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Estimation of 1/f noise. (English) Zbl 0905.94009
Summary: Several models have emerged for describing 1/f γ noise processes. Based on these, various techniques for estimating the properties of such processes have been developed. This paper provides theoretical analysis of a new wavelet-based approach which has the advantages of having low computational complexity and being able to handle the case where the 1/f γ noise might be embedded in a further white-noise process. However, the analysis conducted here shows that these advantages are balanced by the fact that the wavelet-based scheme is only consistent for spectral exponents γ in the range γ(0,1). This is in contradiction to the results suggested in previous empirical studies. When γ(0,1) this paper also establishes that wavelet-based maximum-likelihood methods are asymptotically Gaussian and efficient. Finally, the asymptotic rate of mean-square convergence of the parameter estimates is established and is shown to slow as γ approaches one. Combined with a survey of non-wavelet-based methods, these new results give a perspective on the various tradeoffs to be considered when modeling and estimating 1/f γ noise processes.
MSC:
94A12Signal theory (characterization, reconstruction, filtering, etc.)
93E10Estimation and detection in stochastic control
42C15General harmonic expansions, frames
62M05Markov processes: estimation