Math 414-501 — Final Exam 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.

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, section 2.3, and 3, section 3.2.1, 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, and to keys to assignment 6, assignment 7, problem 4.2 in assignment 7, and problem 4 on test 2.

Topics Covered

Fourier series. Be able to calculate the Fourier series for a $2\pi$ periodic function, using either the real form or, if required, 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-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. Be able to deive 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 $V_j$ (only orthonomality will be required). Be able to derive the scaling relation. Be able to briefly explain how to get wavelet relation.

• 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.

• Signal processing. 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 a^j_k ≈ 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 5.23.

• Daubechies' wavelets. Know how the Daubechies wavelets are classified using $N$, where $2N$ is the length of the four filters — high pass decomposition and reconstuction, and low pass decomposition and reconstruction. Be able to use the $b_k^j$'s, given $\int_{-1}^2\psi(x)dx =\int_{-1}^2x\psi(x)dx=0$, to explain how the approximate form of the wavelet coefficients (equation (6.13)) can be used in singularity detection and data compression.

Updated 4/28/2020.