Download Lectures on Wiener and Kalman Filtering by T. Kailath PDF
By T. Kailath
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Sample text
The method of Bode-Shannon and Zadeh-Ragazzini was based on two observations: i) the Wiener-Hopf equation is trivial to solve if the observed process is white noise, ii) for stationary lumped processes it is easy to transform, without Ioss of intormation, an arbitrary observed process into a white-noise process. Let us consider these points separately. i) Estimation Basedon White-Noise Observations Suppose that we are to estimate x( t + A ), given observations of a white with noise process {v(r),- oo < r < t} Rv(t) = o(t) , Sv(f) = 1 The corresponding Wiener-Hopf equation for the optimum estimating filter is I g(r)Rv(t- r) dr, (1 0) t;;;;.
Through the linear filter 1/S~ (s), sxv = sxy(s);s;(- s) = Sxy(s)/Sy(s) . 53 y( . ) i<. v( . ) 1 -- + 'A) {S XV (s)es~} + s+(s) y Canonical Whitening Filter Optimum Filter Based on White Noise Observations Optimum Filter Fig. 4: The optimum filter by the innovations method Therefore, 1 H(s) = s;(s). + which is the same as the formula (4) we obtained by the Wiener-Hopf method in Sec. 3 An Alternative More Time-Domain Approach The innovations method of the previous section can also be described in more purely time-domain terms.
4. Repeat the previous problern for a process with covariance R(t, s) ~ min (t, s) =) t, t ~s s, t >s . 5. 1. Innovations. Given a sequence of random variables {y0 , y1' ... e. orthogonal random variables. (Hint: recall GramSchmidt) b) Show that if E e; > 0, all i, then the vectors Y~ =, Yn, · .. , Y1' y0 , E'n = , e n , ... , e 0_ ' can be related, for all n, by a nonsingular triangular matrix, say yn = ~nEn. c) If JC n is the linear operation that yields Y0 from Y n, sl- ,w that d) Let Rye(i,j) = E y(i) e'(j) and so on.