Note #02 Continuous-Time Delta and Step Functions

# Note#02 - Note#2 Continuous-Time(t and u(t ECSE-2410 Signals Systems(Wozny 1 The delta distribution(impulse function(t Motivation the limit of a

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Note #2. Continuous-Time δ ( t ) and u ( t ) ECSE-2410 Signals & Systems (Wozny) 1. The delta distribution (impulse “function”) , ) ( t . Motivation: the limit of a sequence of functions: Note#2. p.1 Form a finite pulse 1 then let 0 Δ thus ) ( lim ) ( 0 t t Δ Δ = 0 ) ( t t ) ( t Δ 0 2 Δ 2 Δ Δ 1 1 t ) ( t Δ 0 Δ 1 t Note the characteristics of ) ( t : (1) zero width (Occurs instantaneously, i.e., in zero time!) (2) infinite magnitude (Consequently, it is not a function according to the classical definition of a function. However, engineers still call it a function, or sometimes called a “generalized” function) (3) finite area (Area =1) Symbol for a “shifted” delta distribution with area=1 : “Sampling property” of delta distribution . Now find the area , A , of the product graph, ) ( ) ( ) ( 1 t x t x t Δ Δ = , as 0 Δ . First form Δ Δ Δ Δ = = 2 2 ) ( ) ( ) ( 1 dt t x dt t t x A , and taking the limit, () ) 0 ( ) 0 ( lim ) ( lim lim 1 0 1 0 0 2 2 x x dt t x A = = = Δ Δ Δ Δ Δ Δ Δ Δ . But Δ Δ Δ Δ Δ = = = dt t x t dt t x t dt t x t A ) ( ) ( ) ( ) ( lim ) ( ) ( lim lim 0 0 0 1 Functions other than pulses will also work, as shown later. The pulse function was originally used by Dirac. Rigorous proof of the delta distribution led to a mathematical theory of distributions.

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## This note was uploaded on 05/03/2009 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.

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Note#02 - Note#2 Continuous-Time(t and u(t ECSE-2410 Signals Systems(Wozny 1 The delta distribution(impulse function(t Motivation the limit of a

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