Question

Explain why the frequency response of discrete-time filters is always $LaTeX: 2\pi$ periodic.

Answer 1

Frequency Response of Discrete time signal is always periodic with period 20 :- let consider a Continous signal a(t) then Continous Fourier transform given by tox () = f as aciet en gust de la &(t) = fox (sco) event du * In similar way Lees Consider a diccrefime time signal xC ni then Discrete time Fourier transfun given by (Frequency Response obtained by transfam). xresw) - E xon) eston nio 2(m) = 2 x Cao) esan du . Continuous + In x time domain Response varies from co to to while calculating signal from Response but in di screte filme domain Response varies from oto za only as discreta Signats are sequences possents only for interger values of t but not all other values lie 1=0,112,-..... ; 4= 0,0...--.~). * Therbeste for discrete firme Signal can be expalessed as sum of individual 20 periodic fonctions zjum ejufh+20)

psiwn (m+22) = Sionedeta FireSamw. Cos qaw + s sin zow] esw.(nt 202). 5ejun (Cosząw.tisin 2.avu). [ cosaaw = ! & gîn aww = o) then esw(n42r) - eSon.co. + esa infan) = swn لا .. لا * In continuous time because of real variable ť! the above relation not applicable revet & escutzitt) | * Therfore, frequency Response - the transfolored signal amplitude and phase of discrete forme filters are alencys periodic curth period equal to 2a,