GEL4200/7014
Examen partiel
Hiver 2017
Wednesday 1 March 2017; Duration: 13:30 to 15:20
Two pages of documentation provided; a calculator is permitted.
Problem 1 (15 points out of 100)
Consider the following (non-square) 16QAM constellation shown in signal space.
5
2 ES
27
3
2 ES
27
2 ES
27
−5
2 ES
27
−3
2 ES
27
−
2 ES
27
2 ES
27
−
2 ES
27
−3
2 ES
27
−5
2 ES
27
3
2 ES
27
5
2 ES
27
A. (5 points)
Give the minimal distance.
B. (8 points)
Give the probability of error using the approximation from the
union bound.
Page 1
GEL4200/7014
Examen partiel
Hiver 2017
Problem 2 (10 points out of 100)
Consider the “Bandwidth Efficiency Plane”. Find the coordinates for 16QAM
(square) et 8FSK (coherent) for a probability of error of 10-6.
Problem 3 (20 points out of 100)
A. (10 points)
Describe the phenomenon known as ISI, intersymbol
interference. Describe circumstances where ISI might be
present. How does a Nyquist pulse combat ISI?
B. (10 points)
What is a priori probability in a communications system? How
can knowledge of the a priori probability be used to improve
system performance?
Problem 4 (30 points out of 100)
For the following set of four signals
1
s1(t)
T/3
1
T
s2(t)
T/3
1
T
s3(t)
T/3
1
T
s4(t)
T/3
A. (20 points)
Find a set of orthonormal basis vectors.
B. (10 points)
Give the signal constellation coordinates in this basis for the
signal space. The coordinates should be in terms of the
average energy per bit, Eb.
Page 2
T
GEL4200/7014
Examen partiel
Hiver 2017
Problem 5 (25 points out of 100)
Identify a communications system from the table that is band limited. Justify your
choice, that is, explain the qualities of a band limited system. Suggest an
appropriate modulation format.
Identify a communications system from the table that is power limited. Justify
your choice, that is, explain the qualities of a power limited system. Suggest an
appropriate modulation format.
Bandwidth
Bit rate
Power
source
3 kHz
10 kb/s
grid
Car door
opener
20 MHz
10 b/s
battery
Microwave
link
30 MHz
100 Mb/s
grid
good; distance chosen for good
signal strength and line of sight
Mars lander
4 GHz
20 kb/s
solar
very weak, even with very high
gain antennas
Cellular data
100 kHz
1 Mb/s
battery
Cable
internet
10 MHz
50 Mb/s
grid
Twisted pair
Page 3
Signal strength
~30 dB
varies greatly with distance;
requires line of sight
varies greatly with distance
~25 dB
GEL10280/64486
10
probabilité d'erreur
10
10
10
10
10
Examen partiel
Hiver 2017
QPSK coherent
0
-2
-4
-6
-8
-10
0
1
2
3
4
5
6
7
8
E /N
b
9
10
11
12
13
14
15
0
Plan de l’efficacité spectrale (Bandwidth Efficiency Plane)
10
QAM, M=16
QAM, M=4
PSK, M=32
PSK, M=8
BPSK
1
R W ( b s Hz )
QAM, M=64
FSK, M=4
FSK, M=16
.1
.1
.01
.01
points indique Pe =10 -5
FSK, M=1024
FSK, M=4096
.001
0
5
Limite de Shannon
10
15
EB N 0
Page 4
20
25
GEL10280/64486
Examen partiel
Récepteur d’échantillonnage
n ( t )
si ( t )
Σ
MAP: i qui maximise p(z|si) p(si)
2
i qui minimise r − si − N 0 ln P ( si )
P ( si ) = probabilité a priori de symbole si
t=T
r (t )
filtre
h(t)
décision
z (t )
Raised cosine v ( t ) =
z (T )
z (T ) > γ
<
m̂
ML:
sin (π t Ts ) cos ( rπ t Ts )
π t Ts 1 − 4r 2t 2 Ts 2
( a , a ) =
Q
n
M ⋅ Es
( a I ) 2 + ( a Q ) 2 
∑
n
 n

i =1 
M
coordonnées,
espace du signal
(a ,a )
I
n
i qui maximise p(z|si)
2
i qui minimise r − si
Énergie moyenne
1 M
2
si
Emoy =
∑
M i =1
1 M
=
∑ [énergie du signal i ]
M i =1
Énergie par bit v. énergie par symbole
Eb log 2 M = Es
Conversion de l’espace I/Q vers espace du signal
I
n
Hiver 2017
Q
n
coordonnées,
espace I/Q
QAM cas rectangulaire (carrée) M=L2
6 log 2 L
1   3log 2 M Eb 

=
Pe 2 1 −
 d min =
 Q 

L2 − 1
M   ( M − 1) N 0 

Borne d’union
Eb 
2 K  Dmin  2 K 
Pe ≈
Q
Q  d min
=

M  2 N 0  M 
N 0 
K est le nombre des paires des signaux
séparés par la distance minimale Dmin
Distance minimale dans l’espace du signal
D
Dmin min si − s k et d min = min
=
i≠k
2 Eb
 2 Eb 
Pe ( BPSK ) = Q 

 N0 
 Eb 
Pe ( OOK ) = Q 

 N0 
 2 Eb 
Pe ( QPSK ) ≈ 2Q 

 N0 
Pour une modulation orthogonale
M 2
Pe ( bit=
) P=b Pe ( symbol )
M −1
Pour une modulation nonorthogonale avec codage de
gray
P ( symbol )
Pe ( bit=
) P=b e
log 2 M
Efficacité spectrale
Perte par rapport à QPSK
d min = x 2 perte = −10 log10 x
=
η
Page 5
Rb 1 1
bits/s
=
W Tb W
GEL10280/64486
Examen partiel
η = log 2 M
MPSK cohérent
†
Hiver 2017
 2 Es
 2 Eb log 2 M
π 
π 
Pe ( M ) ≈ 2Q 
sin  =
2Q 
sin 
M
N0
M
 N0

DPSK incohérent
2 log 2 M
M +1
 E 
 E log M
Pe =
( M − 1) Q  s  =
( M − 1) Q  b 2
N0
 N0 

Séparation minimale 1/2Ts
Processus Gram Schmidt
T
1
s1 ( t ) où E1 ∫ s12 ( t ) dt
0
E1
ψ 1 (t ) =
1
Pe = e − Eb
2
~1 dB de perte entre DPSK et BPSK
θ 2 ( t )  s2 ( t ) − s2 ( t ) ,ψ 1 ( t ) ψ 1 ( t )
N0
T
E2 ∫ θ
0
Relations trigonométriques
i.
cos 2θ = 2 cos θ − 1 = 1 − sin θ
2
ψ 2 (t ) =
θ2 (t )
E2
θ=
si ( t ) − ∑ si ( t ) ,ψ k ( t ) ψ k ( t )
i (t )
T
tan θ = sin θ cos θ
Loi de Shannon
=
C W log 2 (1 + SNR )
( t ) dt
2
k =1
cos (α ± β ) =
cos α cos β  sin α sin β
2
2
i −1
sin (α=
± β ) sin α cos β ± cos α sin β
Eb W C W
=
( 2 − 1)
N0 C
η=
MFSK cohérent
E R
SNR = b b
N0 W
Eb
C
→0 ⇒
→ −1.6dB
W
N0
Ei ∫ θ
0
2
( t ) dt
i
ψ i (t ) =
θi ( t )
Ei
η=
MFSK incohérent
log 2 M
M
L’effet d’ISI
1
Pe ( BFSK ) = e − Eb
2
2 N0
~1 dB de perte entre BFSK cohérente et
incohérente
Séparation minimale 1/Ts
† en supposant une impulsion Nyquist idéale
Page 6
†
†



GEL10280/64486
Examen partiel
Page 7
Hiver 2017