Lctur 2: Discrt-Tim Signals & Systms Rza Mohammadkhani, Digital Signal Procssing, 2015 Univrsity of Kurdistan ng.uok.ac.ir/mohammadkhani 1
Signal Dfinition and Exampls 2 Signal: any physical quantity that varis with tim, spac, or any othr indpndnt variabl/variabls Exampl 1: spch signals
Exampl 2: Black & Whit Pictur 3 Intnsity of brightnss:,, Black and whit vido:,,
Exampl 3: ColorPictur 4,,,,,,,,,
5 full colorimag
Signal Typs 6 Continuous-Tim vs Discrt-Tim Continuous-Valud vs Discrt-Valud Analog: continuous in both tim and amplitud Digital: discrt in both tim and amplitud Dtrministic vs Random
Discrt-Tim Signal 7 A squnc of numbrs dfind for vry intgr valu of variabl n: =, < < Can b sampld from a CT signal: =, < < whr is sampling priod.
8 Discrt-Tim Signal Rprsntations
9 DT sampld from a CT signal
Som Elmntary DT Signals 10 Unit sampl (impuls) squnc 1 = 0 0 0 Unit stp squnc 1 0 0 < 0 Exponntial squncs = Sinusoidal = cos " # +%
Basic Oprations 11 Tim-shifting: # Multiplication: Exampl Mor gnrally
Typs of DT signals 12 ' & ' = ( ) *+' 1, ' = lim 0 ' 23+1 ( ) *+0 Enrgy signal vs Powr signal Enrgy signal: 0 < & ' < Powr signal: 0 <, ' < 0
Typs of DT signals (2) 13 Evn andodd signals Any arbitrary signal Evn: Z: = Odd: Z: = Priodic vs Apriodic signals Priodic: Z, 9 Z +9 =
14 Discrt-Tim Systms
Discrt-Tim Systm 15
Exampls 16 Idal Dlay Systm = # < < Moving Avrag = 0 < 1 3 : +3 ) +1 ( ; =*+0 >
Systm Proprtis 17 Linar Systms Additivity proprty: : + ) = : } +{ ) = : + ) Homognity or scaling proprty A = A = A Tim-Invariant Systms If = thn # = # Mmorylss Systms = ) for all intgr valus of n
Systm Proprtis (2) 18 Causality Systm output squnc valu at vry = # only dpnds on th input squnc valus for #. forward diffrnc systm = +1 backward diffrnc systm = 1 Stability If C D < for all thn C E < for all Boundd-Input Boundd Output (BIBO)
19 Linar Tim-Invariant Systms
LTI Systms 20 Linarity: Tim-invariant:
21 Exampl
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Proprtis of LTI systms 23 Convolution is Commutativ: x[n] h[n] y[n] h[n] x[n] y[n] Convolution is Distributiv: x[n] h 1 [n] h 2 [n] + y[n] x[n] h 1 [n]+ h 2 [n] y[n]
Proprtis of LTI systms (2) 24 Cascad connction of LTI Systms: x[n] h 1 [n] h 2 [n] y[n] x[n] h 2 [n] h 1 [n] y[n] x[n] h 1 [n] h 2 [n] y[n]
Proprtis of LTI systms (3) 25 Causality of LTI systms: h ; = 0for ; < 0 Stability of LTI systms: Impuls rspons is absolut summabl: ' ( h ; < =*+' = h = ' =*+' ' =*+' ; h ; = h = h ; ;
Proprtis of LTI systms (4) 26 Mmorylss LTI systm h = I = h = ' =*+' ' =*+' ; h ; = h = h ; ; Finit Impuls Rspons (FIR) systms Infinit Impuls Rspons (IIR) systms Exampl: h = 1/2
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28 Linar Constant-Cofficint Diffrnc Equations
Linar Constant-Cofficint Diffrnc Equations 29 An important class of LTI systms Th output is not uniquly spcifid for a givn input Th initial conditions ar rquird Linarity, tim invarianc, and causalitydpnd on th initial conditions If initial conditions ar assumd to b zro, systm is linar, tim invariant, and causal
Exampl 30 Diffrnc Equation Rprsntation : =*# A = ; = # L*# K L M
Solution mthod 1 31 Homognous quation:
32 Rcursiv Computation
33 Frquncy-Domain Rprsntations of Discrt-Tim Signals & Systms
Discrt-Tim Fourir Transform 34 Complx xponntial ignfunction H( ω ) is a complx function of frquncy Spcifis amplitud and phas chang of th input [ ] n n x ω = [ ] [ ] [ ] [ ] ) ( k n k k k h k n x k h n y = = = = ω [ ] [ ] ( ) n n k k H k h n y ω ω ω ω = = = ( ) [ ] k k k h H ω ω = =
Frquncy Rspons 35 If input signals can b rprsntd as a sum of complx xponntials [ ] = α ω kn xn w can dtrmin th output of th systm k [ ] ( ) ω = α H yn k k Diffrnt from continuous-tim frquncy rspons Discrt-tim frquncy rspons is priodic with 2π k k ω n ( ω+ 2πr ) ( ω+ 2πr ) k 2πrk ωk ωk ( ) = hk [ ] = hk [ ] hk [ ] H = k = k = ( ω+ 2πr ) ω ( ) = H ( ) H k k =
Discrt-Tim Fourir Transform 36 ω ωn ( ) = xn [ ] (forwardtransform) X 1 = 2π π π ω ωn [ ] X ( ) dω (invrstransform) xn n= X( ω ) is th Fourir spctrum of th squnc x[n] It spcifis th magnitud and phas of th squnc Th phas wraps at 2πhnc is not uniquly spcifid Th frquncy rspons of a LTI systm is th DTFT of th impuls rspons π ω ωk 1 ω ωn ( ) hk [ ] and hn [ ] = H ( ) dω H = 2π k = π
DTFT Pair 37 π ω ωn 1 ω ωn ( ) xn [ ] and xn [ ] = X ( ) dω X = 2π n= π
Existnc of DTFT 38 For a givn squnc th DTFT xist if th infinit sum convrgnc Or ω ωn ( ) = xn [ ] X n= ω ( ) < for all ω X ω ωn ωn ( ) xn [ ] xn [ ] = xn [ ] < X = n= n= n= So th DTFT xists if a givn squnc is absolut summabl All stabl systms ar absolut summabland hav DTFTs
39 DTFT Proprtis
Symmtric Squnc and Functions 40 Conugat-symmtric Conugatantisymmtric Squnc Function [ ] [ ] n x n x * = [ ] [ ] n x n x * o o = [ ] [ ] [ ] n x n x n x o + = [ ] [ ] [ ] ( ) n x xn 2 1 n x * + = [ ] [ ] [ ] ( ) n x xn 2 1 n x * o = ( ) ( ) ω ω = * X X ( ) ( ) ω ω = * o o X X ( ) ( ) ( ) ω ω ω + = o X X X ( ) ( ) ( ) [ ] ω ω ω + = * X X 2 1 X ( ) ( ) ( ) [ ] ω ω ω = * o X X 2 1 X
Rfrncs 41 D. Manolakisand V. Ingl, Applid Digital Signal Procssing, Cambridg Univrsity Prss, 2011. Miki Lustig, EE123 Digital Signal Procssing, Lctur nots, Elctrical Enginring and Computr Scinc, UC Brkly, CA, 2012. Availabl at: http://inst.cs.brkly.du/~123/fa12/ GünrArslan, EE351M Digital Signal Procssing, Lctur nots, Dpt. of Elctrical and Computr Enginring, Th Univrsity of Txas at Austin, 2007. Availabl at: www.c.utxas.du/~arslan/351m.html
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