Note #02 Continuous-Time Delta and Step Functions-1

Note #02 Continuous-Time Delta and Step Functions-1 - Note...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Note #2. Continuous-Time δ ( t ) and u ( t ) ECSE-2410 Signals & Systems (Wozny) Spring 2007 1. The delta distribution (impulse “function”) , ) ( t . Motivation: the limit of a sequence of functions: Form a finite pulse 1 then let 0 thus ) ( lim ) ( 0 t 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 = , as 0 . First form - - = = 2 2 ) ( ) ( ) ( 1 dt t x dt t t x A , and taking the limit, ( 29 ) 0 ( ) 0 ( lim ) ( lim lim 1 0 1 0 0 2 2 x x dt t x A = = = - . But - - - = = = 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.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 5

Note #02 Continuous-Time Delta and Step Functions-1 - Note...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online