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W2188013718 abstract "We give a constructive algorithm for spectral factor- ization of 2-D parahermitian positive definite polync- mial matrices. Such an algorithm is expected to be useful in many problems in multidimensional signal and system theory. While earlier solutions to the problem in mathematical literature deals with exitence type re- sults, our algorithm is also believed to provide an ele- mentary proof of the most nontrivial fact that a factor- ization of a specific type can be always carried out. An example illustrating the main aspects of the algorithm is included. Introduction: While many problems in mul- tidimensional system theory are known to suffer from deficiencies inherent to the mathematics of multidimen- sional systems, the lack of factorizability-more specifi- cally spectral facorability of parahermitian positive def- inite polynomials or matrix polynomials (l)-is a major bottleneck, which makes many developments in multidi- mensional signals and systems infeasible (2). Problems , in which such spectral factoriztion plays a role is as follows: (1) Kalman and Wiener filtering (2) maximum entropy spectral estimation and 2-D moment problem (3) multidimensional wavelet theory or subband coding of multidiemnsional signals (4) optimal and H, con- trol (5) stochastic realization theory and associated is- sues of (recursive least square) parameter estimation etc. (6) network synthesis (or equivalently, some prob- lems of lossless inverse scattering theory) (7) hypersta- bility theory and applications in adaptive control and signal processing (8) design of transducer power gain in lumped distributed networks Very recently, it has been recognized (6, 51 that al- though multidimensional polynomials are not factoriza- ble, 2-D parahermitian positive definite (matrix) poly- nomials can, in fact, be expressed as the product of a rational matrix holomophic in the right half plane (or unit disc in the discrete case) with its own paraconju- gate. The result largely rests on the classical work of Hilbert (3) showing that positive definite 2-D polynomi- als can be expressed as the ratios of sums of squares of polynomials. This result was later extended by Landau (4) to show that 2-D polynomials positive definite poly- nomials can be expressed as the sum of squares (not necessarily two-in fact, at most four) of polynomials in one variable, the coefficients of which are rational func- tion in the other variable. 1." @default.
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W2188013718 date "1992-01-01" @default.
W2188013718 modified "2023-09-27" @default.
W2188013718 title "A CONSTRUCTIVE ALGORITHM FOR SPECTRAL FACTORIZATION OF PARAHERMITIAN" @default.
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