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Solution Manual For Fundamentals Of Signals And Systems Using The Web And Matlab, 3rd Edition

Solution Manual For Fundamentals Of Signals And Systems Using The Web And Matlab, 3rd Edition makes tackling textbook exercises a breeze, with clear and concise answers to every problem.

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    Chapter 1: Fundamental Concepts L1G) ’ @ x(O)= py) +p) af, 11 1 ® =2(1-)py)-L 1-11) 00 304 3 1 © x0=220+2060+2(1-10) oy) 171 @ x0=1p,0-2(1-l)ps) » © H)=Y p-2k+15) k=1 1.1 (ii) Simplest to define # and x as follows, then use the MATLAB command plot (t, x) @) 1=[-3-2-2-1-111223]; x=[00 11221100 i (®) 1=[-5-4-1145]; x=[001100]; (©) t=[-7-6-6-3-3033667]; x=[00224642200], @ 1=[3-220223]; x=[0042400]; © 1=[-10011223344556]; x=[001100110011007; 1.2 (a) @==, so At <w/m = 1; sin(n(Atn)) = 0 for all integers » when Az =2 (®) 0=7,a=0.1,50 At S— == 025 aon? +7? (©) @=/4,a=1,50At < z =06177 we +(x /4) + StudyXY © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. EE

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    Downloaded from StudyXY.com ® + StudyXY Sd Ye. o> \ | iF ’ pr E \ 3 S Stu dy Anything This ContentHas been Posted On StudyXY.com as supplementary learning material. StudyXY does not endrose any university, college or publisher. Allmaterials posted are under the liability of the contributors. wv 8) www.studyxy.com

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    3. continued © delt=.1 1 delt=.5 [ ! Eo = ) WII - AMA By _ 5 10 15 20 “lo 5 10 1 Time (sec) Time (sec) s 20 ; delt=.9 delt=1.5 1 ) Wi * | “0 5 -1 \ 10 15 20 0 5 10 < 15 20 Time (sec) Time (sec) OB) TR x=0.] in \ NA 20S g=a2. _ = : ~ ~. / Ra NEE oe GL go \ 8 . T= BN \ \ 1 \, [0] 2 4 6 8 10 12 14 16 18 20 Time (sec) (©) 1 iL. 05 NS r=2 = c LT kes 0 =. -0.5 [+] 1 2 3 4 5 6 7 8 9 10 Time (sec) | © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in 2 Reva system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. -- etn CE RC CEA AREA EE en oe mere eb re RAE I

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    1.3 (a) (a) t=[-1 0 0 5]; 2 x=[0 0 1 1] ; plot (t,x, "LineWidth',1.5); Ps axis([-1 5 0 2]) Ea xlabel ('Time (sec)') ylabel ('u(t)') 0 title('(a)") [] 2 4 Time (sec) (b) t=[-1 012345]; ® x=[0 012345]; . plot (t,x, 'LineWidth',1.5) ; axis([-1 5 0 5]) eg, xlabel ('Time (sec)') ylabel ('r(t)") title(" (b)') Lo 2 n Time (sec) © t=(-1 00 2 2 3]; (©) x=[0 01100]; 2 plot (t,x, 'LineWidth',1.5); axis([-1 5 0 2]) 4 xlabel ('Time (sec)') > ylabel ('x(t)") title('(c)") 0 0 2 4 Time (sec) (d tl=(-1 0 0]; x1=[0 0 10]; 10 @ £2=0:.01:5; x2=10*exp (-3*t2); t=[t1 t2]; €s x=[x1l x2]; plot (t,x, 'LineWidth', 1.5) ; axis([-1 5 0 101) 0 fn) 2 4 xlabel ('Time (sec)') Time (sec) ylabel ("x(t)'")~ title('(d)") (e) ti=[-1 0 0]; x1=[0 0 31; a (e) t2=0:.01:5; %2=3*exp (~£2) . *cos (2+t2) ; 2 t=[tl t2]; £1 x=[x1l x2]; 0 plot (t,x, 'LineWidth',1.5); axis([-1 5 -1 31) B— 2 4 xlabel('Time (sec)') Time (sec) ylabel ('x(t)') title('(e)') 3 © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. et eee See eAlerts

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    1.3 continued (£) tl=[-1 0 0]; x1=[0 0 3]; @ t2=0:.01:5; 0 x2=3*exp (t2) .*cos (2*t2) ; t=[tl t2]; § -200 x=[x1 x2]; plot (t,x, 'LineWidth',1.5); -400 axis ([-1 5 -450 100]) 0 2 4 xlabel ('Time (sec)') Ti ylabel ("x (t)") me Ee9) title('(£)") (@) t=-1:.01:5; @ %=2*sin (3*t-pi/2)- 2 cos (2*t) ; plot (t,x, 'LineWidth', 1.5) ; go axis ([-1 5 -3 3]) xlabel ('Time (sec)') . 2 ylabel ("x (t)") 0 2 4 title('(g)") Time (sec) (h) t=-1:.01:5; [0 x=sin(5*t)+sin(pi*t); 2 plot (t,x, 'LineWidth',1.5); axis([-1 5 -2 2]) xlabel ("Time (sec)') go ylabel('x(t)') title('(h)") 2 0 2 4 Time (sec) 1.4 (a) xX) 4, ~/ : X(t) J o ~/ Ja 3 ¢ -t + StudyXY ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. SE Ee tee tL eA i i.

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    /.4 Con bin ved cc) ale) xt) * 5 al) 4 A CE 14d) a] ® t={-2 -1-111 33 4]; 2 © x=[0011-1-100]; plot (t,x, 'LineWidth', 1.5) ; 1 axis ([-2 4 -2 2]) go xlabel ("Time (sec)') ylabel ("x(t)") -1 title('(a)') 2 2 [}] 2 4 Time (sec) (b) t=[-101122 4]; (b) x=[00-111 00]; 2 plot (t,x, 'LineWidth',1.5); 1 axis([-1 3 -2 2]) xlabel ('Time (sec)') go ylabel('x(t)') title(' (b)") -2 El [] 1 2 3 Time (sec) (ec) tl=[0 1]; x1=[0 0]; t2=1:.01:2; x2=2* (£21) ; © t3=[2 3); 6 x3=[2 2]; t4=3:.01:5; 4 x4=2%t4-4; = t=[tl t2 t3 ta]; ja x=[x1 x2 x3 x4]; plot (t,x, 'LineWidth',1.5) ; 0 axis([0 5 -1 7]) xlabel ('Time (sec)') [) 1 2 3 4 5 ylabel("x(t)"') Time (sec) title('(c)") I~ StudyXY ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. SE i EE

  7. Page 7
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    1.5 ®@) ti=[-1 0 0]; X1=(0 0 1]; 15 @ £2=0:0.01:2; ’ x2=exp (-t2) ; 1 t3=2:.01:4; X3=exp (t3-4) ; £05 t4=[4 51; x x4=[0 0]; 0 t=[tl t2 t3 t4]; x=[x1 x2 x3 x4]; -05 plot (t,x, 'LineWidth',1.5); ’ [} 2 4 axis([-1 5 -.5 1.5]) Time (sec) xlabel ("Time (sec)') ylabel('x(t)") title(*(a)") (b) tl=-1:.05:pi; xl=cos(t1); 1 (®) t2=pi:.05:3%pi/2; $ x2=-cos (t2) ; 1 t3=[3*pi/2 5); 0.5 x3=[0 0]; s t=[t1 t2 £3]; x 0 x=[x1 x2 x3); 05 plot (t,x, 'LineWidth', 1.5); -1 axis([~1 5 -1.5 1.5]) xlabel ("Time (sec) ') AS ) rl ylabel ("x (t)"') Ti title('(b)") me (sec) 1.6 (a) Since x(1) = w(t -c), replacing £ by £ + ¢ gives Wr) = x(t + c). (©) () v(r)= e720) = 65-2 _ 6p) ) G@) vO) = (+2 - (+2) +1= +3143 | Gi) v(t) = sin(2( + 7 /4)) = sin(21 + 7/2) = cos(21) 17 (a) un) I -5 -¢+ -3 a .y © / a2 3 ¢& sn + StudyXY ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be i obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

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    1.7 continved rn] Te + 3 3 li -5 fF <3 a or oo 7/7 A 3 + £7 tc} x(n) 2.5 to 0.3% 0.135 ~~ n ~-3 -x -t a a 3 rs ) x{" 1 py, 0.3% n -.5 ~~ Lo./as (e) xn) 76 72 [3 Pa ” -3 x oy es a 3 | vs li ” T= StudyXY © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. ES

  9. Page 9
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    1.7 Con dinved 2 7 ° (£) OW SRS / ~ 7 _ 0.707 -¢ 3 a os lel ° EY 3 Pad ~ 0.707 . —* ~! “FO pron 9) J xCn) { -1 n -¢ -3 -3 7 EY 3 - =I xn) b3 . 137 , 0.8! 7 ~~ ~£ -¢ «3 a + 5 °o / a1 3 n -1 -0.456 _ -[ 5 ~ ~ <).23 N - 0.835 ~1.94.7 (4 ) x(n} ( - - E ” -s fo-3 -2 2 / a 32 % 5 | Vv 4 T= StudyXY © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. a

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    17 () (a) for n=1:5; (a) u(n)=0; 2 end; for n=6:21; 1.5 u(n)=1; end; E11 n=-5:15; > stem(n,u, 'filled") 05 axis ([-5 15 0 2]) Xlabel('n'") 0 ylabel('uln]') 5 0 5 10 15 title('(a)") n (b) for n=1:5; (b) X(n)=0; 15 end; for n=6:21; x(n)=n-6; 10 end; = n=-5:15; = stem(n, x, 'filled") 5 xlabel('n") ylabel('r([n]") title('(b)") 3 0 5 10 15 (c) for n=1:5; n x(n)=0; end; 1 © for n=6:21; x (n)=.5"(n-6); 0.8 end; n=-5:15; = 06 stem(n, x, 'filled') 04 xlabel('n') ylabel (*x[n]"') 0.2 title('(c)') o -5 0 5 10 15 (d) for n=1:5; n x(n)=0; end; 1 (d) for n=6:21; x(n)=(-.5)"(n-6) ; end; 0.5 n=-5:15; = stem(n,x, "filled"') = xlabel('n') 0 | ylabel('x[n]"') title('(d)") -05 -5 0 5 10 15 n ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions), write to: Rights and Permissions Department, : Pearson Education, Inc., Upper Saddle River, NJ 07458. Ae EPCRA AeA ABE eerste et ee eee eee

  11. Page 11
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    1.7 (j) continued 4 PE. 10 (e) (¢) for n=1:5; x(n)=0; end; E, for n=6:21; ® x (n)=2%(n-6) ; end; o n=-5:15; " stem(n,x, 'filled") s 0 5 10 15 xlabel('n’") n ylabel('x[n]") ® title('(e)") 1 0.5 (£)n=-5:15; E o x=sin(pi*n/4); = stem(n, x, 'filled') -0.5 xXlabel('n") ' ylabel('x[n]') -1 title(' (£)') -5 0 5 10 15 n 1 [€) (g)n=-5:15; 05 x=sin(pi*n/2); = 0 stem(n, x, 'filled') x Xxlabel('n'") -0.5 ylabel('x[n]") 1 title('(g)") -5 0 5 10 15 n 0) 2 (h) (h)n=-5:15; x=(.9.%n).* (sin (pi*n/4)+cos (p 1 i*n/4)); E o stem(n, x, "filled') x® xlabel('n') -1 ylabel('x[n]") title('(h)"* 2 mn 5 0 5 10 15 (i)u(1)=0; n for n=2:10; " u(n)=1; 2 0 end; for n=11:21; 15 u(n)=0; = end; = 1 n=-5:15; 0.5 stem(n,u, "filled") axis([-5 15 0 2]) 0 xlabel('n') 5 0 5°10 15 ylabel ('x[n]') n title('(i)") + StudyXY Jo ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

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    (a) x(n) r 4 bY 3 + ss & a -r |O n -/ x[n) 3 bY [4 bk Sd @ 7/7 A 3 + 5 & 4 c) x Cn) 2 1a n “a -/ °o FYE] + x Cn) | 30 5.53 ~ 2 20 13.02 ss. \ ENE NON 4 n - =! o a3 - | Vv T= StudyXY 1/ © 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

  13. Page 13
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    oo EE EE 1 2 a AeA So ptember 1.8 (¢) (a) (8) n=-5:6; 2 x=[000001-1-1-100 01; 1 stem(n, x, 'filled’) = axis ([-5 6 -2 2]) = 0 xlabel('n') ylabel ('x[n]"') -1 title('(a)") 2 -5 0 5 n (b) n=-5:6; (b) x=[0 00010123000]; 4 stem(n,x, 'filled") 3 axis([-5 6 -1 4]) xlabel('n") = 2 ylabel ("x[n]") ® 4 title('(b)") 0 1s 0 5 n (c)n=-5:6; © x=[0 00020100000]; 3 stem(n, x, 'filled") axis([-5 6 -1 3]) 2 xlabel ('n') : ylabel('x[n]'") E 1 title('(c)") [1] -1 -5 [1] 5 n (d) for n= 1:4; (d) x(n) =0; 150 end; x(5)=exp(-.8); 100 for n=6:12; = Xx (n)=exp(.8* (n-6))+1; x end; 50 n=-5:6; stem(n,x, 'filled') 0 Xlabel('n') -5 [] 5 10 ylabel('x[n]") n | title("(d)") 1 2 ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. ee rete 055mm emer tres rte

  14. Page 14
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    27 2 L_2_4 19@) [=""=2 GH=-—"—=25 L="2C2 ignal is periodic wi i @ 5 - 2,0 yryr S57 2s 5 » so the signal is periodic with fundamental period equal to 57; = 10. Jo ( — —) a) 2 t=0:.01:30; A il x=cos (pi*t)+cos (4*pi*t/5); plot (t,x, "LineWidth',1.5) = xlabel ("Time (sec)') x 0 ylabel ("x (t)") title('(a)") 2 ! 0 5 10 15 20 25 30 Time (sec) 2% 22_2 Lh _ 1 5 b) [== 5=L=2 lo 2 ignal is periodic wi i (LF py L 5 5 5,208 3 » 50 the signal is periodic with fandamental period equal to . ST =2 ' fea ®) t=0:.01:10; 2 x=cos (2*pi* (t-4))+sin(S*pi*t); plot (t,x, 'LineWidth', 1.5) = xlabel('Time (sec)') x0 | ylabel ("x(t) *) title('(b)") -2 0 2 4 6 8 10 Time (sec) 27 2z_7% 0 1 5. q oe ©) [== 5h="2022 Aa to 2,4 i ignal i © i py 1,7 0S GAB x = for any positive integers ¢ and r, so the signal is not periodic. (©) 2 t=0:.01:20; | X=C0s (2*pi*t)+sin (10%t) ; | plot (t,x, 'LineWidth',1.5) = ol xlabel ('Time (sec)') x ylabel('x(t)") \ title('(c)") 2 J [] 5 10 15 20 Time (sec) (@) 10= = for any positive integers and 7, so the signal is not periodic. n=1:31; 1 @ x=sin(10*n); n=0:30; stem(n,x,"filled") = xlabel('n') = 0 . ylabel(*x[n]") title('(d)") » [4] 5 10 15 20 25 30 n + StudyXY /3 ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. tee e755 5A teres.

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    1.9 continued 107 _27q N . APTI © 3 ==; Withq=>5andr=3, so the signal is periodic with fundamental period equal to r =3. n=1:11; 1 3 (© x=sin(10*pi* (n~-1)/3); n=0:10; stem(n, x, 'filled') xlabel ('n') go ylabel('x[n]") title('(e)") -1 0 2 4 6 8 10 n . (0) Is there a positive integer r such that x[» +7] = x[n]? Since x[n+r]= cos(w(n+7)?)= cos? +n +r) the signal is periodic if 7(2nr +72) = 2m where m is an integer, or (2nr +72) =2m . Ifr=2, m=2(n+1), which is an integer for any integer value of m, so r= 2 is the fundamental period. To generalize, consider 2 2 the periodicity of of =) for some integer i. This is a periodic signal if ror =m. The signal is periodic if is even, in which case the fundamental period is 7 = i, and the signal is periodic if if i is odd, in which case the fundamental period is r = 2i. 2 f- f) n=1:11; 1 ¥=cos (pi*n."2); n=0:10; stem(n, x, "filled") Zo xlabel('n') ylabel('x[n]") title(' (£)") 1 0 2 4 6 8 10 n 1.10 (a) When x[n] = é[n], yn) = 47 (Ont —11+-+-+51n-N +1) 1 n=012. ,N-1 Sim={w" ,1,2,..., 0, otherwise 14 + StudyXY ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. OU

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    1 1.10 (b) When x[n] = u[n], yn] = (utr sn =1J+--+uln—N +1) R Forn<0,yn]=0. 1 For0<n<A-, Hr= FornxN, y[n}=1. 1 (c) When x[n] = rn], y[n]= ryy +r{n=1]+:+r[n—N+1}) Forn<0,y[n]=0. For0<n<N-1, 1 Arl=—(ntn-1+n=2+-+2+1+0) Performing the addition nAnl+p2 + +2 + 1 +0 0+ 1 +2 + +n2+ n+ n n+ n + nto +n + n+ n=nm) shows that 2(n +7 —1+n—2+---+2+1+0) = n(n +1) and thus 1(n@+)\_nn+]) n]=—|——=|=——= for0<n<N-1. Mn] ol ) ) oN for 0 <n<N-1 . 1 Finally, for n> N, yn] = lren—len-24tn-N+2en-N+1). Via an addition process similar to the one given above, we have that stnl= (Hh) =2 Vl torn N. 1.11 By the result in Problem 1.10, ){r] Sei for n> Nand thus N-1 . . Nnl= n-—o =r[n-(N-1)/2] for n> N, which shows that the N-point MA filter delays the ramp r{n] by (V-1)/2 time units. L12 n=1:69; x=5*sin(pi*n/10+pi/4); for n=6:69; y(n)=(1/6) *sum (x (n-5:n)) ; end; n=6:69; plot(n,y(n),n,y(n),'o’, 'LineWidth',1.5) grid xlabel('n") ylabel('y[n]") 15 I~ StudyXY ©2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permissions), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. a —

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