A three-level approach with Fourier series for multi-channel chirp mode decomposition in blind source separation

For blind source separation (BSS) problem of chirp mode signals, methods based on variational optimization framework (such as variational mode decomposition, variational nonlinear chirp mode decomposition) suffer from time-varying features of sources or predefined initial frequency. This paper proposes a novel three-level multi-channel chirp mode decomposition method (TMCMD) with Fourier series to solve the problems. Leveraging the analytic properties of chirp signals, the nonlinear and wide-band multi-channel chirp modes are transformed into a linear system via Fourier series expansion of instantaneous frequency (IF) and instantaneous amplitude (IA). Then the constraint problem formulation for linear mixing model is established. The algorithm comprises three stages. Firstly, a general parameterized time–frequency transform technique extracts time–frequency ridges to estimate IFs. Secondly, linear equations are established and solved through least square method to align frequency components from different channels. Finally, prior knowledge from the first two stages guides the alternating direction method of multipliers (ADMM) to separate source signals and the mixing matrix. TMCMD can deal with time varying signals and eliminate the need for preset initial frequencies. Its superior performance in mode alignment, noise robustness, filter bank structure, quasi-orthogonality of modes, and channel number robustness get investigated successively. In the end, the method is highlighted in modal analysis in simulation and experiments of time-invariant/time-varying vibration systems.