Math 414501 — Test 2 Review
General Information
Test 2 will be given on Wednesday, 4/9/2014. Please bring an
8½×11 bluebook. Office hours: Tuesday, 10:30  3:00.

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.

Structure and coverage. There will be 4 to 6 questions, some
with multiple parts. The test will cover sections 2.3,
2.4, 3.1.13.1.4, (also, 3.2.1; see below), 4.14.4, 5.1 in the text, and any material discussed in class, starting 2/26 to 4/4. The problems will
be similar to ones done for homework, and as examples in class 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:
2002
and 2009
Topics Covered
Fourier Transforms
 Filters. Know what a linear, timeinvariant filter is,
what its connection to the convolution is, and what it's system response
function and it's system function are. Given the system response
function, be able to find the system 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: bandlimited function, Nyquist frequency, Nyquist rate. §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 nperiodic sequences and to show that
the result is also nperiodic. Be able to show that the DFT and
inverse DFT take nperiodic sequences to nperiodic sequences. Be able
to prove that any of the properties in Theorem 3.4, p. 137
hold. (Chapter 3, exercise 2.) Be able to describe the FFT algorithm and to explain why itâ€™s fast. §§3.1.13.1.3.
 FFT and Fourier transform. Know what the connection
between the DFT and FFT is. §§3.1.3.
Multiresolution Analysis
 Haar MRA
 Haar scaling function and approximation spaces. Know
what the Haar scaling function is. Be able to define its corresponding
approximation spaces V_{j} and know the nesting and scaling
properties for the Haar these spaces, and that Theorem 4.6 gives bases
for them. §§4.2.14.2.2
 Haar wavelet and wavelet spaces. (§4.2.4) Know the
definition of the Haar wavelet and Haar wavelet spaces W_{j},
along with their properties. §4.2.3
 Decomposition and reconstruction.
 Discretetime signals. Know what a discretetime signal
is and how discretetime invariant filters are defined via the
(discretetime) convolution and impulse response. This was discussed in class on 3/31. It is also covered in §3.2.1.
 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. §§4.3.14.3.2.
 Filter diagrams. 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
derive the decomposition diagram, including the low pass and high pass
filters involved, as we did in class on Monday, 3/30. §4.3.3
 Implementation Know the steps in the implementing the
wavelet algorithms: initialization (sampling),
decomposition, processing, reconstruction. §4.4
 General MRA
 Mutliresolution Analysis. Be able to define the term multiresolution analysis.
 Shannon MRA Be able to discuss the details of the Shannon MRA, including the approximation spaces (the V_{j}'s) and scaling function, φ. Be able to show that
{φ(xk), k ∈ Z} is an o.n. set.
Updated 4/6/2014.