6.003 Homework #11 Solutions
Engineering Design Problem
1. Image reconstruction
The rows and/or columns of the following images have been blurred. Figure out a way
to sharpen each image, and identify the building. Here are thumbnails of the images:
a1
a2
6.003: Signals and Systems
Lecture 7
6.003: Signals and Systems
October 1, 2009
Mid-term Examination #1
Wednesday, October 7, 7:30-9:30pm, Walker Memorial.
Laplace and Z Transforms
No recitations on the day of the exam.
Coverage:
DT Signals and Systems
Le
6.003: Signals and Systems
Lecture 15
6.003: Signals and Systems
November 3, 2009
Fourier Representations
Representations based on sinusoids.
Fourier Series
signal
in
signal
out
system
To date, we have focused primarily on time-domain techniques,
especial
6.003: Signals and Systems
Fourier Series
November 5, 2009
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)
piano
piano
t
k
v
6.003 Homework 1
Due at the beginning of recitation on Wednesday, February 10, 2010.
Problems
1. Independent and Dependent Variables
Assume that the height of a water wave is given by g (x v t) where x is distance, v is
velocity, and t is time. Assume tha
6.003 Homework 2
Due at the beginning of recitation on Wednesday, February 17, 2010.
Problems
1. Blacks Equation
Consider the general form of a feedback problem:
+
X
F
Y
G
Notice the minus sign on the adder: it indicates that the lower input is subtracted
6.003 Homework 3
Due at the beginning of recitation on Wednesday, February 24, 2010.
Problems
1. Laplace Transforms
Determine the Laplace transforms (including the regions of convergence) of each of the
following signals:
a. x1 (t) = e2(t3) u(t 3)
b. x2 (
6.003 Homework 4
Please do the following problems by Wednesday, March 3, 2010. You need not submit
your answers: they will NOT be graded. Solutions will be posted.
Problems
1. Z transforms
Determine the Z transform (including the region of convergence) fo
6.003 Homework 5
Due at the beginning of recitation on Wednesday, March 10, 2010.
Problems
1. DT convolution
Let y represent the DT signal that results when f is convolved with g , i.e.,
y [n] = (f g )[n]
which is sometimes written as y [n] = f [n] g [n].
6.003 Homework 6
Due at the beginning of recitation on Wednesday, March 17, 2010.
Problems
1. Second-order systems
The impulse response of a second-order CT system has the form
h(t) = et cos(d t + )u(t)
where the parameters , d , and are related to the pa
6.003 Homework 7
Due at the beginning of recitation on Wednesday, March 31, 2010.
Problems
1. CT stability
Consider the following feedback system in which the box represents a causal LTI CT
system that is represented by its system function.
X
+
K
s2 + s 2
6.003 Homework 8
Please do the following problems by Wednesday, April 7, 2010. You need not submit
your answers: they will NOT be graded. Solutions will be posted.
Problems
1. Fourier Series
Determine the Fourier series coecients for each of the following
6.003 Homework 9
Due at the beginning of recitation on Wednesday, April 14, 2010.
Problems
1. Fourier varieties
a. Determine the Fourier series coecients of the following signal, which is periodic in
T = 10.
x1 (t)
1
t
10
3 1
1
3
10
b. Determine the Fouri
6.003 Homework 10
Due at the beginning of recitation on Wednesday, April 21, 2010.
Problems
1. DT Fourier Series
Determine the Fourier Series coecients for each of the following DT signals, which are
periodic in N = 8.
x2 [ n ]
1
x1 [ n]
1
1/ 2
n
n
x3 [ n
6.003: Signals and Systems
Fourier Series
November 3, 2009
Fourier Representations
Representations based on sinusoids.
signal
in
system
signal
out
To date, we have focused primarily on time-domain techniques,
especially with transient signals (e.g., impul
6.003: Signals and Systems
Lecture 14
6.003: Signals and Systems
October 29, 2009
Feedback and Control
Continuous-time feedback has many applications.
CT Feedback and Control
Examples:
improve performance of an op amp circuit.
control position of a motor.
6.003: Signals and Systems
CT Feedback and Control
October 29, 2009
Feedback and Control
Continuous-time feedback has many applications.
Examples:
improve performance of an op amp circuit.
control position of a motor.
reduce sensitivity to unwanted parame
6.003: Signals and Systems
Operator Representations for Continuous-Time Systems
October 6, 2009
Mid-term Examination #1
Tomorrow, October 7, 7:30-9:30pm, Walker Memorial.
No recitations tomorrow.
Coverage:
DT Signals and Systems
Lectures 15
Homeworks 14
H
6.003: Signals and Systems
Lecture 8
6.003: Signals and Systems
October 6, 2009
Mid-term Examination #1
Tomorrow, October 7, 7:30-9:30pm, Walker Memorial.
Operator Representations for Continuous-Time Systems
No recitations tomorrow.
Coverage:
DT Signals a
6.003: Signals and Systems
Second-Order Systems
October 8, 2009
Last Time
We analyzed a mass and spring system.
x ( t)
F = K x(t) y (t) = M y (t)
y (t)
x(t)
+
K
M
y (t)
A
1
K2
Y
MA
=
K
X
1 + M A2
y ( t)
A
y ( t)
Last Time
We also analyzed a leaky tanks sy
6.003: Signals and Systems
Lecture 9
6.003: Signals and Systems
October 8, 2009
Last Time
We analyzed a mass and spring system.
Second-Order Systems
x ( t)
F = K x(t) y (t) = M y (t)
y (t)
x(t)
y (t)
K
M
+
y ( t)
A
A
y ( t)
1
K A2
Y
= MK
X
1 + M A2
Octobe
6.003: Signals and Systems
Convolution
October 15, 2009
Multiple Representations of CT and DT Systems
Verbal descriptions: preserve the rationale.
Dierence/dierential equations: mathematically compact.
y [n] = x[n] + z0 y [n 1]
y (t) = x(t) + s0 y (t)
Blo
6.003: Signals and Systems
Lecture 10
October 15, 2009
6.003: Signals and Systems
Multiple Representations of CT and DT Systems
Convolution
Verbal descriptions: preserve the rationale.
Dierence/dierential equations: mathematically compact.
y [n] = x[n] +
6.003: Signals and Systems
Frequency Response
October 20, 2009
Mid-term Examination #2
Wednesday, October 28, 7:30-9:30pm, Walker Memorial.
No recitations on the day of the exam.
Coverage: cumulative with more emphasis on recent material
lectures 112
home
6.003: Signals and Systems
Lecture 11
6.003: Signals and Systems
October 20, 2009
Mid-term Examination #2
Wednesday, October 28, 7:30-9:30pm, Walker Memorial.
Frequency Response
No recitations on the day of the exam.
Coverage: cumulative with more emphasi
6.003: Signals and Systems
CT Frequency Response and Bode Plots
October 22, 2009
Mid-term Examination #2
Wednesday, October 28, 7:30-9:30pm, Walker Memorial.
No recitations on the day of the exam.
Coverage: cumulative with more emphasis on recent material
6.003: Signals and Systems
Lecture 12
6.003: Signals and Systems
October 22, 2009
Mid-term Examination #2
Wednesday, October 28, 7:30-9:30pm, Walker Memorial.
CT Frequency Response and Bode Plots
No recitations on the day of the exam.
Coverage: cumulative
6.003: Signals and Systems
CT Feedback and Control
October 27, 2009
Mid-term Examination #2
Tomorrow, October 28, 7:30-9:30pm, Walker Memorial.
No recitations tomorrow.
Coverage: cumulative with more emphasis on recent material
lectures 112
homeworks 17
H
6.003: Signals and Systems
Lecture 13
6.003: Signals and Systems
October 27, 2009
Mid-term Examination #2
Tomorrow, October 28, 7:30-9:30pm, Walker Memorial.
CT Feedback and Control
No recitations tomorrow.
Coverage: cumulative with more emphasis on recen
6.003 Homework 11
Please do the following problems by Wednesday, April 28, 2010. You need not submit
your answers: they will NOT be graded. Solutions will be posted.
Problems
1. Impulsive Input
Let the following periodic signal
(t 3m) + (t 1 3m) (t 2 3m)