Math 414 - Test II Review
General Information
Test I (Tuesday, April 23) will have 5 to 7 questions, some with
multiple parts. Please bring an 8½×11
bluebook. Problems will be similar to ones done for
homework. 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 either calculus or linear algebra.
Topics Covered
Fourier transforms.
- Computing Fourier transforms & properties.
(§2.1-2.2) Be able to compute the Fourier transform of functions
similar to ones dealt with in class and in the homework
assignments. You should also be able to establish the simple
properties listed on pg. 100 of the text. You also need to know how to
use the major properties of the Fourier transform: the inversion
formula, the convolution theorem, and the Plancheral formula. You will
be given a table listing properties, so you do not need to memorize
them.
- Linear filters. (§2.3) Know what a linear,
time-invariant filter is, and what its system response function and
system function are. Be able to define the term causal
filter.
- The Sampling Theorem. (§2.4) Be able to state and
prove this theorem. Know what aliasing is, be able to give a
brief description of it.
Discrete Fourier analysis.
- Discrete Fourier transform. (§3.1) Be able to
define the DFT and inverse DFT, and be able to explain how it can be
used to approximate coefficients in a Fourier series. Be able to
use this to derive the DFT approxiamtion to the Fourier transform of a
function.
- FFT. (&3.1.3-3.1.4) Know what the connection between the DFT
and FFT is, and be able to explain why the FFT is "fast".
- Applications. Be able to state the (circular)
convolution theorem for the DFT (Theorem 3.4, pg. 136), and to use it
in applications similar to ones done in the homework (i.e.,
diagonalization of circulants and simple compression algorithms).
- Discrete signals. (§3.2.1) Know what a
discrete-time signal is, and how discrete-time invariant filters are
defined via (discrete-time) convolution.
Haar wavelets and multiresolution analysis.
- MRA. Know the definition of Mallat's multiresolution
analysis, approximation spaces, the scaling relation, the wavelet, and
decomposition and reconstruction formulas, high-pass and low-pass
decomposition and reconstruction filters, downsampling and upsampling.
- Haar scaling function and approximation spaces.
(§4.2). For the Haar case, know what the scaling function
(§4.2.1), the approximation spaces corresponding to it
(§4.2.2), and the two-scale relation (Example 5.7) for it. Be
able to verify nesting and scaling properties for the Haar MRA
(§ 4.2.3)
- Haar wavelet and wavelet spaces. (§4.2.4) Know the
definition of the wavelet and wavelet spaces in the Haar case. Be able
to define these for a general MRA. (§5.1.3)
- Implementation Be able to carry out simple decomposition
and reconstruction algorithms. Be able to read and explain filter
diagrams. Be able to discuss the details of the multiresolution
analysis in the Haar case. Know the steps in the implementing the
wavelet algorithms: initialization (sampling), decomposition,
processing, reconstruction. (§§4.3, 4.4, 5.2)