Math 414-501 Spring 2021
Test 2 Review
General Information
- Time and date. Test 2 will be given on Wednesday, 3/17/21,
at 1:35, in our usual classroom.
- Bluebooks. Please bring an 8½×11
bluebook.
- Office hours. I will have office hours on Monday
(3/15/21), 2:30-4:30 and on Tuesday (3/16/21), 10:30-12:30 and
1:30-3.
-
Calculators. You may use calculators to do arithmetic, although
you will not need them. You may not use any calculator that
has the capability of doing linear algebra, or calculus, or of storing
programs or other material.
-
Other devices. You may not use cell phones, computers, or any
other device capable of storing, sending, or receiving information.
-
Structure and coverage. There will be 4 to 6 questions, some
with multiple parts. The test will cover sections 2.1, 2.2, and
2.3.1. The problems will be similar to ones done
for
homework especially assignments 5-7 , and as examples
in class and in the text. A
small
table of some Fourier transform properties and integrals
will be provided. Here are links to practice
tests:
2003
and 2010.
Be aware that these tests cover some material that will not be on the
test for this class.
Topics Covered
Fourier Transforms
- Computing Fourier transforms & properties
- Be able to compute Fourier transforms and inverse
Fourier transforms. Be able to use the properties in Theorem 2.6 to do
this. §2.1-2.2.1
- Be able to prove the simple properties listed in Theorem
2.6 of the text. Know how to use them. You will be given a
small
table listing some of these properties. §2.2.1
- Know and be able to use the "quartet table" below to
find Fourier transforms and inverse Fourier transforms.
"Quartet"
Function |
Fourier Transform |
$f(t)$ |
$\hat f(\lambda)$ |
$\hat f(t)$ |
$f(-\lambda)$ |
- Convolutions
- Directly finding convolutions. Be able to find the
convolution of two function directly from the definition. §2.2.2
- Convolution Theorem. Be able to state and prove this
theorem. Be able to use it to find Fourier transforms of
convolutions and inverse Fourier transforms of products of
functions.
- Plancheral's Theorem. Be able to state
and use Plancherel's Theorem to do integrals of functions.
- Filters
- LTI filter. Be able to define the
term linear, time-invariant filter. Know what its
connection to the convolution is, and what impulse response
functions and frequency response (system) functions
are. Be able to prove Lemma 2.16 in the text.
- Examples. Butterworth filter, "running average filter"
(see assignment 7, problem 4).
- Finding $L[f]$. Given a filter $L$, be able to find
$L[f]$ in either the time domain or the frequency domain or both.
Updated 3/13/2021