6.003: Signals and Systems · 6.003: Signals and Systems Collaboration Policy • Discussion of...
Transcript of 6.003: Signals and Systems · 6.003: Signals and Systems Collaboration Policy • Discussion of...
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6.003: Signals and Systems
Signals and Systems
February 2, 2010
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6.003: Signals and Systems
Today’s handouts:
Lecturer:
Instructors:
Website:
Text:
Single package containing
• Slides for Lecture 1
• Subject Information & Calendar
Denny Freeman
Peter Hagelstein
Rahul Sarpeshkar
mit.edu/6.003
Signals and Systems – Oppenheim and Willsky
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6.003: Homework
Doing the homework is essential for understanding the content.
• where subject matter is/isn’t learned
• equivalent to “practice” in sports or music
Weekly Homework Assignments
• Conventional Homework Problems plus
• Engineering Design Problems (Python/Matlab)
Open Office Hours !
• Stata Basement
• Mondays and Tuesdays, afternoons and early evenings
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6.003: Signals and Systems
Collaboration Policy
• Discussion of concepts in homework is encouraged
• Sharing of homework or code is not permitted and will be re
ported to the COD
Firm Deadlines
• Homework must be submitted in recitation on due date
• Each student can submit one late homework assignment without
penalty.
• Grades on other late assignments will be multiplied by 0.5 (unless
excused by an Instructor, Dean, or Medical Official).
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spring spring spring spring spring
finals finals finals finals finals
6.003 At-A-GlanceTuesday Wednesday Thursday Friday
Feb 2 L1: Signals and
Systems
R1: Continuous &
Discrete Systems
L2: Discrete-Time
Systems
R2: Difference
Equations
Feb 9 L3: Feedback,
Cycles, and Modes
HW1
due
R3: Feedback,
Cycles, and Modes
L4: CT Operator
Representations R4: CT Systems
Feb 16 Presidents Day:
Monday Schedule
HW2
due
R5: CT Operator
Representations
L5: Second-Order
Systems
R6: Second-Order
Systems
Feb 23 L6: Laplace and Z
Transforms
HW3
due
R7: Laplace and Z
Transforms
L7: Transform
Properties
R8: Transform
Properties
Mar 2 L8: Convolution;
Impulse Response EX4
Exam 1
no recitation
L9: Frequency
Response
R9: Convolution
and Freq. Resp.
Mar 9 L10: Bode
Diagrams
HW5
due
R10: Bode
Diagrams
L11: DT Feedback
and Control
R11: Feedback and
Control
Mar 16 L12: CT Feedback
and Control
HW6
due
R12: CT Feedback
and Control
L13: CT Feedback
and Control
R13: CT Feedback
and Control
Mar 23 Spring Week
Mar 30 L14: CT Fourier
Series HW7
R14: CT Fourier
Series
L15: CT Fourier
Series
R15: CT Fourier
Series
Apr 6 L16: CT Fourier
Transform
EX8
due
Exam 2
no recitation
L17: CT Fourier
Transform
R16: CT Fourier
Transform
Apr 13 L18: DT Fourier
Transform
HW9
due
R17: DT Fourier
Transform
L19: DT Fourier
Transform
R18: DT Fourier
Transform
Apr 20 Patriots Day
Vacation HW10
R19: Fourier
Transforms
L20: Fourier
Relations
R20: Fourier
Relations
Apr 27 L21: Sampling EX11
due
Exam 3
no recitation L22: Sampling R21: Sampling
May 4 L23: Modulation HW12
due R22: Modulation L24: Modulation R23: Modulation
May 11 L25: Applications
of 6.003 EX13 R24: Review
Breakfast with
Staff Study Period
May 18 Final Examination Period
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6.003: Signals and Systems
Weekly meetings with class representatives
• help staff understand student perspective
• learn about teaching
One representative from each section (4 total)
Tentatively meet on Thursday afternoon
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The Signals and Systems Abstraction
Describe a system (physical, mathematical, or computational) by
the way it transforms an input signal into an output signal.
systemsignal
in signal out
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Example: Mass and Spring
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Example: Mass and Spring
x(t)
y(t)
mass & spring system
x(t) y(t)
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Example: Mass and Spring
x(t)
y(t)
x(t) y(t)
mass & spring system
t t
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r0(t)
t
Example: Tanks
r0(t)
h1(t) r1(t)
h2(t) r2(t)
r0(t) tank system
r2(t)
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Example: Tanks
r0(t)
h1(t) r1(t)
h2(t) r2(t)
r0(t) r2(t)
tank system
t t
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Example: Cell Phone System
sound out
sound in
sound in cell
phone system
sound out
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Example: Cell Phone System
sound out
sound in
sound in sound out
cell phone system
t t
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Signals and Systems: Widely Applicable
The Signals and Systems approach has broad application: electrical,
mechanical, optical, acoustic, biological, financial, ...
x(t) y(t)
t mass & spring system
t
r0(t)
r1(t)
r2(t)
h1(t)
h2(t) tank system
r0(t) r2(t)
t t
cell phone system
sound in sound out
t t
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tower E/M
tower sound E/M optic cell sound
in phone out cell
phone fiber
focuses on the flow of information, abstracts away everything else
Signals and Systems: Modular
The representation does not depend upon the physical substrate.
sound in
sound out
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Signals and Systems: Hierarchical
Representations of component systems are easily combined.
Example: cascade of component systems
cell phone
tower tower cell
phone sound
in
E/M optic fiber
E/M sound out
Composite system
cell phone system sound in
sound out
Component and composite systems have the same form, and are
analyzed with same methods.
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Signals and Systems
Signals are mathematical functions.
• independent variable = time
• dependent variable = voltage, flow rate, sound pressure
mass & spring system
x(t) y(t)
t t
r0(t) r2(t)
tank system
t t
sound in sound out
cell phone system
t t
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Signals and Systems
continuous “time” (CT) and discrete “time” (DT)
x(t) x[n]
nt 0 2 4 6 8 10 0 2 4 6 8 10
Many physical systems operate in continuous time.
• mass and spring
• leaky tank
Digital computations are done in discrete time.
• state machines: given the current input and current state, what
is the next output and next state.
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Signals and Systems
Sampling: converting CT signals to DT
x(t) x[n] = x(nT )
nt 0T 2T 4T 6T 8T 10T 0 2 4 6 8 10
T = sampling interval
Important for computational manipulation of physical data.
• digital representations of audio signals (e.g., MP3)
• digital representations of pictures (e.g., JPEG)
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Signals and Systems
Reconstruction: converting DT signals to CT
zero-order hold
x[n] x(t)
n t 0 2 4 6 8 10 0 2T 4T 6T 8T 10T
T = sampling interval
commonly used in audio output devices such as CD players
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Signals and Systems
Reconstruction: converting DT signals to CT
piecewise linear
x[n] x(t)
n t 0 2 4 6 8 10 0 2T 4T 6T 8T 10T
T = sampling interval
commonly used in rendering images
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Check Yourself
Computer generated speech (by Robert Donovan)
t
f(t)
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true?
• f1(t) = f(2t) • f2(t) = −f(t) • f3(t) = f(2t) • f4(t) = 2f(t)
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Check Yourself
Computer generated speech (by Robert Donovan)
t
f(t)
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true?
• f1(t) = f(2t) • f2(t) = −f(t) • f3(t) = f(2t) • f4(t) = 2f(t)
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Check Yourself
Computer generated speech (by Robert Donovan)
t
f(t)
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true?
• f1(t) = f(2t) • f2(t) = −f(t) • f3(t) = f(2t) • f4(t) = 2f(t)
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Check Yourself
Computer generated speech (by Robert Donovan)
t
f(t)
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true?
• f1(t) = f(2t) • f2(t) = −f(t) • f3(t) = f(2t) • f4(t) = 2f(t)
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Check Yourself
Computer generated speech (by Robert Donovan)
t
f(t)
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true?
• f1(t) = f(2t) • f2(t) = −f(t) • f3(t) = f(2t) • f4(t) = 2f(t)
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Check Yourself
Computer generated speech (by Robert Donovaf(t)
n)
t
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true?
• f1(t) = f(2t) • f2(t) = −f(t) • f3(t) = f(2t) • f4(t) = 2f(t)
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Check Yourself
Computer generated speech (by Robert Donovan) f(t)
t
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true?
• f1(t) = f(2t) • f2(t) = −f(t) • f3(t) = f(2t) • f4(t) = 2f(t)
Computer generated speech (by Robert Donovan)
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Check Yourself
Computer generated speech (by Robert Donovan)
t
f(t)
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true?
• f1(t) = f(2t) • f2(t) = −f(t) • f3(t) = f(2t) • f4(t) = 2f(t)
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Check Yourself
Computer generated speech (by Robert Donovan)
t
f(t)
Listen to the following four manipulated signals:
f1(t), f2(t), f3(t), f4(t). How many of the following relations are true? 2
• f1(t) = f(2t) √
• f2(t) = −f(t) X • f3(t) = f(2t) X • f4(t) = 2f(t)
√
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Check Yourself
y f(x, y)
−250 0 250 −2
50
0 25
0 x
How many images match the expressions beneath them?
−250
0
250
y
x
−250
0
250
y
x −2
50
0 25
0
y
x−250 0 250 −250 0 250 −250 0 250
f1(x, y)=f(2x, y) ? f2(x, y)=f(2x−250, y) ? f3(x, y)=f(−x−250, y) ?
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Check Yourselfy y y y
−250
0
250
−250
0
250
−250
0
250
−250
0
250
−250 0 250 x −250 0 250 x −250 0 250 x −250 0 250
f(x, y) f1(x, y) = f(2x, y) ? f2(x, y) = f(2x−250, y) ? f3(x, y) = f(−x−250, y) ?
√ x = 0 → f1(0, y) = f(0, y)
x = 250 → f1(250, y) = f(500, y) X
√ x = 0 → f2(0, y) = f(−250, y) √ x = 250 → f2(250, y) = f(250, y)
x = 0 → f3(0, y) = f(−250, y) X
x = 250 → f3(250, y) = f(−500, y) X
x
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Check Yourself
y f(x, y)
−250 0 250 −2
50
0 25
0 x
How many images match the expressions beneath them?
−250 0 250
−250
0
250
y
x
f1(x, y)=f(2x, y) ? −250 0 250
−250
0
250
y
x
f2(x, y)=f(2x−250, y) ? −250 0 250
−250
0
250
y
x
f3(x, y)=f(−x−250, y) ?
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The Signals and Systems Abstraction
Describe a system (physical, mathematical, or computational) by
the way it transforms an input signal into an output signal.
systemsignal
in signal out
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Example System: Leaky Tank
Formulate a mathematical description of this system.
r0(t)
h1(t) r1(t)
What determines the leak rate?
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Check Yourself
The holes in each of the following tanks have equal size.
Which tank has the largest leak rate r1(t)?
3. 4.
1. 2.
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Check Yourself
The holes in each of the following tanks have equal size.
Which tank has the largest leak rate r1(t)? 2
3. 4.
1. 2.
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Example System: Leaky Tank
Formulate a mathematical description of this system.
r0(t)
r1(t) h1(t)
Assume linear leaking: r1(t) ∝ h1(t)
What determines the height h1(t)?
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Example System: Leaky Tank
Formulate a mathematical description of this system.
r0(t)
r1(t) h1(t)
Assume linear leaking: r1(t) ∝ h1(t)
Assume water is conserved: dh1(t) ∝ r0(t) − r1(t)dt
Solve: dr1(t) ∝ r0(t) − r1(t)dt
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Check Yourself
What are the dimensions of constant of proportionality C?
dr1(t) dt
= C � r0(t) − r1(t)
�
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Check Yourself
What are the dimensions of constant of proportionality C?
inverse time (to match dimensions of dt)
dr1(t) dt
= C � r0(t) − r1(t)
�
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Analysis of the Leaky Tank
Call the constant of proportionality 1/τ .
Then τ is called the time constant of the system.
dr1(t) r0(t) r1(t)= − dt τ τ
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Check Yourself
Which of the following tanks has the largest time constant
τ?
3. 4.
1. 2.
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Check Yourself
Which of the following tanks has the largest time constant
τ? 4
3. 4.
1. 2.
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Analysis of the Leaky Tank
Call the constant of proportionality 1/τ .
Then τ is called the time constant of the system.
dr1(t) r0(t) r1(t)= − dt τ τ
Assume that the tank is initially empty, and then water enters at a
constant rate r0(t) = 1. Determine the output rate r1(t).
r1(t)
time (seconds) 1 2 3
Explain the shape of this curve mathematically.
Explain the shape of this curve physically.
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Leaky Tanks and Capacitors
Although derived for a leaky tank, this sort of model can be used to
represent a variety of physical systems.
Water accumulates in a leaky tank.
r0(t)
h1(t) r1(t)
Charge accumulates in a capacitor.
ii io + vC −
dv ii − io dh = ∝ ii − io analogous to ∝ r0 − r1dt C dt
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6.003 Signals and Systems Spring 2010
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