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- Distributions
| Content Provider | Scilit |
|---|---|
| Author | Corinthios, Michael |
| Copyright Year | 2018 |
| Description | A = n−1∑ f(ti)∆ti. (17.3) FIGURE 17.1 Integration as a sum of increments. As the largest time increment ∆ti −→ 0 the number of slices n −→ ∞, the area A approaches the true area under the curve and we can write f(t)dt = lim ∆ti−→0 f(ti)∆ti. (17.4) Such interpretation of the integration operation implies that the surface area and hence the value of the integral is not affected if at an isolated point, say t = τ in the interval of integration, a < τ < b, the value of the function f(τ) is changed arbitrarily. As long as f(τ) is finite, and as long as f(t) is the same everywhere else, whether f(τ) = 0 or f(τ) = 100, the value of the integral remains the same. This is due to the fact that the contribution to the area A of such an isolated point such as t = τ is a strip of height f(τ) and zero width. We note that the same applies if the value of the function changes at a set of isolated points. The area remains the same. The integral of a function is thus the same whatever is its value over “a set of measure zero,” that is, a set of isolated points. Consider now the usual definition of the Dirac-delta impulse: δ(t) = 0, t 6= 0 (17.5) ˆ ∞ −∞ δ(t)dt = 1. (17.6) The impulse, being nil everywhere except at the isolated point t = 0, its integral should be zero since the function is nil almost everywhere and the value of the integral, as just deduced, is unaffected by the value of a function at an isolated point. The fact that it should have a nonzero integral implies that it is not an ordinary function. In fact if its integral is to be other than zero its value f(0) at t = 0 cannot be finite. It is for this reason that the delta function is not an ordinary function and is a main subject of the theory of distributions. In what follows a basic overview of this theory is presented. 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=10.1201/9781315218533-21&format=pdf |
| Ending Page | 1253 |
| Page Count | 44 |
| Starting Page | 1210 |
| DOI | 10.1201/9781315218533-21 |
| 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 |
| Content Type | Text |
| Resource Type | Chapter |
