Math 414-501 — Spring 2023

Test 3 Review

General Information

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 2.2.2, 2.2.4, 2.3, 4.2, 4.3.1, 4.3.2 and 5.1.1 in the text. The problems will be similar to ones done for homework, and as examples in class and in the text. A short table of integrals and Fourier transform properties 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

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

• 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 9, problem 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.

Haar Wavelet Analysis

• 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 $V_j$.

• Haar wavelet and wavelet spaces. Know the definition of the Haar wavelet and Haar wavelet spaces W_j, along with their properties.

• Decomposition and reconstruction formulas. Be able to derive the decomposition and reconstruction formulas, and to do problems similar to the ones done for homework.

Multiresolution Analysis (MRA)

• Mallat's MRA. Be define Mallat's multiresolution analysis, including the approximation spaces (V's), the scaling relation (in terms of $p_k$'s), the wavelet spaces (W's), and the wavelet itself (again, in terms of $p_k$'s).

• Haar & Shannon MRA's. Be able to describe in detail both the Haar and Shannon MRA's. For each, state what the approximation spaces, wavelet spaces, scaling function and wavelets are. Be able to verify that the properties of an MRA are satisfied for both cases. In the Shannon case, able to state -- but not prove -- and use the sampling theorem; see p. 120, Theorem 2.23.

• $\mathbf {p_k}$'s in Scaling relation. Be able to derive simple properties of the the scaling function, including that $\{2^{j/2}\phi(2^j-k)\}_{k=-\infty}^\infty$ is an o.n. basis for $V_j$. Be able to derive the scaling relation. Be able to briefly explain how to get the wavelet relation.

Updated 4/23/2023.