Blind Deconvolution of EEG Signals Using the Stochastic Calculus A. Abutaleb, Aya F. Ahmed, Khaled S. Sayed Systems and Biomedical Engineering Department, Faculty of Engineering, Cairo University, Giza, Egypt. ataleb@mcit.gov.eg, aya_fawzy@eng.cu.edu.eg, ksayed@eng.cu.edu.eg Abstract—a new tool, in the blind deconvolution, for the estimation of both the source signals and the unknown channel dynamics has been developed. The framework for this methodology is based on a multi-channel blind deconvolution technique that has been reformulated to use Stochastic Calculus. The convolution processes is modeled as Finite Impulse Response (FIR) filters with unknown coefficients. Assuming that one of the FIR filter coefficients is time-varying, we have been able to get accurate estimation results for the source signals, even though the filter order is unknown. The time-varying filter coefficient was assumed to be a stochastic process. A stochastic differential equation (SDE), with some unknown parameters, was developed that described its evolution over time. The SDE parameters have been estimated using methods in stochastic calculus. The method was applied to the problem of two chatting persons and the problem of EEG contaminated by EOG. Comparisons to existing methods are also reported. Index Term—EEG, EOG, Blind Deconvolution, Convolutive BSS, Ornstein-Uhlenbeck process, Ito Calculus I. INT RODUCT ION: Blind deconvolution (BD) is very active research domains of the signal processing community. The application range of BD includes, but not limited to, EEG analysis [1], EKG analysis [2], [3], EMG analysis [4] and many more. In BD one is interested in the estimation of n unknown signals, given just a set of filtered mixtures observed at m sensors (in this study m>n). The term "blind" refers to our incomplete knowledge of the mixing operator. The BD has been extensively studied and a number of efficient algorithms have been developed [5], [6]- [8]. One of the most popular and effective techniques is to assume a finite impulse response filter (FIR) model for the modulating channels/paths and to estimate the coefficients of this model. Through the inversion of the FIR filter we get the original source signals. A major drawback of this model and the others is that if the filter order is incorrect, the estimated source signals are far from the true values [9]. This report is an extension to our previous work presented in [10]. The sources are now more than one (specifically and for the sake of simplicity we present the case of two sources and three measurements). The proposed method does not assume that the sources are independent. This is a big advantage over existing methods. It is also assumed that FIR filters are adequate to describe the channels though their orders and values are unknown. In this report, we suggest making one of the FIR filter parameters changing over time. This way, the ambiguity in the filter order is compensated by the time variations of one of the parameters. The time-varying parameter is estimated through the stochastic calculus [11]. In Sec. 2, we describe the blind deconvolution problem, and we propose a method based on regression analysis. In Sec. 3, we modify this method by assuming that one of the filter parameters is following what is known as Ornstein-Uhlenbeck (OU) process. In Sec. 4, the estimated sources are given, using the proposed method and conventional methods, for the case of two mixed voices and EEG corrupted signal. The same approach could be used in other signals as well. Finally in Sec. 5, we provide summary and conclusions. II. PROBLEM FORMULATION: For exposition purposes assume that we have two unknown sources () () and three measurements () () (). The measurements are the convolution of the sources with unknown linear time invariant finite impulse response (FIR) filters. This could be represented by the equation: ) ( ) ( ) ( ) ( ) ( * ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 3 2 1 2 1 32 31 22 21 12 11 3 2 1 t t t t u t u t h t h t h t h t h t h t y t y t y (1) Where "*" is the convolution operation, ()is the ith error or noise that is assumed to be white and Gaussian. The objective is to find an estimate of the source signals () and () . Equation (1) could be written in the Z domain as: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 3 2 1 2 1 32 31 22 21 12 11 3 2 1 z z z z U z U z h z h z h z h z h z h z Y z Y z Y (2) After some manipulations we get: z h z h z h z h z Y z h z Y z h z U z h z h z h z h z Y z h z Y z h ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 32 11 12 31 3 11 1 31 2 22 11 12 21 2 11 1 21 i.e . ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( 3 11 1 31 22 11 12 21 2 11 1 21 32 11 12 31 z Y z h z Y z h z h z h z h z h z Y z h z Y z h z h z h z h z h (3) Similar equations could be obtained for the source signals