Contrivance of convolution arithmetic in Excel

Abstract

We have coded programs for one-dimensional convolution using a spreadsheet program (Excel; Microsoft Corp.). The program computes discrete convolution using values of given functions at 32 points as one period. If the input to a linear shift invariant system and the response function can be measured, the output is obtained by inputting these to the Excel worksheet. Moreover, for computer simulation study, calculations are possible for the convolution between a step function and exponential function, rectangular function and Gauss function, straight line and exponential function, low-pass filter and high-pass filter, projection of circle and reconstruction filter, and so on. Use of the program requires no detailed knowledge of Excel. Users can calculate convolutions using this program in the sense of a tool kit with a paper and a pencil. The program's reliability was verified through comparison with continuous convolution. In addition, we compared convolution by the program and that computed by the fast Fourier transform implemented in Excel. Numerical results obtained by both methods agree well: the "convolution theorem" was confirmed numerically. The program facilitates understanding of the mathematical basis and the actual arithmetic of convolution. It is useful for education and research about input/output of linear systems.