ENEE 408E First Examination 2009. Solutions
(1) The system can be "unfolded" and then can be represented by the product of three
matrices
1
A B
=
0
1
C D
0
A B
=
2 d
C D
2 d
1
Put in values
2 d
f
2 d
1
f
f
1
f
d := 0.5
2 d
1
f
f
1
f
1
2 d
1
0
1
1
f
1
1 2
2 Signals and Systems: Part I
Solutions to
Recommended Problems
S2.1
(a) We need to use the relations w = 21rf, where f is frequency in hertz, and
T = 2w/w, where T is the fundamental period. Thus, T = 1/f.
(i)
f==
(ii)
f =
(iii)
f =
1
1
w /3
=Hz, T21
I
Introduction
Solutions to
Recommended Problems
S1.1
(a) Using Euler's formula,
e
cos
,r4
r
S
2
\IE
F
1r
+ j sin 4
iej"/4
Since z =
'2 +
2
2
Re
Recfw_z=
2
(b) Similarly,
Imcfw_z= Im
4
(c) The magnitude of z is the product of the magnitudes of
I =
6.003: Signals and Systems
Lecture 5
6.003: Signals and Systems
September 22, 2011
Concept Map: DiscreteTime Systems
Multiple representations of DT systems.
Z Transform
Delay R
Block Diagram
X
+
+
Delay
System Functional
Y
Delay
1
Y
= H(R) =
X
1 R R2
Uni
6.003: Signals and Systems
Lecture 4
September 20, 2011
6.003: Signals and Systems
Multiple Representations of DiscreteTime Systems
ContinuousTime Systems
DiscreteTime (DT) systems can be represented in dierent ways
to more easily address dierent types
6.003: Signals and Systems
Lecture 3
6.003: Signals and Systems
September 15, 2011
Homework
Doing the homework is essential to understanding the content.
Feedback, Poles, and Fundamental Modes
Weekly Homework Assigments
tutor (examtype) problems:
answers
6.003: Signals and Systems
Lecture 2
6.003: Signals and Systems
September 13, 2011
Homework
Doing the homework is essential to understanding the content.
DiscreteTime Systems
Weekly Homework Assigments
tutor (examtype) problems:
answers are automaticall
6.003: Signals and Systems
Lecture 1
September 8, 2011
6.003: Signals and Systems
6.003: Signals and Systems
Signals and Systems
Todays handouts: Single package containing
Slides for Lecture 1
Subject Information & Calendar
Lecturer: Denny Freeman (free
6.003 (Fall 2011)
November 16, 2011
Quiz #3
Name:
Kerberos Username:
Please circle your section number:
Section
Time
2
3
4
11 am
1 pm
2 pm
Grades will be determined by the correctness of your answers (explanations
are not required).
Partial credit will be
6.003 (Fall 2011)
November 16, 2011
Quiz #3
Name:
Kerberos Username:
Please circle your section number:
Section
Time
2
3
4
11 am
1 pm
2 pm
Grades will be determined by the correctness of your answers (explanations
are not required).
Partial credit will be
6.003 (Fall 2011)
October 26, 2011
Quiz #2
Name:
Kerberos Username:
Please circle your section number:
Section
Time
2
3
4
11 am
1 pm
2 pm
Grades will be determined by the correctness of your answers (explanations
are not required).
Partial credit will be
6.003 (Fall 2011)
October 26, 2011
Quiz #2
Name:
Kerberos Username:
Please circle your section number:
Section
Time
2
3
4
11 am
1 pm
2 pm
Grades will be determined by the correctness of your answers (explanations
are not required).
Partial credit will be
4 Convolution
Solutions to
Recommended Problems
S4.1
The given input in Figure S4.11 can be expressed as linear combinations of xi[n],
x 2[n], X3 [n].
x,[ n]
0 2
Figure S4.11
(a) x 4[n] = 2x 1 [n]  2x 2 [n] + x3[n]
(b) Using superposition, y 4[n] =
6 Systems Represented by
Differential and Difference
Equations
Solutions to
Recommended Problems
S6.1
We substitute ya(t) = ay1(t) + Oy2(t) into the homogeneous differential equation
dya(t)
dt
dt
d
+ ay3(t) d [ay 1(t) + #y2(t)] + a[ayi(t) + #y 2(t)]
=
7 ContinuousTime Fourier Series
Solutions to
Recommended Problems
S7.1
(a) For the LTI system indicated in Figure S7.1, the output y(t) is expressed as
h(r)x(t  r) dr,
y(t) =f
where h(t) is the impulse response and x(t) is the input.
LTI
x(t)
ON h(t)
y
ENEE 408E Fall 2009 Final Examination Solutions
This was a final taken at home because the University was closed by snow
(1) Ray transfer matrix for Galilean telescope
1
A B
C D
=
0
1
1
f2
C D
=
d
1
1
f1
0 1
d
1
A B
0
1
1 d
d
f1
1
f2
f1
1
1
f2
d
f2
For i
ENEE 408E Optical System Design 2012
First Examination
Tuesday, October 30, 2012. 9:30 10:45

ANSWER 3 QUESTIONS
(if more than 3 are answered best 3 will count)
(1) Prove that the ray transfer matrix for a slab of material of length d and refractive
inde
ENEE 408E 2009. Second Examination Solutions
(1)
If the reflected light is pure Spolarized, this must mean that all the Ppolarized light is
transmitted thorough the boundary, so the boundary must be at Brewster's angle.
For the Pwaves incident at angle
ENEE 408E Optical System Design, Fall 2012
Second Examination
Tuesday, December 4, 2012 9:30  10:45 pm
ANSWER 3 QUESTIONS
(if more than 3 are answered best 3 will count)
(1) A Gaussian beam is a lot like a plane wave except that instead of just having a
ENEE 408E Examination 1, 2014. Solutions
Ray transfer matrix of unfolded system is
(1)
1 0 1 0 1 d
A B 1 d
=
1 1 1 1
0 1
C D 0 1
f
f
A B
C D
1 2 d d d 2 d 1
f
f
=
2
2 d
1
f
f
3
d := 500 10
3
f := 200 10
1 2 d d d 2 d 1
f
f
4 1.5
note tha
ENEE 408E Fall 2010. Solutions to first examination questions
(1)
Ray transfer matrix for thin lens
1 0
A B
=
C D
1
1
f
Since at a thin lens the ray height does not change the first row of the matrix must 1 0.
The C element is defined as 1/f. The D eleme
ENEE 408E 2011 Final Examination Solutions
(1)
The focal length of the convex mirror is
fM :=
0.4
m
2
Unfold the system. The ray transfer matrix from input back to input is
1 0 1 d 1 0 1 d 1 0
1
A B
1
= 1 1
1 0 1
1
C D
0 1
f
f
fM
d
d d
f 1
ENEE 408E Fall 2014. Solutions to Examination 3
(1)
Ray transfer matrix from input to output plane is
1 0
1 0
A B 1 1 d 1
=
C D f 1 0 1 f 1
2
1
d
1
d
f1
A B
=
C D d 1 1 1 d
f f
f2
1 2 f1 f2
For an input ray parallel to the axis the output a
ENEE 408E Solutions to Examination 2 Fall 2014
(1) The two lenses together by the mirrors can be replaced by a single lens of focal length f/2
in the unfolded system, since
1 0 1 0 1 0
1 1 = 2
1
1
1
f
f
f
The unfolded system is just a series of i
ENEE 408E 2010. Solutions to second examination
(1) Unfold system to show unit cell. The two lenses together are equivalent to a single
lens of focal length f/2. The concave mirror is also equivalent to a lens of focal length
f/2.The ray transfer matrix f
ENEE 408E First Examination 2011. Solutions
(1) For derivation of the ray transfer of a spherical interface see the book or notes on line.
0
1
A B
= 1n 1
C D
n R n
The problem can be unfolded as
0
0
1
1
A B =
1 2 d 1 n 1
n1 n
C D R 1 0 1 n R n
ENEE 408E Second Examination 2011 Solutions
(1)
Find ray transfer matrix from source to point P
1 0 1 0.5
A B := 1 0.2
1 1
0 1
C D 0 1
0.4
A B 0.5 0.45
=
C D 2.5 0.25
R in := 0
R out :=
A R in + B
R out = 1.8
C R in + D
Check entrance pupi
6.003 (Fall 2011)
October 5, 2011
Quiz #1
Name:
Kerberos Username:
Please circle your section number:
Section
Time
2
3
4
11 am
1 pm
2 pm
Grades will be determined by the correctness of your answers (explanations
are not required).
Partial credit will be g
6.003 (Fall 2011)
October 5, 2011
Quiz #1
Name:
Kerberos Username:
Please circle your section number:
Section
Time
2
3
4
11 am
1 pm
2 pm
Grades will be determined by the correctness of your answers (explanations
are not required).
Partial credit will be g