Abstract
The introduction to signals starts with a clear distinction between “physical signals” and “mathematical signals”, the model for physical signals. Considering the variety of signals encountered in applications (continuous-time, discrete-time, one-dimensional, two-dimensional, etc.), the model of deterministic signals is applied, with the conclusion that a first unified model has the form s(t), t∈I, where s is a complex function defined on an appropriate domain I. But this is only a notational convenience. The second step to unification concerns the mathematical structure the domain I should have, with the conclusion that I must be an Abelian group. The final step is the identification of a linear functional that permits the introduction of the signal fundamental operations, such as convolution, Fourier transformation and linear filtering. It is shown that the right functional is provided by the Haar integral, which permits defining the signal operations in a unified form.
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Notes
- 1.
A similar synoptic theory may be found in [4].
- 2.
In the Unified Theory, I is a pair of Abelian groups which represent both the domain and the periodicity of the signals. But, in these preliminaries it is sufficient to consider only the domain.
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Cariolaro, G. (2011). Introduction. In: Unified Signal Theory. Springer, London. https://doi.org/10.1007/978-0-85729-464-7_1
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DOI: https://doi.org/10.1007/978-0-85729-464-7_1
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