DeltaAppx - Appendix D Appendix: Impulses, samples, and...

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Unformatted text preview: Appendix D Appendix: Impulses, samples, and delta's, Oh My! D.1 The Dirac delta The Dirac delta is best described as a distribution, rather than as a function, since, strictly speaking, it is not a function. A distribution is de ned as follows A distribution maps a function to a number. With that out of the way, we can now de ne the Dirac delta, or impulse as follows The Dirac delta is the distribution operating on the function f ( t ) , where f ( t ) is assumed to be continuous near t = 0 , is given by f ( t ) , f (0) . We refer to the Dirac delta interchangeably as an impulse, and colloquially as a delta function, even though it is not a function, but rather a distribution. Using this slight abuse of terminology, we also often will replace the form f ( t ) with the less convenient, but more natural form Z- ( t ) f ( t ) dt , f (0) . (D.1) In this form, we can imagine the impulse as having a sifting property whereby when placed within an integral, it sifts out the value of the integrand at the precise value of t = 0 . Note that this special form does not provide any additional properties to the impulse other than its de nition within an integral. Outside of an integral, an impulse is meaningless, since it is not a function and is only de ned by how it operates on a function when placed within an integral of the form above. For the special case offunction when placed within an integral of the form above....
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DeltaAppx - Appendix D Appendix: Impulses, samples, and...

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