Gibbs phenomenon removal and digital filtering directly through the fast Fourier transform

The Gibbs phenomenon in the Fourier transform is due to a periodicity discrepancy between the waveform and its sinusoidal representations in the time domain. Data flipping, baseline tilting, or digital comparison completes the periodicity of the waveform and thus removes the Gibbs phenomenon. This facilitates digital filtering directly through the fast Fourier transform. Such a filtering is unique and exact, with its stop-band totally zeroed out and passband fully passed. It accomplishes lowpass, bandpass, highpass, bandstop, notch, or single-frequency-pass simply by manipulating the band limits