Kalman1960 - R E KALMAN Research Institute for Advanced...

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Introduction A N IMPORTANT class of theoretical and practical problems in communication and control is of a statistical nature. Such problems are: (i) Prediction of random signals; (ii) separa- tion of random signals from random noise; (iii) detection of signals of known form (pulses, sinusoids) in the presence of random noise. In his pioneering work, Wiener [1] 3 showed that problems (i) and (ii) lead to the so-called Wiener-Hopf integral equation; he also gave a method (spectral factorization) for the solution of this integral equation in the practically important special case of stationary statistics and rational spectra. Many extensions and generalizations followed Wiener’s basic work. Zadeh and Ragazzini solved the finite-memory case [2]. Concurrently and independently of Bode and Shannon [3], they also gave a simplified method [2] of solution. Booton discussed the nonstationary Wiener-Hopf equation [4]. These results are now in standard texts [5-6]. A somewhat different approach along these main lines has been given recently by Darlington [7]. For extensions to sampled signals, see, e.g., Franklin [8], Lees [9]. Another approach based on the eigenfunctions of the Wiener- Hopf equation (which applies also to nonstationary problems whereas the preceding methods in general don’t), has been pioneered by Davis [10] and applied by many others, e.g., Shinbrot [11], Blum [12], Pugachev [13], Solodovnikov [14]. In all these works, the objective is to obtain the specification of a linear dynamic system (Wiener filter) which accomplishes the prediction, separation, or detection of a random signal. 4 ——— 1 This research was supported in part by the U. S. Air Force Office of Scientific Research under Contract AF 49 (638)-382. 2 7212 Bellona Ave. 3 Numbers in brackets designate References at end of paper. 4 Of course, in general these tasks may be done better by nonlinear filters. At present, however, little or nothing is known about how to obtain (both theoretically and practically) these nonlinear filters. Contributed by the Instruments and Regulators Division and presented at the Instruments and Regulators Conference, March 29– Apri1 2, 1959, of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS. NOTE: Statements and opinions advanced in papers are to be understood as individual expressions of their authors and not those of the Society. Manuscript received at ASME Headquarters, February 24, 1959. Paper No. 59—IRD-11. Present methods for solving the Wiener problem are subject to a number of limitations which seriously curtail their practical usefulness: (1) The optimal filter is specified by its impulse response. It is not a simple task to synthesize the filter from such data. (2) Numerical determination of the optimal impulse response is often quite involved and poorly suited to machine computation.
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This note was uploaded on 02/06/2012 for the course ECON 101 taught by Professor Tan during the Spring '11 term at The Petroleum Institute.

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Kalman1960 - R E KALMAN Research Institute for Advanced...

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