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Unformatted text preview: ECE2311 EXAM # 3 B09
Name 2 S0 AW ECE Box Number: _ Clearly write the ﬁnal answer on the line provided on the exam sheets. When you
ﬁnish your exam, be sure that your name and ECE box number is on this sheet and turn in
this entire exam with the answers placed where requested. Problem 1 w ’I‘rigonometric Fourier Series In this problem you will construct the Trignometric Fourier Series for a periodic function. The
function 1'(t) has a period of 4 seconds and can be described over the full cycle that takes place
from —2 to 2 second by the following function: 0 —2<tg—1 1 —1<t§0
W): —1 0<tgi 0 1<t§2 ALO it) Find the a” Fourier trigonometric series coefﬁcient:
L 0 i .L ‘ , er
a0: WACMt: q 4++ #5135112“ q[l l1 0 0 l b) Find the a" Fourier trigonometric series coefﬁcient expression (that is, the function that gives an
for any n : 1r 2, Don’t leave this in the form of an unevaluated integral. Show your evaluation and simplification of the integral clearly for partial credit: I
7— 0 an f S 7({i)wg(mwokM+ 3% (CAY(V\MJ0‘E\J,{ _'{wr(nwo 900+ ~\
'/ l/ at (MUQ‘E‘) _ 9,»\(w.)aJc)
Z Ado muJo
Ji 0
w :W— '2“, 1r "‘2 ‘Z 0  “(@4433 calm/v0 — 0 1
L) r I " ’ ,
1’0 Ll L fﬂﬁ/Z' M1171
_'__ L 5')“ (KIT/1.) — :0
V 2' "7 MIT/7. an : c) Find the bn Fourier trigonometric series coefficient expression for any n r: 1, 2, Don’t not leave
this in the form of an unevaluated integral Show your evaluation and simpliﬁcation of the integral clearly for partial credit: l
'1 O l
bk“ 3 7((t‘)3\k(mdulc)d‘b 1’ E n
.. l
'L 0
: L _ Cajun/04:) _. ’ M)
2’ WW0 deo
(l
: L .' +c/oS(_h7T/zx (4)5(viTT/ZJ‘l l/Zwsrﬁl 2 Smcg
L  : ’l“ ,
wwo 4” WW0 Z ’7' n We ws(x3’cﬂ3l«sé) d) Are either the set of all an I 0 or all bn : 0‘? If so, which set? (you may have to simplify your
orginal answer to ﬁnd out if either is true.) 9’“) ﬂu «‘10 e) Explain how your result noted in part ((1) could have predicted from any symmetry (if there is
any) that the function 1'( t) possesses. l" N“ MUL “#66 a.“ V" La & ( Problem 2 »— Exponential Fourier Series
In this problem you will construct the Trignometric Fourier Series for a periodic function The
function 33(t) has a period of 4 seconds and can be described over the full cycle that takes place
from —2 to 2 second by the following function: —l —2<tS—l
—l l<t§2 a) Find the DR Fourier trigonometric series coefﬁcient expression (that is7 the function that gives an for any n : — 2,vel,0,1,‘2,...) : I W chum + ku, 21/'0 7,
52: “I \ 2 ~
' r J t .
I .L jhh’ot— c I / JWUJ erUd—‘Dk '“vot
an , To 7W) 6 on? H— e 4433 e a“, __ 6 6H
(1 “a {I l 
\ l ’2'
W.) = ‘3 a W : L view?“ 'JWDJC rIWo+
Ta 3‘ 9’ v 6 + 3 e _ r U
,;V\M)o ‘3““20 dim o
,2 v‘ I
a ()VWLU 32M) , ‘ “J  I an , no}
,‘ﬁ 6 D e o '3 Jvnao +36%)“ 0 + 6 J 0 ‘— e J o
i ]“ VD L/W \/—\r\_,/
1?; 59» (Mao) _ 23 {NR (inc/J03 b) Find the numerical value of D0, Showing work: 3
I . ) \,
>0 :: m»; Av '27 (1ch + if 3:“ 4a Q Do : ‘ C) Find the numerical value of D1, showng work: /
’1‘. gm (IT/'2.) q
’n’ ’U’ <7
H d) Find Del from D1 by applying a property of the Fourier series for that applies to signals like
this (If you have no idea how to answer this question as stated, then calculate the answer the
way you did in and just lose points for part (e) only): Shoe 276) is ﬂea/Q w lww M11) D“ 0M9... ( ‘11<
W3u‘6aykq‘ ga/MWNL‘lrh% , l, e D_n 5‘ Dun I kw " h 7 ‘35 an 1),1 : What was the property of that you used to produce the answer in (d) a 0m <2 :wwbam D .2 5:“ ...
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This note was uploaded on 04/21/2010 for the course ECE 2311 taught by Professor Hakim during the Winter '08 term at WPI.
 Winter '08
 HAKIM

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