Math 414-501 Spring 2024
Test 2 Review
General Information
Test 2 will be given on Wednesday, 4/10/2024. Please bring an
8½×11 bluebook. Extra office hours: TBA.
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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 2.1-2.4,
Discrete Fourier Transforms, parts 1, 2 and 4, to the extent these
were covered in class on 3/27/2024, 4.1-4.3.2 in the text, and any
material discussed in class, 2/26/24 and 3/4/24 to 4/5/24. The
problems will be similar to ones done in class or
for homework,
and in the text. In addition, you may be asked to define a term or
state a theorem from those listed below. A short
table
of integrals and Fourier transform properties will be provided. Here
are links to practice
tests: 2012 and
2014
Topics Covered
Fourier Transforms
- Finding Fourier transforms Be able to find Fourier
transforms; inverse Fourier transforms; convolutions; and integrals
via Plancherel's Theorem. You may use any property of the Fourier
transform to do the calculation. As mentioned above, a table of
integrals and properties of the Fourier transform will be supplied.
- Be able to prove the convolution theorem.
- Filters. Know what a linear, time-invariant filter is,
what its connection to the convolution is, and what it's impulse
response function and it's frequency response (system) function
are. Given the impulse response function, be able to find the
frequency response function. Know what a causal filter
is. Be able to filter a simple signal. §2.3.
- The Sampling Theorem. Be able to state and prove this
theorem and to define these terms: band-limited
function, Nyquist frequency, Nyquist
rate. §2.4.
Discrete Fourier Analysis
- Discrete Fourier transform. Be able to define the DFT,
the inverse DFT. Know the connection between coefficients in a
Fourier series and the DFT approximation to them. 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. Be able
to describe the FFT algorithm and to explain why it’s fast. See the
notes
on
Discrete Fourier Transforms.
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, $\phi(x)=\phi(2x)+\phi(2x-1)$. Be able to
define its corresponding approximation spaces $V_j$. Be able to show
that $\{2^{j/2}\phi(2^jx-k), k\in \mathbb Z\}$ is an orthonormal 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 the Haar wavelet spaces
Wj, along with their properties. Be able to show that
$\psi(x)=\phi(2x)-\phi(2x-1)$. Be able to show that
$\{2^{j/2}\psi(2^jx-k), k\in \mathbb Z\}$ is an orthonormal basis for
$W_j$. §4.2.3
- Orthogonal spaces. Decomposition and reconstruction. Be able to do simple
decomposition and reconstruction problems similar to the ones done
for homework.
Updated 4/4/2024.