Blind Identification of Non-minimum Phase FIR Systems Using the Kurtosis

In this paper we present a method to estimate nonminimum phase finite impulse response (FIR) system, using Moving-Average (MA) model. It is based on maximum kurtosis properties. The spectrally equivalent minimum phase (SEMP) filter is estimated from second order statistics of the output system. The kurtosis allow us first to localise the zeros of the associated transfer_ function from the zeros of its SEMP filter, then to estimate the true order of the MA model. On simulated seismic data we compare the proposed method to Gianakis and Mendel's algorithm and Tugnait's algorithm. The results obtained confm the robustness of the method to hard conditions of process. INTRODUCTION The classical approach to solve the problem of blind identification of linear time invariant system only uses second order statistics (autocomelation or spectrum). This approach does not provide a complete statistical description. It only allows to identify minimum phase, maximum phase or zero phase system. Recently Higher order statistics (HOS) than two (multicorrelation or polyspectrum) ($1) have received the attention of the statistics signal processing, and theory literature, for processing non-gaussian linear or non-linear processes. For gaussian processes all their HOS are identically zero. Furthermore, all odd order statistics are identically zero for processes with symmetric Probability Density Function (PDF), that is why we choose to use fourthorder statistics. The use of HOS in time domain using parametric approach based on AR, MA, or ARMA model, has provided different solutions to non-minimum phase blind identification problem (92). To identify finite impulse response (FIR) system, our purpose is to estimate using second order statistics, the spectrally equivalent minimum phase (SEMP) systcm, and using the maximum kurtosis properly to recover the true system (03). For given order of the MA, we compare the method lo Gianakis-Mendel's algorithm and Tugnait's algorithm. This comparison is made on simulated seismic dah, with hard condition : short data Icngth and high order of die MA (54.1). Using the same data we show the capacity of the method to estimate the true order (94.2). 1) HIGH ORDER STATISTICS The description of HOS for random variables is essentially made using the cumulants. Let us take ( Xl , . . , X,, ) n-real valued random variable, their crosscumulants of order "m" can be defined from the Taylor series expansion of their second characteristic function by: