# Solutions to Problems from Chapter 2

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1 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,.

2 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(--) u(-5) Figure.5

3 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) r(-) Figure.7 3

4 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] iscree ime k Figure.8 Problem. This signl is ploed in Figure [k] ri4*u[ k+] p[k ]*u[k ].4. Signl [k] iscree ime k Figure.9 4

5 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

6 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

7 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 e e 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

8 Problem. + sin() + e h() () + 3 CHAPTER + 3( ) + 4 () ( )id + 4() () () sin h i () () + 3( ) + 4 () ( ) d + e 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) e sin ( 3)( 5)d ()( 5)d (5) e sin () 3 (d) e ( + 3)d ()( + 3)d (3) e Problem.3 cos ()( ) + i e 3h () ( ) + ( 3) d + + n ()( 3)d ( + )d cos ()( ) + () 9e 3 + e3( 3 ) ( ) + 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

9 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 / 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

10 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) / () (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

11 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 () r()u(-+) -r() Figure.3 r(--) () u(+) u(-+) - Figure.4 -r(-4) The corresponding generlized derivives re given by () 8>< ; ( + ); ; < < undened; ; < < undened; ; > ; () 8 >< ; < undened; ; < < ( + ); < < ( ); ; < < 4 undened; 4 > 4

12 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) (-) - (-) - (-) - (-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.

13 SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC d() d 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

14 4 4 u()g 3 CHAPTER 3 u()g n o () () () () Problem 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 e u() e u() + e () e u() + () 8 e u()9 8 9 e u() 8 e u() + ()9 e u() + () + () () e 3 u() 8 u()9 e n o e u() + () + () () 3 e u() + () + () () + () () 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

15 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 Surion uncion 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 iscree ime.5 Shied sep signl iscree ime Figure.8 This igure is obined vi he ollowing MATLAB code. >> k; k; k3; >> kk:k 5

16 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 Time Rel nd posiive exponen 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

17 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 ) Mgniude Time 4 Phse 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([ ]); 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

18 CHAPTER.5 Trin o recngulr pulses Time Figure. Coninuous sinc signl Coninuous ime.8 iscree sinc signl 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

19 SOLUTION MANUAL or LINEAR YNAMIC SYSTEMS AN SIGNALS by ORAN GAJIC.5 Coninuousime signlrom FIGURE Coninuous ime Figure. 9

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