| 1. |
- Friedlander, B, et al.
(författare)
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Extended Levinson and Chandrasekhar Equations for General Discrete-Time Linear Estimation Problems
- 1978
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Ingår i: IEEE Transactions on Automatic Control. - : IEEE Control Systems Society. - 0018-9286 .- 1558-2523. ; 23:4, s. 653-659
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Tidskriftsartikel (refereegranskat)abstract
- Recursive algorithrms for the solution of linear least-squares estimation problems have been based mainly on state-space models. It has been known, however, that recursive Levinson-Whittle-Wiggins-Robinson (LWR) algorithms exist for stationary time-series, using only input-output information (i.e, covariance matrices). By introducing a way of classifying stochastic processes in terms of an "index of nonstationarity" we derive extended LWR algorithms for nonstationary processes We show also how adding state-space structure to the covariance matrix allows us to specialize these general results to state-space type estimation algorithms. In particular, the Chandrasekhar equations are shown to be natural descendants of the extended LWR algorithm.
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| 2. |
- Friedlander, Benjamin, et al.
(författare)
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Lattice Implementation of the Recursive Maximum Likelihood Algorithm
- 1981
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Ingår i: Proceedings of the 20th IEEE Conference on Decision and Control including the Symposium on Adaptive Processes. - Linköping : Linköping University. ; , s. 1083-1084
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A lattice implementation of the extended least-squares algorithm was recently developed. The convergence of that algorithm depends on a positive real condition, a fact which limits its usefulness in some applications. In this note the lattice equivalent of the recursive maximum likelihood algorithm is derived. This technique does not require the positive real condition and is suitable for modeling and prediction of general ARMA time-series.
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| 3. |
- Friedlander, B, et al.
(författare)
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Levinson and Chandrasekhar-Type Equations for a General Discrete-Time Linear Estimation Problem
- 1976
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Ingår i: Proceedings of the 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes. ; , s. 910-915
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Konferensbidrag (refereegranskat)abstract
- Recursive algorithms for the solution of linear least-squares estimation problems have been based mainly on state-space models. It has been know, however, that such algorithms exist for stationary time-series, using input-output descriptions (e.g., covariance matrices). We introduce a way of classifying stochastic processes in terms of their "distance" from stationarity that leads to a derivation of an efficient Levinson-type algorithm for arbitrary (nonstationary) processes. By adding structure to the covariance matrix, these general results specialize to state-space type estimation algorithms. In particular, the Chandrasekhar equations are shown to be the natural descendants of the Levinson algorithm.
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| 4. |
- Friedlander, Benjaming, et al.
(författare)
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New Inversion Formulas for Matrices Classified in Terms of their Distance from Toeplitz Matrices
- 1979
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Ingår i: Linear Algebra and its Applications. - : Elsevier. - 0024-3795 .- 1873-1856. ; 27, s. 31-60
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Tidskriftsartikel (refereegranskat)abstract
- The problem of solving linear equations, or equivalently of inverting matrices, arises in many fields. Efficient recursive algorithms for finding the inverses of Toeplitz or displacement-type matrices have been known for some time. By introducting a way of characterizing matrices in terms of their “distance” from being Toeplitz, a natural extension of these algorithms is obtained. Several new inversion formulas for the representation of the inverse of non-Toeplitz matrices are also presented.
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| 5. |
- Kailath, Thomas, et al.
(författare)
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Fast Time-Invariant Implementations of Gaussian Signal Detectors
- 1978
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Ingår i: IEEE Transactions on Information Theory. - : IEEE Information Theory Society. - 0018-9448 .- 1557-9654. ; 24:4, s. 469-477
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Tidskriftsartikel (refereegranskat)abstract
- A new implementation is presented for the optimum likelihood ratio detector for stationary Gaussian signals in white Gaussian noise that uses only two causal time-invariant filters. This solution also has the advantage that fast algorithms based on the Levinson and Chandrasekhar equations can he used for the determination of these time-invariant filters. By using a notion of "closeness to stationarity,' there is a natural extension of the above results for nonstationary signal processes.
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| 6. |
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| 7. |
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| 8. |
- Kailath, Thomas, et al.
(författare)
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Recursive Input-Output and State Space Solutions for Continuous-Time Linear Estimation Problems
- 1981
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Rapport (övrigt vetenskapligt/konstnärligt)abstract
- A general linear least-squares estimation problem is considered. It is shown how the optimal filters for filtering and smoothing can be recursively and efficiently calculated under certain structural assumptions about the covariance functions involved. This structure is related to an index known as the displacement rank, which is a measure of non-Toeplitzness of a covariance kernel. When a state space type structure is added, it is shown how the Chandrasekhar equations for determining the gain of the Kalman-Bucy filter can be derived directly from the covariance function information; thus we are able to imbed this class of state-space problems into a general input-output framework.
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| 9. |
- Kailath, Thomas, et al.
(författare)
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Recursive Input-Output and State Space Solutions for Continuous-Time Linear Estimation Problems
- 1983
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Ingår i: IEEE Transactions on Automatic Control. - : IEEE Control Systems Society. - 0018-9286 .- 1558-2523. ; 28:9, s. 897-906
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Tidskriftsartikel (refereegranskat)abstract
- A general linear least-squares estimation problem is considered. It is shown how the optimal filters for filtering and smoothing can be recursively and efficiently calculated under certain structural assumptions about the covariance functions involved. This structure is related to an index known as the displacement rank, which is a measure of non-Toeplitzness of a covariance kernel. When a state space type structure is added, it is shown how the Chandrasekhar equations for determining the gain of the Kalman-Bucy filter can be derived directly from the covariance function information; thus we are able to imbed this class of state-space problems into a general input-output framework.
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| 10. |
- Kailath, Thomas, et al.
(författare)
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The Factorization and Representation of Operators in the Algebra Generated by Toeplitz Operators
- 1979
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Ingår i: SIAM Journal on Applied Mathematics. - : SIAM. - 0036-1399 .- 1095-712X. ; 37:3, s. 467-484
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we study the factorization and the representation of Fredholm operators belonging to the algebra $\mathcal{R}$ generated by inversion and composition of Toeplitz integral operators. The operators in $\mathcal{R}$ have the interesting property of being close to Toeplitz (in a sense quantifiable by an integer-valued index $\alpha $) and, at the same time, of being dense in the space of arbitrary kernels. By using these properties, we derive a set of efficient algorithms (generalized fast-Cholesky and Levinson recursions) for the factorization and the inversion of arbitrary Fredholm operators. The computational burden of these algorithms depends on how close (as measured by the index $\alpha $) these operators are to being Toeplitz.We also obtain several important representation theorems for the decomposition of operators in $\mathcal{R}$ in terms of sums of products of lower times upper triangular Toeplitz operators. These results can be used to approximate operators corresponding to noncausal and time-variant systems in terms of operators representing causal and anticausal time-invariant systems, a property that has a large number of potential applications in signal processing problems.
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