Math 414 - Test II Review

General Information

Test II (Thursday, April 24) will have 5 to 7 questions, some with multiple parts. Please bring an 8½×11 bluebook. Problems will be similar to ones done for homework. 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 either calculus or linear algebra.

Topics Covered

Discrete signals. Know what a discrete-time signal is, and how discrete-time invariant filters are defined via (discrete-time) convolution. Know what the Z transform of a discrete-time signal is.

Haar wavelets. Know the Haar scaling function, wavelet, approximation spaces (V's), and wavelet spaces (W's). Using the Haar wavelet and scaling function, be able to carry out simple decomposition and reconstruction algorithms. Know what the various high pass and low pass filters associated with these algorithms are, what downsampling and upsampling are, and finally be able to use filter diagrams to describe the decomposition and reconstruction algorithms.

Multiresolution analysis (MRA). Be able to define Mallat's multiresolution analysis, approximation spaces (V's), the scaling relation, wavelet spaces (W's), and the wavelet. Know the decomposition and reconstruction formulas, high-pass and low-pass decomposition and reconstruction filters, downsampling and upsampling. Be able to discuss the details of the multiresolution analysis in the Haar case or for the Shannon MRA (exercise 8 in § 5.4). Know how to implement both decomposition and reconstruction algorithms. Be able to obtain the approximate form of the top-level scaling coefficients (Theorem 5.12).

Fourier transform criteria for an MRA. Be able to find the Fourier transformed (i.e., frequency space) form of the scaling relation (Theorem 5.19). Be able to outline how the scaling function and the wavelet are derived from the function P(z) that satisfies the conditions in Theorem 5.23. Know these conditions. Using the iterative scheme from Theorem 5.23, be able to explain why the support of the N=2 Daubechies scaling function is [0,3].

Daubechies' wavelets. Know how the Daubechies wavelets are classified using N, the largest power of z+1 that divides P(z), and also how N relates to the number of vanishing moments of a wavelet. Be able to briefly describe how the simplest Daubechies N = 2 wavelet is constructed (See the notes for April 15.) Be able to explain how the approximate form of the wavelet coefficients (b's) (equation (6.13)) is used in singularity detection and data compression. Be able to explain various ways of handling the problem of overspill; see section 6.3.