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Unformatted text preview: ECSE—2410 SIGNALS AND SYSTEMS SPRING 2097
Rensselaer Polytechnic Institute EXAM #2 (1 hour and 50 minutes ) February 27, 2087 NAME: Section: 1
Do all work on these sheets.
One page of crib notes allowed. Calculators allowed.
Label and Scale axes on all sketches and indicate ail key values. Show all work for full credit.
N 0 partial credit for multiple choice questions and no credit for integration of the deﬁning formulas. Total Grades for Exam: [ Grades for I Points I Score Part II 1 TOTAL I 100 l PART I
Problem Points Grade
1 11
2 6
3 6
4 6
5 15
6 6
TOTAL 50 1(12). The exponential Fourier series representation for the periodic signal x50) shown below is x¢(t)= Zakeﬂm‘“ . x60)
k=ww b.)(1) The signal xii) is (circle one) i.) real i ii.) purely imaginary iz'i.)complex 3i.
c.)(3) The signal y(t) = xc(t) — g/fis (circle one). (No partial credit.)
if 1‘.) odd " at.) even iii) neither (1.) (3) suppose xc(r) is represented by its trigonometric Fourier series, x60) = a0 + ihk cos(ka)0z)+ ﬂk sin(km0t)]. k=l Recall that the relationship between the exponential and trigonometric Fourier series is xc (I) = a0 + i2me{a§jkw9’} ha; iv, 2(6). The ac vs. [C plot shown below represents the exponential Fourier series coefﬁcients, ask, of a real periodic signal, x(t) , with fundamental frequency, wg = r7: rad/see. Find the equation for x(t)
expressed as a trigonometric Fourier series. The series should contain only the nonzero terms. x0) x 3‘ (6) Circle one answer for each question below. No partial credit. a.) (3) The exponentﬁal Fourier series coefﬁcients of the two periodic waveforms shown are related by (circle one.) i.) bk =wak for all k iv.) bk=ew 2Vark 3. (6) Continued, Circle one answer for the question below. No partial credit. b.)(3) The exponential Fourier series coefﬁcients of the two periodic waveforms shown x(t) e ak 4. (6) Find the exponential} Fourier series coefﬁcients, bk , expressed in terms of ask for all k. DO NGT ﬁnd the speciﬁc values for the ak terms. 5(15). x0) is a periodic signal. Express the exponential Fourier series coefﬁcients, ak , for aEI k, of
x(t) in terms of ck , the exponential Fourier series coefﬁcients of the periodic square wave, pa),
shown beiow. DO NOT evaluate the ck. x(t) <—> ak
“ i —9 —8 —7 —6 5 4 3 2 I 0 i 2 3 4 5 6 7 8 9 l0 5'3 a gageﬁgem Intermediate graph if needed Simply? Wherever possibie and ﬁnd (2)0 . You must use properties. No creéit for integrating the deﬁning integral for ak . 6. (6)
a.) (3) IfX (a) (I ‘ jw
+1wx2 +10) ’ ﬁnd [X5 (ml ECSE2410 SIGNALS AND SYSTEMS SPRING 2007
Rensselaer Polytechnic Institute EXAM #2 (1 hour and 50 minutes) February 27, 2007 Section: 1 2 Do all work on these sheets.
One page of crib notes allowed. Calculators allowed.
Label and Scale axes on all sketches and indicate all key values. Show all work for full credit.
No partial credit for multiple choice questions and no credit for integration ofthe deﬁning formulas. PART II 7 10 7 (10). Find the Fourier transform X(a)) , of the signal, x(t) w (EMU) , where Note. You must use properties. No credit for integrating the
deﬁning Fourier integral} for X (60) . AISO Simplzﬁ/ whenever possible. We 2M 2”” 7
w ‘W g ‘ f {j '” 5 ,
{f f {KW M E
X grog}; :. xw em “‘3
W 2 e: “h” ’"r in s r f E
w as??? W. e: “if?”
P 2. I"? g W 2 M f if e r g
r e: w“ {1: i: {J g x“ j
:3 if” 5
3 4,3;— m 5
a; 1] 8 (10). Given x(r) <—> X ((0) , express the Fourier transform, Y(a)), of 320‘) m x(1—t)
in terms of X (co) . Use properties. WM 12 9 (20). Find x0), the inverse Fourier transform of X(co) = ja) ejwz jw+1 I E3 10 (10). a.)(5) If x(t) (—9 X(w), find 2(a)), the Fourier transform of z(:) m x(t)cos(a)0t). gka? wa?
4 s . ' n .i ,» ./ “W;
2%? : a3; “5 mg; mg? 5ft? J
" 3} g: b.) (5) Find the inverse Fourier transform of X (co) = 7:50;?)  47:) + 271580) + 7:6(59 + 47a) . d/ f . g .5 vs?” ' 6”" I?" ﬂ 2%)”: my»)? 41 ; 3f =»f:?='_.r§ 7/) {41,3 3: i;
raw3 ﬁzzy—"€535? 5/5”” f ,J
5’? w" i  4. 14 1} (10). —jm W —jw2 e e sin(x) a.)(5) If X (co) = , express X(w) in terms ofsinc functions, where Sinc(x)= ja’ ‘ x E5 Problem 11 (continued), b.)(5) The exponential Fourier coafﬁcients for a given periodic square wave, x(t), are i ' E
ak = zsmcp‘ 2} “0. Plot ak fork=4, 3, 2,1,0, 1,2, 3, 4.
E, k = 0
Use the deﬁnition, sinc(x) = Sln(x) .
x ﬁ‘ End.
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 Spring '07
 WOZNY
 Fourier Series, Rensselaer Polytechnic Institute, Fourier series coefficients, exponential Fourier series

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