ECE 351 Linear Systems II
Homework #12
1. Find the minimum order of Butterworth filter to meet the following specifications:
(i)
3 dB cutoff frequency at 1 rad/sec.
(ii)
The passband requirement is
H(j)2 0.99 for 0 .25 rad/sec
(iii)
The stopband require
State Variable Equations
y
k1
k
k
= CA q
0
+
!CA
k1i
Bx
i=0
h
k
=
D; k=0
CA
k1
B ; k>0
H(r ) = C[rI ! A] B + D
!1
H(e j! ) = C[e j! I " A] B + D
"1
i
+ Dx
k
ECE 351 Linear Systems II
MATLAB Tutorial #10
FIR Filter Design with MATLAB
This tutorial will use various features of MATLAB (including the use of the Signal Processing
Toolbox) to design and analyze FIR filters. In class, a low pass FIR filter was studi
Table of Summatio ns
!
"a
i
=
i=0
k
!a
i=0
i
!i
=
i=0
k
!i
i=0
!ia
i=0
i
=
a <1
k +1
1a
=
1a
k
k
1
;
1# a
a
1a
2
=
;
k(k+1)
2
k(k +1)(2k +1)
6
k
2
a1
1  (k +1)a + ka
k +1
a! 1
EE 351 Linear System II
MATLAB Tutorial #12
Using the Butterworth Filter Design Functions in MATLAB
This tutorial describes some of the functions contained in the Signal Processing Toolbox that can
be used for the design of Butterworth filters. Similar fu
ECE 351 Linear Systems II
MATLAB Tutorial #11
Using the Fast Fourier Transform (FFT) in MATLAB
This tutorial introduces the use of the Fast Fourier Transform (FFT) for determining the
frequency spectrum of a signal.
The Discrete Fourier Transform (DFT)
Th
ECE 351 Linear Systems II
Homework #11
1. A sequence of length 4 is given by:
f0 = 1, f1 = 2, f2 = f3 = 3.
Find the DFT for this sequence.
2. Compute the DFT (length N=5) of:
fk
1
1
1
k
1
2
3
4
3. Compute the DFT (length N=5) of:
fk
1
1
1
k
1
2
3
4
4. Fin
EE 351 Linear Systems II
Homework #5
1. Write the complete set of state variable equations in matrix form for the discrete time system shown
below.
6
+
xk
+
unit
delay

5


unit
delay
3
3
+
4
yk

" 6 5 %
"0 %
Answer: A = $
B = $ ' C = [ 3 0 ] D = [ 4
ECE 351  Linear Systems II
Homework #1
1. Sketch the following sequences:
a. fk = (1/2)k uk
b. fk = (1/2)k uk+2
c. fk = uk+1  uk3
d. fk = k+2 + k + k2
e. fk = 2 k [ uk+2 uk2 ]
f. fk = (1/3)ku3k
Answers a.
b.
c.
fk
fk
fk
4
1
1/2
1
1
1/4 1/8
1/16 . .
ECE 351 Linear Systems II
Homework #3
1. For a system described by the linear difference equation:
yk
1
yk 1 = xk
2
a. Find an expression for H(ej)
b. Find an expression for H
[Answer:
1
]
1 j
1 e
2
1
[Answer:
"1
%
1
(1 cos )2 + $ sin '
#2
&
2
2
]
1
si
ECE 351  Linear Systems II
Homework #7
1. Find F(z), the ztransform of each of the following sequences:
a.
k
(
! 1$
k+
fk = * 5 + 2 # & 3( 4 )  uk
" 2%
*)
,
b.
! 1$
fk = # &
" 2%
Answer: F(z) =
k2
uk 2
Answer: F(z) =
k
c.
! 1$
fk = # & u k 2
" 2%
Answ
ECE 351 Linear Systems II
MATLAB Tutorial #7
Using the residuez Command in MATLAB
This tutorial will describe one of the MATLAB commands that can be used to perform the
partial fraction expansion of Ztransforms. Note that the Symbolic Math Toolbox has fu
ECE 351 Linear Systems II
MATLAB Tutorial #6
Solving State Variable Equations for Discrete Time Systems
This tutorial will discuss the use of MATLAB in the solution of state variable equations. The
methods described here only use MATLAB commands that are
ECE 351 Linear Systems II
MATLAB Tutorial #3
Finding the Frequency Response of Discrete Time Systems in MATLAB
In class we discussed a method for obtaining the frequency response function, H(ej ) of a discrete
time system. MATLAB can be used to plot the m
ECE 351 Linear Systems II
Solutions to Homework #12
1.
We need to have a 7th order filter:
1
2 n ! 0.99 " n ! 2
1 + ( 0.25 )
1
1 + (2)
2n
# 0.0001 " n ! 7
2. We need to have a 32nd order filter:
!
$
1
10 log #
2 n & > '0.1 ( n ) 12
" 1 + ( 0.85 ) %
!
$
1
Name _
ECE 351 Quiz #1
September 19, 2011
Problem 1
Find the linear difference equation describing the system shown below. Do NOT
solve this equation.
2
+
xk
+
Unit
Delay
Unit
Delay
+
5
6
If we define the top input to the right summer to be yk(1) and the
Name _
ECE 351 Quiz #2
October 3, 2011
Problem 1
A discrete time linear system has an impulse response:
k
"1%
hk = !k!1 + $ ' uk
#2&
and an input:
xk = uk
Find the output yk, for all k.
j
j
.
.
(
"1% +
"%
(! j!1uk! j + + / $ 1 ' u j uk! j
yk = / h j xk!
Name _
ECE 351 Quiz #4
November 9, 2011
Problem 1
A discrete time linear system has a transfer function:
5z 2 + 6 z + 9
H (z) =
z2 + 9
Find the impulse response, hk.

H ( z ) 5z 2 + 6 z + 9 A Bz + C
=
= +2
! A = 1, B = 4, C = 6
z
z z +9
z ( z 2 + 9)
z (
Name _
ECE 351 Quiz #5
November 28, 2011
The block diagram below represents a connection of three subsystems in terms of
their frequencyresponse functions.
a. Find hk, the impulse response of the entire system.
If we let z = e j! , then the transfer func
Name _
ECE 351 Quiz #3
October 12, 2011
Problem 1
Write the complete set of state variable equations in matrix form for the oneinput,
twooutput system shown below.
+
+
x
k
+
Unit
Delay
Unit
Delay
1
k
y
2
k
y
5
6
If we define qk(1) as the output of the
Discrete Time Fourier Transform (DTFT)
j!
F(e
!
)=
! fk
e
jk!
k =0
f
k
=
"
1
2"
F(e
j!
)e
jk!
d!
"
Discrete Fourier Transform (DFT)
N 1
F
n
=
!
f
k
e
j
2!
nk
N
; 0"n"N1
k =0
f
k
=
1
N
N 1
!
n=0
F
n
e
j
2!
nk
N
; 0"k"N1
Butterworth Filter Design Equations
nth order low pass Butterworth filter:
1
1 + ! 2n
1
H (s ) H ( " s ) =
1 + ("1)n s 2 n
H ( j!) =
2
Low Pass to Low Pass Filter Transformation:
s!
s
"c
"!
"
"c
Low Pass to Band Pass Filter Transformation:
s!
2
s 2 + "o
B
ECE 351 Linear Systems II
MATLAB Tutorial #1
Representation of Discrete Time Signals In MATLAB
This tutorial will discuss methods for representing and graphing discrete time signals (sequences)
in MATLAB. It assumes that the reader has a basic knowledge o
ECE 351 Linear Systems II
MATLAB Tutorial #4
Convolution and Impulse Response in Discrete Time Systems
MATLAB has a builtin function, conv that can be used to perform discrete time
convolutions. For two finite sequences:
x = [ 3, 11, 7, 0, 1, 4, 2]
and
ECE 351 Linear Systems II
MATLAB Tutorial #2
Solving Linear Difference Equations In MATLAB
In class we have studied methods for determining closed form solutions for linear difference
equations. This tutorial will discuss two numerical methods for solving
ECE 351 Linear Systems II
MATLAB Tutorial #5
Modeling Discrete Time Systems in Simulink
This tutorial describes the use of Simulink, a graphical user interface within MATLAB. Using
Simulink, the block diagrams of discrete time systems may be entered graph
EE 351 Linear Systems II
Homework #6
1. Find Ak for the following matrices:
"
k
1
$ 5 (2)
k
Answer: A = $
5
$# 0
"2 1 %
a. $
'
# 0 3&
" 1 2 %
b. $
'
# 5 5 &
c.
" 3 1 %
$ 1 1'
#
&
Answer: A k =
k
(2) (3)
5 (3)
k
k
"
%
2sin k
1 ( 1 +
' where = tan * ) 2,
co