The fast Fourier transform for experimentalists. Part II. convolutions

When undergraduate students first compute a fast Fourief transform (FFT), their initial impression is often a bit misleading. The process all seems so simple and transparent: the software takes care of the computations, and it's easy to create the plots. But once they start probing, students quickly learn that like any rich scientific expression, the implications, the range of applicability, and the associated multilevel understandings needed to fully appreciate the subtleties involved take them far beyond the basics. Even professionals find surprises when performing such computations, becoming aware of details that they might not have fully appreciated until they asked more sophisticated questions. In the first of this five-part series, we discussed several basic properties of the FFT. In addition to some fundamental elements, we treated zero-padding, aliasing, and the relationship to a Fourier series, and ended with an introduction to windowing. In this article, we'll briefly look at the convolution process.