Fourier theory is one of the most important tools used ubiquitously for
understanding the spectral content of a signal, extracting and
interpreting information from signals, and transmitting, processing, and
analyzing the signals and systems. Undergraduate engineering students
are exposed to these concepts, usually in their second year, to build
their foundation in the areas of signal processing and communication
engineering. However, the popular signal processing literature
[book1, book2, book3, book4] does not offer a clear explanation regarding the
convergence or the existence of Fourier representations for certain
well-known signals. Because of this subtle gap, it becomes hard for
young students to assimilate the Fourier theory with clarity, and they
are forced to be familiar with some of these concepts without
understanding them. To bring clarity to the existence and the
convergence of Fourier representation, including Fourier series and
transform, lecture notes were published recently in IEEE Signal
Processing Magazine’s September 2022 issue [PSDCs]. This
lecture note is in the continuation with technical details added from
yet another mathematical topic of distribution theory that connects
delta Dirac functions with the Fourier theory. The distribution theory by Schwartz in 1945 is one of the great
revolutions in mathematical function analysis. It is considered as a
completion of differential calculus, similar to how the revolutionary
measure theory or Lebesgue integration theory proposed in 1903, is
considered as a completion of integral calculus. Both these theories
unlocked new paradigms of mathematical development. Although distribution theory is a powerful tool for understanding
Fourier theory, it is ignored in engineering textbooks. In this lecture
note, we utilize the concepts of this theory to show how some signals
that fail to exhibit FT in the conventional sense can have FT in the
distributional sense.