrical Engineering at the University of Sciences and Technology of Oran, then his DEA degree in Automatic and Signal Processing at University Paris 11 and his Ph.D. degree in Biomedical Engineering from the University Paris 12 in 1998. Since 1999 he is an Associate Professor at the same university. His research interests are focused on physiological signal processing, processes modelling and medical signal and
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Amine Nait-Ali was born in 1972 in Oran (Algeria); he received his B.Sc degree in Electrical Engineering at the University of Sciences and Technology of Oran, then his DEA degree in Automatic and Signal Processing at University Paris 11 and his Ph.D. degree in Biomedical Engineering from the University Paris 12 in 1998. Since 1999 he is an Associate Professor at the same university. His research interests are focused on physiological signal processing, processes modelling and medical signal and image compression.
Patrick Siarry was born in France in 1952. He received his Ph.D. degree from the University Paris 6, in 1986 and the Ability to Manage Research from the University Paris 11, in 1994.
He was first involved in the development of analog and digital models of nuclear power plants at Electricité de France (E.D.F.). Since 1995 he is a professor in automatics and informatics. His main research interests are the adaptation of new stochastic global optimization heuristics to continuous variable problems and their application to various engineering fields. He is also interested in the fitting of process models to experimental data, the learning of fuzzy rule bases and of neural networks.