Time and Frequency Characterization of Signals & Systems

Transcription

1 Tim ad Frqucy Characrizaio of Sigals & Sysms Frqucy Domai Characrizaio hrough muliplicaio of Fourir Trasform of ipu sigal ad sysm frqucy rspos Trasfr Fucio Tim Domai Characrizaio hrough covoluio of ipu sigal ad sysm impuls rspos Covi o us frqucy domai bcaus asy opraio of Muliplicaio as oppos o opraio of covoluio i im domai

2 2 Magiud ad Phas Rprsaio of Fourir Trasforms θ θ Phas Magiud Trasform Fourir im Discr Phas Magiud Trasform Fourir im Coiuous

3 Magiud-Phas Rprsaio of Th Frqucy Rspos of LTI Sysms h H yh* YΗ Y Η Y H + [] h[] H y[][]*h[] Y H 3

4 Liar Phas ad Group Dlay of LTI Sysms Y h H Y Η H yh* YΗ + Liar Phas, if H 0 i H ad H 0 Wha his sysm dos, is o shif dlay h ipu by Group dlay is dfi as τ d { H } d i 0 y 0 4

5 Log-Magiud ad Bod plos Th absolu valus of h magiud of h rasfr fucio of a sysm ar ormally covrd io dcibls dfi as 20 log0 H Plos of 20 log 0 H ad H vrsus log 0 ar kow as Bod plos 5

6 6 Coiuous-im Filrs Dscribd By Diffrial Equaios Simpl RC Lowpass Filr R C vs vr vc RC H H H d d RC H Vc Vs Vs Vc d dvc RC ] [, d dvc C i d dvc C i h H vs Vch*vs Vs VcΗVs

7 vs Coiuous-im Filrs Dscribd By Diffrial Equaios Simpl RC Lowpass Filr + vr - R + vc C - Alraiv mhod of gig a h rasfr fucio or frqucy rspos of h Filr dvc RC + vc vs d Takig FT o boh sid of quaio RC Vc + Vc Vs Vc H Vs + RC 7

8 H H y[ ] ay[ [], ] a - Firs-ordr Rcursiv Discrim Filr [ ] From Eig Fucio Propry of LTI sysm h y[] H - Subsiuig i h prvious quaio ah 8

9 Firs-ordr Rcursiv Discr- [] + im Filr y[] ay[-] D a y[-] y[ ] Takig DTFT of Y - a Y ay[ - H ] Y [ ] abov quaio : a [] - h[] H - y[][]*h[] YΗ 9

10 Firs-ordr Rcursiv Discrim Filr F T h[ ] H a u[ ] F T a - 0

11 Impuls rspos of Firs ordr rcursiv D-T lowpass filr 0 ], [ ] [ < < a u a h 0 a a u a H ] [ 0

12 Frqucy Rspos of Firs ordr rcursiv lowpass filr h[ ] a u[ ], 0 < a < H a Phas H -π a + a a 0 a / a 2 H π 2π -π π 2π 2 a a / a 2

13 Impuls-Trai Samplig + p δ T + p T δ T p 3

14 4 Muliplicaio/Modulaio Propry θ θ θ π d P p T F p 2 p P + + k s p k s p k T k T P d P w * δ δ π θ θ θ π ] * [ 2 π P p p p

15 Covoluio i Frqucy Domai 5

16 6 Samplig Thorm / / 2 2 ; cov 0 lim 2,, 0, ] [ sc sc M s M s M T priod T Samplig i T whr ra Nyquis h ra sampl a w if rd r ca b uiquly for i id bad is T vry T sampld is π π < > > ± ±

17 Violaig Samplig Thorm rsulig i aliasig 7

18 Rcosrucio usig a idal lowpass filr 8

19 9 Rcosrucio lookig from h im domai-covolvig h H YΗ p * h y p } } { }* { * y, π Sic π T h frqucy cuoff for, H filr lowpass a idal For c c c c T h T y T p uig d h T y h T y h y T T p p p τ τ τ τ δ τ δ δ δ τ τ p p

20 Coiuous o discr-im sigal covrsio 20

21 Discr-im Procssig of Coiuous-im Sigals 2

22 Rcosrucio of a sampld sigal wih a zro-ordr hold 22

23 Compariso of Frqucy Rsposs Trasfr Fucios of idal lowpass rcosrucio filr ad zro-ordr hold rcosrucio filr 23

24 Rcosrucio of a sampld sigal wih a firs-ordr hold 24

25 Compariso of Frqucy Rsposs Trasfr Fucios of idal lowpass rcosrucio filr, zro-ordr hold rcosrucio filr ad firs-ordr hold rcosrucio filr 25

26 Rcosrucio of a sampld sigal wih idal lowpass filr 26