Math 414-501 — Spring 2024

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.

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 7 or 8 questions, some with multiple parts. The test will cover the topics listed below. (Material from Chapter 0 will be covered only in connection with other topics — e.g., projections and orthogonal/orthonormal sets are used in every topic listed below.) The problems will be similar to ones done in class or for homework, and in the text. Approximately 70% of the test will come from Chapters 4 through 6. 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 transforms. Be able to calculate Fourier transforms, inverse Fourier transforms and convolutions — directly and via the convolution theorem. Be able to define linear time-invariant filters and determine whether they are causal. Be able to filter simple functions, given the impulse response function or frequency response function. Finally be able to state and use the Sampling Theorem.

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 derive simple properties of the the scaling function, including that $\{2^{j/2}\phi(2^j x-k)\}_{k=-\infty}^\infty$ is an o.n. basis for $V_j$. (Equivalently, $\{\phi(2^jx-k)\}_{k=-\infty}^\infty$ is an orthogonal basis.) Be able to derive the scaling relation. Be able to state he formula for the wavelet.

• 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, and the corresponding decomposition and reconstruction filter diagrams

• 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 $a^j_k \approx m f(2^{-j}k)$, where $m=\int_{-\infty}^\infty \phi(x)dx$. (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 in §5.3.3, Theorem 5.23:

  Theorem 5.23 Suppose $P(z) := \frac12 \sum_{k\in \mathbb Z}p_kz^k$ is a polynomial that satisfies the following conditions:

  • $P(1)=1$.
  • $|P(z)|^2+|P(-z)|^2=1$, for $|z|=1$.
  • $|P(e^{it})|>0$ if $|t|\le \frac{\pi}{2}$.

  Then $\hat\phi(\xi)=\frac{1}{\sqrt{2\pi}}\Pi_{k=1}^\infty P(e^{-\xi i/2^k})$ is the Fourier transform of a scaling function.

• Daubechies' wavelets. Know how the Daubechies wavelets are classified using $N$, the largest power of $z+1$ that divides $P(z)=\sum_{k=0}^{2N-1}\frac{p_k}{2}z^k$, and also know that $N$ is the number of vanishing moments $\psi_N(x)$ has. In addition, $2N$ is the length of the four filters — high pass decomposition and reconstruction, and low pass decomposition and reconstruction. For $N=2$, 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 4/30/2024.