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Iteration of mappings and fixed points in mixed spectrum analysis. (English) Zbl 0793.60044

Summary: Here we will analyze the mixed spectrum model \[ Z_ t=\sum^ p_{j=1}A_ j\cos(\omega_ jt+\varphi_ j)+\varepsilon_ t=X_ t+\varepsilon_ t, \quad\text{ for } t\in\mathbb{Z}, \] where \(p\) is not necessarily known and, for each \(j\in\{1,2,\dots,p\}\), \(A_ j\) is an unknown constant, \(\omega_ j\) is an unknown frequency with value in \((- \pi,\pi]\) and the phase \(\varphi_ j\) is a random variable uniformly distributed in \((-\pi,\pi]\) independent of each other and of the noise component. We assume that the noise component is Gaussian white noise such that \(\varepsilon_ t\sim{\mathcal N}(0,\sigma^ 2_ \varepsilon)\). Observe that the process \(\{Z_ t\}_{t\in\mathbb{Z}}\) is not Gaussian. Here we present a recursive method of updating parameters for estimating the frequencies \(\omega_ j\), \(1\leq j\leq p\). The cosines of the frequencies are obtained as attracting fixed points of a certain map.

MSC:

60G35 Signal detection and filtering (aspects of stochastic processes)
62M15 Inference from stochastic processes and spectral analysis
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