An Adaptive Approach to Gibbs’ Phenomenon
Gibbs’ Phenomenon, an unusual behavior of functions with sharp jumps, is encountered while applying the Fourier Transform on them. The resulting reconstructions have high frequency oscillations near the jumps making the reconstructions far from being accurate. To get rid of the unwanted oscillations, we used the Lanczos sigma factor to adjust the Fourier series and we came across three cases. Out of the three, two of them failed to give us the right reconstructions because either it was removing the oscillations partially but not entirely or it was completely removing them but smoothing out the jumps a little too much. Even though the remaining one successfully removed the oscillations and gave us the right reconstruction, consistency needed to be gained. Taking this into account, we have developed an automated scheme that produces the right reconstruction each time. This scheme has been very efficient in reconstructing the signals quite accurately and consistently, leading to a new approach to signal processing.
Lanczos sigma factor, Convergence of Fourier Series, Fast Fourier Transform, Signal processing, Least squares problem, Sigma approximation, Applied Mathematics, Numerical Analysis and Computation
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