DISCRETE TIME FOURIER TRANSFORM (DTFT) Th dicrt-tim Fourir Tranform x x n xn n n Th Invr dicrt-tim Fourir Tranform (IDTFT) x n Not: ( ) i a complx valud continuou function = π f [rad/c] f i th digital frquncy maurd in [ C/S] n d Lctur # 4
EAMPLE 3 n Conidr th ignal xn 5 Un a- Dtrmin it DTFT b- Evaluat ( ) at =, 5π, 5π, 75 π and π c- Plot it magnitud, angl, ral part and imaginary part Solution: a- Sinc th ignal x(n) i abolutly ummabl, thrfor, DTFT xit: n xn 5 n n 5 5 co j in co j in 5 n n Lctur # 4 co j in co 5 j in 5
co 5 j in co 5 in co j in 5 co in jin co inco 5in co in 5 co co 5co j5in 5 5co co 5in 5 co 5in tan 5 co 5co R 5 co 5in Im 5 co Lctur # 4 3
Magntud Magntud Magntud Magntud Magnitud Magntud Rpon Part Magntud Ral Part Part 8 8 6 6 4 4 8 8 6 5 Frquncy 5 [rad\c] 5 3 35 frquncy Pha in pi unitud Angl Rpon Part 6 5 5 5 3 35 Frquncy [rad/c] in Pi unit Imaginary Imaginary Part Part - - - - -3-3 -4-4 -5-5 -6-6 -7 5 5 5 3 35 frquncy Frquncy in pi unitud [rad/c] Lctur # 4 4-7 5 5 5 3 35 Frquncy frquncy in [rad/c] pi unitud
EAMPLE 3 n For th ytm dcribd by it impul rpon hn 9 Un Do th folloing a- Dtrmin th frquncy rpon b- Plot th magnitud and pha rpon Solution a- Uing DTFT find : n 9 n H Hnc n hn 9 n n n 9 9co j9in H 9co 9 in Lctur # 4 5 n
Pha in Radian H H 9co 9in 9 co 9 co in H 88 co and H 9in arctan 9 co b- Th magnitud and pha plot Magnitud Rpon Pha Rpon 9-5 8 - -5 7 6 - -5-3 -35 5 4 3-4 3 4 5 6 7 8 9 Frquncy in pi unit 3 4 5 6 7 8 9 Frquncy in pi unit Lctur # 4 6
EAMPLE 33 A digital ytm i pcifid by th diffrnc quation a: a- Dtrmin H( ) y n 8yn xn Solution : a- Tak th DTFT for both id or apply th abov quation: H Y 8 Y Y 8 Lctur # 5 7
H Pha in Radian y EAMPLE 34 (HW) A third ordr lo pa filtr i dcribd by th diffrnc quation: n 8xn 543xn 543xn 8xn 3 76yn 89yn 784yn 3 a- Dtrmin H( ) b- Plot th magnitud and pha rpon of th filtr Solution 4 Magnitud Rpon 4 Pha Rpon 3 8 6-4 - -3 3 4 5 6 7 8 9 Frquncy in pi unit -4 Lctur # 5 8 3 4 5 6 7 8 9 Frquncy in pi unit
34 SAMPLING OF ANALOG SIGNALS Th qunc x(nt) i obtaind from a continuou tim ignal x(t) by ampling Thi i don multiplying a priodic impul train (t) [ampling function] by x(t) Th priod T i calld th ampling priod and F =/T i th ampling frquncy xt t T x nt t F x t xt t t t t n nt xt Sampling Function t x t T T 3T 4T 5T nt Lctur # 5 9
Thrfor, th ampld ignal i givn by : x t xnt t nt n From th multiplicativ proprty, Thrfor, It i knon that and j S j S T k k i th convolution of d Sinc, th convolution ith an impul imply hift a ignal o j o T k j k Lctur # 5
M T j k k That i uprpoition of hiftd rplica of M I a priodic function of coniting of : S T M T M cald by T M M M M M M M M M 4 M M Lctur # 5 T M S T 4 3 4 3 M M
If Th ignal can b rcovrd xactly from t M by man of Lo pa filtr ith gain x T p t t n nt Cut off frquncy gratr than M xt x p t p p M M T H H c f T M M M c M x f M c t M H T othri Lctur # 5
35 SAMPLING THEOREM Lt x t b a band limitd ignal ith for M Thn x t I uniquly dtrmind by it ampl xnt, n,,, if Whr, M and T Givn th ampl, can rcontruct x(t) by gnrating a priodic impul train Th ucciv impul hav amplitud that ar ucciv ampl valu Thi impul train i thn procd through an idal lo pa filtr ith gain T and cut off frquncy gratr than and l than M Th rulting output ignal ill xactly qual x(t) Lctur # 5 3