ISSN:
1572-9265
Keywords:
Least squares estimations
;
Toeplitz matrix
;
circulant matrix
;
preconditioned conjugate gradient method
;
signal prediction
;
linear prediction
;
covariance matrix
;
windowing methods
;
finite impulse response (FIR) system identification
;
65F10
;
65F15
;
43E10
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
Notes:
Abstract Least squares estimations have been used extensively in many applications, e.g. system identification and signal prediction. When the stochastic process is stationary, the least squares estimators can be found by solving a Toeplitz or near-Toeplitz matrix system depending on the knowledge of the data statistics. In this paper, we employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Our proposed circulant preconditioners are derived from the spectral property of the given stationary process. In the case where the spectral density functions(θ) of the process is known, we prove that ifs(θ) is a positive continuous function, then the spectrum of the preconditioned system will be clustered around 1 and the method converges superlinearly. However, if the statistics of the process is unknown, then we prove that with probability 1, the spectrum of the preconditioned system is still clustered around 1 provided that large data samples are taken. For finite impulse response (FIR) system identification problems, our numerical results show that annth order least squares estimator can usually be obtained inO(n logn) operations whenO(n) data samples are used. Finally, we remark that our algorithm can be modified to suit the applications of recursive least squares computations with the proper use of sliding window method arising in signal processing applications.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02142678
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