The preceding chapter has dealt with the class of LCA groups, which will represent the domain/periodicity of signals (and also those of Fourier transforms). This chapter deals with the Haar integral of signals specified on LCA groups, which represents the fundamental functional of the Unified Signal Theory. Using the Haar integral, the classes of signals are formulated in a Hilbert space, where also a theory of symmetries is developed. The convolution operation is then introduced and its basic properties are developed in a unified form. Associated with convolution operation is the impulse, which extends the well known properties of the “delta function” to every signal class.
KeywordsHaar Measure Quotient Group Periodic Repetition Symmetry Theory Symmetry Pair
- 1.M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970) Google Scholar
- 4.G. Cariolaro, Teoria dei Segnali Multidimensionali e Applicazioni alla HDTV (Edizioni Scientifiche Telettra, Milan, 1991) Google Scholar
- 7.D. Gabor, Theory of communications. J. Inst. Electr. Eng. 93, 429–457 (1946) Google Scholar
- 11.G. Sartori, Lezioni di Meccanica Quantistica, 2nd edn. (Ed. Cortina, Padova, 1997) Google Scholar
- 15.A. Weil, L’Integration dans les Groupes Topologiques (Hermann, Paris, 1940) Google Scholar