Application of a Resolvent Identity to a Linear Smoothing Problem

Kailath, T.

doi:10.1137/0307006

Cited by 34 publications

(3 citation statements)

References 4 publications

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“…(Also see [22].) Here the differentiation is equivalent to changing the order of integration in a stochastic integral which is permitted since the integrand is bounded.…”

Section: ~(Tlr) = :9(t]t) + F~ Gs(t S)[dz(s)-~(s]s)ds]mentioning

confidence: 99%

On Fredholm integral equations, Toeplitz equations and Kalman-Bucy filtering

Lindquist

1975

Appl Math Optim

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In this paper we consider some new algorithms for computing the Kalman-Bucy gain in stationary systems requiring a number of equations of order n (rather than r/2) whenever the order n of the system is much larger than the dimension of the output. These equations were independently obtained by Kailath and Lindquist in continuous and discrete time respectively. We briefly discuss the relations with some recent related results due to Casti, Kalaba & Murthy and Rissanen. Some of the reasons for these reductions are inherent in the properties of general stationary processes, and therefore a considerable portion of the paper is devoted to exploring the connections with some previous work by Levinson, Whittle and Wiggins & Robinson, and also with the Szeg6 theory of polynomials orthogonal on the unit circle and some continuous analogs due to Krein. We demonstrate that the Bellman-Krein formula is the fundamental relation in continuous time, the trick being to introduce a "reversed time" counterpart of the weighting function (Fredholm resolvent). This is suggested by the "forward and backward innovation" approach in a previous paper by the author, the essential relations of which we reformulate in terms of Fredholm integral equations (in continuous time) and Toeplitz equations (in discrete time). Therefore we also derive the discrete-time Bellman-Krein formulas of which there are actually two--one corresponding to the one-step predictor and one to the pure filter. In this way we shall be able to pin down the reasons for the striking discrepancies between the continuous-time and the discrete-time cases. Finally we clarify the relations between Levinson's equations and Chandrasekhar's X-and Y-functions.

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“…(Also see [22].) Here the differentiation is equivalent to changing the order of integration in a stochastic integral which is permitted since the integrand is bounded.…”

Section: ~(Tlr) = :9(t]t) + F~ Gs(t S)[dz(s)-~(s]s)ds]mentioning

confidence: 99%

On Fredholm integral equations, Toeplitz equations and Kalman-Bucy filtering

Lindquist

1975

Appl Math Optim

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“…Connections between the infinite-dimensional solution [30] of (1) and the finitedimensional solution are touched upon briefiy in an important paper of %hurnitzky [34]. That the solution of (1) is closely related to generalizing the Kalman filter tc make it a smoother is a point of view taken by Kailath [35], for whose valuable insights we are immensely grateful. Kailath also drew our attention to the existence of [36], which contains results indispensable in a consideration of singular problems; Geesey has made use of these results in [31].…”

Section: When R(t T)mentioning

confidence: 99%

Spectral factorization of time-varying covariance functions

Anderson

Moore

Loo

1969

IEEE Trans. Inform. Theory

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Abstract-The determinationof the state-spaceequationsof a time+uying tide-dimensional linear system with a prescribed outputcovariancematrixis consideredwhen the systemis excited by Gaussian white-noise inputs. It is shown that a symmetric state covariancematrix provides the key link between the statespace equations of a system and the system output covariance matrix.Furthermore, such a matrixsatisfiesa linearmatrixdifferentialequationif the state-spaceequationsof the systemareknown, and a matrixRiccatiequationif the outputcovariancematrixof the systemis given.Existenceresultsare givenfor the Riccatiequation solution,and discussionof asymptoticsolutionsof the differential equationsis also included.

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“…where the prime denotes transpose here and hereafter, and noise inputs, given that the output covariance of the system is 6 1 + K. The stochastic process overtones are also described in various papers by Kailath [2]- [4]. There are also applications to control theory, at least if one poses the problem of finding k from K with the order of the two factors on the right of (3) reversed, but we shall not dwell on these here.…”

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confidence: 99%

Covariance factorization via Newton-Raphson iteration

Anderson¹

1978

IEEE Trans. Inform. Theory

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FELLOW, IEEE ~b~t~~~~-~h~ solution of an integral equation arising in a eo.of finding the impulse response of a causal linear invertible variance factorization problem is obtained by a Newton-Raphson system with a vector of unit intensity independent white iteration that is almost always globally convergent. Interpretations of the iterates are given, and the result is shown to specialize to known algorithms when the covariance is stationary with a rational Fourier transform.

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Application of a Resolvent Identity to a Linear Smoothing Problem

Cited by 34 publications

References 4 publications

On Fredholm integral equations, Toeplitz equations and Kalman-Bucy filtering

On Fredholm integral equations, Toeplitz equations and Kalman-Bucy filtering

Spectral factorization of time-varying covariance functions

Covariance factorization via Newton-Raphson iteration

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