EE 341
Spring 2006
Prof. Atlas
Final Exam
June 6, 2006
Your Name:
Solutions
Exam Instructions: 1. Open book and notes. But as listed in our course syllabus: No electronic devices (calculators, laptops, Pilots, cell phones, beepers, etc.) are allowed for e
3/1/16
EE235: Lecture 15
Fourier Transform
FT Basic Idea
Fourier Transform Formulas
Signals that have a FT: Dirichlet Conditions
Examples
1
Fourier Transform: Basic Idea
Sound: air molecules
quickly bouncing back and
forth
You dont feel the air hittin
2/6/16
EE235: Lecture 13
Fourier Series
Analysis: decomposing a periodic signal into
Fourier Series coefficients
e.g., given x(t), find F.S. coefficients
1
Fourier Series Analysis:
Finding Fourier Coefficients
x(t) =
c e
jk0t
k
T0 = 2 / 0
Synthesis
Equa
4/19/2016
EE235: Continuous time linear systems
Introduction to signals and systems
Lecture #14
UWEE TC Chen
FINDING THE IMPULSE RESPONSE
Example: Find () when =
(
1
)
Two ways:
1) Plug in for ()
=
1
= ( + 1)
2) Change into convolution form:
=
1
=
5/3/2016
EE235: Continuous time linear systems
Introduction to signals and systems
Lecture #20
UWEE TC Chen
HARMONIC SERIES
=
0
=
0 is the DC (constant) component.
1 0 + 1 0 is the fundamental frequency (first
harmonic) component.
0 for 2 are the harm
4/29/2016
EE235: Continuous time linear systems
Introduction to signals and systems
Lecture #18
THE IMPULSE RESPONSE VS. THE
NATURAL RESPONSE
Example: For an LTI nth order LCCDE, 2 + 3 +
2 = = 10 3 . What is ?
Setting = 0; (impulse response)
with special
2/21/16
EE235: Lecture 20
LCCDE (Linear Constant Coefficient
Differential Equation)
LTI filters described by LCCDE
Summary of methods to solve I/O analysis
with LTI filters
An application of the FT to analyze systems
1
LCCDE
In the physical world, inpu
2/24/16
EE235: Lecture 21
Amplitude Modulation
Demodulation for sinusoidal AM
Frequency-division multiplexing
An application of the FT to analyze systems
f (t)g(t)
1
F( j ) *G( j )
2
1
Communication Systems
All wireless communication transmits data t
2/17/16
EE235: Lecture 19
Linear Phase LTI Systems
LTI Filters
An application of the FT to analyze systems
1
Frequency Response of LTI Systems
()
H ( j ) =
LTI
()() = Y()
Y ( j )
= | H ( j ) | H ( j )
X( j )
Note: |H(j0)| for =0 is the DC gain
|H(j)| is
4/27/2016
EE235: Continuous time linear systems
Introduction to signals and systems
Lecture #17
UWEE TC Chen
ZERO-STATE OUTPUT OF LTI SYSTEM
Total response(t)=Zero-input response (t)+Zero-state
output(t)
Response to our input x(t)
LTI system: characteri
3/11/16
EE235: Lecture 27
Course Review
EE235 Topics
Signal
x(t)
a x1(t)+ b x2(t)
x(t-t0)
x(t)
Eigenfunction
est
Fourier Series
ckejk0t
System
Output
y(t)
LTI
a y1(t)+b y2(t)
y(t-t0)
h(t)
Convolution
x(t) * h(t)
h(t)
h(t)
Time
domain
H(s) est
H(jk0) ck
3/9/16
EE235: Lecture 26
LCCDE and the LT
Interconnected LTI Systems
Pole-Zero Plots
Eect of zeros
1
LCCDE & LT: Example #1
Given the DE: y'(t) + 3y(t) = x(t)
Find H(s), ROC, and h(t).
LT both sides to get algebraic expression:
s
EE235
Name:
Student ID:
Final Exam
University of Washington EE235, Winter 2016
March 14th, 2016
Exam Information:
The test is closed book, and no calculators/devices are allowed. You are allowed ONE 8.5x11 (twosided) page of notes.
Please show all work.
EE 235, Winter 2016, Homework 5: More LTI Systems
Due Friday February 5, 2016 via Canvas Submission
Write down ALL steps for full credit
HW5 Topics:
Impulse Response and Step Response
LTI System Properties
Exponential Response
HW5 Course Learning Goals
EE 235, Winter 2016, Homework 10: Laplace Transforms
Due Friday March 11, 2016 via Canvas Submission
Write down ALL steps for full credit
HW10 Topics:
Laplace Transform and Inverse Laplace Transform
Laplace Transform ROC and Signal Properties
Laplace T
EE 235, Winter 2016, Homework 8: Fourier Transforms, LTI Systems, and Filters Due
Wednesday February 24, 2016 in class via Canvas Submission
Write down ALL steps for full credit
HW8 Topics:
Fourier Transforms: LTI
LTI Filters
HW8 Course Learning Goals S
EE 235, Winter 2016, Homework 9: Sampling and Modulation
(Due Friday March 4, 2016 in class)
Write down ALL steps for full credit
HW9 Topics:
Sampling Theorem and Aliasing
Modulation and Demodulation
HW9 Course Learning Goals Satisfied:
Goal 5: Underst
EE 235, Winter 2016, Homework 10 Supplementary Notes
LAPLACE TRANSFORM PROPERTIES:
PROPERTY
LAPLACE TRANSFORM
ROC
A x1 (t) + B x2 (t)
A X1 (s) + B X2 (s)
ROC = at least ROC1 ROC2
x(t to )
esto X(s)
ROC = ROCx
x(t), > 0
1
s
X( )
ROC = ROCx
x(t)
X(s)
ROC =
EE 235, Winter 2016, Homework 6: Fourier Series
Due Friday February 12, 2016 via Canvas Submission
Write down ALL steps for full credit
HW6 Topics:
Fourier Series: Analysis, Synthesis, Properties, and LTI
HW6 Course Learning Goals Satisfied:
Goal 1: Des
EE235
Name:
Student ID:
Midterm Exam #1
University of Washington EE235, Winter 2016
January 29th, 2016
Exam Information:
The test is closed book, and no calculators/devices are allowed. You are allowed ONE 8.5x11 (twosided) page of notes.
Please show al
EE 235, Winter 2016, Homework 6 Supplementary Notes
1. For Problems 1a: Partial Fraction Expansion. This week, we will only consider the case where we have
complex roots in the denominator or possibly repeated roots, but with the order of the numerator st
EE 235, Winter 2016, Homework 1 Supplementary Notes
1. Complex Number Representation. Any complex number z can be represented in rectangular (Cartesian)
form or in polar form.
(a) In rectangular form, z = x + jy, where x = Recfw_z and y = Imcfw_z .
i. Com
EE 235, Winter 2016, Homework 7: Fourier Transforms
Due Friday February 19, 2016 in class
Write down ALL steps for full credit
HW7 Topics:
Fourier Transforms: Analyze (Transform) and Synthesize (Inverse Transform)
Fourier Transforms: Periodic Signals
P
EE 235, Winter 2016, Homework 7 Supplementary Notes
1. Partial Fraction Expansion. When the order of the numerator is greater than or equal to the order of
the denominator, you must perform a polynomial division first before doing a partial fraction expan
EE 235, Winter 2016, Homework 5 Supplementary Notes
1. Magnitude and Phase for a Ratio of Complex Numbers.
Consider X = X1
X2 , where X1 and X2 are complex.
(a) Evaluating Magnitude. We can evaluate the magnitude of X by taking the magnitude of the
numera
4/20/2016
EE235: Continuous time linear systems
Introduction to signals and systems
Lecture #15
UWEE TC Chen
LTI SYSTEMS WITH AND WITHOUT
MEMORY
Memory: An LTI system is memoryless iff () is a scaled
(unshifted) Dirac.
= ()
UWEE TC Chen
1
4/20/2016
INVER