Professor: Mihnea UDREA DIGITAL SIGNAL PROCESSING. Grading: Web: MOODLE. 1. Introduction. General information

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1 Geeral iformatio DIGITL SIGL PROCESSIG Profeor: ihea UDRE B29 Gradig: Laboratory: 5% Proect: 5% Tet: 2% ial exam : 5% Coure quiz: ±% Web: OODLE 2 alog igal proceig ytem. Itroductio Dicrete Time Sigal ad Sytem 3 4

2 Digital igal proceig ytem Dicrete-Time Sigal x () a t T 2T T x( ) t a x T x t t T periodically amplig a cotiou time igal at time iterval T. x x T a T amplig period T amplig frequecy S 6 Baic igal The dicrete impule,, The dicrete uit tep u,, Periodical igal ( ) u ( ) dicrete igal i ample periodical if: Z x x Z 7 The dicrete-time iuoid a t - i x t t T 2 i x x T a S f ormalied frequecy f t TS 2 S T S agular ormalied frequecy T 2 f S S 8

3 The dicrete-time iuoid x() i ample periodical if there exit iteger ad : 2 2 The dicrete-time iuoid x() i ample periodical if there exit iteger ad : 2 2 = Hz; S = Hz 2.2 S i.2 x.5 3 =.5 Hz; S = Hz i.3 x =, = 9 - =2, =3 The dicrete-time igal eergy requecy aalyi of dicrete-time igal def 2 E x x(t) x T x t t T x t t T E T S E a 2 E T x T a S a S The average power T 2T 3T T d(t) d(t - T ) P lim 2 2 x t T t 2

4 requecy aalyi of dicrete-time igal 2 xt x t t T Deote: xt X xt X a x T x t t T X X X 2 T a a 2 2 T 3 X X T a -2Ω -2Ω -Ω Ω /T Coditio: ) X a (Ω) =, for Ω >Ω 2) Ω 2Ω Samplig Theorem (yquit) X a (Ω) X(Ω) -Ω /2 Ω /2 -Ω Ω Ω 2Ω /T aliaig -Ω -Ω Ω Ω 2Ω 4 Ω Ω The miimum amplig rate i achieved for Ω =2Ω =2 = yquit rate (yquit frequecy) X() X(f) f ( ) X() f f - ormalized frequecy X(f) f -4π -2π -π 2π π 2π 4π (Ω ) [ ; ), f [.5;.5) (baic iterval) 6

5 requecy aalyi of dicrete-time igal requecy aalyi of dicrete-time igal Z traform: X z Z x x z 2 x Z X z X z z dz D z z R, R R R ot importat propertie: C Im{z} Re{z} R - R + Z traform: a) igal with left limited upport, upp x, particular cae: x X z Z x x z D z z R Im{z} R - Re{z} x X z x z X z x x X z X z requecy aalyi of dicrete-time igal requecy aalyi of dicrete-time igal Z traform: b) igal with right limited upport, upp x, x X z Z x x z D z z R Im{z} Re{z} R + Z traform: bilateral equece ca be decompoed: x x ( ), ele D z z R x x x x x ( ), ele D z z R D D D z R z R 9 2

6 requecy aalyi of dicrete-time igal Z traform: Let: ad: The: ad: X z ' x x Z x ' X ' z Z x X z D' z z R R D z R z R X ' z x z x z X z z R z R z R z R 2 requecy aalyi of dicrete-time igal Dicrete time ourier traform (DTT): Xe xe DTTx 2f 2 x Xe e d IDTTXe 2 Dicrete ourier traform (DT): x up, X DT x x W xidtx XW W 2 e 2 22 requecy aalyi of dicrete-time igal requecy aalyi of dicrete-time igal Z traform X z x z Dicrete time ourier traform (DTT): Xe DTT DT z e X e x e Xz Xe X x e 2 2 X 23-3π -2π -π π 2π 3π Dicrete ourier traform (DT): X 2,,, 2-2 X e X 24

7 Example: The rectagular widow The rectagular widow frequecy characteritic I defied by: w ( ) D ,, wd ( ), i ret z WD ( z) z z WD e The pectrum i: e WD ( e ) e e i 2 2 i 2 WD ( ) requecy aalyi of dicrete-time igal i ( ) x u u requecy aalyi of dicrete-time igal i ( ) x u u DTT X e -π - π 2π- 2π 2π+ D X e W e -π - π 2π- 2π 2π+ WD e 2 π 2π Widowig effect 27 DTT 2 DT X e X W D e X π 2π- 2π π 2π 28

8 requecy aalyi of dicrete-time igal i ( ) x u u.2 Dicrete-time Sytem y T x x() T { } y() DTT 2 DT X e X X π 2π- 2π WD e 2 2 π 2π Spectral leaage Liear ytem atify the uperpoitio priciple: T a x a x at x a T x a y a y The impule repoe: h T x x Tx T x xt xh y 29 3 T h Dicrete-time ytem Time ivariat ytem have the followig property: T x y T x y Z y xh h T h Dicrete time liear covolutio: x x x x x x x() h() y hx xh Dicrete-time ytem Cauzal ytem. xa xb petru ya yb petru LTI cauzal ytem: h, Stable ytem the impule repoe ha to be abolutely umable. h 3 32

9 LTIS differece equatio: y x If deote: a b y b x a y IR fiite impule repoe ytem ( = ): y b x h b x a IIR ifiite impule repoe ( > ). b, [, ], i ret 33 LTIS trafer fuctio x() y() h() impule repoe X(z) Y(z) H(z) ytem fuctio y x h X zhz Y z Similar aalyi with ourier traform: z e H z H e Z h Y e X e H e 34 LTIS trafer fuctio ume a LTIS havig the differece equatio: y b x a y pplyig the Z traform: Y z b X z z a Y z z bz Y z B( z) X( z) ( ) z az Z x ( ) X z z Z y ( ) Y z z 35 LTIS trafer fuctio Bz ( ) b b z... b z z ( ) az... a z z the ytem zero: p the ytem pole: the ytem order (umber of pole).... Bz ( ) b z z z z z z z ( ) pz pz... p z H p Stable ytem the pole mut be le tha (i abolute value). p 36

10 LTIS trafer fuctio IIR ytem: IR ytem: a b z b bz... b z h ( ) b ( ) Bz ( ) z ( ) bz az bz Z b ( ) Example: The movig average filter The mea of a legth igal: m What if igal legth i ifiite? Ue frame (widow) of lat ample m x( ) x( +) x( +2) x( +3) x() x(+) x(+2) x( ) h() =6 h ( ) b h( ) z 37 m m m 2 38 The movig average filter.8.6 Deoiig a igal: m ( ) x ( ) = oiy igal = origial igal origial igal deoied igal -.8 deoied igal The movig average filter The i a IR filter of legth the impule repoe: h( ) ( ) h () = /

11 The movig average filter The movig average filter The i a IR filter of legth the impule repoe: h ( ) ( ) /,,, i ret H e h( ) wd ( ) h () =8 / H ( z) z z z The movig average filter The movig average filter The movig average filter output y ( ) x ( ) x ( )... x ( ) x ( ) x ( )... x ( ) y ( ) x ( ) y( ) y( ) x( ) x( ) The recurive equatio y( ) y( ) x( ) x( ) Correpod to Z traform Y( z) z X( z) z Y( z) z H( z) X ( z) z 43 44

12 LTIS aalyi i the frequecy domai The ytem trafer fuctio: H e h e arg H e H e e H e e Sytem havig a real h(): H e H e H e LTIS aalyi i the frequecy domai x() y() h() co x e e e e 2 2 y He e e 2 ( ) y e He e e 2 e H e e e 2 y He co ( ) LTIS aalyi i the frequecy domai x() y() h() y He co H e i the filter gai ad repreet the magitude frequecy characteritic; i the phae delay of the ytem ad repreet the phae frequecy characteritic; d The time group delay: d 47