Math 414501 — Spring 2017
Test 2 Review
General Information
 Time and date. Test 2 will be given on Wednesday, 3/22/17,
at 10:20, in our usual classroom.
 Bluebooks. Please bring an 8½×11
bluebook.
 Office hours. I will have office hours on Monday
(3/20/17), 11:101:30 and on Tuesday (3/21/17), 10:3012:30 pm.

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 (except
2.2.3), 2.3 and 2.4. The problems will be similar to ones
done
for
homework — especially assignments 46 —, 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:
2002 and 2009. 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. §2.12.2.1
 Be able to establish the simple properties listed in Theorem 2.6
of the text, and 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 (or Parseval's) Theorem. Be able to state and use
Plancherel's Theorem. (See assignment 5, problem 3.)
 Filters
 LTI filter. Be able to define the term linear,
timeinvariant filter. Know what its connection to the
convolution is, and what impulse response functions
and frequency response (system) functions are. Given one of
them, be able to find the other.
 Examples. Butterworth filter, "running average filter"
(see assignment 6, problem 2).
 Finding outputs. Given an impulse response function, be able to directly apply the
definition to to find the output of the filter for an input
function.
 Causal filter. Be able to define the term causal
filter, and be able to determine whether an LTI filter is
causal. §2.3.
 The Sampling Theorem. Be able to state and prove this
theorem, and to define these terms: bandlimited
function, Nyquist frequency, and Nyquist rate
§2.4.
Updated 3/13/2017.