Proper Definitions of Dirichlet Conditions and Convergence of Fourier Representations [Lecture Notes

Abstract

Fourier theory is the backbone of signal processing (SP) and communication engineering. It has been widely used in almost all branches of science and engineering in numerous applications since its inception. However, Fourier representations such as Fourier series (FS) and Fourier transform (FT) may not exist for some signals that fail to fulfill a predefined set of Dirichlet conditions (DCs). We note a subtle gap in explaining these conditions as available in popular SP literature. For example, original DCs require a signal to have bounded variations (BVs) over one time period for the convergence of FS, where there can be, at most, a countably infinite number of maxima and minima and, at most, a countably infinite number of discontinuities of finite magnitude. However, a large body of the literature replaces this statement with the requirements of a finite number of maxima and minima over one time period, and a finite number of finite discontinuities over one time period. Due to the latter, some signals fulfilling the original DCs are incorrectly perceived as not having convergent FS (CFS) representation. A similar problem holds in the description of DCs for the FT. Likewise, although it is easy to relate the first DC with the finite value of FS or FT coefficients, the relevance of second and third DCs, as specified in SP literature, is hard to assimilate.