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Estimation of $1/f$ noise. (English) Zbl 0905.94009
Summary: Several models have emerged for describing $1/{f}^{\gamma }$ 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 $1/{f}^{\gamma }$ 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 $\gamma$ in the range $\gamma \in \left(0,1\right)$. This is in contradiction to the results suggested in previous empirical studies. When $\gamma \in \left(0,1\right)$ 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 $\gamma$ 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 $1/{f}^{\gamma }$ noise processes.
##### MSC:
 94A12 Signal theory (characterization, reconstruction, filtering, etc.) 93E10 Estimation and detection in stochastic control 42C15 General harmonic expansions, frames 62M05 Markov processes: estimation