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Note #02 Continuous-Time Delta and Step Functions

Note #02 Continuous-Time Delta and Step Functions - Note#2...

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Note #2. Continuous-Time δ ( t ) and u ( t ) ECSE-2410 Signals & Systems (Wozny) Fall 2006 1. The delta (or impulse) function , ) ( t δ , is motivated as follows: Form a finite pulse 1 then let 0 thus ) ( lim ) ( 0 t t = δ δ Note the characteristics of ) ( t δ : (1) zero width (Occurs instantaneously, i.e., in zero time!) (2) infinite magnitude (Not a function in the classical sense. Sometimes called a “generalized” function) (3) finite area (Area =1) Symbol “fires” when argument is zero i.e. , 0 0 = - t t , or when 0 t t = . “Sampling property” of delta functions . Now find the area , A , of the product graph, ) ( 1 t x , as 0 . First form - - - = = = 2 2 2 2 ) ( ) ( ) ( ) ( 1 1 dt t x dt t x dt t t x A δ , and taking the limit, ) 0 ( ) 0 ( lim ) 0 ( lim ) ( lim lim 1 0 1 0 1 0 0 2 2 2 2 x x dt x dt t x A = = = = - - . But - - = = dt t x t dt t x t A ) ( ) ( ) ( ) ( lim lim 0 0 δ δ 1 Other functions could be used. Theory of Distributions. The square pulse was used by Dirac.
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