Math 414-502 Test 2 Review
General Information
Test 2 will be given on Wednesday, 4/11/2012. Please bring an
8½×11 bluebook. Extra office hours: Monday,
2-4:30; and Tuesday, 1:30-4:30.
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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.
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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 2.2.2, 2.2.4, 2.3,
2.4, 3.1.1-3.1.4, 3.2.1, 3.2.2, 4.1-4.4 in the text. The problems will
be similar to ones done for homework, and as examples in class and in
the text. A short table of integrals and Fourier transform properties
will be provided. Here are links to practice
tests:
2002
and 2009
Topics Covered
Fourier Transforms
- Convolutions. Be able to find the convolution of two
functions and to use it to find Fourier transforms. §2.2.2
- Filters. Know what a linear, time-invariant filter is,
what its connection to the convolution is, and what system response
functions and system functions are. Given the system response
function, be able to find the system function. Know what
a causal filter is and be able to explain why it is
important. §2.3.
- The Sampling Theorem. Be able to state this theorem and
to define these terms: Nyquist frequency, Nyquist
rate, and aliasing. Be able to give a brief description
of aliasing. §2.4.
Discrete Fourier Analysis
- Discrete Fourier transform
- Definition & properties. Be able to define the DFT,
the inverse DFT. Know the connection between coefficients in a
Fourier series and the DFT approximation to them, as well as the DFT
approximation to the Fourier transform of a function. Be able to
define the convolution of two n-periodic sequences and to show that
the result is also n-periodic. Be able to show that the DFT and
inverse DFT take n-periodic sequences to n-periodic sequences. Be able
to prove that any of the properties in Theorem 3.4, p. 137
hold. (Chapter 3, exercise 2.) §§3.1.1, 3.1.2, 3.1.4
- FFT and Fourier transform. Know what the connection
between the DFT and FFT is, and be able to explain why the FFT is
"fast." Also, be able to briefly explain the ODE application
given in the text. §§3.1.3, 3.1.6.
- Discrete-time signals. Know what a discrete-time signal
is and how discrete-time, invariant filters are defined via the
(discrete-time) convolution and impulse response. In the case of
filters used for the Haar multiresolution analysis, be able to the
impulse responses. Know what the Z-transform is, what its connection
to Fourier series is. and be able to state the discrete-time
convolution theorem. §§3.2.1-3.2.2
Haar Wavelet Analysis
- Haar scaling function and approximation spaces. Know
what Haar the scaling function is. Be able to define its corresponding
approximation spaces Vj and know the nesting and scaling
properties for the Haar these spaces, and that Theorem 4.6 gives bases
for them. §§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 algorithms. Be able to
derive the decomposition algorithm and to state the reconstruction
algorithm. Be able to carry these out in simple cases similar to the
ones given for homework. §§4.3.1-4.3.2.
- Filter diagrams. Know the low pass and high pass impulse
response filters used in decompositin and reconstruction. Know
what up sampling and down sampling are. Be able to
derive the decomposition diagram, including the low pass and high pass
filters involved, as we did in class on Wednesday, 4/4/12. §4.3.3
- Implementation Know the steps in the implementing the
wavelet algorithms: initialization (sampling, "wavelet crime"),
decomposition, processing, reconstruction. §4.4
Updated 4/10/2012.