Math 414-501 — Test 2 Review
General Information
Test 1 will be given on Wednesday, 4/8/09. Please bring an
8½×11 bluebook. I will have extra office hours
on Tuesday, 4/7/09 (TBA). Material covered includes all of chapter 2
and chapter 3, except for sections 3.1.5 and 3.1.6.
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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. There will be 4 to 6 questions, some with multiple
parts. The problems will be similar to ones done for homework, and as
examples in class and in the text.
Topics Covered
Fourier Transforms
- Definitions. Be able to define these: Fourier transform,
inverse Fourier transform, convolution, shift operator
Ta, linear time-invariant filter, causal filter, ideal
filter, response function, system function, Nyquist frequency,
Nyquist rate, dispersion of f about t = a.
- Theorems.
- Be able to state and prove (or sketch a proof, as required) of
the following theorems: the convolution theorem for the Fourier
transform, the Sampling Theorem (sketch only), the Uncertainty
Principle (sketch only, for a= α = 0).
- Be able to state and use Plancherel's Theorem (Theorem 2.12), and
the conditions under which a filter is time invariant (Theorem 2.17),
or time invariant and causal (Theorem 2.19).
- Calculations. Be able to find Fourier transforms;
inverse Fourier transforms; convolutions; outputs of filters - given
system response function and signal; 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.
Discrete Fourier Analysis
- Definitions. Be able to define these: discrete Fourier
transform (DFT); FFT algorithm; discrete periodic
signals Sn; convolution of discrete periodic
signals; discrete bi-infinite signals - i.e. ℓ2 ;
shift operator (periodic discrete signals and bi-infinite signals);
convolution of discrete bi-infinite signals; time-invariant,
discrete filters; impulse response (IR), IIR, FIR; Z-transform;
system (transfer) function.
- Theorems.
- Be able to state and prove (or sketch a proof, as required) of
the following theorems: convolution theorems for the DFT and the
Z-transform; the DFT maps Sn; to itself (Lemma
3.2); the inversion theorem for the DFT (sketch); the approximation
of the FT via FFT (eqn. (3.6), p. 157).
- Be able to state these; the conditions under which a discrete
linear filter is time invariant (Theorem 3.10); the number of
operations needed to calculate the FFT vs. matrix multiplication;
the DFT in matrix form (eqn. (3.2) and preceeding equation on
p. 148).
- Calculations. Be able to calculate the DFT and inverse
for SMALL n (n ≤ 4). Be able to calculate the Z-transform of a
SIMPLE discrete-time signal (see example done in class on 4/3/09 and
problem 16, p. 168). Problems will be similar to those done for
homework and to the examples done in class.
- Comparisons. Know the similarities among the various
transforms and be able to discuss them. (See handout given on
Friday, 4/3/09).
Updated 4/4/2009.