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1 Chpter Chpter : Signls nd Systems Introduction Signls Smpling Periodic Signls Discrete-Time Sinusoidl Signls Rel Exponentil Signls Complex Exponentil Signl The Unit Impulse Simple Mnipultions o Discrete-Time Signls Systems Summry Chpter : Problem Sheet Chpter
2 Chpter : Signls nd Systems. Introduction The terms signls nd systems re given vrious interprettions. For exmple, system is n electric network consisting o resistors, cpcitors, inductors nd energy sources. Signls re vrious voltges nd currents in the network. The signls re thus unctions o time nd they re relted by set o equtions. Exmple: i(t) R + - i(t) C + v C (t) - Figure.: An electric circuit The objective o system nlysis is to determine the behviour o the system subjected to speciic input or excittion. It is oten convenient to represent system schemticlly by mens o box s shown in Figure.2. Input System Output Figure.2: Generl representtion o system. Chpter 2
3 .2 Signls There re two types o signls: () Continuous time signls (b) Discrete time signls In the cse o continuous-time signl, x(t), the independent vrible t is continuous nd thus x(t) is deined or ll t (see Figure.3). t Continuous time -independent vrible (- < t < ) On the other hnd, discrete-time signls re deined only t discrete times nd consequently the independent vrible tkes on only discrete set o vlues (see Figure.3). A discrete-time signl is thus sequence o numbers. n discrete time - independent vrible (n = -2, -, 0,, 2, ) Exmples:. A person s body temperture is continuous-time signl. 2. The prices o stocks printed in the dily newsppers re discrete-time signls. 3. Voltges & currents re usully represented by continuoustime signls. They re represented lso by discrete-time signls i they re speciied only t discrete set o vlues o t. Chpter 3
4 Figure.3: Above: An exmple o continuous-time signls. Below: An exmple o discrete-time signls..2. Smpling A discrete-time signl is oten ormed by smpling continuous -time signl x(t). I the smples re equidistnt then x n xt xnt t nt Squre brckets [ ] Discrete time signls Round Brckets ( ) Continuous signls (.) Anlogue Signl x t s T Digitl signl nt xn x The constnt T is the smpling intervl or period nd the smpling requency. Hz. s T Chpter 4
5 Figure.4: An exmple o cquiring discrete-time signls by smpling continuous-time signls. x[n] = { 3.5, 4, 3.25, 2, 2.5, 3.0 } n=- n=0 n=2 n=4 It is importnt to recognize tht x[n] is only deined or integer vlues o n. It is not correct to think o x[n] s being zero or n not n integer, sy n=.5. x[n] is simply undeined or noninteger vlues o n. Chpter 5
6 .2.. Smpling Theorem I the highest requency contined in n nlogue signl x(t) is mx nd the signl is smpled t rte s 2 mx then x(t) cn be exctly recovered rom its smple vlues using n interpoltion unction. Exmple: Audio CDs use smpling rte, s, o 44. khz or storge o the digitl udio signl. This smpling requency is slightly more thn 2 mx [ mx = 20kHz], which is generlly ccepted upper limit o humn hering nd perception o music sounds. Exmple: u( t) 0 t t 0 0 (.2) A continuous-time unit step unction u(t) is deined by Figure.5. Note tht the unit step is discontinuous t t = 0. Its smples u[n] = u(t) t=nt orm the discrete-time signl nd deined by u[ n] 0 n n 0 0 (.3) Chpter 6
7 Anlogue Signl Discrete - time Signl Figure.5: Top: Continuous-time unit step unction. Bottom: Discrete-time unit step unction. Chpter 7
8 Exmple: Sketch the wve orm: yn un un u[n] () u[n-] (b) n y[n] (c) n n Exmple: Sketch the wveorm or y t u t 2u t u t Chpter 8
9 Exmple: Anlogue Signl Discrete-time signl. x(t) = e t time Smple number [0,,2,3, ] t=nt Smpling Period (T) x[n] = e nt s T Hz smpling requency 2. x(t) = 0e -t 5e t t=nt x[n] = 0 -nt nt 5e smple number 3. x(t) = Acos( t) Anlogue requency in rdins = 2 t=nt x[n] = A cos( nt) Acos(2 Acos(2 n s s ) n) = digitl requency 2 = T s Acos( n ) Chpter 9
10 .2.2 Periodic Signls An importnt clss o signls is the periodic signls. A periodic continuous-time signl x(t) hs the property tht there is positive vlue o P or which or ll vlues o t. In other words, periodic signl hs the property tht is unchnged by time shit o P. In this cse we sy x(t) is periodic with period P. Exmple x t xt P (.4) x(t) period = P -P 0 P 2P t Figure.6: An exmple o periodic signls Periodic signls re deined nlogously in discrete time. A discrete-time signl x[n] is periodic with period N, where N is positive integer, i or ll vlues o n. x n xn N (.5) Chpter 0
11 Exmple: Figure.7: x[n] with Period = 3 smples Chpter
12 .2.3 Discrete-Time Sinusoidl Signls A continuous-time sinusoidl signl is given by x t Asin t Asin2 t (.6) = nlogue requency A discrete - time sinusoidl signl my be expressed s x[n] = x(t) t=nt = x(nt) x[ n] Asin( n T) x[ n] Asin( n ) Asin(2 s n) (.7) Smpling requency - Digitl requency s T 2 s T (.8) A discrete-time signl is sid to be periodic with period length N, i N is the smllest integer or which Asin which cn only be stisied or ll n i xn N xn n N Asinn N=2k (where k is n rbitrry integer) 2k N 2k 2 s Chpter 2 see eq. (.8)
13 N s k (.9) So i = 000Hz nd s = 8000 Hz then 8000 N smples An exmple o sinusoidl sequence is shown in Figure.8. Figure.8: An exmple o sinusoidl sequences. 2n The period, N, is 2 smples. x[ n] cos 2 Chpter 3
14 Exmple: Determine the undmentl period o x[n], x [ n ] 0cos 2 n 5 5 digitl requency 2 5 The undmentl period is thereore (see eqution (.9)) N 2k where k is the smllest integer or which N hs n integer vlue. This is stisied when k =. N smples Exmple: The sinusoidl signl x[n] hs undmentl period N=0 smples. Determine the smllest or which x[n] is periodic: 2k 2 k N 0 Smllest vlue o is obtined when k = 2 rdins/ cycle 0 5 Chpter 4
15 .2.4 Rel Exponentil Signls The continuous-time complex exponentil signl is o the orm t x t ce (.0 ) where c nd re, in generl complex numbers. Depending upon the vlues o these prmeters, the complex exponentil cn exhibit severl dierent chrcteristics. x(t) Growing exponentil >0. c t x(t) Decying exponentil <0 c Figure.9: Chrcteristics o rel exponentil signls in terms o time, t. Top: For >0, the signl grows exponentilly. Bottom: For <0, the signl decys exponentilly. t Chpter 5
16 .2.5 Complex Exponentil Signl Consider complex exponentil, ce t where c is expressed in j polr orm, c c e, nd in rectngulr orm, r j0. Then ce t c e j e c e e ( r j0 ) t rt j( 0t ) c e rt cos( t 0 ) j c e rt sin( t 0 ) (.) Thus, or r = 0, the rel & imginry prts o complex exponentil re sinusoidl. For r > 0 Sinusoidl signls multiplied by growing exponentil For r < 0 Sinusoidl signls multiplied by decying exponentil [ dmped sinusoids] x(t) r >0 x(t) r<0 t t Growing sinusoidl signl Decying sinusoidl signl Figure.0: Chrcteristics o complex exponentil signls. Chpter 6
17 In discrete time, it is common prctice to write rel exponentil signl s x[n] = c n (.2) I c nd re rel nd i > the mgnitude o the signl grows exponentilly with n, while i < we hve decying exponentil. Figure.3: Exmples o discrete-time exponentil signls..2.6 The Unit Impulse An importnt concept in the theory o liner systems is the continuous time unit impulse unction. This unction, known lso s the Dirc delt unction is denoted by (t) nd is represented grphiclly by verticl rrow. Chpter 7
18 (t) Mgnitude 0 t Frequency Figure.: Chrcteristic o the continuous-time impulse unction nd the corresponding mgnitude response in the requency domin. The impulse unction (t) is signl o unit re vnishing everywhere except t the origin. ( t) dt (t)=0 or t0 (.2) The impulse unction (t) is the derivtive o the step unction u(t). du( t) ( t) dt (.3) u(t) ( t) du( t) dt The discrete-time unit impulse unction [n] is deined in mnner similr to its continuous time counterprt. We lso reer [n] s the unit smple. t t Chpter 8
19 n 0 [ n] 0 n 0 (.4) Figure.2: Chrcteristic o discrete-time impulse unction. Chpter 9
20 .2.7 Simple Mnipultions o Discrete-Time Signls A signl x[n] my be shited in time by replcing the independent vrible n by n-k where k is n integer. I k>0 the time shit results in dely o the signl by k smples [ie. shiting signl to the right] I k<0 the time shit results in n dvnce o the signl by k smples. x[n-2] x[n+] k=2 k=- Figure.3: Top let: Originl signl, x[n]. Top right: x[n] is delyed by 2 smples. Bottom let: x[n] is dvnced by smple. Advnce: Shiting the signl to the let Dely: Shiting the signl to the right Chpter 20
21 .3 Systems A continuous-time system is one whose input x(t) nd output y(t) re continuous time unctions relted by rule s shown in Figure.4. x(t) t x(t) Continuous Time System y(t) Figure.4: Generl representtion o continuous-time systems. y(t) t A discrete system is one whose input x[n] nd output y[n] re discrete time unction relted by rule s shown in Figure.5. x[n] n x[n] Discrete Time System y[n] y[n] n Figure.5: Generl representtion o discrete-time systems. An importnt mthemticl distinction between continuous-time nd discrete-time systems is the ct tht the ormer re chrcterized by dierentil equtions wheres the ltter re chrcterized by dierence equtions. Chpter 2
22 Exmple: The RC circuit shown in Figure.6 is continuoustime system output e(t) input i(t) R + - i(t) C + v C (t) - Figure.6: A digrm o RC circuit s n exmple o continuous-time systems. I we regrd e(t) s the input signl nd vc(t) s the output signl, we obtin using simple circuit nlysis dvc ( t) dt RC v C ( t) RC e( t) (.5) From eqution (.5), discrete -time system cn be developed s ollows: I the smpling period T is suiciently smll, dvc ( t) vc ( nt ) vc ( nt T ) dt T tnt (.6) Chpter 22
23 v C (t) P v C (nt) v C (nt)-v C (nt-t) T nt-t nt t Bckwrd Euler pproximtion [Assuming T is suiciently smll] Figure.7: An pproximtion o discrete-time systems rom the continuous-time systems. By substituting eqution (.6) into (.5) nd replcing t by nt, we obtin: v nt v nt T T The dierence eqution is: RC nt ent C C vc RC v C [ n] v T C [ n ] RC v C [ n] RC e[ n] v C RC T [ n] vc[ n ] e[ n] RC T RC T (.7) dierence eqution output previous output input Chpter 23
24 Summry: Continuous-Time System Anlogue input Dierentil Equtions Anlogue output Digitl input Dierence Equtions Digitl output Discrete-Time System.4 Summry At the end o this chpter, it is expected tht you should know: The dierence between signls nd systems The smpling theorem, its limittions (e.g. lising), nd the smpling requency ( s ) How to distinguish between continuous (nlog) nd discrete time (digitl) signls How to distinguish between dierentil nd dierence equtions Continuous nd discrete periodic signls nd their deinitions The reltionship between nlog nd digitl requency 2 s The number o smples in period: = Digitl requency Mnipultion o discrete-time signls The unit impulse nd its properties 2 k N sk, Chpter 24
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