Math 414-501 Spring 2020
Test 2 Review
General Information
- Time and date. Test 2, which will be proctered,
will be given on Wednesday, 4/8/20, at 12:40, online. The specific
procedures for giving the test online will be announced.
- Office hours. I will schedule online office hours. I will announce them shortly.
-
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.
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Structure and coverage. There will be 4 to 6 questions, some
with multiple parts. The test will cover sections 2.1, 2.2, 2.3, 4.2
and 4.3.1 and 4.3.2. The problems will be similar to ones done
for
homework, and as examples in class and online (3/25-4/3) for and
in the text. A
short
table of Fourier transform properties and integrals will be
provided. Here are links to practice
tests:
2014
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 and §2.2.1
- 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 and the convolution theorem Be able to find
the convolution of two functions, from the definition of
convolution. Also able to use the convolution theorem to find
Fourier transforms. §2.2.2
- 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. 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.
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 $V_j$. §§4.2.1-4.2.2
- Haar wavelet and wavelet spaces. (§4.2.4) Know the
definition of the Haar wavelet and Haar wavelet spaces Wj,
along with their properties. §4.2.3
- Decomposition and reconstruction. Be able to do simple
decomposition and reconstruction problems similar to the ones done
in the online notes for 3/27/20-4/3/20. §4.3.1, §4.3.2
Updated 4/4/2020.