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- Generalization of Distributions Theory, Extending Laplace-, z- and Fourier-Related Transforms
| Content Provider | Scilit |
|---|---|
| Author | Corinthios, Michael |
| Copyright Year | 2018 |
| Description | IG [Φ (s)] = < G (s) , Φ (s) >ℜ[s]=σ = ˆ σ+j∞ σ−j∞ G (s)Φ (s) ds. (18.1) The test function Φ (s) has derivatives of any order along such a contour line in the s plane, and tends to zero more rapidly than any power of |s|. In what follows to lighten the notation we will sometimes write < G (s) , Φ (s) >, meaning < G (s) , Φ (s) >ℜ[s]=σ. As proposed in [22] and [24], the Dirac-delta impulse may be generalized, leading to a distribution that is a generalized function of a complex variable. The generalized impulse may be denoted ξ(s) being a function of the complex variable s. We may define such a generalized Dirac-delta impulse by writing IG [Φ (s)] = < ξ (s) , Φ (s) >ℜ[s]=σ = ˆ σ+j∞ σ−j∞ ξ (s)Φ (s) ds = { jΦ(0), σ = 0 0, σ 6= 0. (18.2) The following properties are generalizations of properties of the usual real-variable distributions, and can be proven similarly to the corresponding proofs of the well-known theory of generalized functions. Book Name: Signals, Systems, Transforms, and Digital Signal Processing with MATLAB |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2009-0-07491-7&isbn=9781315218533&doi=10.1201/9781315218533-22&format=pdf |
| Ending Page | 1285 |
| Page Count | 32 |
| Starting Page | 1254 |
| DOI | 10.1201/9781315218533-22 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2018-09-03 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Signals, Systems, Transforms, and Digital Signal Processing with MATLAB Distribution |
| Content Type | Text |
| Resource Type | Chapter |
