  The objective of this paper is to introduce an innovative approach for the
recovery of non-stationary signal components with possibly cross-over
instantaneous frequency (IF) curves from a multi-component blind-source signal.
The main idea is to incorporate a chirp rate parameter with the time-scale
continuous wavelet-like transformation, by considering the quadratic phase
representation of the signal components. Hence-forth, even if two IF curves
cross, the two corresponding signal components can still be separated and
recovered, provided that their chirp rates are different. In other words,
signal components with the same IF value at any time instant could still be
recovered. To facilitate our presentation, we introduce the notion of
time-scale-chirp_rate (TSC-R) recovery transform or TSC-R recovery operator to
develop a TSC-R theory for the 3-dimensional space of time, scale, chirp rate.
Our theoretical development is based on the approximation of the non-stationary
signal components with linear chirps and applying the proposed adaptive TSC-R
transform to the multi-component blind-source signal to obtain fairly accurate
error bounds of IF estimations and signal components recovery. Several
numerical experimental results are presented to demonstrate the out-performance
of the proposed method over all existing time-frequency and time-scale
approaches in the published literature, particularly for non-stationary source
signals with crossover IFs.
