Abstract
Signals are often destroyed by various kinds of noises. A common way to statistically assess the significance of a broad spectral peak in signals and the synchronization between signals is to compare with simple noise processes. At present, wavelet analysis of red noise is studied limitedly and there is no general formula on the distribution of the wavelet power spectrum of red noise. Moreover, the distribution of the wavelet phase of red noise is also unknown. In this paper, for any given real/analytic wavelet, we will use a rigorous statistical framework to obtain the distribution of the wavelet power spectrum and wavelet phase of red noise and apply these formulas in climate diagnosis.
Highlights
Signals are often destroyed by various kinds of noises during the process of generation, transportation, and processing [1, 2]
Red noise can describe climatic background noise with relatively enhanced low-frequency fluctuations arising from the interaction of white noise forcing with the slow-response components in the earth system
The red noise model has always provided a reasonable description of the noise spectra for a variety of climatic and hydrological time series [5,6,7,8,9]. e red noise is an important noise model in outputs of the feedback system [4], the neural network coupled with genetic algorithm [10], the optimization system in random scenarios [11, 12], and the big data processing system [5, 13]
Summary
Signals are often destroyed by various kinds of noises during the process of generation, transportation, and processing [1, 2]. There is no general formula on the distribution of the wavelet power spectrum of red noise. For any given real/analytic wavelet, we will use a rigorous statistical framework to derive a general formula for the distribution of the wavelet power spectrum and wavelet phase of red noise. By equation (1), the wavelet transform of an AR (1) red noise can be expressed as [15]
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