On the influence of sampling and observation times on estimation of the bandwidth parameter of a Gauss-Markov process

• Published in: IEEE transactions on signal processing Vol. 54; no. 1; pp. 127 - 137
• Main Authors: Li Wen, Sherman, P.J.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-01-2006; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Active noise cancellation; Aerospace engineering; Applied sciences; autoregressive model; Bandwidth; bandwidth (BW) parameter; Bias; continuous-time stochastic process; Detection, estimation, filtering, equalization, prediction; Estimators; Exact sciences and technology; Frequency estimation; Gauss-Markov (GM) process; Gaussian processes; Guidelines; Information, signal and communications theory; Mathematical analysis; Noise levels; Parameter estimation; prediction; Sampling; Sampling methods; Signal and communications theory; Signal processing algorithms; Signal sampling; Signal, noise; Statistics; Studies; Taylor series; Telecommunications and information theory; Variance; Markov process; Parameter estimation; bandwidth (BW) parameter; Gaussian process; Noise reduction; Prediction; Active noise cancellation; Gauss-Markov (GM) process; Signal processing; continuous-time stochastic process; Autoregressive model; Sampling
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2005.861089

The statistical problem of estimating the bandwidth parameter of a Gauss-Markov (GM) process from a realization of fixed and finite duration T at selectable sampling interval /spl Delta/ is addressed in this paper. As the observation time, T, is fixed and finite, the variance of estimated autocorrelation and continuous-time parameter does not vanish as /spl Delta/ approaches 0. This necessitates a second-order Taylor expansion in deriving the parameter estimator bias and variance. The second-order Taylor expansion produces better bias and variance results than a first-order one does. The distribution of the estimator is also discussed. According to the gradient change of the variance, a key result is that three sample size regions, which are termed finite, large, and very large, corresponding to substantial, gradual, and very slight decrease in the variance of the parameter estimator, respectively, are quantified. In terms of analysis BW, the three regions are (-23,-35),(-35,-55), and (-55,-/spl infin/) dB. The characterization of the tradeoff between the variance decrease and sampling rate results in a practical guideline for choosing sampling rate. These results are applied to the prediction problems of a time invariant GM process to show their value.