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Goals: To study signal analysis, linear systems, and frequency analysis. Tobegin learning and using MATLAB for signal analysis in the time andfrequency domains Grading policies:
Grading policiesHW (20%) + Lab (30%) + Midterm (20%) + Final exam (30%)
Textbooks:
Signals Systems and Transforms by Philips, Parr, Riskin, 4th edition
(Chapters 1-7)
References:
• Interactive notes
http://www.ee.washington.edu/class/235dl/
• Textbook webpage:
http://www.ee.washington.edu/class/SST_textbook/textbook.html
Acknowledgement: courses materials are supported by EE, UW
Course Administration
Policy & Schedule Homeworks
– A HW will be due every week
– I will accept HWs up to 24 hours late. (Penalty 20%?)
– Collaboration is good. Copying verbatim BAD!
Please turn in your own work.
Labs―You must have a lab partner
― You must read the lab beforehand
― During each lab session: teamwork but each person
must program yourself, discussion is encouraged
― Ending lab session: both show your works to TA,
answer TA’s questions (not always that you & your
lab partner will get the same grade)
Lecture Overview
• A Gentle Introduction to Signal Processing
Applications of Signal Processing
What is a Signal
What is a System
How Signals and Systems are Ubiquitous
What is a Signal?
Signals
• A signal is a function of one or more
independent variables (usually time or space).
• Signals often carry information.
Example: Velocity of a car
t (seconds)
v(t)
velocity
(mph)
v(t)
30
1 60
v(1) = 30
v(60) = 0
Systems
A system is a relationship between input and
output signal(s).
Example: Airbag deployment system
What is a System?
t (seconds)
v(t)
t (seconds)
y
n
y(t)Deploy
Airbag?velocity
v(t) y(t)A
A
y(t) =
½y if d
dtv(t) < a
n otherwise
Many signals are oscillatory
Signals due to
vibrations
Signals with cycles
speech signal
Examples
• Geological: temperature, rainfall
• Electromagnetic: AM/FM radio
• Sound: music, speech.
Signals in the Frequency Domain
Each signal can be represented by its frequency content
196
g(t) G(f)
g(t) = sin(2¼ f0t)
f0= 196Hz
t (seconds) f (Hz)
Note on Convention
• Time Domain signals are represented as lower case.
• Frequency Domain signals are represented with upper
case.
Signals in the Frequency Domain
196
293.66
440
659.26
Each signal can be represented by its frequency content
g(t)
d(t)
e(t)
G(f)
D(f)
A(f)
E(f)
t (seconds)f (Hz)
a(t)
Other Example Time Signals• Charge on a capacitor over time
• Power consumed over time by a household
• Signal transmitted by a radio station
• Sound of a heartbeat over time
• Solar power hitting a space shuttle along its
flight path.
• Others? Class?
2D signals
Some signals are defined over space, generally called
“images”
hyperspectral image:
224 spectral bands, 2D
Concepts so far
• Signals as functions over time or space
• Signals carry information
• Signals in the frequency (Fourier) domain
What about Systems?
Systems
A system is a relationship between input and
output signal(s).
Example: Airbag deployment system
What is a System?
t (seconds)
v(t)
t (seconds)
y
n
y(t)Deploy
Airbag?velocity
v(t) y(t)A
A
y(t) =
½y if d
dtv(t) < a
n otherwise
Systems
A system inputs and/or outputs signal(s)
Player outputs signal
via mouth trumpet
sound wave signal leaves the trumpet
microphone
electrical signal speakers
sound
wave
signal
systems
in blue
Some systems are filters
• Filters decrease or increase certain
frequencies (through multiplication)
Hz
LOW PASS Filter
Low freq High freq
1
1 .33
0
Some systems are filters
Filters decrease or increase certain
frequencies (through multiplication)
Hz200 400
Hz200 400
ORIGINAL
SIGNAL
LOW PASS SIGNAL
time
time
multipl
y
Cochlea
Unrolled
View of the
Cochlea
Audio Processing in the Brain
Art from Scientific American http://128.200.122.84/weinberger/publications/Weinberger,%202004a.pdf
The Brain as a Filter bank
Groups of brain
cells respond to
frequencies
differently. Each
cell filters out a
particular range.
Art from Scientific American http://128.200.122.84/weinberger/publications/Weinberger,%202004a.pdf
Band Pass
Filter!
Lecture 2-3 Overview
Signals and Signals…then more Signals
• Types of signals
• Some useful signals
• Common signal operations
What is a Signal?
Signals
A signal is a function of one or more
independent variables (usually time or space).
Signals often carry information.
Example: Velocity of a car
Systems
A system is a relationship between input and
output signal(s).
Example: Airbag deployment system
types of signals
continuous-time vs. discrete-time
• A Continuous Time signal is specified at all values of
time (all Real numbers ).
Must be a functional form
• A Discrete Time signal is specified for only discrete
values of (only on Integers ).
Can be functional OR specified at a set of values
R
Z
f¡1; 0; 1; 7;¡586; 358; : : :gf1; 2:5; ¼; e;
p10; 3; : : :g
types of signals
• A Continuous Time signal
• A Discrete Time signal
f [n] = [0; 1; ¼; 0]
STEM PLOT
0 1 2 3
¼
1
NOTE: Discrete Time but Analog signal!
types of signals
continuous-time vs. discrete-time
1) 88.5 FM radio transmitted signal (continuous)
2) stock’s daily close price (discrete)
3) a capacitor’s charge over time (continuous)
4) local news broadcast to your tv (continuous)
5) picture taken by a digital camera (discrete)
analog vs. digital
analog vs. digital
or or
where is a set of discrete values.
• An Analog signal whose amplitude can take any value in
a continuous interval (all Real numbers ).
• A Digital signal whose amplitude can take only a
discrete set of values of (from some arbitrary set ).
R
G
analog vs. digital
or
1) 88.5 FM transmitted signal (analog)
2) stock market daily close price (digital)
3) a capacitor’s charge over time (analog)
4) local news broadcast to your tv (analog)
5) picture taken with a digital camera (digital)
6) Comedy Central’s the Daily Show on your hard drive (digital)
7) your voice (analog)
or
where is a finite set of values.
analog vs. digital
signals to know
constant signal
t0
a
unit step signal
t0
1
unit ramp signal
t0
1
or: r(t) = f{u(t)}
Find function f ?
Signal operations
signal arithmetic: add, subtract, multiply, divide signals pointwise
Delay: x(t-n) is x(t) delayed by n time units
Amplify: Ax(t) is x(t) amplified by some constant A
Speed-up: x(at) is x(t) sped up by a factor of a
Slow-down: x(t/a) slows x(t) down by a factor of a
playing with signals
signal arithmetic: add, subtract, multiply, divide signals pointwise
t0
a
t
1
Given
looks like: t
and
then
a+1
a
playing with signals0
unit pulse signal
t0
1
1
unit step signal
t
1
Describe p(t) in terms of u(t). Hint: use signal arithmetic and delays.
Hey, I didn’t
get an intro
slide!
playing with signals
unit pulse signal
t0
1
1
What does look like?
t p(t-3)
t
1-5 0
0 0
3 1
4 1
4.1 0
Delay: x(t-n) is x(t) delayed by n time units
What does look like?
3 4
Time Scaling /
Time Shifting
Time Scaling
• Speed-up
• Slow-Down
Time Shifting
• Delay
• Advance
10
p(t)
p(at)
01
a
a0
p(t=a)
y(t) = p(at)
y(t) = p(t=a)
b0
-b 0
p(t¡ b)
p(t+ b)
y(t) = p(t¡ b)
y(t) = p(t+ b)
Combining Time Scaling and
Time Shifting
So how do we deal with: f(t) = p(at – b) ?
Convert to form f(t) = p(a(t – b/a))
10
p(t)
Scale/Shift
0
x(t) = p(at)
1a 0
f1(t) = x(t¡ b
a)
ba
¡ba+ 1
a
¢
In words
Mathematically analyze
Combining Time Scaling and
Time Shifting
So how do we deal with: f(t) = p(at – b) ?
Another solution: x(t) = p(t – b), f(t) = x(at) ?
10
p(t)
Scale/Shift
In words
Combining Time Scaling and
Time Shifting – Dealing with Speech
Let s(t) be a speech signal
Describe in words: y(t) = 2s(5t - 4) ?
Apply both methods for your explanation
Time reversal
t
1
What does look like?
2
1
-2
This is also a form of Time Scaling!
Only by a negative number!
= w((-1)t)
playing with time
Describe z(t) in terms of w(t)
t
1 1
-2 12 3 t
Double check:
Pick out features to match
1) Short Side of Triangle 2) Tall Side of Triangle
playing with time, example 2
Describe z(t) in terms of w(t)
t
1 1
-2 12 5
Doublecheck:
1) Short Side of Triangle 2) Tall Side of Triangle
more fun with signals
t
8
3 5
in terms of unit pulse p(t)
y(t) =
8<:0 if t < 3
1 if 3 · t · 50 if 5 < t
One definition (but not in terms of p(t)):
more fun with signals
t
8
3 5
in terms of unit pulse p(t)
t
8
2
first step:
t
8
2nd step:
3 5
3rd step:
replace t by t-3:
Double-check!pulse starts:
pulse ends:
The Dirac delta
The Dirac delta is denoted:
•also known as the unit impulse or impulse
•it is a signal we will use a lot.
•it is not a function, so do not treat it as a function.
•but we do have rules for dealing with it.
“a spike of signal at time 0”
0
Exponential signals
General Form: Ceat; where a = ¾ + j!
Equivalently: Ce¾tej!t
Euler’s Relationej!t = cos(!t) + j sin(!t)
In Parts:
amplitude
Increasing
Constant
Decaying
Exponentialt
¾ > 0
¾ < 0
¾ = 0
sinusoidal w/
angular freq. !
Ce¾tej!t
Exponential signals
t
1
-1
1 2
t
1
-1
1 2
f(t) = Ce¾tej!t
where C = 1, ¾ = ¡1, ! = 2¼
Reff(t)g Imff(t)g
Periodic Signals I
)(tx is periodic if there exists a T such that x ( t)
repeats itself every T seconds.
Fundamental Frequency:
t (seconds)
T
f0=1
THz !
0=2¼
Trad !
0= 2¼f
0
x(t) = x(t+ nT ) for all integers n
Periodicity of (sums of periodic signals)
x1(t) = x
1(t+ nT
1) for all integers n
z(t) = x1(t) + x
2(t)
x2(t) = x
2(t+mT
2) for all integers m
Given:
Find T s.t. z(t) = z(t+ rT ) for all integers r
Periodicity of (sums of periodic signals)
-1 0 1 2 3 4 5 6 7 8 9 10 11 12
1
2
1
1
T1
T2
Ex: x1(t) has period T
1= 2 and x
2(t) has period T
2= 3.
What is the Period T of z(t) = x1(t) + x
2(t)?
Periodicity of (sums of periodic signals)
-1 0 1 2 3 4 5 6 7 8 9 10 11 12
1
2
1
1
T1
T2
Ex: x1(t) has period T
1= 2 and x
2(t) has period T
2= 3.
What is the Period T of z(t) = x1(t) + x
2(t)?
4
Periodicity of (sums of periodic signals)x1(t) = x
1(t+ nT
1) for all integers n
z(t) = x1(t) + x
2(t)
x2(t) = x
2(t+mT
2) for all integers m
Given:
Find T s.t. z(t) = z(t+ rT ) for all integers r
Solution: T = LCM of T1; T2.
Least
Common
Multiple
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