4/12/2017
EE 341:Discrete time linear signals and
systems
Lecture #11
UWEE TC Chen
DETERMINING STABILITY
Bounded inputs yield bounded outputs
If [] < for all then
[] = [] < for all
UWEE TC Chen
1
4/12/2017
STABILITY TEST
For positive proof: show analyt
BEE 341 Discrete-Time Linear Systems
Lesson Week 6: The Discrete-Time Fourier
Transform (Part I)
The Discrete-Time Fourier Transform (DTFT):Definition
The discrete-time Fourier transform (DTFT) is the (conventional) Fourier
transform of a discrete-time si
BEE 341 Discrete-Time Linear Systems
Lesson Week 6A: LTI Systems (Continuation)
Learning Goals
After completing this lecture students will be able to
Solve a difference equation using a classical method.
Determine the steady state response of an LTI syste
BEE 341 Discrete-Time Linear Systems
Lesson Week 9: The Z-transform (Part I)
The z-Transform
The DTFT (Discrete-Time Fourier Transform) describes systems in
frequency domain using the frequency response function H(ej)
The steady state response of an LTI s
BEE 341 Discrete-Time Linear Systems
Lesson Week 9: Discrete Fourier Transform
(DFT)
The Discrete Fourier Transform
Recall that DTFT is calculated from a discrete-time signals, x[n], and the
resulting function X(ej) is a continuous function, and, therefor
BEE 341 Discrete-Time Linear Systems
Lesson Week 7: The Discrete-Time Fourier
Transform (Part II)
Discrete-Time Fourier Transform: Properties
Paresevals Theorem: Provides the relationship of the energy expression in time
domain and frequency domain.
| x (
BEE341DiscreteTimeLinearSystems
LessonWeek10:TheZtransform(PartII)
TheSystemfunctionofLTISystem
The output of an LTI system y[n] to an input x[n] can be obtained by:
Where * is the convolution operation, and h[n] is the impulse response of
the system.
Exp
BEE341DiscreteTimeLinearSystems
LessonWeek8:TheZtransform(PartII)
TheSystemfunctionofLTISystem
The output of an LTI system y[n] to an input x[n] can be obtained by:
Where * is the convolution operation, and h[n] is the impulse response of
the system.
Expr
BEE341DiscreteTimeLinearSystems
LessonWeek7:TheZtransform(PartI)
ThezTransform
The DTFT (Discrete-Time Fourier Transform) describes systems in
frequency domain using the frequency response function H(e j)
The steady state response of an LTI system to any
BEE341DiscreteTimeLinearSystems
LessonWeek7:TheDiscreteTimeFourierTransform
(PartII)
DiscreteTimeFourierTransform:Properties
Symmetric properties of real sequences: If x[n] is real then its DTFT X(ej) is
conjugate symmetric. That is
X (e j) = X * (e j)
Pr
BEE341DiscreteTimeLinearSystems
LessonWeek8:DiscreteFourierTransform(DFT)
TheDiscreteFourierTransform
Recall that DTFT is calculated from a discrete-time signals, x[n], and the
resulting function X(ej) is a continuous function, and, therefore, cannot be
e
BEE 341 Discrete-Time Linear Systems
Lesson Week 10: The Z-transform (Part II)
The System function of LTI System
The output of an LTI system y[n] to an input x[n] can be obtained by:
y[ n ] x[ n ] * h[ n ]
Where * is the convolution operation, and h[n] is
University of Washington Bothell
Electrical Engineering Program
Due 2/13/2017 (Monday)
BEE 341 Discrete-time Linear Systems
Homework 3
Instruction: Work out the problems and show all steps of your solutions. When plotting graphs, put label
on all axes. Wr
4/18/2017
EE 341:Discrete time linear signals and
systems
Lecture #14
UWEE TC Chen
STEPS TO PERFORM CONVOLUTION
Two important rules for performing convolution
Flip the easy function
Draw a picture
Something to memorize
= []
0 = 0
UWEE TC Chen
1
4/18/
EE 341 Discrete Time Linear Systems
Problem Set #1 Solution
Problem 1
Problem 2
100
2
a)
b)
c)
k 1
k
1 2
1 2101
2
2 1
1
1 2100 1
1 2
3
3
k 0
101
100
k
2
k
k 2
k 0
3k 3 k 3 k 3 9
k
1
1
1
3
12
27
2
k
1
1
1
3
1
1
j
1
1
j 1
1
3 k 0 3
3
2 j
1
k
4/19/2017
EE 341:Discrete time linear signals and
systems
Lecture #15
UWEE TC Chen
DISCRETE TIME SIGNAL CONVOLUTION
Example: Given [] and [], find = []
[ ]
[]
3
Shift [ ] (leading edge is from = 0 to 2; end edge
is from = 3 to 5)
Multiply and sum both f
EE 341 Discrete Time Linear Systems
Problem Set #3 Solution
Problem 1
(b)
This system is time invariant, linear, causal and stable.
Proofs:
1. The system output depends on the previous inputs
2. cfw_[ 0 ] = [ 0 2] 2[ 0 8] = [ 0 ]. Thus, the
system is time
EE 341 Discrete Time Linear Signals and Systems
Problem Set #2 Solution
Problem 1
(a) Suppose [] is a periodic signal.
6
6
6
[] = [ + ] = sin ( + 1) = sin ( +
+ 1)
7
7
7
Therefore,
6
7
= 2, is an intrger. =
7
3
So the fundamental period is 0 = 7, which
4/14/2017
EE 341:Discrete time linear signals and
systems
Lecture #12
UWEE TC Chen
TEST TIME INVARIANCE
Example 3: test if the system is time-invariant.
Given = =0 []
1. Find 0
0 =
0
[]
=0
2. Find 0
0
=
[ 0 ]
=0
Let = 0, then
0
=
0
=0
[ ]
3. Compare,
EE 341 Discrete Time Linear Systems
Problem Set #1
Problem 1
Recall that complex numbers can be written either as + or as . Convert the
following to + and plot in the complex plane, be sure to draw the unit circle for
reference in your plots.
. , b.
1 2
EE 341 Discrete Time Linear Signals and Systems
Problem Set #2
Problem 1
Oppenheim & Willsky 1.26 (a), (c), (d)
Problem 2
Plot the even and odd parts of the following signals:
a. [] as shown in Figure 2.1
b. [] as shown in Figure 2.2
c. cos (2 ) ([] [ 5])
EE 341 Discrete Time Linear Signals and Systems
Problem Set #3
Problem 1
Oppenheim & Willsky 1.28 (b), (e), (g)
Problem 2
Let us define a system by the input/output relationship [] = [] + 1
a. Find the output of the system when the input is [] = [ + 2] [
University of Washington Bothell
Science and Technology Program
B EE 341 Discrete-time Linear Systems
Homework 6 (Due 3/10/2017 9:00pm through Dropbox)
1. Given the following two sequences:
x[n]= cfw_1, 0, 0, 1 for n=0,1,2, and 3
y[n]= cfw_0, 1, 0, -1 for
University of Washington Bothell
Science and Technology Program
B EE 341 Discrete-time Linear Systems
Homework 5
Due: 3/1/2017 5:45pm (Wednesday)
1. A causal first-order infinite impulse response (IIR) low-pass filter is described by the following
differe
University of Washington Bothell
Science and Technology Program
B EE 341 Discrete-time Linear Systems
Homework 4
due 2/22/2017 Wednesday 5:45pm
1.
Determine the DTFT of each of the following sequences if |<1.
(10 points each)
a) x[n]= [n]+[n-3]
b) x[n]=u[
BEE341DiscreteTimeLinearSystems
LessonWeek6:DiscreteFourierTransform(DFT)
TheDiscreteFourierTransform
Recall that DTFT is calculated from a discrete-time signals, x[n], and the
resulting function X(ej) is a continuous function, and, therefore, cannot be
e
BEE 341 Discrete-Time Linear Systems
Lesson Week 4: LTI Systems (Continuation)
Tadesse Ghirmai
Learning Goals
After completing this lecture students will be able to
Determine the output of an LTI system from its impulse response using
convolution.
Determi
B EE 341: Discrete-Time Linear Systems
Introduction
Course Overview
In B EE 341 you will study the mathematical representation of discrete-time signals and systems,
solve linear difference equations using classical techniques and z-transform, analyze the