Math 414-501 — Spring 2022

Test 3 Review

General Information

Time and date. Test 3 will be given on Wednesday, 4/27/22, at 11:30, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Office hours. I will have office hours on Monday (4/25/22), 12:30-1:1:30 & 2-3, and on Tuesday (4/26/22), 11:30-1:30 & 2-3.

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 calculus, or of 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 2.2-2.4, 4.2, 4.3. The problems will be similar to ones done for homework assignments 7-9 homework and as examples in class and in the text. You will be required to derive properties of the FT and/or prove various results. (See below for a list.) A small table of some Fourier transform properties and integrals will be provided. Here are links to practice tests: Here are links to practice tests: 2002 and 2009 and 2016 Be aware that these tests cover some material that will not be on the test for this class.

Topics Covered

Fourier Transforms

  1. Computing Fourier transforms & properties Be able to compute Fourier transforms and inverse Fourier transforms.

  2. Convolutions
    1. Directly finding convolutions. Be able to find the convolution of two functions directly from Definition 2.9.
    2. Convolution Theorem. Be able to state and prove this theorem. Be able to use it to find Fourier transforms of convolutions and inverse Fourier transforms of products of functions.
    3. Plancheral's (or Parseval's) Theorem. Be able to state Plancherel's Theorem and to use it to find integrals, as in assignment 7, problem 3.

  3. Filters
    1. LTI filter. Be able to define the term linear, time-invariant filter. (Remember that a definition is not a lemma, proposition or theorem!) Know what its connection to the convolution is, and what impulse response functions and frequency response (system) functions are. Given one of them, be able to find the other. Know the what the Butterworth and running average filters are.
    2. Causal filter. Be able to define the term causal filter, and be able to determine whether an LTI filter is causal.

  4. Sampling Theorem. Be able to state and prove the sampling theorem.

Haar Wavelet Analysis

  1. Haar scaling function and approximation spaces. Know what the Haar scaling function, $\phi$, is and be able to derive its two-scale 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 $\{\phi(2^jx-k\}_{k=-\infty}^\infty$ basis for $V_j$.

  2. Haar wavelet and wavelet spaces. Know the definition of the Haar wavelet and Haar wavelet spaces Wj, along with their properties.

  3. Decomposition, reconstruction and filters. Be able to do simple decomposition and reconstruction problems similar to the ones done for homework. 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 a problem similar to Assignment 9, problem 3(b).

Updated 4/23/2022.