NOTES:

• Graded out of 100 points
• Expected effort: 1-2h, 50', 30', 40', 40'; respectively.
• Please hand in a **TYPED** report

1) SVD [45 pts]

Consider the points (1,2), (3,1), (-1,3), (5,0), (2.1,1.5).

[30 pts] Give the unit vector that (a) starts from the center of gravity and (b) gives the best direction to project these points on. 'Best' means minimizing the sum of squared errors.

[15 pts] Plot the original points, along with your best vector

2) SVD [15 pts]

Compute the SVD for the following 2x5 matrix 'A', ie., give the U, Lambda, V matrices of the decomposition.
 

4	1	3	4	6	10
6	1.5	4.5	6	9	15

 Specifically:

• [2 pts] what is the rank r of the input matrix A?
• [1 pt] what are the dimensions of matrix U?
• [4 pts] what are the entries of matrix U?
• [1 pt] what are the dimensions of Lambda?
• [2 pts] what are the entries of matrix Lambda?
• [1 pt] what are the dimensions of V?
• [4 pts] what are the entries of matrix V?

Hint: use a commercial SVD package (like 'mathematica', matlab, Splus, etc, or the freeware 'R' at  http://www.r-project.org/ )

3) DWT and DFT [ 10 pts]

Hint: for the Haar DWT, use the definitions in the textbook, p. 115, eq (C.1-C.3); you are welcome to use/adapt open-source code. Similarly, for the DFT,  use the definition on p. 103, eq. (B.1).

• Some signals compress better under DWT (Haar) than under DFT. Give such a signal, of duration 16:

• [1  pts] describe it in English or with equations
• [1 pt] give its plot
• [1 pts] give its non-zero DWT coefficients (specify the level and the time-offset) and
• [2 pts] give its non-zero DFT coefficients
• The reverse: Give a signal of length 16, that compresses better under DFT than DWT (Haar). Again:
• [1 pts] describe it in English or with equations
• [1 pt] give its plot
• [2 pts] give its non-zero DWT coefficients (specify the level and the time-offset) and
• [1 pts] give its non-zero DFT coefficients

4) DFT [15 pts]

• [10 pts] Give the 16-point DFT coefficients X_f  (f=0, ... 15), for the impulse function: x₀= 1, x_t= 0 (t = 1, 2, ... 15).
• [5 pts] Verify that Parseval's theorem holds

5) DWT [15 pts]

• [10 pts] Repeat, for the Haar DWT: On a scalogram (= time/frequency tiling), give the 16-point DWT(Haar) coefficients d_l,t (l is the level; t is the time offset), for the impulse function: x₀= 1, x_t = 0 (t = 1, 2, ... 15).
• [5 pts] Verify that Parseval's theorem holds