A#07_Solutions - Assignment #7 ~ Solutions - p.l ECSE-2410...

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Unformatted text preview: Assignment #7 ~ Solutions - p.l ECSE-2410 Signals & Systems — Spring 2006 Due Tue 09/26/06 , . ‘1}. . 1(5). Text3.21 Tug mg: 3.2% C “:7; 53;: at"? :7 fig“; Quiz: 3‘. 139%? ~ +- ‘ - t #9:: e. “W” 91/ 3%“5 . 55% we? .nvo . mt 74$ch) 2: "*9 T £61 +~ 6" Z: +3) we? “J 23 2 Z( < 9%?) W9 '3 ‘gr “3561‘; "2 5"”??? 3% “1+ .. sIACJQt} ‘ Nyzzwsgmwg + 2—} w3(%{7) I 0’) W: mm = wig? H\ p"; GWEN rm. __._. (C) W‘L‘H: @ 2: 3*“) H \EI‘ 51 $443.1. {9:54 K 3(5). Text 3.42 (c) M :58 W9 acid '53? 50AM airfield) AM!) an: :11. WW” Kg) weafi‘ a"? Keri-291+) “‘7aia ""“ 6:4. "7 alf- =5L“K/‘ifl7 BK“ 4* " H“ K gm) M :7 szfldaq akwa Ffizflgkpal‘ Wm. p— —2+§J: =57 E2" lav—7 fine x 3 W Fmdw‘fi 4,) r. .H w“? . N a gmgVL Jlég‘g w - .3362ng M311,“ 2w” T3?) «5?? 4;} "~63 50L ‘7 "” 5%" g Assignment #7 w Solutions « p.2 ECSE~2410 Signals 85 Systems - Spring 2006 Due Tue 09/26/06 5(20). For the periodic waveforms shown, (a) Find the equations for ak and bk . (b) Use MATLAB to plot the spectra, ak vs. k and bk vs. k , out to roughiy the 10th harmonic. close all narrow putse 08h ‘ A I é p A A k=[~10:10]; ' ’’’’’’’ 77777777 777777777 kkkk ________ """"" """"""" VVVVVVVV 77777777 777777777 a=(1/3)*Sin0(k/3); 0 6t ________ 6666666 ,,,,,,,,, ,,,,,, _________ _________ _________ nnnnnnnnnn ,,,,,,,,, ,,,,,,,,, subplot(2,1,1) ' g g g g g g g g g stern(i<,a,'g','filled'); -------- ---------- 777777777 iiiiiiii --------- --------- --------- --------- iiiiiiiii 777777777 grid fl a titleCnarrow pulse') 0.2;— ........ ....... iiiiiii 7777777 ))))))) ..; ......... ......... .......... 777777777 777777777 axis([-10,10,-.2,.81]) : j fl 1 g t _ % 3 g s ; hold on o Mu—fimemmfuwi : é it it as b—(2/3)*sinc(3*id3); 5 J : ‘ E i ; ' E ‘ ' subpiot(2,1,2) “0'2 5 L.“ I > em“ 2 i ammo > i . .. , , 40 e e »4 —2 o 2 4 6 a as “imam b ’ filled )’ grid . titleCwide pulse’) Wide poise axis([~10,10,-.2,.81]) 03 T ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ......... ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, __ 1 in v v r r r r . r r r L 7 7 7 r 7 r 7 7 7 7 A 7 7 7 7 7 7 7 r r r A u , , , u , u u , ......... .4: _ _ _ _ _ _ _ . . . i . . . . V . . . t V , r . V , 7 7 7 7 7 7 7 i , , , , , , 7 , , , 7 , , , 7 , , u u H 0.4 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ________ ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, _. {)2 77777777777777777777777777777777777777777 ........ iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii .. e womrm—r». 4w. emf. ~o o .0 2 L-.- a, W s w \ sh mi” I t- _________ M1 VVVV W_—n ,,,,,,,,,, _.| (0) Study the plots and answer the foilowing question: Which periodic signal (the wide pulse train or the narrow one) has almost ali of its harmonic content at DC? From the bottom spectrum, we see that the DC value is the dominant frequency, while the narrow pulse has significant first and higher harmonics. in a relative sense, this means that the wide pulse changes “more slowly”. Assignment #7 w Solutions - p.3 ECSEw2410 Signals & Systems — Spring 2006 Due Tue 09/26/06 6(20)‘ For the periodic waveforms shown, x(t) <—> ak z: 5%; Sxkfllk T171} (a) Finé the equations for ak and bk. 7M MSW we ‘m glam-e ma wav-LWS. \ 7769 [Mala SJWQI éfiém be? (fa-11mm: or by We mmww fan Fwng M, NH ymwflzéék. Lat; 501% 5+ [20% Lox-415$ 095% Cowipme MWI/ eff flacng fiwwag W xix/iii” WW aLwa—y. \ ‘ ‘ 9L0 pmmduw ‘ D WJEMKE céixgibwjkwbagc :EKW‘QL Assignment #7 m Solutions m 13.4 BOSE-2410 Signals & Systems - Spring 2006 Due Tue 09/26/06 6. Continued “w Him #619“. W 3W3 F awmcéwgs é: mama) [am—«>33: L 1‘ ». t Assignment #7 —— Soiutions - p.5 ECSE-2410 Signaks & Systems ~ Spring 2006 Due Tue 09/26/06 MM 6. Continued 4 M PM “‘15 Pix-3 :9: £23) gzkcégfli r: SJALCHEJ va wubwhhx Fwfimfl Z “I”: W w” aflm == 2(25 WW = 5 wd“ 55:21 52r- M ‘9 A $655!) _. .L 31%;? kg) Assignment #7 w Solutions ~ p.6 ECSE~24IO Signals & Systems — Spring 2006 Due Tue 09/26/06 6. Contiinued. (1)) Compare their spectra by simply calcuiating their DC vaiues (a, ,be ), and the magnitude of their first harmonics Ga, Liblf). No piot is necessary. (Find values only for k = +1.) What do you think is the reason for the “values you obtained? Clearly, the two waveforms are different? Actually, we do need plots to fully expiain what is happening. Note that the two waveforms (p3) have different periods. We need to take this into account so that we are comparing the waveforms on the same basis. The x(t) <—~> a, plot (see below) has values at harmonic frequencies, keen m k (22”) = k]: rad/sec. So its first harmonic occurs at frequency, 7: rad/sec. The y(r) (we bk piot (also below) has values at harmonic frequencies, keno = k (if) : kw; rad/sec. (due to T = 4) which is a lower frequency than the first harmonic of x(t) (—> 52,. Thus we expect to see more 10w~frequency components for the y(t) <——> b,t waveform, because it changes “more slowly.” Both waveforms are stiil triangular, because the respective harmonics maintain the same ratio for all At. All of this becomes apparent in the following MATLAB plot of CI, ,b,C vs. [£609. The ak for x(t)~waveform, with T:2 05L """"""""""" --------------- “T """""""""""" ************* ': --------------- "r ************* {)_4 1 ,,,,,,,,,,,,,,, ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ,3 ..................................... ,L ,,,,,,,,,,,,,,,,, L 0‘3 """""""""" """"""""" ””””””””” """""""""""" """""""""""" ””” " k=['424}*Pi; g g 3‘ g ; aIFO.5*(sinc(0.5*k))."2; 02; """"""""""" """"""""" ””””””””” """""""""" """""""""" ””” ” subplot(2,1,i) 01? ________________ ______________ ,,,,,,,,,,, _________________ ,,,,,,,,,,,,,,,, ,,,,,, ,, stemtkak) ' i ‘ Q Q titie('The ak for x(t)wwavefonn, with T3?!) 0 m m i I r p (D i 5 (P r) mi A grid -1 5 ~10 '5 0 5 10 mm[-s:s]*(pi/2); brn=(sinc(0.5*m)).’\2; bm(9)m0.5; subplot(2,1,2) stem{m,bm) titieCThe bk for y(t)*wavefonn, with Tmi') axis([-15,15,0,0,5]) arid Assignment #7 —Solutions - 13.7 ECSE-2410 Signals & Systems — Spring 2006 Due Tue 09/26/06 7(35). The pulse width is i]? . (a) Caiculate the Fourier coefficients for the three cases. close aii k={-25 :25]; w12k*0.5; Taiwsincfwi); subplot{3,1,1) hizstem(pi*k,’?al ,3"); set(hi ,‘MarkerFaceCoIor','red‘) grid titieCTm’Z') axis([-20,20,-.4, E }) w2:k*0.2§; Ta23sinc(w2); subpiot(3,1,2) E thstem(O.5*pi*k,TaZ,‘b'); set(h2,'MarkerFaceCoIor’,'blue') grid titleCTzé') axis({-20,20,—.4,1]) w3xk*0.125; Ta3fisinc(w3); subplotB ,1,3) h3=stem(0.25*pi*k,Ta3,‘g'); set(h3,‘MarkerFaceColor‘,‘green') grid title('Tw8') axis({—20,20,—.4, 1]) (0) Note, as T —> 00 the plot becomes more dense and approaches a continuous enveiope (i.e., the outer boundary). ...
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This homework help was uploaded on 04/10/2008 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.

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A#07_Solutions - Assignment #7 ~ Solutions - p.l ECSE-2410...

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