Math 414501 — Test 2 Review
General Information
Test 2 will be given on Friday, 4/17/2015. Please bring an
8½×11 bluebook. Extra office hours: TBA.

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 2.12.4,
3.1.13.1.4, 3.2.1, 3.2.2 (pgs. 149150), 4.14.3.1 in the text, and
any material discussed in class, starting 2/23 to 4/13. 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
 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. A brief table of integrals will be supplied. The
problems will be similar to those done in class or for homework.
 Filters. Know what a linear, timeinvariant 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: 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 FT is. §§3.1.3.
 Discretetime signals Know what a discretetime signal
is. Be able to calculate Ztransforms in simple cases. Be able to state the
discretetime convolution theorem. §§3.2.13.2.2
Haar Wavelet Analysis
 Haar scaling function and approximation spaces. Know
what the Haar scaling function, $\phi$, is and be able to derive its
twoscale 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 $\{\phi(2^jxk\}_{k=\infty}^\infty$ basis
for $V_j$. §§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. Be able to do simple
decomposition and reconstruction problems similar to the ones done
for homework.
Updated 4/12/2014.