  Recently the synchrosqueezed transform (SST) was developed as an empirical
mode decomposition (EMD)-like tool to enhance the time-frequency resolution and
energy concentration of a multi-component non-stationary signal and provides
more accurate component recovery. To recover individual components, the SST
method consists of two steps. First the instantaneous frequency (IF) of a
component is estimated from the SST plane. Secondly, after IF is recovered, the
associated component is computed by a definite integral along the estimated IF
curve on the SST plane. The reconstruction accuracy for a component depends
heavily on the accuracy of the IFs estimation carried out in the first step.
More recently, a direct method of the time-frequency approach, called signal
separation operation (SSO), was introduced for multi-component signal
separation. While both SST and SSO are mathematically rigorous on IF
estimation, SSO avoids the second step of the two-step SST method in component
recovery (mode retrieval). The SSO method is based on some variant of the
short-time Fourier transform. In the present paper, we propose a direct method
of signal separation based on the adaptive continuous wavelet-like transform
(CWLT) by introducing two models of the adaptive CWLT-based approach for signal
separation: the sinusoidal signal-based model and the linear chirp-based model,
which are derived respectively from sinusoidal signal approximation and the
linear chirp approximation at any time instant. A more accurate component
recovery formula is derived from linear chirp local approximation. We present
the theoretical analysis of our approach. For each model, we establish the
error bounds for IF estimation and component recovery.
