Abstract
Fourier theory is a popular tool for analyzing various signals and
interpreting their spectral contents. It finds applications in a wide
variety of subject areas. However, some of the popular signals, such as
sinusoidal signals, Dirac delta, signum, and unit step, fail to have a
convergent Fourier representation (FR) in the conventional sense. Hence,
it becomes imperative to utilize the distribution theory to understand
and build a suitable representation for these signals. The signal
processing and communication engineering literature does not explain
these concepts clearly. As a result, many of the concepts of FR for
signals that do not conform to the conventional derivations remain
obscure to researchers. We attempt to bridge this gap and provide a
comprehensive explanation regarding the existence of FR. Further, we
have proposed a new linear space of Gauss–Schwartz (GS) functions and
corresponding tempered superexponential (TSE) distributions. It is shown
that the Fourier transform (FT) is an isomorphism on the GS space of
test functions and hence by duality is an isomorphism on TSE
distributions. The space GS is smallest in the sense that its dual
space, a set of TSE distributions, is the largest linear space over
which FT can be defined by duality.