Chapter 1 Signal and Systems

Transcription

1 ELG 3 Sigals ad Sysems Chaper Chaper Sigal ad Sysems. Coiuous-ime ad discree-ime Sigals.. Examples ad Mahemaical represeaio Sigals are represeed mahemaically as fucios of oe or more idepede variables. Here we focus aeio o sigals ivolvig a sigle idepede variable. For coveiece, his will geerally refer o he idepede variable as ime. There are wo ypes of sigals: coiuous-ime sigals ad discree-ime sigals. Coiuous-ime sigal: he variable of ime is coiuous. A speech sigal as a fucio of ime is a coiuous-ime sigal. Discree-ime sigal: he variable of ime is discree. The weekly Dow Joes sock marke idex is a example of discree-ime sigal. x() x[ x[] x[-] x[-] x[] x[] Fig.. Graphical represeaio of coiuousime sigal. Fig.. Graphical represeaio of discree-ime sigal. To disiguish bewee coiuous-ime ad discree-ime sigals we use symbol o deoe he coiuous variable ad o deoe he discree-ime variable. Ad for coiuous-ime sigals we will eclose he idepede variable i pareheses ( ), for discree-ime sigals we will eclose he idepede variable i bracke [ ]. A discree-ime sigal x [ may represe a pheomeo for which he idepede variable is iherely discree. A discree-ime sigal x [ may represe successive samples of a uderlyig pheomeo for which he idepede variable is coiuous. For example, he processig of speech o a digial compuer requires he use of a discree ime sequece represeig he values of he coiuous-ime speech sigal a discree pois of ime. / Yao

2 ELG 3 Sigals ad Sysems Chaper.. Sigal Eergy ad Power If v () ad i () are respecively he volage ad curre across a resisor wih resisace R, he he isaaeous power is p ( ) = v( ) i( ) = v ( ). (.) R The oal eergy expeded over he ime ierval is ( ) d = p v ( ) d, (.) R ad he average power over his ime ierval is p( ) d = v R ( ) d. (.3) For ay coiuous-ime sigal x () or ay discree-ime sigal x [, he oal eergy over he ime ierval i a coiuous-ime sigal x () is defied as x ( ) d, (.4) where x deoes he magiude of he (possibly complex) umber x. The ime-averaged power is x( ) d ierval is defied as. Similarly he oal eergy i a discree-ime sigal x [ over he ime x [ (.5) The average power is + x[ I may sysems, we will be ieresed i examiig he power ad eergy i sigals over a ifiie ime ierval, ha is, for + or +. The oal eergy i coiuous ime is he defied E T T = T = lim x( ) d x( ) d, (.6) / Yao

3 ELG 3 Sigals ad Sysems Chaper ad i discree ime E + N = lim x[ = N N + x[. (.7) For some sigals, he iegral i Eq. (.6) or sum i Eq. (.7) migh o coverge, ha is, if x () or x [ equals a ozero cosa value for all ime. Such sigals have ifiie eergy, while sigals wih E < have fiie eergy. The ime-averaged power over a ifiie ierval T P = lim x( ) d T T (.8) T P N = lim x[ (.9) N N + + N Three classes of sigals: Class : sigals wih fiie oal eergy, E < ad zero average power, P E = lim = T T (.) Class : wih fiie average power P. If P >, he E =. A example is he sigal x [ = 4, i has ifiie eergy, bu has a average power of P =6. Class 3: sigals for which eiher P ad E are fiie. A example of his sigal is x ( ) =.. Trasformaios of he idepede variable I may siuaios, i is impora o cosider sigals relaed by a modificaio of he idepede variable. These modificaios will usually lead o reflecio, scalig, ad shif... Examples of Trasformaios of he Idepede Variable 3/ Yao

4 ELG 3 Sigals ad Sysems Chaper x[ x[- ] (a) (b) Fig..3 Discree-ime sigals relaed by a ime shif. x() x(-) Fig..4 Coiuous-ime sigals relaed by a ime shif. x[ x[- (a) (b) Fig..5 (a) A discree-ime sigal x [; (b) is reflecio, x[ abou =. x() x(-) (a) (b) Fig..6 (a) A coiuous-ime sigal x () ; (b) is reflecio, x( ) abou =. 4/ Yao

5 ELG 3 Sigals ad Sysems Chaper x() x() (a) x(/) (b) (c) Fig..7 Coiuous-ime sigals relaed by ime scalig... Periodic Sigals A periodic coiuous-ime sigal x () has he propery ha here is a posiive value of T for which x ( ) = x( + T ) for all (.) From Eq. (.), we ca deduce ha if x () is periodic wih period T, he x ( ) = x( + mt ) for all ad for all iegers m. Thus, x () is also periodic wih period T, 3T,. The fudameal period T of x () is he smalles posiive value of T for which Eq. (.) holds. x() Fig..8 Coiuous-ime periodic sigal. 5/ Yao

6 ELG 3 Sigals ad Sysems Chaper A discree-ime sigal x [ is periodic wih period N, where N is a ieger, if i is uchaged by a ime shif of N, x [ = x[ + N] (.) for all values of. If Eq. (.) holds, he x [ is also periodic wih period N, 3 N,. The fudameal period N is he smalles posiive value of N for which Eq. (.) holds. x[ Fig..9 Discree-ime periodic sigal...3 Eve ad Odd Sigals I addiio o heir use i represeig physical pheomea such as he ime shif i a radar sigal ad he reversal of a audio ape, rasformaios of he idepede variable are exremely useful i examiig some of he impora properies ha sigal may possess. Sigal wih hese properies ca be eve or odd sigal, periodic sigal: A impora fac is ha ay sigal ca be decomposed io a sum of wo sigals, oe of which is eve ad oe of which is odd. x() x() (a) (b) Fig.. A eve coiuous-ime sigal; (b) a odd coiuous-ime sigal. 6/ Yao

7 ELG 3 Sigals ad Sysems Chaper EV = (.3) { x( ) } [ x( ) + x( ) ] which is referred o as he eve par of x (). Similarly, he odd par of x () is give by OD = (.4) { x( ) } [ x( ) x( ) ] Exacly aalogous defiiios hold i he discree-ime case. x[, x[ =, < x[ EV{ x[ }, =,, < = > (a) (b) x[ OD (c) { x[ }, =,, < = > Fig.. The eve-odd decomposiio of a discree-ime sigal..3 Expoeial ad siusoidal sigals.3. Coiuous-ime complex expoeial ad siusoidal sigals The coiuous-ime complex expoeial sigal x = a ( ) Ce (. 5) where C ad a are i geeral complex umbers. 7/ Yao

8 ELG 3 Sigals ad Sysems Chaper Real expoeial sigals x() x() C C (a) (b) a Fig.. The coiuous-ime complex expoeial sigal x ( ) = Ce, (a) a > ; (b) a <. Periodic complex expoeial ad siusoidal sigals If a is purely imagiary, we have jω x( ) e = (.6) A impora propery of his sigal is ha i is periodic. We kow x () is periodic wih period T if jω jω ( + T ) jω jω T = = (.7) e e e e For periodiciy, we mus have jω T e = (.8) For ω, he fudameal period T is π T = (.9) ω Thus, he sigals e jω ad e jω have he same fudameal period. A sigal closely relaed o he periodic complex expoeial is he siusoidal sigal x ( ) = Acos( ω + φ) (.) Wih secods as he ui of, he uis of φ ad ω are radias ad radias per secod. I is also kow ω = πf, where f has he ui of circles per secod or Hz. 8/ Yao

9 ELG 3 Sigals ad Sysems Chaper The siusoidal sigal is also a periodic sigal wih a fudameal period of T. x( ) = A cos( ω + φ ) π T = ω A A cosφ Fig..3 Coiuous-ime siusoidal sigal. Usig Euler s relaio, a complex expoeial ca be expressed i erms of siusoidal sigals wih he same fudameal period: e = cosω + j si (.) jω ω Similarly, a siusoidal sigal ca also be expressed i erms of periodic complex expoeials wih he same fudameal period: A A Acos( ω e e e e jφ jω jφ jω + φ) = + (.) A siusoid ca also be expresses as j { } ( ω + φ e ) A cos( ω + φ) = ARe (.3) ad j { } ( ω+ φ e ) A si( ω + φ) = A Im (.4) Periodic sigals, such as he siusoidal sigals provide impora examples of sigal wih ifiie oal eergy, bu fiie average power. For example: E period = jω e d = d = T T T (.5) P period = T T T jω e d = d = (.6) 9/ Yao

10 ELG 3 Sigals ad Sysems Chaper Sice here are a ifiie umber of periods as rages from o +, he oal eergy iegraed over all ime is ifiie. The average power is fiie sice P T = lim e T T T jω d = (.7) Harmoically relaed complex expoeials: jkω φ k ( ) = e, =, ±, ±,... k (.8) ω is he fudameal frequecy. Example: j j 3 j.5 j.5 j.5 j.5 Sigal x( ) = e + e ca be expressed as x( ) = e ( e + e ) = e cos(.5), he magiude of x () is x ( ) = cos(.5), which is commoly referred o as a full-wave recified siusoid, show i Fig..4. x() 4 π π π 4π Geeral complex Expoeial sigals Fig..4 Full-wave recified siusoid. Cosider a complex expoeial a Ce expressed i recagular form. The, where a = r + jω is jθ C = C e is expressed i polar ad Ce a = C e jθ e ( r+ jω ) r j ( ω+ θ ) r = C e e = C e θ cos( ω + θ) + j C e si( ω + r ). (.9) Thus, for r =, he real ad imagiary pars of a complex expoeial are siusoidal. For r >, siusoidal sigals muliplied by a growig expoeial. For r <, siusoidal sigals muliplied by a decayig expoeial. Damped sigal Siusoidal sigals muliplied by decayig expoeials are commoly refereed o as damped sigal. / Yao

11 ELG 3 Sigals ad Sysems Chaper x() x() (a) (b) Fig..5 (a) Growig siusoidal sigal; (b) decayig siusoidal sigal..3. Discree-ime complex expoeial ad siusoidal sigals A discree complex expoeial or sequece is defied by x [ = Cα, (.3) where C ad α are i geeral complex umbers. This ca be aleraively expressed x β [ = Ce, (.3) β where α = e. Real Expoeial Sigals If C ad α are real, we have he real expoeial sigals. x[ x[ x[ (a) (b) x[ (c) (d) / Yao

12 ELG 3 Sigals ad Sysems Chaper Fig..6 Real Expoeial Sigal x [ = Cα : (a) α >; (b) <α <; (c) <α <; (d) α <-. Siusoidal Sigals jω x[ e = (.3) e jω = cosω + j si ω (.33) Similarly, a siusoidal sigal ca also be expresses i erms of periodic complex expoeials wih he same fudameal period: A A A cos( jφ jω jφ jω ω + φ) = e e + e e (.34) A siusoid ca also be expresses as j { } ( ω+φ e ) A cos( ω + φ) = ARe (.35) ad j { } ( ω+φ e ) A si( ω + φ) = AIm (.36) The above sigals are examples of discree sigals wih ifiie oal eergy, bu fiie average jω power. For example: every sample of x[ = e coribues o he sigal s eergy. Thus he oal eergy < < + is ifiie, while he average power is equal o. / Yao

13 ELG 3 Sigals ad Sysems Chaper Fig..7 Discree-ime siusoidal sigal. Geeral Complex Expoeial Sigals Cosider a complex expoeial Cα, where α = α e jθ jω C = C e ad, he θ Cα = C α cos( ω + θ ) + j C α si j( ω + ). (.37) Thus, for α =, he real ad imagiary pars of a complex expoeial are siusoidal. For α <, siusoidal sigals muliplied by a decayig expoeial. For α >, siusoidal sigals muliplied by a growig expoeial. 3/ Yao

14 ELG 3 Sigals ad Sysems Chaper (a) (b) Fig..8 (a) Growig siusoidal sigal; (b) decayig siusoidal sigal..3.3 Periodiciy Properies of Discree-Time Complex Expoeials There are a umber of impora differeces bewee coiuous-ime ad discree-ime jω siusoidal sigals. The coiuous-ime sigals e are disic for disic values of ω. For discree-ime sigals, however, hese values are o disic because he sigal wih ω is ideical o he sigals wih frequecies ω ± π, ω 4π ±, ad so o, j ( ω j j e ± π ) ( ω ± 4π e ) ω = = e. (.38) I cosiderig discree-ime expoeials, we eed oly cosider a frequecy ierval of π. I mos occasios, we will use he ierval ω < π or π ω < π. jω The discree-ime sigal x[ = e does o have a coiuously icreasig rae of oscillaio as ω is icreased i magiude, bu as ω is icreased from, he sigal oscillaes more ad more rapidly uil ω reaches π, ad whe ω is coiuously icreased, he rae of oscillaio 4/ Yao

15 ELG 3 Sigals ad Sysems Chaper decreases uil ω reaches π. We coclude ha he low-frequecy discree-ime expoeials have values of ω ear, π, ad ay oher eve muliple of π, while he high-frequecies are locaed ear ω = ± π ad oher odd muliples of π. I order for he sigal jω x[ e = o be periodic wih period N >, we mus have jω N ) j e ( + = e ω, (.39) or equivalely j e ω N =. (.4) For Eq. (.4) o hold, ω N mus be a muliple of π. Tha is, here mus be a ieger m such ha ω N = πm, (.4) or equivalely ω π = m N. (.4) From Eq. (.4), oherwise. jω x[ e = is a periodic if ω / π is a raioal umber ad is o periodic The fudameal frequecy of he discree-ime sigal jω x[ e = is π ω =, (.43) N m ad he fudameal period of he sigal ca be π N = m. (.44) ω The compariso of he coiuous-ime ad discree-ime sigals are summarized i he able below: 5/ Yao

16 ELG 3 Sigals ad Sysems Chaper Table Compariso of he sigals e jω ad e j ω. e jω Disic sigals for disic values of ω Ideical sigals for values of ω separaed by muliples of π Periodic for ay choice of ω Periodic oly if ω = πm / N for some iegers N > ad m. Fudameal frequecy ω Fudameal frequecy ω / m Fudameal period Fudameal period ω = : udefied ω = : udefied ω : π ω ω : π m ω e jω Example: Suppose ha we wish o deermie he fudameal period of he discree-ime sigal x[ j (π / 3) j(3π / 4) = e + e (.45) Soluio: The firs expoeial o he righ had side has a fudameal period of 3. The secod expoeial has a fudameal period of 8. For he eire sigal o repea, each of he erms i Eq. (.45) mus go hrough a ieger umber of is ow fudameal period. The smalles icreme of he accomplished his is 4. Tha is, over a ierval of 4 pois, he firs erm will have goe hrough 8 of is fudameal periods, ad he secod erm hrough hree of is fudameal periods, ad he overall sigal hrough exacly oe of is fudameal periods. Harmoically relaed periodic expoeials jk (π / N ) φ k [ = e,,,... k = ± (.46) I he coiuous-ime case, all of he harmoically relaed complex expoeials k =, ±,..., are disic. Bu his is o he case for discree-ime sigals: jk e ( π / N ) j( k + N )(π / N ) j( k π / N ) j π φ k + N [ = e = e e = φk [ (.47) There are oly N disic period expoeials i he se give i Eq. (.46)., 6/ Yao

17 ELG 3 Sigals ad Sysems Chaper.4 The Ui Impulse ad Ui Sep Fucios The ui impulse ad ui sep fucios i coiuous ad discree ime are cosiderably impora i sigal ad sysem aalysis..4. The discree-time Ui Impulse ad Ui Sep Sequeces Discree-ime ui impulse is defied as, δ [ =, (.48), = δ [ Discree-ime ui sep is defied as Fig..9 Discree-ime ui impulse., < u [ =, (.49), u [ Fig.. Discree-ime ui sep sequece. The discree-ime impulse ui is he firs differece of he discree-ime sep δ [ = u[ u[ ], (.5) The discree-ime ui sep is he ruig sum of he ui sample: 7/ Yao

18 ELG 3 Sigals ad Sysems Chaper u [ = δ [ m], (.5) m= I ca be see ha for <, he ruig sum is zero, ad for, he ruig sum is. If we chage he variable of summaio from m o k = m we have, u[ = δ [ k]. k= The ui impulse sequece ca be used o sample he value of a sigal a =. Sice δ [ is ozero oly for =, i follows ha x[ δ [ = x[] δ[. (.5) More geerally, a ui impulse δ ], he [ x δ [ ] = x[ ] δ[ ] (.53) [ This samplig propery is very impora i sigal aalysis..4. The Coiuous-Time Ui Sep ad Ui Impulse Fucios Coiuous-ime ui sep is defied as, < u ( ) =, (.54), u() Fig.. Coiuous-ime ui sep fucio. The coiuous-ime ui sep is he ruig iegral of he ui impulse u( ) = δ ( τ ) dτ. (.55) The coiuous-ime ui impulse ca also be cosidered as he firs derivaive of he coiuousime ui sep, 8/ Yao

19 ELG 3 Sigals ad Sysems Chaper du( ) δ ( ) =. (.56) d Sice u () is discoiuous a = ad cosequely is formally o differeiable. This ca be ierpreed, however, by cosiderig a approximaio o he ui sep u (), as illusraed i he figure below, which rises from he value of o he value i a shor ime ierval of legh. u () δ () (a) (b) Fig.. (a) Coiuous approximaio o he ui sep u (); (b) Derivaive of u (). The derivaive is du ( ) δ ( ) =, (.57) d, < δ ( ) =, (.58), oherwise as show i Fig... Noe ha δ () is a shor pulse, of duraio ad wih ui area for ay value of. As, δ () becomes arrower ad higher, maiaiig is ui area. A he limi, δ ( ) = limδ ( ), (.59) u( ) = limu ( ), (.6) ad 9/ Yao

20 ELG 3 Sigals ad Sysems Chaper du( ) δ ( ) =. (.6) d Graphically, δ () is represeed by a arrow poiig o ifiiy a =, ex o he arrow represes he area of he impulse. δ () kδ ( ) k Fig..3 Coiuous-ime ui impulse. Samplig propery of he coiuous-ime ui impulse: x( ) δ ( ) = x() δ ( ), (.6) Or more geerally, x ) δ ( ) = x( ) δ ( ) (.63) ( Example: Cosider he discoiuous sigal x () x( ) x&() Fig..4 The discoiuous sigal ad is derivaive. / Yao

21 ELG 3 Sigals ad Sysems Chaper Noe ha he derivaive of a ui sep wih a discoiuiy of size of k gives rise o a impulse of area k a he poi of discoiuiy..5 Coiuous-Time ad Discree-Time Sysems A sysem ca be viewed as a process i which ipu sigals are rasformed by he sysem or cause he sysem o respod i some way, resulig i oher sigals as oupus. Examples R + + v s () - C v ( ) i() - (a) f () (b) Fig.. 5 Examples of sysems. (a) A sysem wih ipu volage v s () ad oupu volage v ( ). (b) A sysem wih ipu equal o he force f () ad oupu equal o he velociy v (). A coiuous-ime sysem is a sysem i which coiuous-ime ipu sigals are applied ad resuls i coiuous-ime oupu sigals. Coiuous-ime x () y() sysem A discree-ime sysem is a sysem i which discree-ime ipu sigals are applied ad resuls i discree-ime oupu sigals. Discree-ime x [ y[ sysem / Yao

22 ELG 3 Sigals ad Sysems Chaper.5. Simple Examples of Sysems Example : Cosider he RC circui i Fig. 5 (a). The curre i () is proporioal o he volage drop across he resisor: vs ( ) vc ( ) i( ) =. (.64) R The curre hrough he capacior is dvc ( ) i( ) = C. (.65) d Equaig he righ-had sides of Eqs..64 ad.65, we obai a differeial equaio describig he relaioship bewee he ipu ad oupu: dvc ( ) + vc ( ) = vs ( ), (.66) d RC RC Example : Cosider he sysem i Fig. 5 (b), where he force f () as he ipu ad he velociy v() as he oupu. If we le m deoe he mass of he car ad ρ v he resisace due o fricio. Equaig he acceleraio wih he e force divided by mass, we obai dv( ) dv( ) = [ f ( ) ρv( ) ] + ρ v( ) = f ( ). (.67) d m d m m Eqs..66 ad.77 are wo examples of firs-order liear differeial equaios of he form: dy( ) + ay( ) = bx( ). (.66) d Example 3: Cosider a simple model for he balace i a bak accou from moh o moh. Le y [ deoe he balace a he ed of h moh, ad suppose ha y [ evolves from moh o moh accordig he equaio: y [ =.y[ ] + x[, (.67) or y [.y[ ] = x[, (.68) where x [ is he e deposi (deposis mius wihdraws) durig he h moh.y [ ] models he fac ha we accrue % ieres each moh. / Yao

23 ELG 3 Sigals ad Sysems Chaper Example 4: Cosider a simple digial simulaio of he differeial equaio i Eq. (.67), i which we resolve ime io discree iervals of legh ad approximae dv ( ) / d( ) a = by he firs backward differece, i.e., v( ) v(( ) ) Le v [ = v( ) ad f [ = f ( ), we obai he followig discree-ime model relaig he sampled sigals v [ ad f [, m v[ v[ ] = f [. (.69) ( m + ρ ) ( m + ρ ) Comparig Eqs..68 ad.69, we see ha hey are wo examples of he firs-order liear differece equaio, ha is, y [ + ay[ ] = bx[. (.7) Some coclusios: Mahemaical descripios of sysems have grea deal i commo; A paricular class of sysems is referred o as liear, ime-ivaria sysems. Ay model used i describig ad aalyzig a physical sysem represes a idealizaio of he sysem..5. Iercoecs of Sysems Ipu Sysem Sysem Oupu (a) Sysem Ipu + Oupu Sysem (b) 3/ Yao

24 ELG 3 Sigals ad Sysems Chaper Sysem Sysem Ipu + Oupu Sysem 3 (c) Fig..6 Iercoecio of sysems. (a) A series or cascade iercoecio of wo sysems; (b) A parallel iercoecio of wo sysems; (c) Combiaio of boh series ad parallel sysems. Ipu + Sysem Oupu Sysem Fig..7 Feedback iercoecio. R s + V s ± V i - A Vo V f R R R L (a) v s + + v i = v v s f BASIC AMPLIFIER v L = A v i A - Feedback Sigal v = β f v L FEEDBACK NETWORK FB v L (b) 4/ Yao

25 ELG 3 Sigals ad Sysems Chaper Fig..8 A feedback elecrical amplifier..6 Basic Sysem Properies.6. Sysems wih ad wihou Memory A sysem is memoryless if is oupu for each value of he idepede variable as a give ime is depede oly o he ipu a he same ime. For example: y[ = (x[ x [ ]), (.7) is memoryless. A resisor is a memoryless sysem, sice he ipu curre ad oupu volage has he relaioship: v ( ) = R i( ), + (.7) v() where R is he resisace. Oe paricularly simple memoryless sysem is he ideiy sysem, whose oupu is ideical o is ipu, ha is y ( ) = x( ), or y [ = x[ A example of a discree-ime sysem wih memory is a accumulaor or summer. i() - y[ = x[ k] = x[ k] + x[ = k = k = y[ ] + x[, or (.73) y [ y[ ] = x[. (.74) Aoher example is a delay y [ = x[ ]. (.75) A capacior is a example of a coiuous-ime sysem wih memory, i() + v ( ) = i( τ ) dτ, (.76) C v() - 5/ Yao

26 ELG 3 Sigals ad Sysems Chaper where C is he capaciace..6. Iveribiliy ad Iverse Sysem A sysem is said o be iverible if disic ipus leads o disic oupus. x[ Sysem y[ Iverse sysem w[=x[ x() y()=x() y() w()=.5y() w()=x() x[ y() y [ = x[ k] w [ = y[ y[ ] w [ = x[ k= Examples of o-iverible sysems: y [ =, Fig..9 Cocep of a iverse sysem. he sysem produces zero oupu sequece for ay ipu sequece. y ( ) = x ( ), i which case, oe cao deermie he sig of he ipu from he kowledge of he oupu. Ecoder i commuicaio sysems is a example of iverible sysem, ha is, he ipu o he ecoder mus be exacly recoverable from he oupu..6.3 Causaliy A sysem is causal if he oupu a ay ime depeds oly o he values of he ipu a prese ime ad i he pas. Such a sysem is ofe referred o as beig oaicipaive, as he sysem oupu does o aicipae fuure values of he ipu. The RC circui i Fig. 5 (a) is causal, sice he capacior volage respods oly o he prese ad pas values of he source volage. The moio of a car is causal, sice i does o aicipae fuure acios of he driver. 6/ Yao

27 ELG 3 Sigals ad Sysems Chaper The followig expressios describig sysems ha are o causal: y [ = x[ x[ + ], (.77) ad y ( ) = x( + ). (.78) All memoryless sysems are causal, sice he oupu respods oly o he curre value of ipu. Example: Deermie he Causaliy of he wo sysems: () y[ = x[ () y ( ) = x( )cos( + ) Soluio: Sysem () is o causal, sice whe <, e.g. = 4, we see ha y [ 4] = x[4], so ha he oupu a his ime depeds o a fuure value of ipu. Sysem () is causal. The oupu a ay ime equals he ipu a he same ime muliplied by a umber ha varies wih ime..6.4 Sabiliy A sable sysem is oe i which small ipus leads o resposes ha do o diverge. More formally, if he ipu o a sable sysem is bouded, he he oupu mus be also bouded ad herefore cao diverge. Examples of sable sysems ad usable sysems: R + - v s ( ) C ( ) i() - v + f () (a) (b) The above wo sysems are sable sysem. The accumulaor bouded. y [ = x[ k] is o sable, sice he sum grows coiuously eve if x [ is k = 7/ Yao

28 ELG 3 Sigals ad Sysems Chaper Check he sabiliy of he wo sysems: S; y ( ) = x( ) ; S: y ( ) = e x( ) S is o sable, sice a cosa ipu x ( ) =, yields y ( ) =, which is o bouded o maer wha fiie cosa we pick, y () will exceed he cosa for some. S is sable. Assume he ipu is bouded ha y () is bouded B B e < y( ) < e. x ( ) < B, or B < x( ) < B for all. We he see.6.5 Time Ivariace A sysem is ime ivaria if a ime shif i he ipu sigal resuls i a ideical ime shif i he oupu sigal. Mahemaically, if he sysem oupu is y () whe he ipu is x (), a imeivaria sysem will have a oupu of y ) whe ipu is x ). ( ( Examples: The sysem y ( ) = si[ x( )] is ime ivaria. The sysem y [ = x[ is o ime ivaria. This ca be demosraed by usig couerexample. Cosider he ipu sigal x [ = δ[ ], which yields y [. However, = he ipu x[ = δ [ ] yields he oupu y[ = δ [ ] = δ[ ]. Thus, while x [ ] is he shifed versio of x [ ], y [ ] is o he shifed versio of y [ ]. The sysem y ( ) = x() is o ime ivaria. To check usig couerexample. Cosider x ( ) show i Fig..3 (a), he resulig oupu y ( ) is depiced i Fig..3 (b). If he ipu is shifed by, ha is, cosider x ) = x ( ), as show i Fig..3 (c), we obai ( he resulig oupu y ( ) = x () show i Fig..3 (d). I is clearly see ha y ) y ( ), so he sysem is o ime ivaria. ( 8/ Yao

29 ELG 3 Sigals ad Sysems Chaper x ( ) y ( ) x ( ) = x ( ) (a) (b) (c) y ( ) y ( ) (d) (e) 3 Fig..3 Ipus ad oupus of he sysem y ( ) = x()..6.6 Lieariy The sysem is liear if The respose o x ) + x ( ) is y ) + y ( ) - addiiviy propery ( ( ay ( ( The respose o ax ) is ) - scalig or homogeeiy propery. The wo properies defiig a liear sysem ca be combied io a sigle saeme: Coiuous ime: ax ) + bx ( ) ay ( ) + by ( ), ( [ + bx[ ay[ by[. Discree ime: ax + ] Here a ad b are ay complex cosas. Superposiio propery: If x k [, k =,, 3,... are a se of ipus wih correspodig oupus y k [, k =,, 3,..., he he respose o a liear combiaio of hese ipus give by x[ = ak xk [ = a x[ + ax[ + a3x3[ +..., (.79) is k 9/ Yao

30 ELG 3 Sigals ad Sysems Chaper y[ = ak yk [ = a y[ + a y[ + a3 y3[ +..., (.8) k which holds for liear sysems i boh coiuous ad discree ime. For a liear sysem, zero ipu leads o zero oupu. Examples: The sysem y ( ) = x( ) is a liear sysem. The sysem y ( ) = x ( ) is o a lier sysem. The sysem y [ = Re{ x[ }, is addiive, bu does o saisfy he homogeeiy, so i is o a liear sysem. The sysem y [ = x[ + 3 is o liear. y [ = 3 if x [ =, he sysem violaes he zeroi/zero-ou propery. However, he sysem ca be represeed as he sum of he oupu of a liear sysem ad aoher sigal equal o he zero-ipu respose of he sysem. For sysem y [ = x[ + 3, he liear sysem is x[ x[, ad he zero-ipu respose is y [ = 3 as show i Fig..3. y ( ) x() Liear sysem + y() Fig..3 Srucure of a icremeally liear sysem. y ( ) is he zero-ipu respose of he sysem. The sysem represeed i Fig..3 is called icremeally liear sysem. The sysem respods liearly o he chages i he ipu. The overall sysem oupu cosiss of he superposiio of he respose of a liear sysem wih a zero-ipu respose. 3/ Yao