TAILIEUCHUNG - Adaptive Filtering Part 3

Tham khảo tài liệu 'adaptive filtering part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 50 Adaptive Filtering while a stochastic random process is a rule or function that assigns a time function to every outcome of a random experiment. The elements of a stochastic process X n z for different value of time-index n are in general complex-valued random variable that are characterized by their probability distribution functions. A stochastic random process is called stationary in the strict sense if all of its single and joint distribution function is independent of a shift in the time origin. In this subsection we will be reviewing some useful definitions in stochastic process Stochastic Average mx E xw 8 where E is the expected value of X n . Autocorrelation function for a stochastic process X n x n rn E xwxm 9 where x denotes the complex conjugate of x and the symmetry properties of correlation function is k fxx k 10 Furthermore a stochastic random process is called stationary in the wide sense if mx and 0xx n m are independent of a shift in the time origin for any k m and n therefore mx n mx n k constant for all n 11 and ộxx n m ộxx n k m k 12 which means ộxx n m depends only on the difference n - m or k E fx x -k 13 Special case xx 0 E lxw mean square ofx X- 14 The Z-transform of ộxx k is given by TO m Z ộxx k Z 15 k -TO where it can be easily shown that m Z f-1-1 16 xx xx I Z I 51 The Ultra High Speed LMS Algorithm Implemented on Parallel Architecture Suitable for Multidimensional Adaptive Filtering Equation 16 implies that if xx Z is a rational function of z then its poles and zeros must occur in complex-conjugate reciprocal pairs. Moreover the points that belong to the region of convergence of xx Z also occur in complex-conjugate pairs which suggest that the 1 region of convergence of xx Z must be of the form a z j- which will cover the unit H circle z 1. By assuming that xx Z is convergent on the unit circle s contour C and by letting z eiw then 1 n L i í xx e dw 17 2n - -n and if mx 0 we will have E lxw I 2 í xx e w dw 18 - -1 -n .

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