i. previously on iet
DESCRIPTION
I. Previously on IET. QPSK Modulation over AWGN Channels. E S Energy per Symbol. Symbol Error given transmitted : Noise on Cosine axis < or Noise on Sine axisTRANSCRIPT
I. Previously on IET
© Tallal Elshabrawy
QPSK Modulation over AWGN Channels
ES Energy per Symbol S S S S S S S SS E 2, E 2 , E 2, E 2 , E 2, E 2 , E 2, E 2
( )2E,2E SS S SE 2, E 2
S SE 2, E 2 S SE 2, E 2
S SI S Q Se E 2, E 2
P Pr n E 2 Pr n E 2
2
2
S S
S
n
2σe E 2, E 2 2
E 2
1P 2 e dn
2πσ
( ) ( )2S2E,2Ee
σ2EQ×2=PSS
( ) ( )2S2E,2Ee
σ4Eerfc=PSS
Symbol Error given transmitted :Noise on Cosine axis <or Noise on Sine axis <
SE 2( )2E,2E SS
SE 2
© Tallal Elshabrawy
BER of QPSK over AWGN Channels
Notes: Define N0 Total Noise Power
N0/2 Noise Power over Cosine axis, i.e., σ2=N0/2
N0/2 Noise Power over Sine axis, i.e., σ2=N0/2
Each symbol MOST LIKELY corresponds to a single bit (Gray Coding) Eb = ES/2
Pb ≈ Pe/2
S S S S S S S S
e e E 2, E 2 e E 2, E 2 e E 2, E 2 e E 2, E 2
2e S
1 1 1 1P P P P P
4 4 4 4
P erfc E 4σ
( )0bb NEerfc21
=P
1101
00 10
Gray Coding: Neighbor constellations points vary in only one bit
© Tallal Elshabrawy
Union Bound for QPSK BER Performance
ES Energy per Symbol
( )2E,2E SS S SE 2, E 2
S SE 2, E 2 S SE 2, E 2
12
4 3
Given transmitted :
( )2E,2E SS
© Tallal Elshabrawy
Union Bound for QPSK BER Performance
ES Energy per Symbol
( )2E,2E SS S SE 2, E 2
12
Given transmitted :
( )2E,2E SS
S12 E2=d
d12 : Euclidian distance between constellation points 1 & 2P12 : Probability received signal closer to constellation point 2 than constellation point 1 given constellation point 1 transmitted
[ ]2d<nPr=P 1212 -
2
2
12
n
2σ12 2
d 2
1P e dn
2πσ
( )σ2dQ=P 1212
( )σ22derfc21
=P 1212
2N=σ 02
( ) ( )0S01212 N2E2erfc21
=N2derfc21
=P
( )0S12 N2E2erfc21
=P
( )0S12 N2Eerfc21
=P
2z
2
x
1Q x e dz
2π
1Q x erfc x 2
2
© Tallal Elshabrawy
Union Bound for QPSK BER Performance
ES Energy per Symbol
( )2E,2E SS
1
Given transmitted :
( )2E,2E SS
S13 E2=d
d13 : Euclidian distance between constellation points 1 & 3P13 : Probability received signal closer to constellation point 3 than constellation point 1 given constellation point 1 transmitted
13 13P Pr n d 2
2
2
13
n
2σ13 2
d 2
1P e dn
2πσ
( )σ2dQ=P 1313
( )σ22derfc21
=P 1313
2N=σ 02
( ) ( )0S01313 N2E2erfc21
=N2derfc21
=P
( )0S13 N2E2erfc21
=P
( )0S13 N2Eerfc21
=P
S SE 2, E 2
3
2z
2
x
1Q x e dz
2π
1Q x erfc x 2
2
© Tallal Elshabrawy
Union Bound for QPSK BER Performance
ES Energy per Symbol
( )2E,2E SS
1
Given transmitted :
( )2E,2E SS
S14 E2=d
d14 : Euclidian distance between constellation points 1 & 4P14 : Probability received signal closer to constellation point 4 than constellation point 1 given constellation point 1 transmitted
14 14P Pr n d 2
2
2
14
n
2σ14 2
d 2
1P e dn
2πσ
( )σ2dQ=P 1414
( )σ22derfc21
=P 1414
2N=σ 02
( ) ( )0S01414 N2E2erfc21
=N2derfc21
=P
( )0S14 NEerfc21
=P
( )0S14 NEerfc21
=P
S SE 2, E 2
4
2z
2
x
1Q x e dz
2π
1Q x erfc x 2
2
© Tallal Elshabrawy
Union Bound for QPSK BER Performance
ES Energy per Symbol
( )2E,2E SS S SE 2, E 2
S SE 2, E 2 S SE 2, E 2
( ) 1413122E,2EeP+P+P<P
SS
Given transmitted :
( )2E,2E SS
12
4 3
S S
4
1k 0e E 2, E 2k 2
1P erfc d 2 N
2
( )ikkmin dmin=d
1k 0 min 0erfc d 2 N erfc d 2 N
S S
4
min 0e E 2, E 2k 2
1P erfc d 2 N
2
S S
min 0e E 2, E 2
3P erfc d 2 N
2
( ) ( )0min0mine N2derfc23
<N2derfc23
×4×41
<P
© Tallal Elshabrawy
Union Bound for BER Performance
M
ike ik 1k i
P P
M
min 0e ik 1k i
1P erfc d 2 N
2
M
e e ii 1
1P P
M
M M
e min 0i 1 k 1
k i
1P erfc d 2 N
2M
M
e min 0i 1
1P M 1 erfc d 2 N
2M
e min 0
1P M 1 erfc d 2 N
2
© Tallal Elshabrawy
Union Bound for 8 PSK Modulation BER
e min 0
1P M 1 erfc d 2 N
2
( )2E,2E SS S SE 2, E 2
S SE 2, E 2 S SE 2, E 2
000
ES Energy per Symbol
001
011
010
110
111
101
001
( )0,ES SE ,0
S0, E
( )SE,0
dmin
Θ
πθ
M
min Sd 2 E sin π M
e S 0
7P erfc sin π 8 E N
2
bS E3=E
b b 0
7P erfc sin π 8 3E N
6
3P
=P eb
M=8
© Tallal Elshabrawy
Union Bound for 16 QAM Modulation BER
e min 0
1P M 1 erfc d 2 N
2
10E3 S
ES Energy per Symbol
0000
min Sd 2 E 10
e S 0
15P erfc E 10N
2
S bE 4E
b b 0
15P erfc 4E 10N
8
eb
PP
4
M=16
000100110010
0100010101110110
1100110111111110
1000100110111010
10ESSE 10S3 E 10
10E3 S
10ES
SE 10
S3 E 10
dmin
© Tallal Elshabrawy
Exact vs Union Bound BER Performance
12
0 1 2 3 4 5 6 7 8 9 1010
-6
10-5
10-4
10-3
10-2
10-1
100
Eb/N
0
Pb
Theoretical QPSKUnion Bound QPSKUnion Bound 8 PSKTheoretical 8 PSK