Instantaneous Frequency-Embedded Synchrosqueezing Transform for Signal Separation

The synchrosqueezing transform (SST) and its variants have been developed recently as an alternative to the empirical mode decomposition scheme to model a non-stationary signal as a superposition of amplitude- and frequency-modulated Fourier-like oscillatory modes. In particular, SST performs very well in estimating instantaneous frequencies (IFs) and separating the components of non-stationary multicomponent signals with slowly changing frequencies. However its performance is not desirable for signals having fast-changing frequencies. Two approaches have been proposed for this issue. One is to use the 2nd-order or high-order SST, and the other is to apply the instantaneous frequency-embedded SST (IFE-SST). For the SST or high order SST approach, one single phase transformation is applied to estimate the IFs of all components of a signal, which may yield not very accurate results in IF estimation and component recovery. IFE-SST uses an estimation of the IF of a targeted component to produce accurate IF estimation. The phase transformation of IFE-SST is associated with the targeted component. Hence the IFE-SST has certain advantages over SST in IF estimation and signal separation. In this article, we provide theoretical study on the instantaneous frequency-embedded short-time Fourier transform (IFE-STFT) and the associated SST, called IFE-FSST. We establish reconstructing properties of IFE-STFT with integrals involving the frequency variable only and provide reconstruction formula for individual components. We also consider the 2nd-order IFE-FSST.