Spectral Factorization of Time - Varying Covariance Functions BRIAN
نویسندگان
چکیده
Abstruct-The determination of the state-space equations of a time-varying finite-dimensional linear system with a prescribed output covariance matrix is considered when the system is excited by Gaussian whitenoise inputs. It is shown that a symmetric state covariance matrix provides the key link between the statespace equations of a system and the system output covariance matrix. Furthermore, such a matrix satisfies a linear matrix difforential equation if the state-space equations of the system are known, and a matrix Riccati equation if the output covariance matrix of the system is given. Existence results are given for the Riccati equation solution, and discussion of asymptotic solutions of the differential equations is also included.
منابع مشابه
Spectral factorization of time-varying covariance functions
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 covariancematrixprovidesthe key link between the statespace equationsof a system and the system output covariance matrix.Furthermore,such a matrixsatisfiesa lin...
متن کاملStationary discrete-time covariance factorization using Newton-Raphson iteration
The solution to the problem of factorization of the covariance function of a stationary, discrete-time process is obtained by using a Newton-Raphson procedure which converges quadratically in I1 provided the initial iterate is chosen suitably. The existence of a suitable initial iterate is guaranteed by an approximation result. An application to error localization in spectral factorization is s...
متن کاملOn Spectral Factorization and Riccati Equations for Time-Varying Systems in Discrete Time
It is known that positive operators Ω on a Hilbert space admit a factorization of the form Ω = W∗W, where W is an outer operator whose matrix representation is upper. As upper Hilbert space operators have an interpretation of transfer operators of linear time-varying systems in discrete time, this proves the existence of a spectral factorization for time-varying systems. In this paper, the abov...
متن کاملToeplitz and Hankel kernels for estimating time-varying spectra of discrete-time random processes
For a nonstationary random process, the dual-time correlation function and the dual frequency Loéve spectrum are complete theoretical descriptions of second-order behavior. That is, each may be used to synthesize the random process itself, according to the Cramér–Loève spectral representation. When suitably transformed on one of its two variables, each of these descriptions produces a time-vary...
متن کاملCovariance factorization via Newton-Raphson iteration
The solution of an integral equation arising in a covariance factorization problem is obtained by a Newton-Raphson 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.
متن کامل