a note on interpolation and convolution

The convolution step can be performed in a number of different ways; hence, as has been often repeated in PHYS 542, a choice needs to be made. In particular, since the experimental data need not be evenly spaced on the energy-axis, it may be sensible to first interpolate the data so as to perform the convolution on an evenly spaced grid. Otherwise, convolution might erroneously skew the data in a way that is related to the inconsistency of horizontal spacing. I wanted to test this hypothesis, and therefore tried doing the convolution both with and without interpolation. In the future I also hope to test between different kinds of interpolation, such as between linear and spline interpolations.

The results can be seen most clearly by enabling Lorentzian broadening and choosing a meaningfully sized FWHM (choose, say, FWHM \( = 4\) eV, keeping alpha \( =0 \) and \( t=1 \)) then using the buttons to switch between a mode using no interpolation and one that uses linear interpolation. A slight change can be observed, especially in the region around 5440 eV. When interpolation is turned off, an erroneous feature arises near 5440 eV, which we can be sure is illegitimate - not a real consequence of performing the convolution - since Lorentzian convolution certainly could not give rise to such a feature in that location. (Lorentzian convolution tends to 'smooth out' a data set; it does not typically create new features, and certainly not for our data set with FWHM \(\approx 4\) eV.) It turns out that the data is approximately 5 times more densely spaced in the energy range from about 5445 to 5465 eV than in other regions; apparently, it is this change in horizontal spacing that accounts for the erroneous behavior near 5440 eV.

This post on Stackoverflow suggests how irregular spacing could give rise to a potential false positive. The accepted answer in that post does not perform any interpolation. The algorithm it uses is identical to mine in the case of no interpolation, except that its output grid is evenly spaced (even though its input grid is not) whereas both my input and output grids are unevenly spaced. But the spacing of the output grid cannot seem in my imagination by any means to so significantly affect the end result. Thus, that post on Stackoverflow prescribes a solution opposite to what I have seen in this investigation to constitute the preferable procedure for convolving an irregularly spaced data set.

This project has suggested that Lorentzian convolution on an irregularly spaced data set encourages some kind of interpolation. Further investigations would attempt the convolution again with no interpolation, but on an evenly-spaced output grid, and would then illustrate comparisons between different interpolation schemes.