N. Weiner, Extrapolation, Interpolation and Smoothing of Stationary Time Series, John Wiley and Sons, New York., 1949.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the numbers j , # j : j = 1, 2, I , and n 0 = 2I when values x(m) m = 0, 1, N 1 (observations) are known. The Wiener Levinson method, formulated in terms of Szego polynomials, can briefly be described as follows (the original ideas of the method can be found in [12, 20]) An absolutely continuous measure #N is defined on [ #, #] or on the unit circle T through the transformation # z = e ) by the formula d#N d# 1 2# # # # N 1 x(m)e im# # # # . 1.2) Here N is an arbitrary natural number. The measure gives rise to a positive definite inner ....

N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, The Technology Press of the Massachusetts Institute of Technology and John Wiley and Sons, 1949. 12

....in H and in its terms defines an independent of the future class of the causal continuous operators H. In this case the problem is called causal. If H are finite dimensional and D is the identity matrix, the problem (1) 2) 3) is equivalent to the problem first being solved in [6] [9]. The causal operators become the upper triangular matrices and the solution is unique and can be efficiently represented in terms of the factorization of the correlation matrix R y of y by the so called Bode Shannon formula ( 1] Spectral factorization coincides with the well known Holetsky ....

.... of the correlation matrix R y of y by the so called Bode Shannon formula ( 1] Spectral factorization coincides with the well known Holetsky factorization of matrices in this case ( 2] The result was immediately applied in various branches of the filtering theory, for some applications see [9], 7] However, in many applications one estimates only the specific components of x or their combination, which is represented by the degenerate matrix D in the quality functional (3) and the solutions of the generalized finite dimensional problems can be found in [8] In this case the solution ....

Wiener, N. The Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications. New York, 1949.

....and . The functions g k (t) are the integrands of the (normalized) gamma function. Hence the name gamma model for structures that utilize tap variables of type to store the past of x(t) here denotes the convolution operator) Closely related are Laguerre functions, that were proposed by Norbert Wiener (1949) as a very convenient basis for decomposition of linear systems in a signal processing context. In fact, the functions g k (t) k = 1, K, can be easily written in terms of Laguerre functions. Consequently, the functions g k (t) are complete in . Gz( z 1 = Gz( Gz; Gz( z 1 ( ....

Wiener N., Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, New York, Wiley, 1949.

....that need to be answered in the optimization problem of (2) First, what should be the optimum value of the weights, and second, what should be the optimum value of the time scale l so that the mean square error J is minimized. The answer to the first problem was given by Wiener early this century [8]. It is embedded in the normal equations. Here we will be attacking a generalization of this equation due to the time scale parameter l involved in the Gamma basis functions. As shown in [7] l yields an extra degree of freedom that improves the approximation by rotating the manifold so that it ....

....the manifold so that it becomes as parallel as possible to the signal. Due to the parameter l, the problem is a parametric least squares approximation. Optimum Value of the Weights For a fixed l, the best choice of weights can be found from the condition which yields the normal equations [8] (4) P is the crosscorrelation vector while R is the Gram matrix whose (n,m) th element is g n (l,t)g m (l,t) 7] 5) For a given l, the mean squared error J is quadratic in the weights. Therefore W opt has a single solution and that solution depends on l. That s why this is called a ....

N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, with Engineering Applications. New York : Wiley, 1949.

....long, and computationally expensive. In this paper we present a new filter class that we call the generalized feedfoward structure that extends directly the results of Widrow s LMS [1] to a special class of ARMA systems. The adaptation process can still be described by the Wiener Hopf solution [2], the filters have trivial stability conditions, and can be adapted by a modified LMS algorithm with the same complexity as the LMS. Moreover, they decouple the length of the impulse response from the filter order. Therefore, the filter order can be set by the number of degrees of freedom of the ....

....[3] The choice of G(s) z 1 is simply one solution, out of a possibly large set of functions. The Discrete Time Gamma Filters We have investigated a set of basis functions g k that are complete in , and that has been proposed by Wiener as very convenient for decomposition of linear systems [2]. These functions are the integrands of the gamma integral, and can therefore be called the gamma functions (although they are also known as the Laguerre functions [2] In order to obtain a digital equivalent of the gamma functions, the change of variables is utilized (that corresponds to the ....

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Wiener N., Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, New York, Wiley , 1949.

....analysis. Two frameworks arise naturally for the study of deterministic spectral analysis: bounded power signals and bounded energy (l 2 ) signals. 12 4. 1 Bounded Power Signals There is a long historical tradition in a non stochastic theory of white noise, going as far back as Wiener (see [28]) who considered ergodicity properties to build a spectral theory of stationary signals devoid of probability. For disturbance rejection problems, this approach was followed in Zhou et al. 30] who considered the class of bounded power signals, defined by BP = x(t) r x ( lim N 1 1 2N ....

Wiener N., Extrapolation, Interpolation and Smoothing of Stationary Time Series, Wiley, New York, 1950.

....the name of the Bode Shannon formula. However, in many applications one estimates only the specific components of x or their combination, whichis represented by the degenerate matrix D in the quality functional (3) and the solutions of the generalized finite dimensional problems can be found in [10]. In this case the solution need not be unique and there are conditions on the degeneracy of D for which the Bode Shannon formula still defines an optimal filter. On the other hand, the infinite dimensional applica tions required the development of the filtering theory in Hilbert ( 3] and ....

....h is not bounded and the problem (24) is not solvable in H . 5 Bode Shannon representation of optimal filters First we will briefly review the results on the spectral factorization of the operators which we need in order to discuss the application of Bode Shannon theory (cf. 1] 4] 9] [10]) in our setting. The detailed discussion on various types of spectral factorization and separation of the operators can be found in [4] Let P T be a Hermitian resolution of the identity in H . As in the previous section we denote by H a t completion of H and by t a discrete linearly ordered ....

N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications, Academic Press, New York, 1949.

....states are also Gaussian, and the problem is solved by directly computing the conditional expectation and covariance of the state. The stationary linear ltering problem (that is, when F, G, and H are constant matrices, was essentially completely solved as a result of the pioneering work Wiener (Wiener, 1949), and the nonstationary case was solved by Kalman (Kalman, 1960) Kalman and Bucy (Kalman and Bucy, 1961) and Kalman (Kalman, 1963) resulting in the well known Kalman Bucy Filter. For nonlinear systems of the form (1) and (2) the ltering problem becomes much more dicult because the entire ....

Wiener, N. (1949). The Extrapolation, Interpolation and Smoothing of Stationary Time Series.

....et al. 86] for their inspiration, the actual techniques were well known to some other scientific communities, in particular the community of control theory. The starting point for estimation theory is commonly thought to be the independent developments of Kolmogorov [Kolmogorov 41] and Weiner [Weiner 49] Bucy [Bucy 59] showed that the method of calculating the optimal filter parameters by differential equation could also be applied to non stationary processes. Kalman [Kalman 60] published a recursive algorithm in the form of difference equations for recursive optimal estimation of linear ....

N. Weiner, Extrapolation, Interpolation and Smoothing of Stationary Time Series, John Wiley and Sons, New York., 1949.

....OF MEMORY DEPTH IN THE GAMMA FILTER Jyh Ming Kuo, Samel Celebi Computational Neuroengineering Lab. CSE 447 University of Florida, Gainesville FL32611, USA ABSTRACT Gamma filter is a special class of generalized feedforward filters where feedbacks are allowed only locally. We present the conditions for the selection of optimal parameters which are the weights and the memory depth of the filter. The conditions for these two set of ....

....the block diagram of the generalized feedforward filters. For the gamma filter G(s) is given by ( 1) which means that the system transfer function is ( 2) In the identification of a signal d(t) the optimal coefficients can be obtained by minimizing the output mean square error, i.e. solving for (3) Gs( l s l = Ys( w i Gs( i i 1 = K = J w i 0 = i 1. K = and J l 0 = where J is the mean square error defined as the average of (d(t) y(t) 2 over all t s. For a given l the optimal value for Figure 1 Generalized feedforward filter the weights is trivially obtained by solving ....

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N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, with Engineering Applications. New York : Wiley, 1949

....the convolution operation. The MSE between the stimulus and the estimate is given by E = D [m(t) Gamma m(t) 2 E = oe 2 m D m 2 (t) E Gamma 2 h m(t) m(t) i (6) The mathematical formulation and solution of the optimal linear estimation problem was originally carried out by Wiener (Wiener, 1949). Bialek and his colleagues introduced the reconstruction approach to theoretical neuroscience as a technique to quantify the amount of information single neurons can transmit about random inputs in the form of their spike outputs (Bialek et al. 1991; Bialek Rieke, 1992; Rieke et al. 1997) ....

Wiener N., 1949. Extrapolation, Interpolation and Smoothing of Stationary Time Series. MIT Press: Cambridge, Massachusetts.

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N. Wiener, Extrapolation, interpolation and smoothing of stationary time series, MIT press, 1949.

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N. Weiner, Extrapolation, Interpolation and Smoothing of Stationary Time Series, John Wiley and Sons, New York., 1949.

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N. Weiner, Extrapolation, Interpolation and Smoothing of Stationary Time Series, John Wiley and Sons, New York., 1949.

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N. Wiener. Extrapolation, Interpolation and Smoothing of Stationary Time Series with Engineering Applications. New York, Wiley, 1949.

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N. Wiener. The Extrapolation, Interpolation and Smoothing of Stationary Time Series. John Wiley & Sons, New York, N.Y., 1949.

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N. Wiener. Extrapolation, Interpolation and Smoothing of Stationary Time Series. M.I.T. Press, 1949.

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, Extrapolation, Interpolation and Smoothing of Stationary Time Series, M.I.T. Press, 1949.

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N. WIENER, Extrapolation, Interpolation and Smoothing of Stationary Time Series, M.I.T. Press, 1949.

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Wiener, N. (1942), Extrapolation, Interpolation and Smoothing of Stationary Time Series, MIT Press, Cambridge, MA.

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N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, M.I.T. Press, 1949.

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Wiener, N. The extrapolation, Interpolation and Smoothing of Stationary Time Series. Wiley, New York, 1949. Darryl R. Morrell is with the Department of Electrical Engineering, Arizona State University, Tempe, Arizona, USA. E-mail Morrell@asu.edu Wynn C. Stirling is with the Department of Electrical and Computer Engineering, Brigham Young University, Provo, Utah, USA. E-mail wynn@ee.byu.edu

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N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series. With Engineering Applications, J. Wiley, New York, 1949.

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N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series, New York, NY: John Wiley & Sons, 1949.

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WIENER, N. The extrapolation, Interpolation and Smoothing of Stationary Time Series. Wiley, New York, 1949. Darryl R. Morrell is with the Department of Electrical Engineering, Arizona State University, Tempe, Arizona, USA. E-mail Morrell@asu.edu Wynn C. Stirling is with the Department of Electrical and Computer Engineering, Brigham Young University, Provo, Utah, USA. E-mail wynn@ee.byu.edu

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