IMPERIAL COLLEGE
LONDON
[E303/ISE3.3]
DEPARTMENT of ELECTRICAL and ELECTRONIC ENGINEERING
EXAMINATIONS 2004
EEE/ISE PART III/IV: M.Eng., B.Eng. and ACGI
COMMUNICATION SYSTEMS
There are FOUR questions (Q1 to Q4)
Answer question ONE (in separate booklet) and TWO other questions.
Question 1 has 20 multiple choice questions numbered 1 to 20, all carrying equal marks. There is only
one correct answer per question.
Distribution of marks
Question-1: 40 marks
Question-2: 30 marks
Question-3: 30 marks
Question-4: 30 marks
The following are provided:
ì A table of Fourier Transforms
ì A “Gaussian Tail Function" graph
Examiner: Dr A. Manikas
2nd Marker: Dr P. L. Dragotti
Information for candidates:
The following are provided on pages 2 and 3:
-
a table of Fourier Transforms;
a graph of the 'Gaussian Tail Function'.
Question 1 is in a separate coloured booklet
which should be handed in at the end of the
examination.
You should answer Question 1 on the separate
sheet provided. At the end of the exam, please
tie this sheet securely into your main answer
book(s).
Special instructions for invigilators:
Please ensure that the three items mentioned
below are available on each desk.
-
the main examination paper;
the coloured booklet containing Question 1;
the separate answer sheet for Question 1.
Please remind candidates at the end of the exam
that they should tie their Answer Sheet for
Question 1 securely into their main answer
book, together with supplementary answer
books etc.
Please tell candidates they must NOT remove
the coloured booklet containing Question 1.
Collect this booklet in at the end of the exam,
along with the standard answer books.
Communication Systems
Page 1 of 7
The graph below shows the Tail function TeBfwhich represents the area from B to _ of the
Gaussian probability density function N(0,1), i.e.
Tail Function Graph
TeBf œ '
B
_
"
È#1 .expŠ

C#
# ‹.C
pB
TeBf
TeBf
pB
Note that if B  'Þ& then TeBf may be approximated by TeBf ¸
"
È#1 Þ B Þexpš

B#
# ›
FOURIER TRANSFORMS
DESCRIPTION
1
Definition
FUNCTION
gÐtÑ
TRANSFORM
GÐfÑ ='
_
gÐtÑ.ej#1ft dt
-
TABLES
DESCRIPTION
FUNCTION
14
Rectangular function
rect{t} ´ œ
_
TRANSFORM
"
#
"
!
if ± t ± 
9>2/<A3=/
sin 1f
sincÐfÑ= 1f
2
Scaling
gÐ Tt Ñ
|T| . GÐfTÑ
15
Sinc function
sincÐtÑ
3
Time shift
gÐt  TÑ
GÐfÑ . ej#1fT
16
Unit step function
uÐtÑ =œ
4
Frequency shift
gÐtÑ . e j#1Ft
GÐf  FÑ
17
Signum function
5
Complex conjugate
g* ÐtÑ
G* Ð  fÑ
18
Temporal derivative
Decaying exponential
Ðtwo-sidedÑ
e|t|
6
d8
dt8
#
"+Ð#1fÑ#
7
Spectral derivative
Ð  j21tÑ8 .gÐtÑ
d8
df 8
19
Decaying exponential
Ðone-sidedÑ
e|t| .uÐtÑ
"j#1f
"+Ð#1fÑ#
8
Reciprocity
GÐtÑ
gÐ  fÑ
20
Gaussian function
e1t
9
Linearity
A . gÐtÑ  B . hÐtÑ
A . GÐfÑ  B . HÐfÑ
21
Lambda function
10
Multiplication
gÐtÑ . hÐtÑ
GÐfÑ * HÐfÑ
A{t} ´ œ
22
Repeated function
11
Convolution
gÐtÑ * hÐtÑ
GÐfÑ . HÐfÑ
12
Delta function
$ ÐtÑ
"
23
Sampled function
13
Constant
"
$ ÐfÑ
. gÐtÑ
8
Ðj#1fÑ . GÐfÑ
. GÐfÑ
rectÐfÑ
 "ß
!ß
sgnÐtÑ = œ
>!
>!
 "ß
 "ß
>!
>!
#
"
# $ ÐfÑ

j
1f
e1 0
1t
1t
0ŸtŸ1
if
if  " Ÿ t Ÿ 0
repT egÐtÑf = gÐtÑ * repT e$ ÐtÑf
combT egÐtÑf = gÐtÑ.repT e$ ÐtÑf
j
 #1f
#
sinc# ÐfÑ
| T" |.comb "T eGÐfÑf
| T" |.rep "T eGÐfÑ f
The Questions
1.
This question is bound separately and has 20 multiple choice questions
numbered 1 to 20, all carrying equal marks .
You should answer Question 1 on the separate sheet provided.
Circle the answers you think are correct .
There is only one correct answer per question.
There are no negative marks.
Communication Systems
Page 4 of 7
2.
A signal 1Ð>Ñ having the probability density function (pdf) shown below is
bandlimited to ) kHz.
pdf:
¼
-4
0
4
g (Volts)
The signal is sampled at the Nyquist rate and is fed through a 4-level uniform
quantizer.
a) Calculate the end points b3 and the quantizer levels m3 of the quantizer.
[5]
b) Calculate the average signal to quantization noise power ratio (SNRq ).
[6]
c) Calculate the average information per quantization level.
[3]
d) Design a prefix source encoder to encode the output levels from
the quantizer.
[7]

e) Find the average codeword length per symbolß i.e. l ß at the output of the
source encoder.
[3]
f) Calculate the information rate and data rate associated with above single-level
source encoding approach.
Communication Systems
[6]
Page 5 of 7
3.
A discrete channel is modelled as follows:
Ô Šm1 ,0.25‹ ×
Ù
ŠŒ,p‹=Ö
Õ Šm2 ,0.75‹ Ø
Estimate:
a) The probability of error at the output of the channel.
[6]
b) The conditional entropy H(‘ ± Œ), where
Š‘,q‹=Šr1 ,Pr(r1 )‹,Šr2 ,Pr(r2 )‹Ÿ
denotes the ensemble at the channel output.
c) The amount of information delivered at the output of the channel.
Communication Systems
[12]
[12]
Page 6 of 7
4.
1
2
3
For the convolutional encoder shown in the above figure find
a) the code rate and the constraint length
[5]
b) the generator polynomials
[6]
c) the Generator Matrix †-
[9]
d) the encoded output sequence for the input sequence "!""!"!!ÞÞÞ
where the first (oldest) input bit is on the left
[10]
[END]
Communication Systems
Page 7 of 7