Note #02 Continuous-Time Delta and Step Functions

Note #02 Continuous-Time Delta and Step Functions - 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) Fall 2006 1. The delta (or impulse) function , ) ( t , is motivated as follows: 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 (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 , 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 1 Other functions could be used. Theory of Distributions. The square pulse was used by Dirac.
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.

This note was uploaded on 04/10/2008 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.

Page1 / 5

Note #02 Continuous-Time Delta and Step Functions - 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