A transformation is a mathematical model that describes signal operations, where the original signal is treated as the input and the resulting signal as the output. It represents a very general concept that can be used to describe operations, however complicated, although it is often convenient to decompose a complicated operation into a sequence of simpler ones. The Unified Signal Theory gives a very general definition, which includes the case where the output domain is different from the input domain. In the multidimensional case, transformations between signals with different dimensionality are also included. In the first part of the chapter, transformations are analyzed in the signal domain, and in the second part in the frequency domain, using the concept of dual transformation. The development is mainly confined to linear transformations, but nonlinear transformations are also briefly considered. Linear transformations will be further developed in the next two chapters with multirate transformations and with sampling and interpolation, which are important examples of transformations between different domains.
KeywordsInput Signal Impulse Response Output Relation Quotient Group Scale Change
- 1.R.N. Bracewell, The Fourier Transform and Its Applications, 2nd edn. (McGraw–Hill, New York, 1986) Google Scholar
- 4.A. Papoulis, Systems and Transforms with Applications in Optics (McGraw–Hill, New York, 1968) Google Scholar
- 5.J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, in Berichte über die Verhandlungen der Königlichen Sächsichen Gesellschaft der Wissenschaften zu Leipzig (Reports on the proceedings of the Saxony Academy of Science), vol. 69 (Mathematisch-Physikalische Klasse, 1917), pp. 262–277 Google Scholar