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The notion of “instantaneous frequency” is utilized in processing signals with time-varying spectral content. Specifically, a variable bandwidth filter (VBF) and some related spectral analysis tools are proposed to handle a multicomponent signal whose components interact with each other in the time-frequency (TF) domain. In the analysis, a multicomponent signal is decomposed into its components by estimating local instantaneous frequencies (IFs). In order to construct the proposed filter, we define a differential operator composed of the derivative operator and a positive time-varying coefficient. An extended Fourier transform (EFT) and its functional relations are derived to exploit the newly defined differential operator and the VBF in some weighted vector spaces. Further, a heterogeneous TF distribution is constructed by a short-time EFT to utilize as a criterion for identifying whether an estimated IF is intrinsic or not and repairing a suspect estimate of IF. To confirm usefulness of the EFT and the VBF, we conduct several numerical simulations for spectral localization of an AM-FM signal and for estimation of local IFs, and then, for decomposition of a multicomponent signal into its components with their resolving powers to be compared.