## Abstract

In this chapter, signals are studied in the frequency domain by means of the where

*Fourier transform*, which is introduced in unified form using the Haar integral as$$S(f) = \int_I \mathrm{d} t\, s(t) \psi^{\ast}(f,t), \quad f\in \widehat{I},$$

*I*is the signal domain, \(\widehat{I}\) is the frequency domain and*ψ*^{∗}(*f*,*t*) is the conjugate of the*kernel*. A preliminary problem is the identification of the frequency domain \(\widehat{I}\) and of the*kernel**ψ*(*f*,*t*). In particular, for the groups of ℝ, we shall find that the kernel has the familiar exponential form*ψ*(*f*,*t*)=exp (i2*πft*). Since the frequency domain \(\widehat{I}\) associated to a given LCA group*I**is still an LCA group*(the*dual*group), all definitions and operations introduced in the signal domain (in particular the Haar integral) are straightforwardly transferred to the frequency domain. For the Fourier transform, several*rules*will be established in a unified form and then applied to specific classes of signals, one-dimensional as well as multidimensional.The chapter also contains two unusual topics: the Fourier transform on multiplicative groups and the *fractional* Fourier transform.

## Keywords

Frequency Domain Discrete Fourier Transform Signal Domain Dual Group Coordinate Change
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