ECE 4213/5213
Test 1
Monday, October 15, 2018
4:30 PM - 5:45 PM
Name: __S_O_L_U_'T_IQ_t\l_ _ _ _ __
Fall 2018
Student Num: _ _ _ _ _ _ _ _ __
Dr. Havlicek
Directions: This test is open book. You may also use a calculator, a clean copy of the course
notes, and a clean copy of the formula sheet from the course web site. Other materials are
not allowed. You have 75 minutes to complete the test. All work must be your own.
Students enrolled for undergraduate credit: work any four problems. Each problem
counts 25 points. Below, circle the numbers of the four problems you wish to have graded.
Students enrolled for graduate credit: work all five problems. Each problem counts 20
points.
SHOW ALL OF YOUR WORK for maximum partial credit!
GOOD LUCK!
SCORE:
1.
(25/20)
2.
(25/20)
3.
(25/20)
4.
(25/20)
5.
(25/20)
TOTAL (100):
On my honor, I affirm that I have neither given nor received inappropriate .aid in the completion of this test.
Date: _ _ _ _ _ _ _ _ _ _ _ _ __
Signed: _ _ _ _ _ _ _ _ _ _ _ __
1
1. 25 /20 pts. A discrete-time LTI system
h[n] =
H has impulse response h[n] given by
(1)n u[n-l].
The system input is given by
x[n] =
(i) n(u[n] - u[n - 6]) = { (¼6,n'
0 ~ n ~ 5,
otherwise.
~(k.1
~)""
~k
\
hCl<--1'\1-:. hf")~l
c~)ti.JcK
fbrrrr.~
\-'r\
----=-=-=-=-==--,,_,_-=--=-~=:::-=~o~__,,,,_,,,>------V\="-l--2 t~)~ - l ~):)}
~ (l)V\ )
2
\ ~ W\ ' t.,
1/\ 71
b
0Tl+~vt WA'l ~
r:
90
~['A)::
More Workspace for Problem 1. ..
~ [\(.J1lC~ --~ J
\c: ~ -IX>
--'-d
_____b____, , l
I)
'-.S-
I}
1
-x (t<.)
lt)~
_Lhfn
___~k.
()
s
0
'2-((~)~-t~)l\1
¥b-ll)~
\
V\<.\
J
,
1
~ ~<.b
"'.,,
b
_______
'=-_ _ _ _ _ __
-- ..
3
~
~ - - _ , _ , _ _ _---=1>
2. 25 /20 pts. The input x[n] and output y[n] of a discrete-time system H are related by
y[n]
= x[2 - n] + 1.
(a) 6/5 pts. Is the system H linear? Justify your answer.·
-v V\
___,_,.
L-e, \-- 1t 2 t\-\) := o 'QI/\
_.,
Le+
'1--t t';() -;. 0
Le~ ~= 2
~~
~ t t't"1 ~ 1- V
V'f
:fl.. YI
r-1
i1-T)1) ,-:-,
b ~ '3.
Le-\- ~ 3 t,A°')-=:- ~~1. ~1/\."1--\- l1 ~z 'C" 1 .:::. ()
5~\A.(e
a.~tl"-JT \o11:CtA1
-rke
J,I V\
---,.
~~LIA)-=-
i
t- ~,; rtA1;
'5'f~etAA
tr t0DT
:pt,an-
4
L-f N~L
n,eo_
a
.
\-I
II\ .
Problem 2, cont . ..
(b) 6/5 pts. Is the system H causal? Justify your answer.
LeJ 61t:t'n:) ~~ ~e.. ~"'- \>'~-:t f ,jlf\."- \ ,
\J
"'-e"'
we.. ~ut. \J e. ,
:f Co1 = --x C:-z.1 -r- i..
J-e,~.e11dt, M +1tt~ -Gv,"h-t\ff
vy·==--0J
-
w"'itVl-
\l/\~l.il~
"'/.
C"2-?,
(c) 6/5 pts. Is the system H BIBO stable? Justify your answer.
Let -x tlA)
be.
tt
t~eJ. , \f\.\>v---\- s,~ Ill~\.
T~eV\ 3'5 ~ t\'l1 B-;, 0 J S\.,d.A ~(d\._..\
NDWJ
\
~ tvt~ \
-:::. \ -X C1.-Y\1
-t<L \
~ \1<..t1.,II\)\ 4r
~
~-ti.
th~" e.Jore. , '1 c. .>
\""'t-(V\'1 \ ~ ~ ~ "'E ~ ..
1-
< .-:-:::,,
,1 ~ti-eA by
1ovV\kd l"'-\)IA.t sl~M\
1V'11t\J c0-s
'B + .i.
P, ,,.•.).
e" ery
a. b o ~ c;u,\~t"
5t~V\~\.
l
\
e
!I-€(/' ~Wl~--)
i
~ S'f f
€
5 1? \Bb
r J:---:::-
. .:. ,.
Sf/1..'e Lf°
Problem 2, cont . ..
(d) 7 /5 pts. Is the system H time invariant? Justify your answer.
--'T~tll\
y, t'I\')-=
,( [ '2.--nj ;- 1..
- u
Le.+ ~" =- 1 ,.
(1'2. - V\) -t'
T~el/\
1-.
ult
~-'nb 1 ::
j
\j
j
~ rl'\-f)
; U[ 2-eJ
-=
l
Ti.
g-;~f
LV\ -1) J +
Lt t_-i. -
::;;. U'C. 3-~j
-t-
i.
1-
2 -----""'111
'--.1
(
.
Now \e:t ')(, lV\1 = % t t'4\-'Ao1
uJl- V',)::
"'l
L
r 2.-'f\1 + 1- -:::
=:
1<.t r~-r)
l
·U. Le-, 1
'::;. \A. ( '2. -
\I\ .- \
{:).:. '2-Y\
1 ~ 1-
=
3
.
IAX'A-r)
.,. .1...
I
2·- - l
~
::; vl'C
\-1/\"")-t-1.
6
-1
----..!!u"P- ti\
~iii
3. 25/20 pts. The input x[n] and output y[n] of a discrete-time LTI system Hare related
by
1
y[n] = x[n]
4
1
1
1
+ 2x[n - 1) + x[n - 2] + 2x[n - 3] + x[n - 4].
4
Find the system group delay r9 .
:lrn===---r : y {d"') ::: X( t'. ,lw) [ ¾- t- "i e.- j""' -1- e.- \'2.-;, ..I- ~ e.,-.i ,..., -\-
Hlei"')::
-
~
YLe:i") = L-..L +½e_-i~+
X(-eS'--')
A:
-\
L4e
jt.w
L1
l
4--i:.e
-1- COS'->
~~
+-
A
\
+-1-+2.
e-J'~ +4l. e-j1.wJ e-12~
~ Cos '2.w] e- ~ '2.~
arj
~"'use.~
e"r>Vt
ar-i
~lei'-'))
==
9 (Lv)
d.e¼y -= l:j :: - JwGltw)
.i...----·- -
::.
-
~
'
7
J
e-~i~~½.e..-33~ ~ke-.i~
'1
~
)
~
SfQr"'1rt'll\
~ e-l '1-w
'2 ~
l-l le jv)
4. 25/20 pts. The input x[n] and output y[n] of a discrete-time LTI system Hare related
by
y[n] -
10
3
y[n - 1] + y[n - 2] = x[n] -
1
2x[n -
l].
(a) 5/4 pts. Find the transfer function H(z) and give a pole zero plot.
l - ½=e.-1
( l-
½2-, ) L.1 - 3 z-~
...
ierc?'> ~ ~ .::. 0 , :z -:: . {
-roLH ~
t=-\ , ~ ~ 3
ff~',
'
l+lB) =
\.,LG
'2..-
--- -
fl= 1e)(t~30)
,- ~e
f,-=-
a
-t-'38
l0:::i
=
--
+B
_
1~56
~
·-, - ~ \ -
-=£.
J..
\ - ls>
\
-
- 1.
~
8
"
1-19
-
_ 1~
-f
-
\
\ l:,
Problem 4, cont ...
(b) 10/8 pts. Assume that the system His causal. Specify the ROC of H(z) and find
the impulse response h[n]. For this assumption, is the system H BIBO stable?
!-\ ca.u.s(.(_.l
-- · - ·
Wttv\S
fU>l-:.
'"t1Ae R,l'L \~ ev...-kBttlvr -h i~e. \&t~ei..\ ~e,\e,
l't.\ '7 3
S'l'l\{e.. +ke \U)L
doe.~
--
~e
~{ft" Cc,t\'t~ill\
t...\
·g
\Ali\l"\7
~,rt
-
~-t lr) :::
\/l 6
-j
----l
l"l
~
l'l' \-, \~
+
lS/i,Ja
-
l-,rt
~
\1:\-,3
-
g
t':t\J'"'-l•e,
~\~0
----
S7k~ t,..i;: •
I
=>--=
Problem 4, cont ...
(c) 10/8 pts. Now assume instead that the system His ·BIBO stable. Specify the
ROC of H(z) and find the impulse response h[n]. For this assumption, is the
system H causal?
5',"<e
-.+"'-.e.
RDL
1s
s'1 ~keVA \s
ut~\ =::
\Jtt~
\--~ :e--\
v-v----)
\~\ ') 1
ttf/\V\\J--\~
~
+
1.
I/'
1
~(vl)
\s
at I.As~ \. ,,
\~~b
\ - "3 ~\
~
\~\ ~ 3
10
·\\,..,p-s,·tled ttV\d. ~e
5. 25/20 pts. The continuous-time LTI system Ha(D.) shown in the figure below is
implemented using an ideal A/D converter, a discrete-time LTI system Hd(eiw), and
an ideal D/ A converter.
f=r ~ 44-1 l ~i-\1
==- 4. 4 1 lot> H~
r-----------------------------
xaCt)
yaCt)
Ha(ejro)
------------------------------'
-fl' -:: '2.. 'Tf °F-r
- 8~, iuo-rr
T :: '2."if
..JlT -
·
r.d:
\
sec.
.L - -....-- 5""'
F,. - 4~>
~c.
As in professional compact disc (CD) audio, the sampling frequency of the A/D and
D/ A converters is Fr= 44.1 kHz. So Or= 88, 2001r rad/sec. All input signals xa(t)
are bandlimited to IOI < Or/2 (this simply ensures that the overall structure shown
in the figure will be an LTI system).
'"°
The impulse response of the continuous-time filter Ha is given by
ha(t)
= sin 70001rt.
1rt
(a) 8/6 pts. Find the continuous-time frequency response Ha(O).
TtthLe
~
s\~~!.
6-=)
\
i :]
-1/J
w::; -tooo,c,
:1-, .{)..
vi
C
--,()Q()'7l
(b) 8/7 pts. Find the discrete-time frequency response Hd(eiw) .
..n.
i
i
)
0
ooo 1(
\w\~ 44J \ob
~ l r.,.j\ < 1t
\
-1'
11
r
--
- 70()01f
44-J lfJO
I
0
l
i
!!f!!JJ
I)
14,t(JP 1\
(..J
Problem 5, cont . . .
(c) 9/7 pts. Find tlie discrete-time impulse response hd[n] .
--------
12