Math 414-501 — Spring 2018

Current Assignment

Assignment 9 - Due Monday, 4/23/2018.

• Read sections 5.2 (Note: we covered this section, especially the decomposition and reconstruction diagrams, in class on 4/16/18. The only new thing is initialization.), 5.3.3 and 5.3.4.

• If you haven't formed a group for a project, do so by Friday. Once you've done that, send me an email with a list of the group members and a short description of the project.

• Problems.
1. Chapter 5: 8(d,e,f), 11

2. Do any two of the problems below.

1. Write two programs in your favorite language to do the following. Consider the $p_k$'s from the Example 5.25, p. 227 (Daubechies' wavelet; db2 in matlab). The first program inputs a signal $a^j$ and outputs $a^{j-1}$ and $b^{j-1}$. It is the decomposition step in a wavelet analysis. The second program inputs $a^{j-1}$ and $b^{j-1}$ and outputs $a^j$. This is the reconstruction step. In matlab, both programs involve the conv (convolution) function, which will be used to filter inputs. In addition, the first program requires a down-sampling part; the second, up sampling.

2. Let $f(t)=\begin{cases} t, & t \le \frac{3}{17}\\ 1.02t +\frac{6.06}{17}, & \frac{3}{17} < t \end{cases}$. Sample $f$ at $k 2^{-10}$, $k=0, \ldots, 1024$. (The time interval is $2^{-10}$, so $a^{10}_k$ corresponds to the time $k2^{-10}$.) Either use matlab's wavemenu or the decomposition program from the previous problem, applied to $a^{10}_k=f(k2^{-10})$ to obtain $b^{-9}$. (The time interval is $2^{-9}$, $b^9_k$ corresponds to the time $k2^{-9}= 2k2^{-10}$.) Keeping in mind the times involved, plot both $a^{10}$ and $b^9$ versus $t$, $0\le t\le 1$. Can you detect the change in slope from the wavelet coefficients?

3. Chapter 5: 17

4. Chapter 6: 2

Updated 4/17/2018