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Unformatted text preview: MODULE 3 GENERALIZED FUNCTIONS AND FOURIER SERIES Impulse Function Generalized Functions Generalized Limits Fourier Series Page 3.1 INDEX Generalized Functions and Fourier Series Impulse as Generalized Function Scaled Impulse Derivatives of Impulse Fourier Transform of Impulse Convolution with Impulses Shifted Impulse, Complex Exponential, Sine and Cosine Unit Step as Generalized Function Fourier Transform of Unit Step Integration Generalized Limit Limit of Fourier Kernel Inversion of Fourier Transform of Impulse RiemannLebesgue Lemma Fourier Series Fourier Transform of Periodic Function Fourier Series Examples MAIN INDEX Page 3.2 3. GENERALIZED FUNCTIONS AND FOURIER SERIES index Reference : Lighthill, Fourier Analysis & Generalized Functions. Impulse Function The impulse function ( t ) is often introduced and defined by the properties R ( t ) dt = 1 and ( t ) = 0 for t 0, (0) = . But this has no mathematical meaning . ( t ) is not a function . Sometimes ( t ) is defined as a limit of a sequence of unit area functions ( t ) = n lim d n ( t ) with n lim d n ( t ) = 0 for t 0. But this definition is not unique ; there are sequences of functions with these properties but do not converge to ( t ). For example, it is easy to construct sequences that converge to ( t ) + ( t ). Page 3.3 Impulse as Generalized Function index The "impulse" ( t ) has meaning only by accepting it as a new concept . One way is to characterize ( t ) is by its integral properties relative to other signals . Such a tool is called a generalized function . We define an admissible signal x ( t ) to be: infinitely and continuously finite differentiable to vanish: x ( t ) 0 as  t  . If x ( t ) is admissible, then so is X ( ). This definition is quite strong, but all properties we develop will hold under it. A generalized function assigns a number to each admissible signal x ( t ). This is expressed by an integral property : R ( t ) x ( t ) dt = x (0) or generally R ( t  t ) x ( t ) dt = x ( t ). A generalized function has no meaning unless it is inside an integral, where it can be treated like a normal function (shifted, scaled, differentiated, etc.) Page 3.4 Scaled Impulse index If a 0, then in the sense of generalized functions: ( at ) = a 1 ( t ) Since ( a > 0) R ( at ) x ( t ) dt = (1/ a ) R ( s ) x ( s / a ) ds = (1/ a ) x (0) and similarly for a < 0. The equality is true only in the sense that ( at ) and a 1 ( t ) have the same integral properties. The product of ( t ) with any admissible signal x ( t ) gives another generalized function....
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This note was uploaded on 05/06/2010 for the course ECE 539 taught by Professor Alanc.bovik during the Spring '04 term at University of New Brunswick.
 Spring '04
 AlanC.Bovik
 Digital Signal Processing, Signal Processing

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