PROBLEM 9.8:
Problem 1
McClellan, Schafer and Yoder, Signal Processing First, ISBN 0130655627.
Pearson Prentice Hall, Inc. Upper Saddle River, NJ 07458. c 2003
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EE 224, Spring 2014 Quiz VIII, page 2 of 6 Tue 8th Apr, 2014
1. Plots of an input signal x(t) and an impulse response h(t) of a continuoustime (CT)
linear timeinvariant (LTI) system are given below.
The output of the LTI system is y(t) 2 Mt) >k x(t) =
Problem 1:
1
Problem 2: ejn4t is a complex sinusoid with frequency 4n. So the output,
corresponding to the input ejn4t will be
H F (4n)ejn4t
Then if the input is
n
ejn4t
1+n2
the output is
n
H F (4n)ejn4t
1 + n2
Now our input is
x(t) =
So the output is
n
BB 224, Spring 2014 Quiz VII, page 2 of 6 Thu 27th Mar, 2014
1. Recall our denition of the unitrectangle function:
1
rect(t)
1/2 0 1/2 (5)
(2) (a) Sketch z(t) = rect
Hint: What is the Width of the pulse z(t)? 2/
(b) Consider the signal x(t) below.
x(t
RE 224, Spring 2014 Quiz VI, page 2 of 5 Thu 13th Mar, 2014
m] LTI : fem]
.A b
(em), h in];
ii
bmu_ 4wqmd.w~.r max 249
1. A linear timeinvariant (LTl) system has the following magnitude and phase frequency
response plots: r
. , I , _ I r. 7 3? 'h
il
BB 224, Spring 2014 Quiz 111, page 2 of 4 Thu 13th Feb, 2014
1. Suppose that a periodic signal x(t) is dened by the plot below (only the section 8s
to 83 is shown):
e To; 35 ~z
87654321 012 3 4 5 6 7 8
Time(sec)
(a) (3 points) Determine the fundamental
BB 224, Spring 2014 Quiz VI, page 2 of 5 Thu 13th Mar, 2014
LTI
Hwnhm
x[n]
1. A linear timeinvariant (LTI) system has the following magnitude and phase frequency
1esponse plots:
Magnitude 01 Frequency Response Function H( e3;
19) E (1.2? 0.4' 0.611
EE 224, Spring 2014 Quiz II, page 3 of 5 Tue 4"1 Feb, 2014
2. For the following short answer questions, write your answers in the space provided or
check the correct answer. Provide a. short justication for your answer.
(a) (3 points) The periodic signal
V1,. (6 pts) For each of the following systems, nd the corresponding impulseresponses h(t),_
Hint: Use the denition of the impulse responSe: replace x with (5.
(a) <3 ms) m = w = KW) d7. , ' _ ,
l?wh Meme
g/   i g?
(b) (3 pigs) = = 56(7) d? for a giv
Problem 1:
Problem 2:
F
Problem 3: Recall that f (x) g(x) F ()G(). Then Y () = X()H().
We know that the Fourier Transform property for time scaling is,
F
f (t)
and that
1
F( )

1
F
sinc(t) rect( )
2
2
Using the (1) and (2) we can nd X() .
1
F
x(t) = si
EE 224
3.
hw 8 solutions
a)
b)
c)
d)
e)
3 of 4
h(t) = et
h(t) = rect( t1 )
2
h(t) = rect(t)
h(t) = (t + 1) (t 1)
h(t) = u(t)ejt
Solution:
a) Since h(t) = 0, for t < 0, then the system is NOT CAUSAL.
Analyzing if it is absolutely integrable,
et dt = .
h(t
Problem 1
1
Unit triangular function is dened as
(t) = 1 + t if 1 < t < 0
= 1 t if 0 < t < 1
= 0 otherwise
And the unit rectangular function is dened as
1
1
rect(t) = 1 if < t <
2
2
= 0 otherwise
Then we found from the picture that
a)
x(t) = 2 t +
1
2
2
Problem 1
a) Magnitude and the phase of frequency response are given in the following picture.
b) Yes, because H(e j ) is an even function of and H(e j ) is an odd
function of .
c)
Digital Frequencies are .
4
Digital Frequencies are same as in previou
Problem 1
a) We found that x(t) = 4p(t 1.5). Now the fundamental period from
the picture is T0 = 4s. Then 0 = 2 = .
4
2
b) As p(t) is a periodic signal then p(t) can be written as p(t) = k= ak e jk0 t .
k=
For the signal p(t), which is a squarewave funct
HW 3 Solutions
Problem 1:
1
Problem 1
(1) even
a k ak
So, x t
ae
k
jk0t
k
a
k
k
e jk0t
If k ' k
x t
a
k'
k '
e jk '0t x t , so x (t) is an even function
(2) odd
a k ak
k
So, x t
k
ak e jk0t ak e jk0t
If k ' k
x t ak 'e jk '0t x t , so x (t) is an
1. (11 pts) We showed in class that the Fourier coefcients 0f
are  ' _ ("53 M
Gk = Sine t _ aviipebww w 1)
To T0 V f?
for all integer k and T 2 0, see also the denition of the sine function on p. 2. \Use this fact
to derive FS representatiOns of si
ee 224: hw 2 solutions
February 3, 2014
1.
s ( t ) = cos 2
(e
( t ) =
0
j0t
+ e j0t
)
4
2
=
e j 20t + e j 20t + 2
4
1 1
1
= + e j 20t + e j 20t
2 4
4
Spectrum of s(t)
1/2
1/4
2w0
0
1/4
2w0 W(rad/s)
2. P 2.7 (a)
3e j 3 + 4ej 6
= 3 cos
+ j sin
+ 4 cos
3
3
1. (14 pts) Consider the following sampling and reconstruction system:
Ideal Ideal ya)
AtOD DtoFI
Converter I r Converter
Ts : z
where
I = :L(n wmax
Consider the following CT input: L
I "16% 4752.310 098(10) +2;C?s<407rzt+vr/> y W. '
(a) (3 pt
1. (12 pts) Plots of an input signal x05) and an impulse response h(t) of a CT linear LTI
system are given below.
Tx) ' (1)
'1
i
, /
/ \
/
v/ .r 1 L fwww 1 A
e 3 g 2" 7 2
I"! .Li;w / 3/
\i ,
The output of this LTI system is * cfw_h(t) = 23(7) h(yt 
_1. (17 pts) The periodic signal x(t) in the gure below has Fourierseries representation
x(t) = 2:1 ak ej Wot where ak are its Fourier coefcients (which we denote as 00(25) +> ak).
m .
(a) (2 pts) What are the fundamental period To (in seconds) and
/ , h 57 A fl \ V; 5 g i
f a 5 l  3 '3" W I, u i
' [/f I 'J N" 5 " v (If 1 ' s " WW 7 M a d s
K #1? 1 05M if sf 6? g
1. Simplify each of the following expressions and sketch the results:
(a) (3+1 pts) (1+ t2) [6(t) 26(t 4)],
(b) (3+1 pts) +00<T2 + 6) 6(
EE 224, Spring 2014 Quiz V, page 2 of 4 Thu 6th Mar, 2014
1. (a) (7 points) Evaluate the convolution:
yn= conv( [10 O 10 O 10], [1 2 3] );
Where a Matlab vector implicitly denes a signal to have its starting point at n = 0.'
Give your answer as a stem plo
Tue 28th Jan, 2014
BB 224, Spring 2014 Quiz 1, page 2 of 7
1. (a) (3 points) In the gure below two sinusoidal signals are shown. Which one has a
phase of +71:/ 3? Check the correct answer: Q x1(t) xt).
A B
75 , . . . i i . , , , . . , , . , . , l H
50
BB 224, Spring 2014 Quiz IX, page 2 of 5 Tue 15th Apr, 2014
1. Plots of an input signal x(t) and an impulse response 11(1) of a CT linear timeinvariant
(LTI) system are given below.
x(t) ha)
t (5]
(1) (a) Make a carefully labeled plot of
z(t) = rec
BB 224, Spring 2014 Quiz IV, page 2 of 5 Thu 27th Feb, 2014
Fourier Series Properties.
,x(t) with Fourier coefcients ak fundamental period To
y (t) with Fourier coefcients bk fundamental period To
YO) = AXU) bk = Aak
y(t)=B+x(t) b0=B+ao andbk=ak,k70
BB 224, Spring 2014 Quiz X, page 2 of 5 Tue 22nd Apr, 2014
1. Consider the continuoustime (CT) signal
. 00
x(t) = smc
see p. 1 for the denition and sketch of the sine function.
(3) (a) Determine coo so that the zeros of x(t) lie at t = 2n for nonzero int