A#06_Solutions

# A#06_Solutions - Assignment#6 — Solutions — p.1...

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Unformatted text preview: Assignment #6 — Solutions — p.1 BOSE-2410 Signals & Systems - Spring 2006 Due Tue 09/19/06 1(10). (3) Write the complex expression, 2: (\$3.33 in polar form i.,e. z- — ]2:[ej “gm Express arg(z) "i” J as pzrradians, Where p is a fraction. Amﬁem l I JCPL‘fZE-‘n’yj W ”.17 ,1 , .._._._ 36ng J26 % ___ J13 3... LéJWL £61ng J““73; __ E: .. f2, N32; 5% j(I+j)e 1-K (‘0) Write the complex number 2— — J[ ,in rectangular form i. e. z m 93:2{2 }+ j‘sm{z}. 1+ e 6:] Mime M t7 W 4.2.: 6h“) ‘1‘ 1 x -_L ﬁg: KMS%*JS’“‘% “Eilj—‘zjﬂ iii] 1(12). Given 22:12:]. ,calculate the following: @115: £1- 1 w (a) 2+2: age—3:313:11 3.2.1.11 9-1111: 7.. 2 ‘~ ”.1. “1&4“ 1... Lia i3 ‘1‘} 14:1 *1} “3 {if 2. 2,. 2 (b) Z‘Re{z} ,_ a’ t~§21+ ”3+3 agate . 32:}, ﬂ “rim a+%"“[email protected]§%£ (c) z 2 We Ztcf z——’....3,:L..i ‘ "a 7;; 77—3?“ M "15.1 (d) lzi Assignment #6 ~— Solutions — p.2 ECSE-2410 Signals & Systems « Spring 2006 Due Tue 09/19/06 -1, 0<r51 t~2, i<r<2 3(8). One period of a periodic signal, x10), is given as R16) = . Sketch x10) for mz+& 2<t<4 —L 4<t<5 several cycles and ﬁnd its average value. Assignment #6 — Soiutions — p.3 ECSE~2410 Signals & Systems - Spring 2006 Due Tue 09/19/06 4(35). For the periodic signal shown, now 2 (a)(10) Find the exponential Fourier series coefﬁcients, ak. 3::th or M regujstg fW EMMW s35 :ﬁ‘m-t- { wﬁijrma bug».- agree Cit—@333: mix— r.‘ __~ \ \ 7F . . are: gig—13 SetLKWQ-E) : 2L3; SMQO'LH’I’Z.) =‘é SMC (1“ 43.») “in all K (b)(IO)Use equation 3.31 in text to express x0) as a Fourier genes in the trigonometric form x(t) = a9 + 2:: A cog/twat) W argm/AFKQiSE Si‘ﬁC—Q, at“ 1: Raﬁ/Q +1an 6"?“ (e)(10)EvaIuate the ﬁrst five harmonies and write the series in (b) in expanded form, ie, £0“) = a0 + 2A1 cos(a)0t) + 2.42 cos(2a)ez) + + 2/15 003(5w0t) Note. Some harmonic terms may be zero. 2W”) -— -~ Z+yws®tj2WwZ Hart )«+.Sm g} 60.28%) Sic???“ J... 91%”???ng ‘V2. 72/ 7‘“ 2?}; w: O t 3111 (SW2, ___L_“., g.” A ST? ﬁre? ‘33, “H‘d‘i— Z. ..«2... ”2:“ 2/ L 2+ Eamsért) ﬁpmggzré) S'cgﬂégFt) Assignment #6 — Solutions .. p.4 ECSEa24IO Signals & Systems ~ Spring 2006 Due Tue 09/ I9/06 (d)(5) Use MATLAB to plot the original x(r) and its Fourier series approximation. t:[—3:.01:3]; xorigﬁt>ﬁ-5/2&t<m3/2Et>=«1f2&t<=E/21t>=3/2&t<=5f2; x0205; ’ xiﬁ{2/pi).*cos(pi.*t}; x3ﬂ-(2/(3 *pi)).*cos(3 *pi.*t); x5\$(2/(5 *pi)). *cos(5 *pik’kt); xapprox=x0+x 1 "Hi3 +x5; plot(t,xorig,t,xapprox} grid On a second graph, plot 2(3) and all the individual harmonics in (c). ﬁgﬁanmm Z Cam-t) t2{—3:.01:3]; x0205; xi:(2/pi).*cos(pi.*t); - x3i—(2f(3*pi)).*cos(3*pi.*t); 5, ﬁne/(5mg).*cos(5*pi.*o; xapprox=x0+x1+x3+x5; ‘ plot(t,xapprox,t,x0,t,x1,’—.',t,x ,’——',t,x5,':') E grid SOL-N 5 Assignment #6 I: page 5 BOSE—2410 Signals & Systems — Spring 2006 Due Tue 09/} 9/06 5(35) (a)(1 5) By direct integration, ﬁnd the exponential Fourier series coefﬁcients, czk , of the wavefom, \$01.53 5 Assignment #6 — page 6 rt ECSE~2410 Signals & Systems - Spring 2006 Due Tue 09/19/06 5. eontinueé dx(t) (b)(5) Sketch the waveform x10) = dr <——> bk = related to ak. (0)00) By direct integration, ﬁnd the exponential Fourier series coefﬁcients, bk , of waveform x1(r). a 0 Z, ' , we 3ng - :Wc LK‘ZIFJXICHE d»? ﬁtgom cit—i: 5% )e‘3 at ”2“ “2. a t . 2* ' .3 r .1. i “E 533K553 0.. 53W _. .3. ;”{s~ék:el+l ..' B7 "— A 4* (ﬁ‘ 2) «z. o ‘4? (JP???) f— : i 9'5) 11?? ~33? L (N; ‘HHUd'g, A} “jI‘WZ. +6) % “JKWL MCL EOmO (d)(5) Using your answers above, ShOW that the two Fourier coefﬁcients, ak ,bk , are related by the formula I) m j.ka)0ak UOLU QM WSMCZO‘JFZJ: Shit/1% \$91 {1%) it? S‘ECészg’ 919:): 9% F751. Subs. veto L)?“ J stitctkgghﬂ‘ﬁ) an) K372. 0‘14» K”? .m biz-21:} ng ka {H30 H‘AQ aj‘ [ﬁr—O @ED ...
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