Math 414501 — Spring 2016
Test 2 Review
General Information
 Time and date. Test 2 will be given on Wednesday, 3/23/16,
at 10:20, in our usual classroom.
 Bluebooks. Please bring an 8½×11
bluebook.
 Office hours. I will have office hours on Monday
(3/21/16), 1:404, and on Tuesday (3/22/15), 13:30 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 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, 2.4, 3.1.13.1.3. The problems will be similar to ones
done
for
homework, and as examples in class and in the text. A
short
table of 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 table
listing these properties plus a few others, so you do not need to
memorize them.) §2.2.1
 Convolutions. Be able to find the convolution of two
functions and to use it to find Fourier transforms. §2.2.2
 Plancheral'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 problem 2.12).
 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.
Discrete Fourier Transforms
 Definition & properties.
 DFT. Be able state the formulas for both the DFT and
inverse DFT. Be able to prove Theorem 3.3, which gives the inversion
formula for the DFT.
 Approximation fo FS coefficients. Know the connection
between coefficients in a Fourier series and the DFT.
 $\mathcal S_n$. This is the space of $n$ periodic sequences. Be
able to show that the DFT and inverse DFT of an $n$ periodic sequence
is an $n$ periodic sequence — i.e, that the DFT and inverse DFT
transform $\mathcal S_n$ into itself.
 FFT. Know what the fast Fourier transform (FFT)
algorithm for computing the DFT is. Be able to compare the number
of multiplications needed to compute the DFT vs. the number for
the FFT.
Updated 3/19/2016.