Rate of Convergence in Approximating the Spectral Factor of Regular Stochastic Sequences

Common methods for the calculation of the spectral factorization rely on an approximation of the given spectral density by a polynomial and a subsequent factorization of this polynomial. It is known that the regularity of the stochastic sequence determines the achievable approximation rate of its sp...

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Bibliographic Details
Published in:IEEE transactions on information theory 2009-12, Vol.55 (12), p.5674-5681
Main Authors: Boche, H., Pohl, V.
Format: Article
Language:English
Subjects:
Applied sciences
Approximation
Approximation error
Approximation methods
Convergence
Density
Error analysis
error bounds
Error correction
Exact sciences and technology
Factorization
Information theory
Information, signal and communications theory
Linear approximation
Linear equations
Mathematical analysis
Polynomials
Regularity
sampled data
Sampling methods
Sampling, quantization
Signal and communications theory
Spectra
spectral factorization
Stochastic models
Stochastic processes
stochastic sequences
Stochasticity
Telecommunications and information theory
Upper bound
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Summary:Common methods for the calculation of the spectral factorization rely on an approximation of the given spectral density by a polynomial and a subsequent factorization of this polynomial. It is known that the regularity of the stochastic sequence determines the achievable approximation rate of its spectrum. However, since the approximative polynomial should be factorized, it has to be positive. It is shown that this restriction on the approximation polynomial implies a limitation on the approximation rate for linear methods whereas for nonlinear methods the optimal approximation rate can still be achieved. This has also consequences for the rate of convergence of the spectral factor, which is investigated in the second part. There, a lower and an upper bound for the error in the spectral factor is derived, which shows the dependency on the approximation degree and on the regularity of the stochastic sequence. Finally, if the spectral density is given only on a finite set of sampling points, no linear approximation method exists such that the error in the spectral factor can be controlled by the approximation degree.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2009.2032808