Automatic Differentiation of Computer Programs in the Time and Frequency Domain

Automatic differentiation of computer programs has been successfully used in a wide variety of application areas. However, in this set of techniques, the differentiation is not carried out on the level of an abstract mathematical representation of some function, but on the level of an actual implementation of this mathematical representation. We consider the resulting subtle differences when automatic differentiation is used to transform functions from the area of digital signal processing. To this end, we apply the software tool ADiMat, implementing automatic differentiation for programs written in Matlab, to an implementation of a simple sinusoidal function in both, the time and the frequency domain. This comparison illustrates that the mechanical process of applying automatic differentiation is currently not capable of recognizing and exploiting the known mathematical connection between derivatives in the time and in the frequency domain.