The application of Fourier analysis to real-world signals often violates convergence requirements, producing distortions such as Gibbs-like oscillations, spectral leakage, and sensitivity to window alignment. Traditional remedies rely on window functions, which taper the signal edges and suppress leakage but inevitably alter the signal and require corrective post-processing. We introduce a preprocessing step that ensures mathematical validity for various Fourier Transforms, without modifying the original signal. This method requires no computation, eliminates corrective post-processing, and naturally removes spurious oscillations, alignment sensitivity and introduces a natural cut-off frequency. When applied to the Short-Time Fourier Transform, our method removes the need for window tapering, yielding faultless spectra at fixed resolution. When combined with adaptive Short-Time Fourier Transform, it ensures discontinuity-free segments, thereby enabling both dynamic resolution and perfect spectral accuracy. Our preprocessing serves as a universal conditioning step, enhancing Fourier-based time–frequency analysis across diverse applications. Its universality and efficiency suggest new directions for signal processing research and applications.
Patrick Lee (Thu,) studied this question.