This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Name Exam 3 363M CM 1. (25 points) Finding the energy in a given bandwidth For the signal x(t) with spectrum shown below: . _ . _ _ _ _ _ _ _. . _ . _ _ _ _ . _ . _ . _ _ . g O
100 £0 60 40 20 0 20 4O 60 80 100
Frequency (radians/sec) Determine the percentage of the total energy in x(t) between 20 and 60 radians/ sec. R AID .no '20 L”) Sro
CT : ’E—ES 313)“) g ‘iz‘m 1% (9250* g Lizacvd #2 110?“)
It! ~90 no ‘20 lo ‘0
7.qu
0 + 401:
:‘LK‘Lqucf(Qp 10+ 2(9qu «HQ: 1 ‘i .m.
2H L” 0
9:0 “\ 4,3431: f0
r“ «{on CINE“ 2 .40 101' 4T?
: , X Li wing 2 u " ——
Lawué 1" 810 q: i 7f? Name CM 2. (30 points) System analysis with the Fourier Transform
Consider a linear time invariant system with impulse response given by h(t) = 3—sinc 22: 27:
with input x(t) = 3 sinc2 cos(t — 3)
Jr 71' The output of the system is y(t) . Show all of your work and draw a BOX around your
ﬁnal answer. a) Determine X (m) . b) Sketch the spectrum of X ((0) (magnitude and phase) accurately labeling the axes
and important points. c) Determine H (w) . d) Sketch the spectrum of H ((0) (magnitude and phase) accurately labeling the axes
and important points. w 2'3“)
6) Determine y(t), the output of the system. Z I‘ 3
2’ (lo
Ca) For 11(9) 3 €m8<$n> Yul”) :TrJu “) D XXI»)
E’V‘IT K: ,3
Kim) = 2 )4 I ‘\ W Name CM 3. (25 points) Fourier Series of a Periodic Signal
The following set of questions refer to the signal below (a) What is the fundamental frequency of x(t) in (rad/s)? (b) Find an expression for the Fourier Series Coefﬁ
answer as much as possible. cients, ck, of x(t). Simplify your @ 7; .5ng w” 7 7:2 ’ r 2
Z t
KW A“ (My
a ‘3’“; fwd 0/é*f‘2a4 at] K 1
2 0 ’0 W0
Kaleb o ﬂ’t’woe‘ a") eqkwo “I ‘F e w ,I 60 / + e a t : i “mi—“wax
 2 #1
2 L7 Milo 1 170*“? 0 )ﬁwo
" ‘ a“ +9?  u a .1 20045103 M2]
2 ~~Mg§gu~y ﬂ~m w A ’ l W 4Kujo
~y<.—A W»~~ Haw/W"Hﬂ“ ~ ~ ‘4.“ ¢¥
e l ws (CW>~/; ‘ lk1 2 {A )K., A 9": ‘Jm ” # A Name CM 4. (20 points) Properties of the Fourier Transform Show that if a signal, x(t), is real and even, then the Fourier Transform of the
signal, X(co), is also real and even. (7/ a m i ’33 K m new
4th z / 90/5  «(13 QM (wt
X/w) = :5398 “’1”; .Lq’tw W d M 'T M avgv‘ don; : UJJ
: {GK/k3 om/wt3Jé Cniiautmg, Over 1‘ T '30
.. :: 40?) 00qu
J/yw) g {53/63 VJ (a 35/43 Jo dA/p’ah/PVVW ...
View
Full Document
 Fall '07
 Throne
 Fourier Series, LTI system theory

Click to edit the document details