Discrete-Time Signals & Systems

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1 Chpter 2 Discrete-Time Sigls & Systems 清大電機系林嘉文 cwli@ee.thu.edu.tw Discrete-Time Sigls Sigls re represeted s sequeces of umbers, clled smples Smple vlue of typicl sigl or sequece deoted s x = {x} with x is defied oly for iteger vlues of d udefied for o-iteger vlues of Represettio of discrete-time sigls: 2, 0 Fuctiol represettio x 3, 0 Tbulr represettio Sequece represettio x() = {,0.2, 2.2,., 0.2, -3.7, 2.9. } 20/3/2 Digitl Sigl Processig 2

2 Discrete-Time Sigls Grphicl represettio 20/3/2 Digitl Sigl Processig 3 Discrete-Time Sigls Smplig speech sigl ( ), xx T 20/3/2 Digitl Sigl Processig 4 2

3 Bsic Sequeces Uit smple sequece -, 0 0, 0 Uit step sequece -, 0 u 0, 0 u 0 u u 20/3/2 Digitl Sigl Processig 5 Bsic Sequeces Uit rmp sigl - u r, 0 0, 0 Rel expoetil sigl - is rel vlue x A 0 < > /3/2 Digitl Sigl Processig 6 3

4 Bsic Sequeces Complex expoetil sigl - x x jx R R cos x A I j j0 j0 x A A e e e 0 0 si 0 A cos j I si x A 0 20/3/2 Digitl Sigl Processig 7 Bsic Sequeces Complex expoetil sigl - x x jx x x R I x A r x 20/3/2 Digitl Sigl Processig 8 4

5 Bsic Sequeces Siusoidl sigls with differet frequecies 20/3/2 Digitl Sigl Processig 9 Bsic Sequeces A rbitrry sequece c be represeted i the timedomi s weighted sum of some bsic sequece d its delyed (dvced) versios p p 20/3/2 Digitl Sigl Processig 0 5

6 The orm of Discrete-Time Sigl Size of Sigl - give by the orm of the sigl L p -orm: where p is positive iteger The vlue of p is typiclly or 2 or L 2 -orm x 2 is the root-me-squred (rms) vlue of {x} L -orm x is the me bsolute vlue of {x} L -orm is the pe bsolute vlue of {x} (why?) x x p x p p x x mx 20/3/2 Digitl Sigl Processig Clssifictio of Discrete-Time Sigls Periodic sigls d periodic sigls A sigl is periodic with period ( > 0) if d oly if x x for ll The smllest vlue of for which the bove coditio holds is clled the (fudmetl) period A sigl ot stisfyig the periodicity coditio is clled operiodic or periodic 20/3/2 Digitl Sigl Processig 2 6

7 Clssifictio of Discrete-Time Sigls Cojugte-symmetric sequece: * x x If x is rel, the it is eve sequece for cojugte-symmetric sequece {x}, x0 must be rel umber 20/3/2 Digitl Sigl Processig 3 Clssifictio of Discrete-Time Sigls Cojugte-tisymmetric sequece: * x x If x is rel, the it is odd sequece for cojugte ti-symmetric sequece {y}, y0 must be imgiry umber 20/3/2 Digitl Sigl Processig 4 7

8 Clssifictio of Discrete-Time Sigls Ay complex sequece c be expressed s sum of its cojugte-symmetric d cojugte-tisymmetric prts: x xcs xc where * x xx * xc xx 2 Ay rel sequece c be expressed s sum of its eve prt d its odd prt: where cs 2 x x x ev 20/3/2 Digitl Sigl Processig 5 od xev x x 2 xod x x 2 Clssifictio of Discrete-Time Sigls Periodic sigls d periodic sigls A sigl is periodic with period ( > 0) if d oly if x x for ll The smllest vlue of for which the bove coditio holds is clled the (fudmetl) period A sigl ot stisfyig the periodicity coditio is clled operiodic or periodic 20/3/2 Digitl Sigl Processig 6 8

9 Clssifictio of Discrete-Time Sigls Eergy sigls d power sigls The totl eergy of sigl x() is defied by E x 2 A ifiite legth sequece with fiite smple vlues my or my ot be eergy sigl (with fiite eergy) The verge power of discrete-time sigl x is defied by P lim x 2 2 Defie the sigl eergy of x() over the fiite itervl s E 2 x 20/3/2 Digitl Sigl Processig 7 Clssifictio of Discrete-Time Sigls Eergy sigls d power sigls The sigl eergy c the be expressed s E lim E The verge power of x() becomes lim P E 2 If E is fiite, P = 0. O the other hd, if E is ifiite, the verge power P my be either fiite or ifiite If P is fiite (d ozero), the sigl is clled power sigl 20/3/2 Digitl Sigl Processig 8 9

10 Clssifictio of Discrete-Time Sigls Eergy sigls d power sigls Exmple Determie the power d eergy of the uit step sequece The verge power of the uit step sigl is lim P lim It s power sigl with ifiite eergy Exmple - Cosider the cusl sequece defied by 3, 0 x 0, 0 ote: x() hs ifiite eergy, its verge power is 20/3/2 Digitl Sigl Processig 9 P lim Clssifictio of Discrete-Time Sigls A ifiite eergy sigl with fiite verge power is clled power sigl Exmple - A periodic sequece which hs fiite verge power but ifiite eergy A fiite eergy sigl with zero verge power is clled eergy sigl Exmple - A fiite-legth sequece which hs fiite eergy but zero verge power 20/3/2 Digitl Sigl Processig 20 0

11 Clssifictio of Discrete-Time Sigls A sequece x is sid to be bouded if Exmple - The sequece x = cos0.3π is bouded sequece s x A sequece x is sid to be bsolutely summble if x cos(0.3 ) B x Exmple - The followig sequece is bsolutely summble 0.3, 0 y 0, 0 x 20/3/2 Digitl Sigl Processig 2 Clssifictio of Discrete-Time Sigls A sequece x is sid to be squre summble if Exmple - The sequece 2 x si(0.4 ) h is squre-summble but ot bsolutely summble 20/3/2 Digitl Sigl Processig 22

12 Mipultio of Discrete-Time Sigls (/5) Trsformtio of idepedet vrible (time) Time shiftig: A sigl x my be shifted i time by replcig the idepedet vrible by 20/3/2 Digitl Sigl Processig 23 Mipultio of Discrete-Time Sigls (2/5) Trsformtio of idepedet vrible (time) Foldig/Reflectio: A sigl x my be folded i time by replcig the idepedet vrible by 20/3/2 Digitl Sigl Processig 24 2

13 Mipultio of Discrete-Time Sigls (3/5) The opertios of foldig d time delyig (or dvcig) sigl re OT commuttive Deote the time-dely opertio by TD d the foldig opertio by FD TD x x ow wheres, 0 FDx x TD FD x TD x x FD TD x FD x x TD FD x 20/3/2 Digitl Sigl Processig 25 Mipultio of Discrete-Time Sigls (4/5) Trsformtio of idepedet vrible (time) Time Sclig or dow-smplig: A sigl x my be scled i time by replcig by 20/3/2 Digitl Sigl Processig 26 3

14 Mipultio of Discrete-Time Sigls (5/5) Trsformtio of idepedet vrible (time) Additio, multiplictio, d sclig of sequeces: Amplitude modifictios iclude dditio, multiplictio, d sclig of discrete-time Amplitude sclig of sigl by costt : y Ax, Sum of two sigls: 2 Product of two sigls: yx + x, yx x 2, 20/3/2 Digitl Sigl Processig 27 Discrete-Time Systems Discrete-time system: A device or lgorithm tht performs some prescribed opertio o discrete-time sigl (iput or excittio) to produce other discretetime sigl (output or respose) We sy tht the iput sigl x is trsformed by the system ito sigl y s expressed below yt x 20/3/2 Digitl Sigl Processig 28 4

15 Iput-Output Descriptio of Systems The iput-output descriptio or discrete-time system cosists of mthemticl expressio or rule, which explicitly defies the reltio betwee the iput d output sigls T x y Exmple: Determie the respose of the followig systems to the iput sigl, 33 x 0, otherwise () y x (b) y x x x 3 (d) y x (c) y medi x, x, x 20/3/2 Digitl Sigl Processig 29 Lier Systems: Accumultor Accumultor - y xl l x l x y x l The output y is the sum of the iput smple x d the previous output y The system cumultively dds, i.e., it ccumultes ll iput smple vlues Iput-output reltio c lso be writte i the form y xl xl y xl, 0 l l0 l0 The secod form is used for cusl iput sequece, i which cse y is clled the iitil coditio 20/3/2 Digitl Sigl Processig 30 5

such tht K umbers from the set hve vlues greter th this umber d the other K umbers hve vlues smller Medi c be determied by r-orderig the umbers i the

16 Lier Systems: Movig Averge M 2 y x M M2 M M M 2 x M x M... x x... x M2 A pplictio: Cosider x = s + d where s = 2(0.9) is the sigl corrupted by rdom oise d 20/3/2 Digitl Sigl Processig 3 olier Systems: Medi Filter (/3) The medi of set of (2K+) umbers is the umber such tht K umbers from the set hve vlues greter th this umber d the other K umbers hve vlues smller Medi c be determied by r-orderig the umbers i the set by their vlues d choosig the umber t the middle Exmple: Cosider the set of umbers {2, 3, 0, 5, } R-order set is give by { 3,, 2, 5, 0} medi{2, 3, 0, 5, } = 2 20/3/2 Digitl Sigl Processig 32 6

17 olier Systems: : Medi Filter (2/3) Medi Filterig Exmple 20/3/2 Digitl Sigl Processig 33 olier Systems: Medi Filter (3/3) Medi Filterig Exmple Origil Imge oisy Imge (pepper-d-slt oise) Filtered Imge 20/3/2 Digitl Sigl Processig 34 7

18 Bloc Digrm Represettio of Discrete-Time Systems Adder Costt multiplier 20/3/2 Digitl Sigl Processig 35 Bloc Digrm Represettio of Discrete-Time Systems Sigl multiplier/modultor Uit dely elemet Uit dvce elemet 20/3/2 Digitl Sigl Processig 36 8

19 Bloc Digrm Represettio of Discrete-Time Systems Exmple: y y x x 20/3/2 Digitl Sigl Processig 37 Sttic vs. Dymic Systems A discrete-time system is clled sttic or memoryless if its output t y time istt depeds t most o the iput smple t the sme time yt x, y y x bx If discrete-time system is ot sttic, it is sid to be dymic or to hve memory y x 3x (fiite memory) y x 0 x 3 (ifiite memory) 20/3/2 Digitl Sigl Processig 38 9

20 Time (Shift) Ivrice Time-ivrit vs. time-vrit systems A system is clled time-ivrit if its iput-output chrcteristics do ot chge with time y Tx Defiitio: A relxed system T is time-ivrit or shift-ivrit if d oly if x( ) T y( ) Implies tht x( ) T y( ) For every iput sigl x() d every time shift. I geerl, we c write the output of time-ivrit system s y (, ) T x ( ) 20/3/2 Digitl Sigl Processig 39 Time (Shift) Ivrice Exmples time ivrit time vrit y T x xx, y, y y x x y x x yt x x y, x y x y, y 20/3/2 Digitl Sigl Processig 40 20

21 Time (Shift) Ivrice Exmples time vrit yt x x y, x y x y, y time vrit T cos0 y, x cos0 cos0 y, y y x x y x 20/3/2 Digitl Sigl Processig 4 Lierity (/3) A lier system is oe tht stisfies the superpositio priciple Defiitio: A system T is lier if d oly if x 2x2 x 2 x2 T T T for y rbitrry iput sequeces x d x 2, d y rbitrry costts d 2. Multiplictive/sclig property: Suppose tht 2 =0 T x T x y Additivity property: Suppose tht = 2 = x x2 x x2 y y2 T T T 20/3/2 Digitl Sigl Processig 42 2

22 Lierity (2/3) Grphicl represettio of the superpositio priciple T is lier if d oly if y =y 20/3/2 Digitl Sigl Processig 43 Lierity (3/3) Lier vs. o-lier systems The lier coditio c be exteded rbitrrily to y weighted lier combitio of sigls where M M T x x y y,,2,, y T x M If system produces ozero output with zero iput, it my be either o-relxed or olier Exmples: ()y =x, (b) y =x 2, (c) y = x 2, (d) y =Ax +B, (e)y =e x 20/3/2 Digitl Sigl Processig 44 22

23 Cuslity Cusl vs. o-cusl systems Defiitio: A system is sid to be cusl if the output of the system t y time depeds oly o preset d pst iputs, but does ot deped o future iputs T y x, x, x2, where T{ } is some rbitrry fuctio. ocusl vs. ticusl If system produces ozero output with zero iput, it my be either o-relxed or olier Exmples: ()y =x x, (b) y =x + 3x+4, (c) y =x 2, (d) y =x2, (e) y =x 20/3/2 Digitl Sigl Processig 45 Stbility Bouded-Iput, Bouded Output (BIBO) stbility If y is the respose to iput x d if x Bx for ll vlues of the y By for ll vlues of Exmple the M-poit movig verge filter is BIBO stble M y x M 0 With bouded iput x Bx M M y x x M M M 0 0 MB x B x 20/3/2 Digitl Sigl Processig 46 23

24 Pssive & Lossless Systems A discrete-time system is defied to be pssive if, for every fiite-eergy iput x, the output y hs, t most, the sme eergy y 2 For lossless system, the bove iequlity is stisfied with equl sig for every iput Exmple - Cosider the discrete-time system defied by y =α x with positive iteger Its output eergy is give by pssive system if ǀαǀ <, d lossless if ǀαǀ = x y x 2 20/3/2 Digitl Sigl Processig 47 Itercoectio of Discrete-Time Systems Cscde itercoectio T x 2 2 y yt y T T x Systems T d T 2 c be combied or cosolidted ito sigle overll system yt x where T TT c I geerl TT 2 TT 2. However, if systems T d T 2 re LTI, the () is time ivrit d (b) TT TT c /3/2 Digitl Sigl Processig 48 24

25 Itercoectio of Discrete-Time Systems Prllel itercoectio Tx T2x TT2x y y y 3 2 Tp x We c use prllel d cscde itercoectio of systems to costruct lrger, more complex systems 20/3/2 Digitl Sigl Processig 49 Techiques for the Alysis of Lier Systems Two bsic methods for lyzig the behvior of lier system: The first is bsed o the direct solutio of the iputoutput equtio y y b x 0 The secod method is to decompose or resolve the iput sigl ito sum of elemetry sigls. The, usig the lierity of the system, the respose of the system to the elemetry sigls re sum to obti the totl respose M 20/3/2 Digitl Sigl Processig 50 25

26 Techiques for the Alysis of Lier Systems Suppose the iput sigl is resolved ito weighted sum of elemetry sigls x cx The respose y of the system to the compoet x is y T x If the system is lier, we hve y T x= T cx c T x c y 20/3/2 Digitl Sigl Processig 5 Why & how to do the sigl decompositio? Resolutio of Discrete-Time Sigl ito Impulses Select the elemetry sigls x to be x where represets the dely of the uit smple sequece Multiply the two sequeces x d? x x 20/3/2 Digitl Sigl Processig 52 26

27 Resolutio of Discrete-Time Sigl ito Impulses Cosequetly x x Exmple - Cosider fiite-durtio sequece give s The sequece c be resolved s x 2, 4,0, x 20/3/2 Digitl Sigl Processig 53 Resolutio of Discrete-Time Sigl ito Impulses The respose of relxed lier system to the uit smple sequece iput: y, h, T If the impulse t the iput is scled by s, x h, ch If the iput is expressed s x x The output becomes y T x T x T, x x h 20/3/2 Digitl Sigl Processig 54 27

28 Respose of LTI Systems to Arbitrry Iputs If the system is time ivrit, d deote the respose of the LTI system to the uit smple sequece s h T The respose of the system to is Cosequetly T h y x h The relxed LTI system is completely chrcterized by sigle fuctio h, the impulse respose. Covolutio is commuttive y x h h x 20/3/2 Digitl Sigl Processig 55 Computig the Covolutio Sum The output of LTI system t = 0 is give by To compute y y x h Foldig. Fold h bout = 0 to obti h Shiftig. Shift h by 0 to the right (left) if is positive (egtive), to obti h 0 Multiplictio. Multiply x by h 0 to obti the product sequece v 0 xh0 Summtio. Sum ll the vlues of v 0 to obti y 0 20/3/2 Digitl Sigl Processig 56 28

29 Computig the Covolutio Sum x, 2,3, h, 2,,,0,0,,4,8,8,3, 2,,0,0, y 20/3/2 Digitl Sigl Processig 57 Computig the Covolutio Sum y x * h x * h x 2 x0 x3 * h x 2 2 x0 x3 3 * h x 2( 2* h ) x0( * h ) x3( 3* h ) x 2 h 2 x0 h x3 h 3 y y y /3/2 Digitl Sigl Processig 58 29

30 30 Computig the Covolutio Sum 20/3/2 Digitl Sigl Processig 59 Tbulr Method of Covolutio Sum Computtio 20/3/2 Digitl Sigl Processig ) ( * h g h g h g h g y

31 3 20/3/2 Digitl Sigl Processig 6 Exmple:, h u x u y y y 2 y lim y y Computig the Covolutio Sum 20/3/2 Digitl Sigl Processig 62 Computig the Covolutio Sum r r r r ote u u u x h h x y u x u u h M M : if, ) ( 0 if ) ( 0 if 0, * 0 0 ) ( 0 0,,

32 Properties of Covolutio (/2) Commuttive Property y x h x h hx hx Idetity d Shiftig Properties y x x x y x 20/3/2 Digitl Sigl Processig 63 Properties of Covolutio (2/2) Associtive Property xh h2 xh2 h xh h2 Distributive Property 2 2 x h h x h x h 20/3/2 Digitl Sigl Processig 64 32

33 Cuslity of LTI Systems (/2) The output of LTI system t = 0 is give by 0 0 y x h Divide the sum ito two sets of terms: y hx h x h0x 0hx 0 hx 0 h2x 0 2 deped o preset d pst iputs deped o future iputs For cusl system, h = 0 for < 0 Sice h is the respose of the relxed LTI system to uit impulse sequece t = 0, LTI system is cusl if d oly if its impulse respose is zero for egtive vlues of 20/3/2 Digitl Sigl Processig 65 Cuslity of LTI Systems (2/2) The output of cusl LTI system becomes y h x x h 0 A sequece x is clled cusl sequece if x = 0 for < 0; otherwise, it s ocusl sequece If the iput to cusl LTI system is cusl sequce, the iput-output equtio reduces to Exmple: Determie the uit step respose of the LTI system with impulse respose h u, y 20/3/2 Digitl Sigl Processig 66 y h x x h

34 Stbility of LTI Systems (/3) BIBO Stbility Coditio - A discrete-time system is BIBO stble if d oly if the output sequece {y} remis bouded for ll bouded iput sequece {x} A LTI discrete-time system is BIBO stble if d oly if its impulse respose sequece {h} is bsolutely summble, i.e. B h h Proof: Assume h is rel sequece Sufficiet coditio: Sice the iput sequece x is bouded we hve x ( ) Bx therefore y hx h x B h BB x x h 20/3/2 Digitl Sigl Processig 67 Stbility of LTI Systems (2/3) Thus, B h < implies ǀyǀ B x B h <, idictig tht y is lso bouded To prove the ecessry coditio, ssume y is bouded, i.e., ǀyǀ B y Cosider the bouded iput give by * h, h 0 x h 0, h 0 For this iput, y t = 0 is 2 h y0 xh Bh h Therefore, if B h =, the {y} is ot bouded sequece 20/3/2 Digitl Sigl Processig 68 34

35 Stbility of LTI Systems (3/3) Exmple - Cosider cusl LTI discrete-time system with impulse respose For this system h Therefore B h < if <, for which the system is BIBO stble If =, the system is ot BIBO stble u Bh u, if 0 20/3/2 Digitl Sigl Processig 69 35

=> PARALLEL INTERCONNECTION. Basic Properties LTI Systems. The Commutative Property. Convolution. The Commutative Property. The Distributive Property

=> PARALLEL INTERCONNECTION. Basic Properties LTI Systems. The Commutative Property. Convolution. The Commutative Property. The Distributive Property Lier Time-Ivrit Bsic Properties LTI The Commuttive Property The Distributive Property The Associtive Property Ti -6.4 / Chpter Covolutio y ] x ] ] x ]* ] x ] ] y] y ( t ) + x( τ ) h( t τ ) dτ x( t) * h(