Wiley signals and systems e book TLFe BO 173

158 Chirwlution and Intpulse Rcsgont;e _I froin (8.5) T h e rcsult is called the delta impulse b j t ) liiii ‘fil-0 r ( t )= 6(/) (8.9) It i s also known as tlit Dirac dclfcc junc Lion, Dirac tmpiilsc and ztnd r m p u l w The delta impulsc~d(t) is no( a iiinction in the usual spiise httcmp(iilig t o tlefinc h ( t ) by a riile that assigiis a v;lluc SjC) Lo e w i y time point t leads to a functioii that tnkw the value m t ) for t # and grows beyoilcl all limits at I = The cz~bualformnlatiori (8.10) although not wrong, is of little iisc a h it d o ~ not s explain how b j r ) caii be niatheiiialictilly coinkhetl with other fiuictions A mathcmat icall~7precise definition of the delta impulse as a reliable fuiiction for which evcry value of an independeiit variable i s assigned a firrctioii value is not posdde It is iicccssary to use tlie idea of di,slr~b?6f~ons i~isteatl[17 191 WP mill refrain from a Ixiat,liematicwlly thorough formulation of the dclta iriipulse arid related distributioiis and instead we will illustrate some iniportant p r opwt that will be iisefiil wheir dealing with sigilals and s ealt with using the aiialogj oi iiiliriitely t lriii ective erty The pririciple for iise of (,he delta impiilst ant1 other dixtiibiitioris is (hat t licy ot described by their undetermineablc properties, rat her by tlreir cffrcts on furictions Instead of clescrihing the tklt a itnpulse by >iri assignment (for instarice (8.10))swe see whether the value of the integral tlescritm thc dfect of tlie delta impulse 011 the iunction f i t ) F’irit we ;~l>aiidoii the delta impulse in favour 01 the rectangle impiilse ~ ( tfrom ) ( ) evaluate the inlegral arid carry out the limit 7;4 With ( X j , f ( t ) J ( t dt ) li =?I* f ( t )d t ‘ F(21,) - F ( ) To (8.11)