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(ELEC317)2002_midterm.pdf
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ELEC 317 Midterm Test (Fall 2002) 31 Oct 2002.
Answer all questions. Some questions may be more difficult than the others.
1. (10%) Linear time invariant system
a.
(5%) Your friend says that a bicycle is a linear time invariant system. You decide that what your friend says is correct. What are your reasons that a bicycle is a linear time invariant system? (Simple answer to explain what the input and output are, why linear, and why time invariant)

b.
(5%) Suppose you decide that what your friend says is wrong. What are your reasons that a bicycle is not a linear time invariant system? (Simple answer)


2. (10%) Random Variable Transformation
Prove that Y =FX()

X has a uniform distribution, where X is a continuous random
variable (e.g. a Gaussian random variable with mean and variance 2 ) and FX is the cumulative distribution of X.
1 3
a.
(5%) Find the eigenvalues and eigenvectors.

b.
(5%) What are the eigenvalues and eigenvectors of A-1.

c.
(5%) Compute A100=AA A (i.e. 100 of matrix A multiplied together).

d.
(5%) What are the eigenvalues and eigenvectors of A100.


. ..
2
.
3. (20%) Suppose A =
..
.

2
d .b
1

.

.

ab
.

.

Hint: If
B

=

, then B.1
=

.

..

..

..

..

.

caad .bc
c d
4. (20%) Linear and circular convolution Suppose x=[1 2 3 4] and h=[1 1 1].
a.
(5%) Compute the linear convolution of x and h.

b.
(5%) Compute the circular convolution of x and h.

c.
(10%) Our lecture notes say that we can use circular convolution to compute linear convolution. Show how you do that for x and h.


5.
(20%) Lloyd-Max quantizer Suppose you have a sample of 10 numbers {0, 5, 5, 10, 15, 25, 30, 35, 85, 95}. You know that the dynamic range of the numbers are from 0 to 100. Using the sample distribution, you decide to find the Lloyd-Max quantizer with 3 cells. Suppose you start with the initial cell boundaries {0, 33, 66, 100}. Find the Lloyd-Max quantizer (with 3 cells). Show all the steps. (Hint: It should converge in less than 5 iterations.)

6.
(20%) Image transform Given 2-dimensional image transform, V =AUAT , where both U and V are 2x2 images,


... ..
1 11
a.
(5%) Show that A is a Hermitian matrix.

b.
(5%) Find the four basis images. Show your steps.


11
.
HVA*
and
. And
A
UA
=
=
.


.
2
..
aa
11 12
c. (10%) If A is Hermitian and
A
=
, what are the four basis images?
....
aa
21 22
End of the test.