Math 414501 — Spring 2018
Test 2 Review
General Information
 Time and date. Test 2 will be given on Wednesday, 3/21/18,
at 10:20, in our usual classroom.
 Bluebooks. Please bring an 8½×11
bluebook.
 Office hours. I will have extra office hours  in addition
to my usual ones  on Monday (3/19/18), 2:304, and on Tuesday (3/20/18),
23:30..

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), and 2.3. The problems will be similar to ones done for
homework assignments
46
homework and as examples in class and in the text. You will be
required to derive properties of the FT and/or prove various
results. (See below for a list.) A
small
table of some Fourier transform properties and integrals
will be provided. Here are links to practice
tests:
2009
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 find Fourier transforms and inverse Fourier
transforms, either by directly doing the integrals involved and/or by
using properties from Theorem 2.6. §2.12.2.1
 Be able to derive any of the 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 functions directly from Definition 2.9. §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
Plancherel's Theorem and to use it to find integrals, as in assignment
6, problem 2.
 Filters
 LTI filter. Be able to define the term linear,
timeinvariant filter. (Remember that a definition is
not a lemma, proposition or theorem!) 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. Below are
examples of filters that we have used.
 Butterworth filter.
 Running average filter. (See assignment 4, problem 2).
 The filter in problem 6, assignment 6.
 The "ideal" filter given in equation (2.21).
 Finding outputs. Given an impulse response function, be
able use the convolution form $L[f]=h*f$ to find the output of the
filter for an input function. In addition, be able to use the
convolution theorem to find the FT of the output of a filter.
 Causal filter. Be able to define the term causal
filter, and be able to prove this: If the impulse
response function $h(t)=0$ for $t< 0$, the the filter is
causal. Know that the converse is true: if a filter is causal,
then $h(t)=0$ for $t<0$. Be able to use these to determine whether
an LTI filter is causal. §2.3.
Updated 3/12/2018.