Discrete Fourier analysis. Know what the connection between the DFT and FFT is, and be able to explain why the FFT is "fast". Be able to explain how to approximate Fourier series and Fourier transforms via the DFT. Be able to state and prove the (circular) convolution theorem for the DFT, and to use it in applications similar to ones done in the homework (i.e., diagonalization of circulants and simple compression algorithms). For time series (bi-infinite sequences), know how discrete-time invariant filters are defined via (discrete-time) convolution. Know how to find the Z transform of a simple sequence, and be able to state and prove the discrete-time convolution theorem.
Haar wavelets. Be able to explain what aspects of signal processing wavelets can do better than Fourier transforms. Using the Haar wavelet and scaling function, be able to carry out simple decomposition and reconstruction algorithms. Be able to read and explain filter diagrams.
Multiresolution analysis and the Daubechies wavelets. Be able to define Mallat's multiresolution analysis, approximation spaces, the scaling relation, the wavelet, and 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. 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). Be able to outline the how the scaling function and the wavelet come from the function P(z) that satisfies the conditions in Theorem 5.23. Know these conditions and the extra conditions on P(z) that must be met to ensure the smoothness of the scaling function and the wavelet, and how these relate to the number of vanishing moments of a wavelet. In particular, be able to briefly describe how the simplest Daubechies N = 2 wavelet is constructed; be able to classify the other Daubechies wavelets. Be able to obtain the approximate form of the wavelet coefficients for the N = 2 Daubechies wavelet (equation (6.13)). Know how this applies to singularity detection and compression. Be able to explain various ways of handling the problem of overspill; see section 6.3.