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 DocumentThis 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 DocumentThis 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 DocumentThis 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: ECSE—2410 SIGNALS AND SYSTEMS SPRING 2007
Rensselaer Polytechnic Institute EXAM #3 (1 hour and 50 minutes) Check One: April 11, 2007 3 Sect. 1 8:30am (Wozny) _ Sect. 2 10:00am (Desrochers)
NAME: 6’ , ‘ Do all work on these sheets. Tables will be handed out separately.
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. Total Grades for Exam: Grades for Points . Score ﬂ
Part I 40
Part II 38 ___ﬂ
Part III 22
TOTAL 100
PART I
Problem Points Grade A
1 w 6
2 6 J
3 .. 8
4 12 V
5 8
TOTAL 40 1(6). Find the Laplace transform, Y (S), of 320‘) = 5(t — 2) * 3 cos (2t) u(t) where * denotes convolution.
W 775 «a
){f/ a #6? K0)
:: 6/2.; 3 5 KW” 2(6). Find the Laplace transform of v(t) 2' (t —1)e'(“‘)u(t). 3(8). Find x0), the inverse Laplace transform of X(s) = e (s + aXS + 2) ,wherea>0and c1962. 1?) l s ”44" + ”’6"
Y‘ ”(Ha/(5+2; 9“” 5’”
/
,1. = j”;
’4' {14.2 j: ~61 4(12). Find x(t), the inverse Laplace transform of X(s)= /3
, x3 :
ﬂ WIN
3;? 1/)
ﬂKx/rC‘ )m
1
?
[2 ,. 9,52
I “3:; {4'2—
..  (5W ; '
’ ’77?
2.
,1.» +~’:’
x(s)> J” [f7
:2 .2»
== ,4 * 3 5’2»? 13 (S + 2)(s‘2 + 9) . 587 ’11,..ﬁ/ 52+? 5(8). A twostage ampliﬁer system is shown below Y where G, (S) = a and G2 (S) = 7 c . Performance Speciﬁcations require that the poles of
S + a S“ + 195 + c '
Y (S) . . . 9,
H (S) = he on a Circle of radius 2 as shown. Jm
X(S) s—plane a.)(2) Is the system 02(3) (circle one) 5
A. overdamped, '9‘ ﬁe B. underdamped, C. or critically damped? b.)(6) Find the constants a, b, and c so that H(s) has the required poles shown in the ﬁgure above. ﬂo/e, keg/22w f «2—, We? ECSE2410 SIGNALS AND SYSTEMS SPRING 2007
Rensselaer Polytechnic Institute EXAM #3 (1 hour and 50 minutes) April 11, 2007
Check One: _ Sect. 1 8:303m (Wozny)
_ Sect. 2 10:00am (Desrochers)
NAME: 301—4) Tl 0N3 Do all work on these sheets. Tables will be handed out separately. 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. PART 11
Problem Points Grade
6 12 T
7 10
8 8
9 . 8
TOTAL 38 lOOja) 6(12). A certain linear system has the transfer function H(a)) == ———————~————~——.
(5+jco)(1+jco/20) (a)(6) If the input to this system is x(t)= 5cos (lOt) ﬁnd the exact, complete steady—state output, y(t).
Complete means that you should include magnitude and phase in your answer. . 1772..
L140”! ”122ml...— _, W (W6)
Given H(a)) = 'K0w+U 20 200
number, not in dB!) so that the magnitude of H(a)) is zero dB when a) = 400 radians/sec. Note. You must use
the Bode approximation. No other answer is acceptable (“+900 0
(W74: K‘U ‘ We Made .Using the Bode magnitude approximation, ﬁnd the value of K (a View" 3;” 1+ J;°°l (75°30) "we: K S)
lA 10(jco+10)
(WHEN)
a described by 1H (60)] >  20 dB for a) > 500 rad/sec. 7(10). Given H(a)) = ﬁnd the largest value of a which avoids the forbidden region, T }H(w) in dB ‘ E ; ; §,\,,F,thidden _
r, , I, :Region 8(8). A signal x(t) With spectrum X (co) (shown below) is ideally sampled (i.€., with delta functions) at l, a) < 5
sampling frequency cos = 10 rad/sec and then ﬁltered with an ideal ﬁlter H (60) = i 1 . Sketch the signal
0, else spectrum Y (0)) at the output of the ﬁlter. Scale vertical axis! X(co) sampling 2” 2” 27f operation —12 —5 5 12 9(8) (a)(4). Find Y“) X( ) , expressed in terms of G1(S), G2 (3), H1 (3), and H 2 (s) for the following feedback system.
s X(S) Y(s) 11 9(8) Continued. (b)(4). For the block diagram shown, if E(s) = X (s) — Y (s) , ﬁnd 1ime(t) when x(t) :2 tu(t) . X“) +_.G(S)=s(::1> W)
1: a“ 25) g 613) 1 >
«t9 5%! m) EQ— , WA 79% My
_ \
_. 313) 1+ to
SLSH) +7960 3+0 Sac 1.9T];
: é » l ==.1031. ]
Saw $4219... W 12 ECSE2410 SIGNALS AND SYSTEMS SPRING 2007 ég
Rensselaer Polytechnic Institute n‘y EXAM #3 (1 hour and 50 minutes) NAME: Do all work on these sheets. Tables will be handed out separately.
One page of crib notes allowed. Calculators allowed. Check One: April 11, 2007 D Sect. 1 8:30am (Wozny)
D Sect. 2 10:00am (Desrochers) Label and Scale axes on all sketches and indicate all key values. Show all work for full credit. PART III
Problem Points Grade
10 6 ﬁ
11 L 8 ,
12 8
TOTAL 22 13 10(6). Suppose the frequency response for a certain system is H1(a)) = 1H1(a))jej‘”' (w) . x(t>———~> m) (a)(3). If the magnitude of H ‘ (60“) at some frequency (on is ~6 dB, then its absolute value (not in dB) is A. :9.
20 B. 6 c. l E. 2. (b)(3). The Bode plot (magnitude and/or phase) of H 2 (co) : (1 / j (0)111 1 (wje’w ‘0”) , compared to the original
Bode plot of H1(a)) , changes as follows: A. The phase plot, AH} (co) , is shifted up 90 degrees. he phase plot is shifted down 90 degrees.
C. The magnitude plot, 1H 1 (co) , shifts up by 10 dB. D. The magnitude plot shifts up by 20 dB. E. There is no change in the magnitude or phase plot. 11(8 ). On the next page, match the step response to the correct Bode magnitude plot. Place the appropriate
number (1 ,2,3,4) in the boxes below. The scales on all step response plots are the same. The scales on all Bode magnitude plots are NOT the same. 14 1‘: Hwy (radius) Tm (soc)
Sup Rama ;
1
z
i
5 150:
00 2‘5 05 Rwy (radius) Tm (use) m an
20 33 €3.23: {.33. . :. iii: _i(., .10 
~20 35%.5 150
menu“
5°T“" IO 10' 10' Tm (lac) nuﬁuiuJiﬁrl “Muununnu “HUME
«:owulx¢r~»nuu....»n»xv
xur+»uoruux.7:wyno as 835.: Frequency (Ind/w) TM (sec) 14 (12)(8)MATLAB Questions. a.)(4) What does the following Simulink model simulate? Circle your answer. [ll—+3 u(t) d2)» dy A.———~ ——— z: t
dtz dt y() u()
dy l B.——+— + z=0
dt 2y “0 C (z)~u(z)[i——1——1) b.)(4). Given the transfer functions H1(S) = 53 ~35 +1 and H2 (S) = 35 + 453 —— 252 +5 + 5, 6(5) = Hl (3)112 (s)
can be calculated by which MATLAB statement? Circle your answer. A.G = H1 * H2;
B. H conv([ll '31 1]! [ll 4! 21 1: 5]); , conv([ll OI 4! 2! ll 5]: [ll 0! '31 1]);
D.G = poly(Hl) * poly(H2); l6 ...
View
Full Document
 Spring '07
 WOZNY

Click to edit the document details