This work presents a novel perfect reconstruction filter bank decomposition (PRFBD) for nonlinear and nonstationary time series and image data representation and analysis. The Fourier decomposition method (FDM), an adaptive approach wholly based on the Fourier representation, is shown to be a special case of the proposed PRFBD. The adaptive Fourier–Gauss decomposition (FGD) proposed in this work is a variation of the FDM, which is based on the FR and Gaussian filtering. Similarly, we also consider Butterworth filtering to develop adaptive Fourier–Butterworth decomposition (FBD). The proposed theory can decompose any signal (time series, image, or other data) into a set of the desired number of Fourier intrinsic band functions (FIBFs), which follow the amplitude-modulation and frequency-modulation (AM-FM) representations. A generic filterbank representation is also provided, where perfect reconstruction can be ensured for any given set of lowpass or highpass filters. We have performed an extensive analysis of both simulated and real-life data (COVID-19 pandemic, Earthquake and Gravitational waves) to demonstrate the efficacy of the proposed method. The resolution results in the time-frequency representation demonstrate that the proposed method is more promising than the state-of-the-art approaches.