# New approach to multichannel linear prediction problems

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2002 IEEE International Conference on Acoustics Speech and Signal Processing ()
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#### Abstract

This paper provides a new approach to the multichannel linear prediction problem. To solve the consequently redefined problem in an order-recursive manner, a new multichannel Levinson algorithm is presented by virtue of the time-varying Wiener filter theory. The new algorithm is computationally efficient relative to the Levinson-Wiggins-Robinson algorithm and its variants since it avoids any matrix operations and needs scalar operations only. Furthermore, the new algorithm is optimum in that it produces the exact minimum prediction error power.

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# New approach to multichannel line...

Introduction AN 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. The situation gets rapidly worse with increasing complexity of the problem. (3) Important generalizations (e.g., growing-memory filters, nonstationary prediction) require new derivations, frequently of considerable difficulty to the nonspecialist. (4) The mathematics of the derivations are not transparent. Fundamental assumptions and their consequences tend to be obscured. This paper introduces a new look at this whole assemblage of problems, sidestepping the difficulties just mentioned. The following are the highlights of the paper: (5) Optimal Estimates and Orthogonal Projections. The Wiener problem is approached from the point of view of condi- tional distributions and expectations. In this way, basic facts of the Wiener theory are quickly obtained the scope of the results and the fundamental assumptions appear clearly. It is seen that all statistical calculations and results are based on first and second order averages no other statistical data are needed. Thus difficulty (4) is eliminated. This method is well known in probability theory (see pp. 75���78 and 148���155 of Doob [15] and pp. 455���464 of Lo��ve [16]) but has not yet been used extensively in engineering. (6) Models for Random Processes. Following, in particular, Bode and Shannon [3], arbitrary random signals are represented (up to second order average statistical properties) as the output of a linear dynamic system excited by independent or uncorrelated random signals (���white noise���). This is a standard trick in the engineering applications of the Wiener theory [2���7]. The approach taken here differs from the conventional one only in the way in which linear dynamic systems are described. We shall emphasize the concepts of state and state transition in other words, linear systems will be specified by systems of first-order difference (or differential) equations. This point of view is A New Approach to Linear Filtering and Prediction Problems1 The classical filtering and prediction problem is re-examined using the Bode- Shannon representation of random processes and the ���state transition��� method of analysis of dynamic systems. New results are: (1) The formulation and methods of solution of the problem apply without modifica- tion to stationary and nonstationary statistics and to growing-memory and infinite- memory filters. (2) A nonlinear difference (or differential) equation is derived for the covariance matrix of the optimal estimation error. From the solution of this equation the co- efficients of the difference (or differential) equation of the optimal linear filter are ob- tained without further calculations. (3) The filtering problem is shown to be the dual of the noise-free regulator problem. The new method developed here is applied to two well-known problems, confirming and extending earlier results. The discussion is largely self-contained and proceeds from first principles basic concepts of the theory of random processes are reviewed in the Appendix. R. E. KALMAN Research Institute for Advanced Study,2 Baltimore, Md. Transactions of the ASME���Journal of Basic Engineering, 82 (Series D): 35-45. Copyright �� 1960 by ASME

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