The Influence of Source Overlapping on Correlation Matrix and Activity Index Rok Isteni, Damjan Zazula Faculty of Electrical Engineering and Computer Science University of Maribor Maribor, Slovenia rok.istenic@uni-mb.si, zazula@uni-mb.si Abstract— In this paper, the influence of overlapping of pulse signal sources on their correlation matrix and activity index is studied. The activity index is defined as a Mahalanobis distance of signal observations. Influences of source overlapping on activity index were simulated for a different number of overlapping sources and degrees of their overlapping. The findings lead to an improved model of activity index, which can support a more reliable estimation of the number of active sources in convolutive mixtures of pulse sources. Keywords— convolutive signal mixtures, correlation matrix of sources, activity index, source overlapping. I. INTRODUCTION One of the challenging goals of the research in telecommunications [1], [2], speech signal processing [3], [4], and biomedical applications [5] is the estimation of the number of active sources in signal observations. Biomedical signals emerge in most cases from sparse pulse sources (electromyography–EMG, electroencephalography–EEG) [5]. A method for the estimation of the number of sources in convolutive mixtures of pulse sources was introduced in [6]. It is based on the Mahalanobis distance of signal observations, also known as activity index. Using a statistical model of activity index [7], the estimation of the number of sources has proved possible, but the approach becomes error-prone when the amount of overlapped sources increases. Therefore we paid special attention to this problem in order to improve the estimation model. A detailed analysis of source overlapping is presented in this paper. The structure of the paper is as follows: in the following section data model and methods are presented, Section 3 describes the simulation study and results, while the last section discusses and concludes the paper. II. DATA MODEL AND METHODS A. Activity Index As mentioned in the introduction, we tackled sparse pulse signals that are common in biomedical applications. Sparse pulse signals are pulse driven, i.e., sources are in the form of pulse trains (trains of Dirac pulses). We consider the convolutive signal mixtures, such as in [3]-[7]. Suppose N source signals s 1 (n),…,s N (n) are convolved by impulse responses of system channels and observed at M sensors, producing M observations x 1 (n),…,x M (n). In vector notation, x(n)=[x 1 (n),…, x M (n)] T stands for the vector of M observations and s(n)=[s 1 (n),…, s N (n)] T for the vector of N sources. Observations x(n) can be expressed in matrix form as () () , n n s H x ∗ = (1) where H stands for a matrix of system-channel impulse responses and operator * stands for convolution. If the system (1) is overdetermined, a positive integer K exists that satisfies M(K+1) > N(L+K), where K is known as an extension factor and stands for the number of shifted replicas of original observations (see [5] for details). A measure of source activity, called activity index, is obtained by multiplying the extended observations () n x and the inverse of their sample correlation matrix R x -1 that was estimated from extended observations () () () . 1 n n n I x T A x R x - = (2) B. Overlapping of Sources Observing multichannel linear systems, source overlapping causes that channel responses overlap correspondingly in the output signal mixtures. For the reason of the analysis of such overlapping, we observed two different levels: the level of source pulse trains and the level of output signal observations or activity index. The overlapping rate at the level of pulse trains is always smaller, because for each source activation only one Dirac pulse exists, while at activity index level, the lengths of the channel impulse responses are taken into account (which are usually longer than one sample), so that the rate of overlapping is higher. In practice it is almost impossible that multiple active sources would never overlap, this is why the source overlapping is one of the major problems related to multichannel signal processing. Define the overlapping of at least two sources if the channel responses they trigger (or their activity index) overlap in time by any number of samples. At the same time, the overlapping rate defines the number of overlapping samples divided by the number of all samples in observations. To designate the amount 2009 International Conference on Signals, Circuits and Systems -1- 978-1-4244-4398-7/09/$25.00 ©2009 IEEE