  The continuous wavelet transform (CWT) is a linear time-frequency
representation and a powerful tool for analyzing non-stationary signals. The
synchrosqueezing transform (SST) is a special type of the reassignment method
which not only enhances the energy concentration of CWT in the time-frequency
plane, but also separates the components of multicomponent signals. The "bump
wavelet" and Morlet's wavelet are commonly used continuous wavelets for the
wavelet-based SST. There is a parameter in these wavelets which controls the
widths of the time-frequency localization window. In most literature on SST,
this parameter is a fixed positive constant. In this paper, we consider the CWT
with a time-varying parameter (called the adaptive CWT) and the corresponding
SST (called the adaptive SST) for instantaneous frequency estimation and
multicomponent signal separation. We also introduce the 2nd-order adaptive SST.
We analyze the separation conditions for non-stationary multicomponent signals
with the local approximation of linear frequency modulation mode. We derive
well-separated conditions of a multicomponent signal based on the adaptive CWT.
We propose methods to select the time-varying parameter so that the
corresponding adaptive SSTs of the components of a multicomponent signal have
sharp representations and are well-separated, and hence the components can be
recovered more accurately. We provide comparison experimental results to
demonstrate the efficiency and robustness of the proposed adaptive CWT and
adaptive SST in separating components of multicomponent signals with fast
varying frequencies.
