Math 414-501 Spring 2022
Test 3 Review
General Information
- Time and date. Test 3 will be given on Wednesday, 4/27/22,
at 11:30, in our usual classroom.
- Bluebooks. Please bring an 8½×11
bluebook.
- Office hours. I will have office hours on Monday
(4/25/22), 12:30-1:1:30 & 2-3, and on Tuesday (4/26/22),
11:30-1:30 & 2-3.
-
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 2.2-2.4, 4.2,
4.3. The problems will be similar to ones done for homework
assignments
7-9
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: Here are links to
practice
tests:
2002
and
2009
and 2016
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.
- Convolutions
- Directly finding convolutions. Be able to find the
convolution of two functions directly from Definition 2.9.
- 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
7, problem 3.
- Filters
- LTI filter. Be able to define the term linear,
time-invariant 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. Know the what the
Butterworth and running average filters are.
- Causal filter. Be able to define the term causal
filter, and be able to determine whether an LTI filter is
causal.
- Sampling Theorem. Be able to state and prove the
sampling theorem.
Haar Wavelet Analysis
- Haar scaling function and approximation spaces. Know
what the Haar scaling function, $\phi$, is and be able to derive its
two-scale relation. Be able to define its corresponding approximation
spaces $V_j$. Know the nesting and scaling properties for these
spaces. Be able to use the $\{\phi(2^jx-k\}_{k=-\infty}^\infty$ basis
for $V_j$.
- Haar wavelet and wavelet spaces. Know the definition of
the Haar wavelet and Haar wavelet spaces Wj, along with
their properties.
- Decomposition, reconstruction and filters. Be able to do
simple decomposition and reconstruction problems similar to the ones
done for homework. Know the low pass and high pass impulse response
filters used in decomposition and reconstruction. Know what up
sampling and down sampling are. Be able to a problem
similar to Assignment 9, problem 3(b).
Updated 4/23/2022.