6.003: Signals and Systems
Applications of Fourier Transforms
November 17, 2011
1
Filtering
Notion of a lter.
LTI systems
cannot create new frequencies.
can only scale magnitudes and shift phases of existing components.
Example: Low-Pass Filtering with
Quiz 1 Details
Quiz I Review
Signals and Systems
6.003
Date: Wednesday March 3, 2010
Time: 7.30pm9.30pm
Content: (boundaries inclusive)
Lectures 17
Recitations 18
Homeworks 14
Massachusetts Institute of Technology
March 1, 2010
(Massachusetts Ins
6.003 (Spring 2010)
May 20, 2010
Final Examination
Name:
Kerberos Username:
Please circle your section number:
Section
1
2
3
4
Instructor
Time
Peter Hagelstein
Peter Hagelstein
Rahul Sarpeshkar
Rahul Sarpeshkar
10 am
11 am
1 pm
2 pm
Grades will b e determ
6.003 (Fall 2009)
Final Examination
December 17, 2009
Name:
Kerberos Username:
Please circle your section number:
Section
1
2
3
4
Instructor
Time
Marc Baldo
Marc Baldo
Elfar Adalsteinsson
Elfar Adalsteinsson
10 am
11 am
1 pm
2 pm
Partial credit will b e g
6.003 (Fall 2009)
Final Examination
December 17, 2009
Name:
Kerberos Username:
Please circle your section number:
Section
1
2
3
4
Instructor
Time
Marc Baldo
Marc Baldo
Elfar Adalsteinsson
Elfar Adalsteinsson
10 am
11 am
1 pm
2 pm
Partial credit will b e g
6.003 (Fall 2007)
17 December 2007
Final exam
Name:
Please circle your section number:
Section
Instructor
Time
1
Jeffrey Lang
10
2
Jeffrey Lang
11
3
Karen Livescu
11
4
Sanjoy Mahajan
12
5
Antonio Torralba
1
6
Qing Hu
2
Partial credit will be given, accord
Final exam answers
1 Matching time and frequency representations
a.
1
A
5
0
C
0
D
10
BC
0
D
B
b.
5
A
c.
0
A
0
B
5
C
0
D
2 Discrete-time periodicity
Answer is 15.
3 Find the output signal
2
1
-2
-1
0
1
2
3
4
5
6
Final exam answers / 6.003: Signals and Syst
Discrete-time Signals and Systems
ii
Discrete-time Signals and Systems
An Operator Approach
Sanjoy Mahajan and Dennis Freeman
Massachusetts Institute of Technology
Typeset in Palatino and Euler by the authors using ConTEXt and PDFTEX
C
Copyright 2009 Sanj
Discrete-time Signals and Systems
ii
Discrete-time Signals and Systems
An Operator Approach
Sanjoy Mahajan and Dennis Freeman
Massachusetts Institute of Technology
Typeset in Palatino and Euler by the authors using ConTEXt and PDFTEX
C
Copyright 2009 Sanj
8
Proportional and derivative
control
8.1
8.2
8.3
8.4
8.5
Why derivative control
Mixing the two methods of control
Optimizing the combination
Handling inertia
Summary
95
96
98
99
103
The goals of this chapter are:
to introduce derivative control; a
7
Control
7.1
7.2
7.3
7.4
Motor model with feedforward control
Simple feedback control
Sensor delays
Inertia
83
85
87
90
The goals of this chapter are to study:
how to use feedback to control a system;
how slow sensors destabilize a feedback system; and
6
The perfect (sine) wave
6.1
6.2
6.3
6.4
Forward Euler
Backward Euler
Leapfrog
Summary
72
76
79
82
The goals of this chapter are:
to analyze several methods for discretizing a continuous-time sys
tem; and
to illustrate complex poles and the signi
5
Repeated roots
5.1
5.2
5.3
5.4
5.5
Leaky-tank background
Numerical computation
Analyzing the output signal
Deforming the system: The continuity argument
Higher-order cascades
64
65
67
68
70
After reading this chapter you should be able
to use a continu
6.003 (Spring 2010)
March 3, 2010
Quiz #1
Name:
Kerberos Username:
Please circle your section number:
Section
1
2
3
4
Instructor
Time
Peter Hagelstein
Peter Hagelstein
Rahul Sarpeshkar
Rahul Sarpeshkar
10 am
11 am
1 pm
2 pm
Grades will b e determined by t
Quiz II Review
Signals and Systems
6.003
Massachusetts Institute of Technology
April 5, 2010
(Massachusetts Institute of Technology)
Quiz II Review
April 5, 2010
1 / 15
Quiz 2 Details
Date: Wednesday April 7th, 2010
Time: 7.30pm9.30pm
Content: (boundar
6.003 (Spring 2010)
April 7, 2010
Quiz #2
Name:
Kerberos Username:
Please circle your section number:
Section
1
2
3
4
Instructor
Time
Peter Hagelstein
Peter Hagelstein
Rahul Sarpeshkar
Rahul Sarpeshkar
10 am
11 am
1 pm
2 pm
Grades will b e determined by t
6.003: Signals and Systems
Relations among Fourier Representations
November 15, 2011
1
Mid-term Examination #3
Wednesday, November 16, 7:30-9:30pm,
No recitations on the day of the exam.
Coverage:
Lectures 118
Recitations 116
Homeworks 110
Homework 10 wil
6.003: Signals and Systems
Discrete-Time Frequency Representations
November 8, 2011
1
Mid-term Examination #3
Wednesday, November 16, 7:30-9:30pm,
No recitations on the day of the exam.
Coverage:
Lectures 118
Recitations 116
Homeworks 110
Homework 10 will
6.003: Signals and Systems
DT Fourier Representations
November 10, 2011
1
Mid-term Examination #3
Wednesday, November 16, 7:30-9:30pm,
No recitations on the day of the exam.
Coverage:
Lectures 118
Recitations 116
Homeworks 110
Homework 10 will not be coll
6.003: Signals and Systems
Fourier Transform
November 3, 2011
1
Last Time: Fourier Series
Representing periodic signals as sums of sinusoids.
new representations for systems as lters.
Today: generalize for aperiodic signals.
2
Fourier Transform
An aperio
6.003: Signals and Systems
Fourier Series
November 1, 2011
1
Last Time: Describing Signals by Frequency Content
Harmonic content is natural way to describe some kinds of signals.
Ex: musical instruments (http:/theremin.music.uiowa.edu/MIS .html )
piano
pi
6.003: Signals and Systems
Fourier Representations
October 27, 2011
1
Fourier Representations
Fourier series represent signals in terms of sinusoids.
leads to a new representation for systems as lters.
2
Fourier Series
Representing signals by their harmo
6.003: Signals and Systems
CT Feedback and Control
October 25, 2011
1
Mid-term Examination #2
Tomorrow, October 26, 7:30-9:30pm,
No recitations on the day of the exam.
Coverage:
Lectures 112
Recitations 112
Homeworks 17
Homework 7 will not be collected or
6.003: Signals and Systems
CT Feedback and Control
October 20, 2011
1
Mid-term Examination #2
Wednesday, October 26, 7:30-9:30pm,
No recitations on the day of the exam.
Coverage:
Lectures 112
Recitations 112
Homeworks 17
Homework 7 will not be collected o
6.003: Signals and Systems
Feedback and Control
October 13, 2011
1
Courtesy of Jason Dorfman MIT / CSAIL. Used with permission.
2
Example: Perching
Can we make a xed-wing UAV land on a perch like a bird?
3
The Perching Problem
Courtesy of Leon van Dommel
6.003: Signals and Systems
Frequency Response
October 6, 2011
1
Review
Last time, we saw how a linear, time-invariant (LTI) system can be
characterized by its unit-sample/impulse response.
DT: y[n] = (x h)[n] =
0
x[k]h[n k]
k=
CT: y(t) = (x h)(t) =
x( )
6.003 (Spring 2010)
April 28, 2010
Quiz #3
Name:
Kerberos Username:
Please circle your section number:
Section
1
2
3
4
Instructor
Time
Peter Hagelstein
Peter Hagelstein
Rahul Sarpeshkar
Rahul Sarpeshkar
10 am
11 am
1 pm
2 pm
Grades will b e determined by
Quiz III Review
Signals and Systems
6.003
Massachusetts Institute of Technology
April 26th, 2010
(Massachusetts Institute of Technology)
Quiz I II Review
April 26th, 2010
1 / 1
Quiz 3 Details
Date: Wednesday April 28th, 2010
Time: 7.30pm9.30pm
Content:
4
Modes
4.1
4.2
4.3
4.4
Growth of the Fibonacci series
Taking out the big part from Fibonacci
Operator interpretation
General method: Partial fractions
52
55
57
59
The goals of this chapter are:
to illustrate the experimental way that an engineer st
3
Block diagrams and operators:
Two new representations
3.1
3.2
3.3
3.4
3.5
3.6
Disadvantages of difference equations
Block diagrams to the rescue
The power of abstraction
Operations on whole signals
Feedback connections
Summary
34
35
40
41
45
49
Th