Unified Theory: Frequency Domain Analysis

  • Gianfranco Cariolaro


In this chapter, signals are studied in the frequency domain by means of the Fourier transform, which is introduced in unified form using the Haar integral as
$$S(f) = \int_I \mathrm{d} t\, s(t) \psi^{\ast}(f,t), \quad f\in \widehat{I},$$
where 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 (i2πft). 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.


Frequency Domain Discrete Fourier Transform Signal Domain Dual Group Coordinate Change 
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