Soluions o Problems rom Chper Problem. The signls u() :5sgn(), u () :5sgn(), nd u h () :5sgn() re ploed respecively in Figures.,b,c. Noe h u h () :5sgn() :5; 8 including, bu u () :5sgn() is undeined..5 u()-.5sgn() u ()-.5sgn() h u ()-.5sgn() no deined.5.5 () (b) ( c) Figure. Problem. This signl is presened in Figure.. shed lines denoe priculr signls. u(+ ) u(-+ ) p () Figure. Problem.3 This signl is presened in Figure.3. The dshed lines denoe priculr signls. p () sgn() u() sgn() - - - - () - -3 Figure.3 Noe h he impornce o he usge o he Heviside uni sep uncion in digil iler design nd smpling is discussed in recen pper: L. Jckson, A Correcion o Impulse Invrince, IEEE Signl Processing Leers, Vol. 7, 73 75,.
CHAPTER Problem.4 The required signls re presened respecively in Figures.4,b,c,d. () u(-+) ( c) r(-+3) 3 (b) u(--) (d) r(--) - - - Figure.4 Problem.5 The grphs o he required signls re ploed respecively in Figures.5,b,c,d. () () (b) () r(-) u(-3) 3 -r(-+) 3 4 -u(-4) 3() ( c) (d) () 4 r() u(-) -r(--) 3 4 - -u(-5) Figure.5
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC Problem.6 The signl () u( + ) + u( ) r( ) + r( 3) is presened in Figure.6 wih dshed lines denoing priculr signls. ()u(+)+u(-)-r(-)+r(-3) r(-3) u(+) u(-) - 3 -r(-) Figure.6 Problem.7 Srigh orwrd soluions re given by () () r() r( ) + r( ); (b )() r( ) r( 3) u( ) (c) 3() r( ) r( ) u( ) Ploing hese uncions we obin he desired grphs. Noe h oher soluions re possible since he presenion is no unique. Problem.8 See commens mde in he soluions o Problem.7. In his problem, one o possible severl soluions is given by () u() r( ) + r( 3) + u( 5) The corresponding signls re ploed in Figure.7 using dshed lines. ()u()-r(-)+r(-3)+u(-5) r(-3) u() u(-5) 3 4 5 -r(-) Figure.7 3
CHAPTER Problem.9 This signl is ploed using MATLAB nd presened in Figure.8. Commen: Noe h he MATLAB uncion sepun(-k,k); k-4::6; does no produces he correc resul. 4 3 [k] p4[k ] r[k]u[ k+] Signl [k] 4 3 3 4 5 6 iscree ime k Figure.8 Problem. This signl is ploed in Figure.9..8.6 [k] ri4*u[ k+] p[k ]*u[k ].4. Signl [k].8.6.4. 4 3 3 4 5 6 iscree ime k Figure.9 4
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC Problem. [3] sinc[3] r[3 + 5] + 6 [3]p 4 [3] sinc[3] r[] + sin 3 3 Problem. Using he noion rom Secion., he nlyicl represenion o he discree-ime ringulr pulse is 8> < m[k] ; k m ; k m + m k; m k m k; k m Noe h m, s deined in ormul (.4) is n even ineger. This signl cn be represened in erms o discreeime rmp signls s ollows m[k] h m r k m i h r[k] + r k + m i Problem.3 I is known rom (.9) h () (m) ()d () m (m) () On he oher hnd, we hve () (m) ()d () m () (m) ()d () m (m) () due o he c h d m ()d m () m d m ()d m or. I ollows h or m even he righ-hnd sides o he bove wo expressions re idenicl, which implies h (m) () (m) (); m n. For m odd we hve () m (m) () () m (m) () (m) (), which implies h (m) () (m) (); m n +. Problem.4 Sring wih he inegrl () () ( )d () d d ( )gd ssuming h >, nd using he chnge o vribles s, we obin + d d () d + For <, he sme derivions imply () () ( )d d d ()d + () ()d + () ()d () () ; > + () ()d () ; < 5
The second inegrl CHAPTER () () (( ))d or > nd wih he chnge o vribles s ( ) produces Similrly or <, we hve Problem.5 Sring wih he inegrl () () (( ))d + () ()d () () ( ); > () (n) ( )d + () ()d () ( ); < () dn d n ( )gd ssuming h >, nd using he chnge o vribles s, we obin + n dn d n () d + d n n n For <, he sme derivions imply The inegrl n + (n) ()d n () n (n) d n ()d ; > () (n) ( )d n + (n) ()d + (n) ()d n () n (n) () (n) (( ))d () dn d n (( ))gd ; < is evlued similrly, ssuming h >, nd using he chnge o vribles s ( ), which leds o + n dn d n () d d n n + d n ()d n + (n) ()d n () n (n) ( ); > 6
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC For <, he sme derivions imply () (n) (( ))d n + (n) ()d n () n (n) ( ); < Problem.6 Using he ormul derived in Problem.5 we hve + e 3 () (3 + 5)d 3() d d 8e 3 9j 5 3 3 9 e 3 5 3 7e 5 The second inegrl ccording o he ormul derived in Problem.4 is zero. This cn be lso jusiied by direc inegrion s 4 () ( + )d 4 4 () d ( )d 4 ()gd 4 d d() 4()j Problem.7 By he siing propery o he del impulse uncion, or given uncion g() (noe we use here o denoe requency), we hve g()()d g() I insed o () we use (!), wih!, we obin rom (.7), in his cse he requency scling propery h g()( )d g() This leds o he conclusion h (!) ( ) (). Problem.8 The curren in his elecricl circui or he consn volge E nd wih no iniil volge on he cpcior is given by i() E R e RC ; In he cse when R!, he mgniude o o he curren ends o ininiy nd he durion o he curren ends o zero since lim ne RC! ; >. Hence, he circui curren ends o he del impulse uncion. R! Problem.9 The curren in his elecricl circui is given by i() C dv C() d C " ; < < " ; > " When "!, he mgniude o he curren ends o ininiy nd is wih ends o zero. Hence, he curren ends o he del impulse uncion. 7
Problem. + sin() + e h() () + 3 CHAPTER + 3( ) + 4 () ( )id + 4() () () + 3 4 + sin h i () () + 3( ) + 4 () ( ) d + e 4 4 + cos () e 4 Problem. sin() + e h (3) () () ( )id h i () (3) () () ( ) d () 3 (3) () () () () 3 sin () cos () e cos () sin () + e cos () + sin () e Problem. () 5 e 4 u()( 4)d e 44 u(4) e 6 (b) 5 3 e ( 6)d since ( 6) is ouside o he inegrion limis (c) 3 5 5 e sin ( 3)( 5)d ()( 5)d (5) e sin () 3 (d) 3 5 5 e ( + 3)d ()( + 3)d (3) e 3 3 3 Problem.3 cos ()( ) + i e 3h () ( ) + ( 3) d + + n ()( 3)d + 4 3 5( + )d cos ()( ) + () 9e 3 + e3( 3 ) + + 5 ( ) + 9e 3 + e( 9 ) + 5 Problem.4* e ( ) + h i sin (( )) () ( ) + ( ) 3 e ( ) + () sin (( )) + sin (( )) + 4 n (4) + 5 e ( ) + 4 d + + n ()( 4)d + 4 5( + )d n (4) + 5 8
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC Noe h he irs erm ws evlued using he resul esblished in (.3), h is ()( ) ( ) nd h we used lso he ollowing resul (see (.)) (b)( )d b + () d b ; > Problem.5 () e 5 sin ( + 4)( ) ()( ) ()( ) e 5 sin (6)( ) (b) () k cos [k ][k 4] [k][k 4] [4][k 4] () 4 cos [7][k 4] Problem.6 () The scled nd shied recngulr pulse signls re deined by ; p(3) 3, 3 3 ; ; p(3 ) 3, 3 ; elsewhere ; elsewhere p4(4( 5)) ; 4( 5), :5 + 5 :5 + 5, 4:5 5:5 ; elsewhere The plos o hese signls re presened in Figure.. p (3) p (3-) p (4(-5)) 4 -/3 /3 /3 4 5 6 Figure. (b) The nlyicl expressions or he scled nd shied sep signls re given by ; u( 3) 3, 3 ; 3 + ; u(3 + ), 3 ; elsewhere ; elsewhere Grphicl presenions o hese signls re given in Figure.. u(-3) u(-3+) /3 Figure. 9
CHAPTER Problem.7 The rmp signl is deined by r() u(); > nd r() ; <. Using his deiniion, we hve Noe lso h r(4( + )) 4( + )u(4( + )); 4( + ) >, < ; elsewhere u(4( + )) ; 4( + ) >, < ; elsewhere u( ) The scled nd shied sep nd recngulr pulse signls re nlyiclly given by u(3 ) ; 3, 3 ; p ( 4) ; elsewhere The bove signls re ploed in Figure.. 4 8; < ; elsewhere ; 4, 5 3 ; elsewhere r(-4(+)) 4 u(-3-) p (--4) -4-3 - - -4 - -/3 - -4-3 - () (b) ( c) Figure. Problem.8 Generlized derivives or he signls deined in Problem.5 re given by () 8 >< ; < undened; ; < < 3 ; ( 3); 3 ; > 3 () 8 >< ; < undened; ; < < 4 ( 4); 4 ; > 4 3 () 8 >< ; < undened; ; < < ; ( ); ; > 4 () 8 >< ; < undened; ; < < 5 ( 5); 5 ; > 5 The generlized derivive or he signl deined in Problem.6 is given by () 8 >< ; < ( + ); ; < < ( ); ; < < 3 undened; 3 ; > 3
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC Problem.9 Using he derivive produc rule nd he propery o he del impulse uncion, which ses h ()( ) ( )( ), we obin () u( ) + ( + )( ) sin ()u( 3) + cos ()( 3) 4e 4 sin () + e 4 cos () u( ) + 4( ) sin ()u( 3) + cos (3)( 3) + e 4 (cos () 4 sin ()) Problem.3 The grphs re presened in Figures.3 nd.4 ()r(-)-r()u(-+)+p () 4 r(-) u(-+) p () 4 - - -r()u(-+) -r() Figure.3 r(--) () u(+) -3 - - 3 4 -u(-+) - Figure.4 -r(-4) The corresponding generlized derivives re given by () 8>< ; ( + ); ; < < undened; ; < < undened; ; > ; () 8 >< ; < undened; ; < < ( + ); < < ( ); ; < < 4 undened; 4 > 4
CHAPTER Problem.3 Signls () nd (g) re nicusl since hey re dieren rom zero or negive ime. Signls (b), (c), (d), (e), (), nd (h) re cusl since ll o hem re equl o zero or <. Problem.3 The required generlized derivives o he signls in FIGURE. re given by () () 8 >< ; < undened; ; < < ( + ) ( ) ( )( ) ( ); ; < < ; ( + ) ( ) ( )( ) ( ); ; < < 3 (3 + ) (3 ) ( ())( 3) ( 3); 3 ; 3 < (b) () 8 >< ; < undened; ; < < ( + ) ( ) ( )( ) ( ); ; < < 3 undened; 3 ; 3 < < 4 (4 + ) (4 ) ( )( 4) ( 4); 4 ; 4 < These generlized derivives re grphiclly represened in Figure.5. () () (b) () (-3) 3 3 - (-) - (-) - (-) - (-4) Figure.5 Problem.33 The regulr derivive d ()d is presened in Figure.6. The generlized derivive () conins, in ddiion, he impulse del uncions ( :3); ( :9); ( ). Noe h FIGURE x.y denoes igure rom he ex book nd Figure x.y denoes igure rom he soluion mnul.
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC d() d...3.4.5.6.7.8.9 - Figure.6 Problem.34 Since n inegrl is equl o he re beween he uncion (signl) nd he horizonl xis, we hve : : ()d + : : : : + : : : Hence, he signl inegrion hs smoohing eec. Problem.35 sin ()u()g cos ()u() + sin ()() cos ()u() + () cos ()u() sin ()u()g sin ()u()g cos ()u()g sin ()u() + () 3 sin ()u()g 3 4 sin ()u()g 3 4 sin ()u()g sin ()u() + ()g cos ()u() + () () n o cos ()u() + () () sin ()u() () + () () 3 sin ()u()g Problem.36 u()g () + u() () + u() u() u()g u()g u()g () 3 3 u()g u()g ()g () () 3
4 4 u()g 3 CHAPTER 3 u()g n o () () () () Problem.37 4 3 cos ()u()g cos ()u()g 3 cos ()u()g 4 cos ()u()g sin ()u() + cos ()() sin ()u() + () 3 cos ()u()g sin ()u() + ()g cos ()u() + () () cos ()u()g cos ()u()g 3 n n o cos ()u() + () () sin ()u() () + () () sin ()u() () + () () o cos ()u() () () + (3) () Problem.38 8 9 e u() e u() + e () e u() + () 8 e u()9 8 9 e u() 8 e u() + ()9 e u() + () + () () 3 8 9 e 3 u() 8 u()9 e n o e u() + () + () () 3 e u() + () + () () + () () 4 8 9 e 4 u() 3 8 u()9 e 3 n o 3 e u() + () + () () + () () 4 e u() + 3 () + () () + () () + (3) () Following he sme pern, we hve n 8 9 e u() n e u() + n () + n () () + + (n) () + (n) (); > n Problem.39 s() r() r( ) r () r ( ); 4
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC MATLAB PROBLEMS Problem.4 The surion signl is presened in Figure.7. This igure is obined vi he ollowing MATLAB code. >> lp; be; be/lp; >> -:; :; 3:; [ 3]; >> s[zeros(,),,be*ones(,9)] >> plo(,s) >> xlbel( Time ); ylbel( Surion signl ) >> grid; xis([-5 5.5.5]).5.5 Surion uncion.5.5 5 4 3 3 4 5 Time Figure.7 Problem.4 The shied discree-ime impulse del nd uni sep signls re presened in Figure.8..5 Shied del impulse signl.5.5 3 4 5 6 7 8 9 iscree ime.5 Shied sep signl.5.5 3 4 5 6 7 8 9 iscree ime Figure.8 This igure is obined vi he ollowing MATLAB code. >> k; k; k3; >> kk:k 5
CHAPTER >> del3[(k-k)] >> subplo(); plo(k,del3, * ) >> xlbel( iscree-ime ) >> ylbel( Shied del impulse signl ) >> grid; xis([ -.5.5]) >> k; sep[(k-k)>] >> subplo() >> plo(k,sep, * ) >> xlbel( iscree-ime ); ylbel( Shied sep signl ) >> grid; xis([ -.5.5]) Problem.4 For rel nd negive exponens <, he dmped sinusoidl signl decys oscillory o zero s ime increses. Such signl or is ploed in Figure.9() in he ime inervl rom o 5. For posiive vlues o he exponen, he dmped sinusoidl signl increses o ininiy s!. The considered signls is lso presened in Figure.9() or..4 Rel nd negive exponen.3....5.5.5 3 3.5 4 4.5 5 Time Rel nd posiive exponen 5 5.5.5.5 3 3.5 4 4.5 5 Time Figure.9 For complex conjuge vlues o, he signl is complex uncion h hs is rel nd imginry prs. For such signls we plo heir mgniude nd phse (ngle). The corresponding plos or + j, obined vi MATLAB, re presened in Figure.9(b). I cn be observed h or his complex signl he mgniude decys o zero, bu he phse is periodic uncion o ime. In Chper 3, where he Fourier rnsorm o signls will be presened, we will sudy complex signls in deil. The igures in his problem re obined vi he ollowing MATLAB code. >> :.:5 >> % Sinusiodl signl wih rel nd negive exponen >> lm- >> exp(lm*).*sin(*) >> igure() >> subplo(); plo(,) >> xlbel( Time ); ylbel( Rel nd negive exponen ) >> % Sinusiodl signl wih rel nd posiive exponen >> lm >> exp(lm*).*sin(*) >> subplo(); plo(,) 6
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC >> xlbel( Time ); ylbel( Rel nd posiive exponen ) >> % Sinusoidl signl wih complex conjuge exponen >> :.:5 >> lm-+j* >> ccexp(lm*).*sin(*) >> igure () >> subplo(); plo(,bs(cc)) >> xlbel( Time ); ylbel( Mgniude ) >> subplo(); plo(,ngle(cc)) >> xlbel( Time ); ylbel( Phse ).7.6.5 Mgniude.4.3.. 5 5 Time 4 Phse 4 5 5 Time Figure.9b Problem.43 The rin o recngulr pulses is ploed in Figure.. This igure is obined vi he ollowing MATLAB code. >> u.;t.5;-.3:.:4.3; >> pusepun(,-u/)-sepun(,u/) >> rinpupu >> or i:8 >> rinpurinpu+sepun(,i*t-u/)-sepun(,i*t+u/) >> end >> plo(,rinpu) >> xlbel( Time ); ylbel( Trin o recngulr pulses ) >> xis([-.3 4.3 -.5.5]); grid Problem.44 The coninuous- nd discree-ime sinc signls re ploed in Figure.. This igure is obined vi he ollowing MATLAB code. >> -5:.:5; >> consincsinc(*-3) >> subplo(); plo(,consinc) >> xlbel( Coninuous ime ); ylbel( Coninuous sinc signl ); grid >> T.; kt-5:t:5; >> discsincsinc(*kt+) >> subplo(); plo(kt,discsinc, o ) >> xlbel( iscree ime ); ylbel( iscree sinc signl ); grid 7
CHAPTER.5 Trin o recngulr pulses.5.5.5.5.5 3 3.5 4 Time Figure. Coninuous sinc signl.8.6.4...4 5 4 3 3 4 5 Coninuous ime.8 iscree sinc signl.6.4...4 5 4 3 3 4 5 iscree ime Figure. Problem.45 The considered signl is ploed in Figure.. This igure is obined vi he ollowing MATLAB code. >> -:.:7; >> sepun(,)+sepun(,5)-(-).*sepun(,)+(-3).*sepun(,3); >> plo(,) >> xlbel( Coninuous-ime ) >> ylbel( Coninuous-ime signl rom FIGURE.8 ); xis([- 7 - ]) 8
SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC.5 Coninuousime signlrom FIGURE.8.5.5.5 3 4 5 6 7 Coninuous ime Figure. 9