Chapter 2 Linear Time-Invariant Systems

Transcription

1 ELG 3 Sigals ad Sysems Caper Caper Liear Time-Ivaria Sysems. Iroducio May pysical sysems ca be modeled as liear ime-ivaria (LTI sysems Very geeral sigals ca be represeed as liear combiaios of delayed impulses. By e priciple of superposiio, e respose y of a discree-ime LTI sysem is e sum of e resposes o e idividual sifed impulses maig up e ipu sigal.. Discree-Time LTI Sysems: Te Covoluio Sum.. Represeaio of Discree-Time Sigals i Terms of Impulses A discree-ime sigal ca be decomposed io a sequece of idividual impulses. Eample: Fig.. Decomposiio of a discree-ime sigal io a weiged sum of sifed impulses. Te sigal i Fig.. ca be epressed as a sum of e sifed impulses:... 3 δ 3 δ δ δ δ δ... or i a more compac form (. δ. (. Tis correspods o e represeaio of a arbirary sequece as a liear combiaio of sifed ui impulse δ, were e weigs i e liear combiaio are. Eq. (. is called e sifig propery of e discree-ime ui impulse. / Yao

2 ELG 3 Sigals ad Sysems Caper.. Discree-Time Ui Impulse Respose ad e Covoluio Sum Represeaio of LTI Sysems Le be e respose of e LTI sysem o e sifed ui impulse δ, e from e superposiio propery for a liear sysem, e respose of e liear sysem o e ipu i Eq. (. is simply e weiged liear combiaio of ese basic resposes: y. (.3 If e liear sysem is ime ivaria, e e resposes o ime-sifed ui impulses are all ime-sifed versios of e same impulse resposes:. (.4 Terefore e impulse respose of a LTI sysem caracerizes e sysem compleely. Tis is o e case for a liear ime-varyig sysem: oe as o specify all e impulse resposes (a ifiie umber o caracerize e sysem. For e LTI sysem, Eq. (.3 becomes y. (.5 Tis resul is referred o as e covoluio sum or superposiio sum ad e operaio o e rig-ad side of e equaio is ow as e covoluio of e sequeces of ad. Te covoluio operaio is usually represeed symbolically as y. (.6..3 Calculaio of Covoluio Sum Oe way o visualize e covoluio sum of Eq. (.5 is o draw e weiged ad sifed impulse resposes oe above e oer ad o add em up. / Yao

3 ELG 3 Sigals ad Sysems Caper Eample: Cosider e LTI sysem wi impulse respose ad ipu, as illusraed i Fig....5 (a Te oupu respose based o Eq. (.5 ca be epressed y (b y.5 (c 3 Fig.. (a Te impulse respose of a LTI sysem ad a ipu o e sysem; (b e resposes o e ozero values of e ipu; (c e overall resposes. 3/ Yao

4 ELG 3 Sigals ad Sysems Caper Aoer way o visualize e covoluio sum is o draw e sigals ad as fucios of (for a fied, muliply em o form e sigal g, ad e sum all values of g. Eample: Calculae e covoluio of ad sow i Fig.. (a..5 -, < -, - - -, - -, 3-, 3 3 -, >3 Fig..3 Ierpreaio of Eq. (.5 for e sigals ad. 4/ Yao

5 ELG 3 Sigals ad Sysems Caper For <, y For, y. 5 For, y.5. 5 For, y.5. 5 For, y 3 For > 3, y Te resulig oupu values agree wi ose obaied i e precedig eample. Eample: Compue e respose of a LTI sysem described by is impulse respose, 6, 4 o e ipu sigal., oerwise, oerwise 3 4, > 3 4 To do e aalysis, i is coveie o cosider five separae iervals: For <, ere is o overlap bewee e ozero porios of ad, ad cosequely, y. For 4,,,, oerwise 5/ Yao

6 ELG 3 Sigals ad Sysems Caper 6/ Yao Tus, i is ierval y For 6 4 <, oerwise, 4, ( ( y. For 6 <, oerwise, 4 6 (, 4 6 y. Le 6 r, ( r r r r y. For 4 6 >, or >, ere is o overlap bewee e ozero porios of ad, ad ece, y. Te oupu is illusraed i e figure below. y

7 ELG 3 Sigals ad Sysems Caper. Coiuous-Time LTI sysems: e Covoluio Iegral Te respose of a coiuous-ime LTI sysem ca be compued by covoluio of e impulse respose of e sysem wi e ipu sigal, usig a covoluio iegral, raer a a sum... Represeaio of Coiuous-Time Sigals i Terms of Impulses A coiuous-ime sigal ca be viewed as a liear combiaio of coiuous impulses: ( ( δ ( d. (.7 Te resul is obaied by coppig up e sigal ( i secios of wid, ad aig sum ( 3 Recall e defiiio of e ui pulse δ ( ; we ca defie a sigal ˆ( as a liear combiaio of delayed pulses of eig ( ˆ( ( δ ( (.8 Taig e limi as, we obai e iegral of Eq. (.7, i wic we ( Te summaio approaces o a iegral ( ad ( ( (3 d (4 δ ( δ ( Eq. (.7 ca also be obaied by usig e samplig propery of e impulse fucio. If we cosider is fied ad is ime variable, e we ave ( δ ( ( δ ( ( ( δ (. Hece 7/ Yao

8 ELG 3 Sigals ad Sysems Caper ( δ ( d ( δ ( d ( δ ( d (. (.9 As i discree ime, is is e sifig propery of coiuous-ime impulse... Coiuous-Time Ui Impulse Respose ad e Covoluio Iegral Represeaio of a LTI sysem Te lieariy propery of a LTI sysem allows us o calculae e sysem respose o a ipu sigal ˆ( usig Superposiio Priciple. Le ˆ ( be e pulse respose of e liear-varyig sysem o e ui pulses δ ( for < <. Te respose of e sysem o ˆ( is y ˆ ( ( (. (. Noe a e respose ˆ ( eds o e impulse respose ( as. Te a e limi, we obai e respose of e sysem o e ipu sigal ( lim ˆ( : y ( lim yˆ( ( ( d. (. For a LTI sysem, e impulse resposes ( are e same as (, ecep ey are sifed by, a is, ( (. Te we may defie e ui impulse respose of e LTI sysem ( (, (. ad a LTI sysem is compleely deermied by is impulse respose. So e respose o e ipu sigal ( ca be wrie as a covoluio iegral: y ( ( d (, (.3 or i ca be epressed symbolically y( ( (. (.4 8/ Yao

9 ELG 3 Sigals ad Sysems Caper..3 Calculaio of covoluio iegral Te oupu y ( is a weiged iegral of e ipu, were e weig o ( is (. To evaluae is iegral for a specific value of, Firs obai e sigal ( (regarded as a fucio of wi fied from ( by a reflecio abou e origi ad a sif o e rig by if > or a sif o e lef by is <. Te muliply ogeer e sigals ( ad (. y ( is obaied by iegraig e resulig produc from o. Eample: Le ( be e ipu o a LTI sysem wi ui impulse respose (, were a ( e u(, a > ad ( u(. Sep: Te fucios (, ( ad ( are depiced ( ( ( < 9/ Yao

10 ELG 3 Sigals ad Sysems Caper ( > Sep : From e figure we ca see a for <, e produc of e produc ( ad ( is zero, ad cosequely, y ( is zero. For > a e, < < ( (, oerwise Sep 3: Compue y ( by iegraig e produc for > : y( e a a a d e ( e. a a Te oupu of y ( for all is a y( ( e u(, ad is sow i figure below. a y( a Eample: Compue e covoluio of e wo sigals below:, < < T ( ad, oerwise, < < T (, oerwise For is eample, i is coveie o calculae e covoluio i separae iervals. ( is seced ad ( is seced i eac of e iervals: / Yao

11 ELG 3 Sigals ad Sysems Caper For <, ad > 3T, ( ( for all e values of, ad cosequely y (. For oer iervals, e produc ( ( ca be foud i e figure o e e page. Tus for ese ree iervals, e iegraio ca be calculaed wi e resul sow below: / Yao

12 ELG 3 Sigals ad Sysems Caper ( T ( T < T ( ( ( T T < < T < < T T ( ( ( T T < < T T T < < T T T ( ( ( T T < < 3T T T T < < 3T T T ( T > 3T T / Yao

13 ELG 3 Sigals ad Sysems Caper 3/ Yao > < < < < < < < T T T T T T T T T T y 3, 3, 3,,, (.3 Properies of Liear Time-Ivaria Sysems LTI sysems ca be caracerized compleely by eir impulse respose. Te properies ca also be caracerized by eir impulse respose..3. Te Commuaive Propery of LTI Sysems A propery of covoluio i bo coiuous ad discree ime is a Commuaive Operaio. Ta is, (.5 d d ( ( ( ( ( ( ( (. (.6 y y.3. Te Disribuive Propery of LTI Sysems ( (.7 for bo discree-ime ad coiuous-ime sysems. Te propery meas a summig e oupus of wo sysems is equivale o a sysem wi a impulse respose equal o e sum of e impulse respose of e wo idividual sysems, as sow i e figure below. y( T T T 3

14 ELG 3 Sigals ad Sysems Caper y y Te disribuive propery of covoluio ca be eploied o brea a complicaed covoluio io several simpler oes. For eample, a LTI sysem as a impulse respose u, wi a ipu u u. Sice e sequece is ozero alog e eire ime ais. Direc evaluaio of suc a covoluio is somewa edious. Isead, we may use e disribuive propery o epress y as e sum of e resuls of wo simpler covoluio problems. Ta is, u, u, usig e disribuive propery we ave ( ( ( ( ( ( ( ( y y y.3.3 Te Associaive Propery ( (. (.8 for bo discree-ime ad coiuous-ime sysems. y * y For LTI sysems, e cage of order of e cascaded sysems will o affec e respose. 4/ Yao

15 ELG 3 Sigals ad Sysems Caper For oliear sysems, e order of cascaded sysems i geeral cao be caged. For eample, a wo memoryless sysems, oe beig muliplicaio by ad e oer squarig e ipu, e oupus are differe if e order is caged, as sow i e figure below. w w y4 w y.3.4 LTI sysem wi ad wiou memory A sysem is memoryless if is oupu a ay ime depeds oly o e value of is ipu a e same ime. Tis is rue for a discree-ime sysem, if for. I is case, e impulse respose as e form Kδ, (.9 were K is a cosa ad e covoluio sum reduces o e relaio y K. (. Oerwise e LTI sysem as memory. For coiuous-ime sysems, we ave e similar resuls if i is memoryless: ( Kδ (, (. y ( K(. (. Noe a if K i Eqs. (.9 ad (., e sysems become ideiy sysems, wi oupu equal o e ipu..3.5 Iveribiliy of LTI sysems We ave see a a sysem S is iverible if ad oly if ere eiss a iverse sysem S - suc a S - S is a ideiy sysem. y 5/ Yao

16 ELG 3 Sigals ad Sysems Caper Sice e overall impulse respose i e figure above is, mus saisfy for i o be e impulse respose of e iverse sysem, amely δ. ideiy sysem y Applicaios - cael equalizaio: for rasmissio of a sigal over a commuicaio cael suc as elepoe lie, radio li ad fiber, e sigal a e receivig ed is ofe processed roug a filer wose impulse respose is desiged o be e iverse of e impulse respose of e commuicaio cael. Eample: Cosider a sysem wi a pure ime sifed oupu y (. ( Te impulse respose of is sysem is ( δ (, sice ( ( δ (, a is, covoluio of a sigal wi a sifed impulse simply sifs e sigal To recover e sigal from e oupu, a is, o iver e sysem, all a is required is o sif e oupu bac. So e iverse sysem sould ave a impulse respose of δ, e δ δ ( δ ( ( Eample: Cosider e LTI sysem wi impulse respose u. Te respose of is sysem o a arbirary ipu is ( y u. Cosiderig a u is for < ad for, so we ave y. Tis is a sysem a calculaes e ruig sum of all e values of e ipu up o e prese ime, ad is called a summer or accumulaor. Tis sysem is iverible, ad is iverse is give as y, I is a firs differece operaio. Te impulse respose of is iverse sysem is δ δ, 6/ Yao

17 ELG 3 Sigals ad Sysems Caper We may cec a e wo sysems are really iverses o eac oer: { δ δ } u u δ * u *.3.6 Causaliy for LTI sysems A sysem is causal if is oupu depeds oly o e pas ad prese values of e ipu sigal. Specifically, for a discree-ime LTI sysem, is requireme is y sould o deped o for >. Based o e covoluio sum equaio, all e coefficies a muliply values of for > mus be zero, wic meas a e impulse respose of a causal discree-ime LTI sysem sould saisfy e codiio, for < (.3 A causal sysem is causal if is impulse respose is zero for egaive ime; is maes sese as e sysem sould o ave a respose before impulse is applied. A similar coclusio ca be arrived for coiuous-ime LTI sysems, amely (, for < (.4 Eamples: Te accumulaor u, ad is iverse δ δ are causal. Te pure ime sif wi impulse respose y ( for is causal, bu is o causal for <. ( >.3.7 Sabiliy for LTI Sysems Recall a a sysem is sable if every bouded ipu produces a bouded oupu. For LTI sysem, if e ipu is bouded i magiude B, for all If is ipu sigal is applied o a LTI sysem wi ui impulse respose, e magiude of e oupu y B (.5 y is bouded i magiude, ad ece is sable if 7/ Yao

18 ELG 3 Sigals ad Sysems Caper <. (.6 So discree-ime LTI sysem is sable is Eq. (.6 is saisfied. Te similar aalysis applies o coiuous-ime LTI sysems, for wic e sabiliy is equivale o ( d <. (.7 Eample: cosider a sysem a is pure ime sif i eier coiuous ime or discree ime. I discree ime, δ, wile i coiuous ime, ( d δ ( d, ad we coclude a bo of ese sysems are sable. Eample: Te accumulaor u is usable because u..3.8 Te Ui Sep Respose of a LTI Sysem Te sep respose of a LTI sysem is simply e respose of e sysem o a ui sep. I coveys a lo of iformaio abou e sysem. For a discree-ime sysem wi impulse respose, e sep respose is s u. However, based o e commuaive propery of covoluio, s u, ad erefore, s ca be viewed as e respose o ipu of a discreeime LTI sysem wi ui impulse respose. We ow a u is e ui impulse respose of e accumulaor. Terefore, s. (.8 From is equaio, ca be recovered from s usig e relaio s s. (.9 I ca be see e sep respose of a discree-ime LTI sysem is e ruig sum of is impulse respose. Coversely, e impulse respose of a discree-ime LTI sysem is e firs differece of is sep respose. 8/ Yao

19 ELG 3 Sigals ad Sysems Caper Similarly, i coiuous ime, e sep respose of a LTI sysem is e ruig iegral of is impulse respose, s( ( d, (.3 ad e ui impulse respose is e firs derivaive of e ui sep respose, ds( ( s'(. (.3 d Terefore, i bo coiuous ad discree ime, e ui sep respose ca also be used o caracerize a LTI sysem..4 Causal LTI Sysems Described by Differeial ad Differece Equaios Tis is a class of sysems for wic e ipu ad oupu are relaed roug A liear cosa-coefficie differeial equaio i coiuous ime, or A liear cosa-coefficie differece equaio i discree-ime..4. Liear Cosa-Coefficie Differeial Equaios I a causal LTI differece sysem, e discree-ime ipu ad oupu sigals are relaed implicily roug a liear cosa-coefficie differeial equaio. Le us cosider a firs-order differeial equaio, dy( y( (, (.3 d were y ( deoes e oupu of e sysem ad ( is e ipu. Tis equaio ca be eplaied as e velociy of a car y ( subjeced o fricio force proporioal o is speed, i wic ( would be e force applied o e car. I geeral, a N -order liear cosa coefficie differeial equaio as e form N a d y( d M b d (, (.33 d 9/ Yao

20 ELG 3 Sigals ad Sysems Caper Te soluio of e differeial equaio ca be obaied if we ave e N iiial codiios (or auiliary codiios o e oupu variable ad is derivaives. Recall a e soluio o e differeial equaio is e sum of e omogeeous soluio of e N d y( differeial equaio a (a soluio wi ipu se o zero ad of a paricular d soluio (a fucio a saisfy e differeial equaio. Forced respose of e sysem paricular soluio (usually as e form of e ipu sigal Naural respose of e sysem omogeeous soluio (depeds o e iiial codiios ad forced respose. dy( 3 Eample: Solve e sysem described by y( (. Give e ipu is ( Ke u(, d were K is a real umber. As meioed above, e soluio cosiss of e omogeeous respose ad e paricular soluio: y( y ( y (, (.34 p dy( were e paricular soluio y p ( saisfies y( ( ad omogeous soluio y ( d saisfies dy( y(. (.35 d For e paricular soluio for >, y p ( is a sigal a as e same form as ( for >, a is y 3 p ( Ye. (.36 3 Subsiuig ( Ke u( ad y dy(, we ge d 3 p ( Ye io y( ( 3 Ye Ye Ke, (.37 Cacelig e facor e 3 o bo sides, we obai Y K / 5, so a y K 5 3 p ( e, > (.38 / Yao

21 ELG 3 Sigals ad Sysems Caper To deermie e aural respose y ( of e sysem, we ypoesize a soluio of e form of a epoeial, y s ( Ae. (.39 Subsiuig Eq. (3.38 io Eq. (3.35, we ge s s Ase Ase, (.4 wic olds for s. Wi is value of s, dy( y( for ay coice of A. d Ae is a soluio o e omogeeous equaio Combiig e aural respose ad e forced respose, we ge e soluio o e differeial dy( equaio y( ( : d y y y Ae ( ( p ( K e 5 3, > (.4 Because e iiial codiio o y ( is o specified, so e respose is o compleely deermied, as e value of A is o ow. For causal LTI sysems defied by liear cosa coefficie differeial equaios, e iiial N ( ( codiios are always ( dy dy y..., wic is called iiial res. N d d For is eample, e iiial res implies a y (, so a soluio is K K y( A A, e 5 5 K 3 y( ( e e, > (.4 5 For <, e codiio of iiial res ad causaliy of e sysem implies a y (, <, sice (, <..4. Liear Cosa-Coefficie Differece Equaios I a causal LTI differece sysem, e discree-ime ipu ad oupu sigals are relaed implicily roug a liear cosa-coefficie differece equaio. / Yao

22 ELG 3 Sigals ad Sysems Caper I geeral, a N -order liear cosa coefficie differece equaio as e form N a M y b, (.43 Te soluio of e differeial equaio ca be obaied we we ave e N iiial codiios (or auiliary codiios o e oupu variable. Te soluio o e differece equaio is e sum of e omogeeous soluio N a y (a soluio wi ipu se o zero, or aural respose ad of a paricular soluio (a fucio a saisfy e differece equaio. y y y, (.44 p Te cocep of iiial res of e LTI causal sysem described by differece equaio meas a, < implies y, <. Eample: cosider e differece equaio y y, (.45 Te equaio ca be rewrie as y y, (.46 I ca be see from Eq. (.46 a we eed e previous value of e oupu, y, o calculae e curre value. Suppose a we impose e codiio of iiial res ad cosider e ipu Kδ. (.47 Sice for, e codiio of iiial res implies a y, for, so a we ave as a iiial codiio: y. Sarig from is iiial codiio, we ca solve for successive values of y for : y y K, / Yao

23 ELG 3 Sigals ad Sysems Caper y y K, y y K, y 3 y 3 K, y y K. 3 Sice for a LTI sysem, e ipu-oupu beavior is compleely caracerized by is impulse respose. Seig K,, δ we see a e impulse respose for e sysem is u. (.48 Noe a e causal sysem i e above eample as a impulse respose of ifiie duraio. I fac, if N i Eq. (.43, e differece equaio is recursive, i is usually e case a e LTI sysem correspodig o is equaio ogeer wi e codiio of iiial res will ave a impulse respose of ifiie duraio. Suc sysems are referred o as ifiie impulse respose (IIR sysems..4.3 Bloc Diagram Represeaios of s -order Sysems Described by Differeial ad Differece Equaios Bloc diagram iercoecio is very simple ad aure way o represe e sysems described by liear cosa-coefficie differece ad differeial equaios. For eample, e causal sysem described by e firs-order differece equaio is y ay b. (.49 I ca be rewrie as y ay b Te bloc diagram represeaio for is discree-ime sysem is sow: 3/ Yao

24 ELG 3 Sigals ad Sysems Caper b y D a y Tree elemeary operaios are required i e bloc diagram represeaio: addiio, muliplicaio by a coefficie, ad delay: adder a muliplicaio by a coefficie a a ui delay D Cosider e bloc diagram represeaio for coiuous-ime sysems described by a firs-order differeial equaio: dy( ay( b(. (.48 d Eq. (.48 ca be rewrie as dy( b y ( b(. a d a Similarly, e rig-ad side ivolves ree basic operaios: addiio, muliplicaio by a coefficie, ad differeiaio: 4/ Yao

25 ELG 3 Sigals ad Sysems Caper ( b / a y( D / a dy ( d ( ( adder ( ( ( a muliplicaio by a coefficie a( differeiaor ( D d ( d However, e above represeaio is o frequely used or e represeaio does o lead o pracical implemeaio, sice differeiaors are bo difficul o implemeed ad eremely sesiive o errors ad oise. A aleraive implemeaio is o used iegraors raer a e differeiaors. Eq. (.48 ca be rewrie as dy( b( ay(, (.49 d iegraig from o, ad assumig y (, e we obai b( ay( d y(. (.5 I is form, e sysem ca be implemeed usig e adder ad coefficie muliplier, ogeer wi a iegraor, as sow i e figure below. 5/ Yao

26 ELG 3 Sigals ad Sysems Caper iegraor ( ( d b ( y( a Te iegraor ca be readily implemeed usig operaioal amplifiers, e above represeaios lead direcly o aalog implemeaios. Tis is e basis for bo early aalog compuers ad moder aalog compuaio sysems. Eq. (.5 ca also epress i e form b( ay( d y( y(, (.5 were we cosider iegraig Eq. (.5 from a fiie poi i ime. I maes clear e fac a e specificaio of y ( requires a iiial codiio, amely y. Ay iger-order sysems ca be developed usig e bloc diagram for e simples firs-order differeial ad differece equaios. ( 6/ Yao