Signal and Image Processing by Computer
Final Exam A Winter 2013-2014
Student ID:
Length of test: 3 Hours.
Allowed material: 2 formula pages written on both sides, calculator.
Instructions:
1.
2.
3.
4.
Fill your student id at the first page of the exam.
All answers should be written on the exam file.
Use clear hand-writing. Unreadable answers will not be checked.
Notations:
a. X  f  ,Y  f  ,H f  are the Fourier transforms of x t  , y t  , h t  .
b. X k ,Y k , H k  are the DFT of x  n  , y  n  , h  n  .
1 n  0
c. u[n] is the discrete step function u  n   
0 else
1 t  0
d. u(t) is the continuous step function u t   
0 else
There are 12 pages (including this page) in the exam. Make sure that no page is missing and that
you have answered all questions.
Lecturer in charge: Alfred M. Bruckstein
Lecturer: Raja Giryes
TA: Raja Giryes
1
Question 1 (25 pts)
a) (20 pts) Given a noisy signal y  x  , we have seen in class that we can use the Wiener filter for
recovering x  R m from y 
m
(denoising). Let x̂ be the recovery result of the Wiener filter and
Exx *  Rx , Ex  0, E *   2 I , E  0 . Calculate the recovery error:
2
*
E  x  xˆ 2   E  x  xˆ   x  xˆ   .




Clue: First calculate the error of the representation in the basis that the Wiener filter uses.
2

b) (5 pts) What is the error in the following cases: trace Exx *
3

m 2 and m 2

trace Exx *

Question 2 (30 pts)
𝑡
0 ≤ 𝑡 ≤ 0.25
Given the following signal φ(𝑡) = { 0.25 0.25 ≤ 𝑡 ≤ 0.5
0.25 − 𝑡 0.5 ≤ 𝑡 ≤ 0.75
𝑡
0.75 ≤ 𝑡 ≤ 1
a) (1 pt) Draw the signal
b) For N divided by 4, we have the Haar family of functions defined with N = 2𝑛 and defined as
𝜓(0,0; 𝑡) = 1
And for 1 ≤ 𝑘 ≤ 𝑛 and 1 ≤ 𝑚 ≤ 2𝑘−1
√2𝑘−1
𝜓(𝑘, 𝑚; 𝑡) =
−√2𝑘−1
{ 0
1
𝑚−2
𝑚−1
≤ 𝑡 ≤ 𝑘−1
2𝑘−1
2
1
𝑚−2
𝑚
≤ 𝑡 ≤ 𝑘−1
𝑘−1
2
2
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Denote by φ(𝑘, 𝑚) the coefficient of 𝜓(𝑘, 𝑚).
1)
(4 pts) What is the signal approximation (its representation) without quantization
using these functions, i.e., what are the values of φ(𝑘, 𝑚) ?
2) (3 pts) What is the error (without quantization)?
3) (3 pts) Explain how increasing N influence the error.
4) (3 pts) How the error behaves in the limit 𝑙𝑖𝑚𝑁→∞ and what is the meaning of this
behavior?
Clue: You can use the equality ∑𝑛𝑖=1 𝑖 2 =
4
2𝑛3 +3𝑛2 +𝑛
6
5
c) (3pts) Explain shortly how you can calculate the coefficients of the standard basis given the
coefficients of the Haar basis from section b.
d) (13 pts) Answer the questions of section b for the Rademacher family of functions. The
Rademacher family is defined as:
𝜓𝑖𝑅 (𝑡) = 𝑠𝑖𝑔𝑛{𝑠𝑖𝑛(2𝑖−1 𝜋𝑡)}
where 1 ≤ 𝑖 ≤ 𝑁.
e)
6
7
Question 3 (20 pts)
a) (10 pts) Calculate the Fourier transform of the following signal
𝑓(𝑡) = cos(2𝜋𝑡)𝑒 −|𝑡|
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b) Given the following impulse response of a system
ℎ(𝑡) = {
𝑡 0.1 ≤ 𝑡 < 𝑇
0
𝑒𝑙𝑠𝑒
and input signal
𝑥(𝑡) = {
1/𝑡
0
0.1 ≤ 𝑡 < 𝑇
𝑒𝑙𝑠𝑒
Calculate the systems’ output: 𝑦(𝑡) = 𝑥(𝑡) ∗ ℎ(𝑡)
9
Question 4 (25 pts)
a) (15 pts) Given the following signal x
-3w
we will sample it as follows:
Transmitter side
x
-2w
-w
w
2w
3w
Sampling
Receiver side
 nT
H2  f 
H1  f 
x̂
What is the minimal sampling rate and the frequency responses H1  f  and H2  f  required
ˆ?
for getting x  x
10
 2 ko n 

 N 
b) (5 pts) Calculate the DFT of cos 
11
  ko  2n  1  

2N


c) (5 pts) Calculate the DCT of cos 
Reminder: the DCT coefficients are calculated using the following formula:
N 1
   2n  1  k 
c  k   w k    n  cos 

2N
n 0


 1
k 0
 N
w k   
 2
0  k  N 1
 N
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