EE 261
Fourier Transform and Applications
June 22, 2016
Maximum score: 100
Homework #1
Due Wednesday, June 29
Note: You can solve any or all of these problems. The maximum score you can obtain is 100 points.
1. (10 points) Let f (t) = sin 3t + cos 5t and
EE 261 The Fourier Transform and its Applications Fall 2009 Solutions to Problem Set One
1. Some practice with geometric sums and complex exponentials (5 points each) Well make much use of formulas for the sum of a geometric series, especially in combinat
EE 261 The Fourier Transform and its
Applications
Fall 2015
Solutions to Midterm Exam
There are five questions for a total of 90 points.
Please write your answers in the exam booklet provided, and make sure that your
answers stand out.
Dont forget to w
EE261
Raj Bhatnagar
Summer 2009-2010
EE 261 The Fourier Transform and its Applications
Midterm Examination
19 July 2010
(a) This exam consists of 4 questions with 12 total subparts for a total of 50 points.
(b) The questions dier in length and diculty. Do
EE 261 The Fourier Transform and its Applications Fall 2009 Solutions to Problem Set Two
1. (10 points) A famous sum You cannot go through life knowing about Fourier series and not know the application to evaluating a very famous sum. Let S (t) be the saw
EE261
Raj Bhatnagar
Summer 2010-2011
EE 261 The Fourier Transform and its Applications
Problem Set 1
Due Wednesday, June 29
1. (10 points) Some practice with complex numbers
(a) Express the following numbers in polar form:
(i)
(ii)
(iii)
(iv)
(b) For
(i)
EE 261 The Fourier Transform and its Applications
Fall 2016
Problem Set One Due Wednesday, October 5
1. Some practice combining simple signals. (5 points each)
The scaled triangle function with a parameter a > 0 is
(
1 a1 |t| ,
a (t) = (t/a) =
0,
|t| a
|t
EE 261 The Fourier Transform and its Applications
Fall 2012
Solutions to Problem Set One
1. Some practice combining simple signals. (5 points each)
The scaled triangle function with a parameter a > 0 is
1
1 a |t| ,
0,
a (t) = (t/a) =
|t| a
|t| > a
The gra
EE 261 The Fourier Transform and its Applications
Fall 2012
Problem Set Eight Due Wednesday, November 28
1. (20 points) A True Story : Professor Osgood and a graduate student were working on a
discrete form of the sampling theorem. This included looking a
EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set Eight Due Wednesday, November 18
1. (10 points) Dierent denitions for the DFT This is an alternate version, in one respect, to Section 6.9 in the notes, on dierent denitions of the DF
EE 261 The Fourier Transform and its
Applications
Fall 2011
Final Exam Solutions
1. (15 points) Recall that the Fourier transform of cos 2at is (1/2)(a + a ). Find the
Fourier transform of the following modied cosine signals.
1.5
1.0
f1 t
0.5
0.0
0.5
1.0
EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set Two Due Wednesday, October 7, 2009
1. (10 points) A famous sum You cannot go through life knowing about Fourier series and not know the application to evaluating a very famous sum. Le
EE 261 The Fourier Transform and its Applications
Fall 2012
Problem Set Eight Due Wednesday, November 28
1. (20 points) A True Story : Professor Osgood and a graduate student were working on a
discrete form of the sampling theorem. This included looking a
EE 261 The Fourier Transform and its Applications
Fall 2012
Problem Set Four Due Wednesday, October 24
1. (10 points) Solving the wave equation
An innite string is stretched along the x-axis and is given an initial displacement described
by a function f (
EE 261 The Fourier Transform and its Applications
Fall 2012
Problem Set Three
Due Wednesday, October 17, 2012
1. (5 points) Equivalent width: Still another reciprocal relationship
The equivalent width of a signal f (t), with f (0) = 0, is the width of a r
EE 261 The Fourier Transform and its Applications
Fall 2012
Solutions to Problem Set Four
1. (10 points) Solving the wave equation
An innite string is stretched along the x-axis and is given an initial displacement described
by a function f (x). It is the
EE 261
Fourier Transform and Applications
February 16, 2011
Handout #13
Homework #5
Due Friday, February 25
1. Exercises on distributions.
a. Let g (t) be a Schwartz function. Show that
g (t) (t) = g (0) (t) g (0) (t) .
b. Let Tf be the distribution induc
EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set One Due Wednesday, September 30
1. Some practice with geometric sums and complex exponentials (5 points each) Well make much use of formulas for the sum of a geometric series, especia
EE 261 The Fourier Transform and its
Applications Fall 2012
Problem Set Seven
Due Wednesday, November 14
1. (20 points) Handels Hallelujah
In this problem we will explore the eects of sampling with or without anti-aliasing
lters. As we saw in lecture ther
EE 261 The Fourier Transform and its Applications
Fall 2012
Problem Set One Due Wednesday, October 3
1. Some practice combining simple signals. (5 points each)
The scaled triangle function with a parameter a > 0 is
1
1 a |t| ,
0,
a (t) = (t/a) =
|t| a
|t|
EE 261 The Fourier Transform and its
Applications
Fall 2011
Solutions to Midterm Exam
1
1. (10 points) Multiplying periodic functions
Let f (t) and g (t) be periodic functions with period 1 and Fourier series expansions
given by
n=
an ei2nt ,
f (t) =
n=
n
EE261
Raj Bhatnagar
Summer 2010-2011
EE 261 The Fourier Transform and its Applications
Problem Set 3
Due Wednesday 13 July
1. (15 points) Convolution and cross-correlation
The cross-correlation (sometimes just called correlation) of two real-valued signal
EE 261 The Fourier Transform and its Applications
Fall 2012
Problem Set Five Due Wednesday, October 31
1. (20 points) Windowing functions
In signal analysis, it is not realistic to consider a signal f (t) from < t < . Instead,
one considers a modied secti
EE 261 The Fourier Transform and its
Applications Fall 2012
Midterm Exam
October 31, 2012
There are ve questions for a total of 85 points.
Please write your answers in the exam booklet provided, and make sure that your
answers stand out.
Dont forget to
EE 261 The Fourier Transform and its
Applications
Fall 2012
Final Exam Solutions
1. (15 points)Finding Fourier transforms: The following two questions are independent.
(a) (5) In communications theory the analytic signal fa (t) of a signal f (t) is dened,
EE 261
Fourier Transform and Applications
March 17, 2011
Handout #21
Final Examination Solutions
1. (15 points) Fourier series. A function f (t) with period 1 has the Fourier series coecients
n
1
n<0
2
cn = 0
n=0
1n
2
n>0
These Fourier series coecients
EE 261 The Fourier Transform and its
Applications Fall 2012
Problem Set Seven
Due Wednesday, November 14
1. (20 points) Handels Hallelujah
In this problem we will explore the eects of sampling with or without anti-aliasing
lters. As we saw in lecture ther
EE 261 The Fourier Transform and its Applications Fall 2009 Solutions to Problem Set Three
1. (25 points) Piecewise linear approximations and Fourier transforms. (a) The stretched triangle function is dened by a (t) = (t/a) = Find F a (s). (b) Find the Fo
EE 261 The Fourier Transform and its
Applications
Fall 2011
Final Exam December 15, 2011
Notes:
There are eight questions for a total of 140
points
Be sure to write your name (neatly) on your
exam booklet(s)
Write all your answers in your exam booklets
Wh
EE 261 The Fourier Transform and its Applications
Fall 2012
Problem Set Two
Due Wednesday, October 10, 2012
1. (10 points) Rayleighs identity and a famous sum
It doesnt matter if youre an engineer, a scientist, or a mathematician, you cannot go through
li
EE 261, Lecture 25
n-dimensional Fourier transform
Let f (x, y) be a complex-valued function of two real variables.
The 2-D Fourier transform of f is
Z Z
Ff (u, v) =
f (x, y) e2i (ux+vy) dx dy
Z
Z
=
f (x, y) e2i (u,v)(x,y) dx dy
In general, the n-D Fo
EE 261, Lecture 26
n-dimensional Fourier transform (review)
The 2D Fourier transform is defined as a double integral:
Z Z
f (x, y) e2i (ux+vy) dx dy
Ff (u, v) =
The inverse transform has the same form but different kernel:
Z Z
F (u, v) e2i (ux+vy) du
EE 261, Lecture 23
Linear time-invariant systems: review
A system is linear if superposition holds.
L a1 v1 + a2 v2 = a1 Lv1 + a2 Lv2
A discrete-time linear system can be represented by a matrix:
Lv = Av
A continuous-time linear system can be represented
EE 261, Lecture 24
Hilbert transform
The Hilbert transform is defined by
1
1
v(x) =
Hv(x) =
x
Z
v(y)
1
dy =
xy
Z
v(y)
dy
yx
(Many books and Matlab use 1/x instead of 1/x.)
Calculating the integral requires finding a Cauchy principal value.
Hilbert tran
EE 261, Lecture 22
Linear time-invariant systems
A (one-dimensional) signal is a (complex-valued) function of a
(continuous or discrete) real variable.
A functional is a (complex-valued) function whose inputs are signals.
A system is a function whose inpu