# Math 414-501 — Final Exam Review

## General Information

The final exam will be given on Friday, 5/8/2015, from 10:30 am to 12:30 pm. 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 5 to 7 questions, some with multiple parts. The test will cover the topics listed below. Points on the test will be approximately distributed this way: chapter 1, 10%; chapters 2 and 3, 20% and, chapters 4-6, 70%. Questions from chapters 1-3 will be involve calculations, but no theory. Any theory from the remaining chapters will only be derivations. In addition, there will be a table of integrals and Fourier transform properties. Here are links to practice tests: 2001 and 2009

## Topics Covered

Fourier series. Be able to calculate the Fourier series for a $2\pi$ periodic function, using either the real form or the complex form, and to sketch the function to which the series converges. Finally, be able to determine whether or not the convergence is uniform.

Fourier transforms & filters. Be able to find Fourier transforms, inverse Fourier transforms, and convolutions. Be able to filter a simple signal. A table of integrals and Fourier transforms will be supplied. §2.3.

Discrete Fourier transform. Be able to define the DFT and to explain how the FFT is related to the DFT. Know how fast (i.e., number of multiplications) the FFT is vs. the DFT.

Discrete-time signals Be able to filter discrete-time signals using the discrete-time convolution. (This really will only be used in conjunction with MRA filters.)

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). Be able to define these quantities, along with the corresponding wavelets, for the Haar and Shannon MRA's.

• $p_k$'s in Scaling relation. Know the properties of the $p_k$'s listed in Theorem 5.9.

• Decomposition & reconstruction. Know the decomposition and reconstruction formulas, high-pass and low-pass decomposition and reconstruction filters, and how to down sample and up sample a signal.

• Know how to implement both decomposition and reconstruction algorithms. In particular, be able to show that the top level coefficients, which are used in the initialization step, have the approximate form ajk ≈ m f( 2 -jk), where m = ∫-∞ φ(x)dx (Theorem 5.12).
• Fourier transform criteria for an MRA. Be able to find the Fourier transformed form of the scaling function and the wavelet. Be able to outline how the scaling function and the wavelet are derived from the function P(z) that satisfies the conditions §5.3.3, Theorem 5.1.9, & notes for 5/4/15.

• Daubechies' wavelets. Know how the Daubechies wavelets are classified using N, where 2N is the length of the four filters and how how N relates to the number of vanishing moments of a wavelet. For N=2, be able to explain how the approximate form of the wavelet coefficients (equation (6.13)) can be used in singularity detection and data compression. Be able to explain various ways of handling the problem of overspill; see section 6.3.

Updated 5/4/2015.