Double wavelet transform of frequency-modulated nonstationary signal

A mathematical model is proposed for a frequency-modulated signal in the form of a system of Gaussian peaks randomly distributed in time. An analytic expression is obtained for continuous wavelet transform (CWT) of the model signal. For signals with time-varying sequence of peaks, the main ridge of the skeleton characterized by frequency $\nu_{\mathrm{max}}^{\mathrm{MFB}}(t)$ is analyzed. The value of $\nu_{\mathrm{max}}^{\mathrm{MFB}}(t)$ is determined for any instant $t$ from the condition of the CWT maximum in the spectral range of the main frequency band (MFB). Double CWT of function $\nu_{\mathrm{max}}^{\mathrm{MFB}}(t)$ is calculated for a frequency-modulated signal with a transition regions of smooth frequency variation (trend) as well as with varying frequency oscillations relative to the trend. The duration of transition periods of the signal is determined using spectral integrals $E_\nu(t)$. The instants of emergence and decay of low-frequency spectral components of the signal are determined. The double CWT method can be used for analyzing cardiac rhythms and neural activity, as well as nonstationary processes in quantum radio physics and astronomy.