chapter 3 ofdm transmission over gaussian channel_modify newage
DESCRIPTION
The Signal Constellations of Different Modulation overan AWGN ChannelTRANSCRIPT
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Chapter 3OFDM Transmission over Gaussian
Channel
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Outline3 OFDM Transmission over Gaussian Channel
3.1 Gaussian Distribution3.2 The AWGN Channel Model3.3 OFDM System Performance over AWGN Channel3.4 The Signal Constellations of Different Modulation over
an AWGN Channel
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3.1 Gaussian DistributionThe PDF of a Gaussian or normally distributed random variable is
( ) ( )2
2
122
xX
x mp x e
σπσ− −
=
The PDF of a Gaussian-distributed random variable
( )Xp x
x
σπ21
1
xm0
1/2
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3.1 Gaussian DistributionThe CDF of a Gaussian or normally distributed random variable is
where
( )
( ) ( )
2
2
2 exp( )
21 exp( )
x
x
erf x z dz
erfc x erf x z dz
−∞
∞
= −
= − = −
∫
∫
π
π
( ) 1 1 112 2 22 2
x xX
x m x mF x erf erfcσ σ
− −⎛ ⎞ ⎛ ⎞= + = −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
xm0
21
( )XF x
xThe CDF of a Gaussian-distributed random variable
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3.2 The AWGN Channel ModelThe received signal in the interval may beexpressed as
where denotes the sample function of an additive white Gaussian noise (AWGN) process.
0 t T≤ ≤
( ) ( ) ( ) , 0mr t s t n t t T= + ≤ ≤
2/2
( )n t
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3.2 The AWGN Channel ModelThe channel is assumed to corrupt the signal by the addition of white Gaussian noise as shown in Figure 3.1.
Figure 3.1 Transmission model for received signal passed through an AWGN channel
ms (t)+
AWGN
Received Signal Transmitted SignalChannel
)(tn
mr(t) s (t) n(t)= +
2/2
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3.3 OFDM System Performance over AWGN Channel
Serial Data Output
OFDM Receiver
Channel Model
OFDM Transmitter
AWGN
S(t)Guard Interval Insertion
Parallel-to-Serial
Converter IFFT
Signal Mapper
Serial-to-Parallel
Converter
Random Data
Generator
Serial-to-Parallel
Converter FFT
Signal Dema-pper
Parallel-to-Serial
Converter
Guard Interval
Removal
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3.3 OFDM System Performance over AWGN Channel
Modulation of OFDM subcarrier is analogous to the modulation in conventional serial systems.The modulation schemes of the subcarriers are generally QAM or PSK in conjunction with both coherent and non-coherent detection.As the additive white Gaussian noise (AWGN) in the time domain channel corresponds to AWGN of the same average power in the frequency domain, an OFDM system performance in an AWGN channel is identical to that of a serial system.Analogously to a serial system, the bit error rate (BER) verses signal-to-noise rate (SNR) characteristics are determined by the modulation scheme used. It can be seen from the figures that the experimental BER performance of the OFDM system is in very good accordance with the theoretical BER curves of conventional serial systems in AWGN channels.
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3.3 OFDM System Performance over AWGN Channel
AWGNChannel
BPSK, QPSK,8PSK, 16PSK
Modulation
0 - 30 dBSNR
Cyclic PrefixGuard Type
1024Subcarrier #
1024FFT size
ValueSimulation parameter
BER versus SNR curves for the OFDM system using BPSK, QPSK, 8PSK,16-PSK under an AWGN channel
0 5 10 15 20 25 3010
-6
10-5
10-4
10-3
10-2
10-1
100
BER vs. SNR
SNR
BE
R
BPSK theoretical result BPSK simulation QPSK theoretical result QPSK simulation 8PSK approximate result 8PSK simulation 16PSK approximate result 16PSK simulation
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3.3 OFDM System Performance over AWGN Channel
PSKBPSK
QPSK with Gray code
M-ary PSK
where
( ),12e B P S Kp er fc= γ
( ),12e QPSKp erfc= γ
( ) 22 exp( )x
erfc x z dzπ
∞= −∫
sE−
2m
3m
4m
5m
6m
7m
8m
2φ
sE−
sE
d
d
MπMπ 1m 1φ
sE
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛=
MNEerfcp s
MPSKeπsin
0,
0
bE SNRN
γ =
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3.3 OFDM System Performance over AWGN Channel
BER versus SNR curves for the OFDM system using BPSK/QPSK, 16QAM, 64QAM, 256QAM under an AWGN channel
AWGNChannel
BPSK, QPSK, 16QAM, 64QAM, 256QAM
Modulation
0 - 30 dBSNR
Cyclic PrefixGuard Type
1024Subcarrier #
1024FFT size
ValueSimulation parameter
0 1 2 3 4 5 6 7 8 9 10 111213 141516 171819 202122 232425 262728 2930
10-5
10-4
10-3
10-2
10-1
100
BER vs.Eb/N0
Eb/N0
BE
R
BPSK/QPSK theorem BPSK/QPSK simulation16QAM theorem 16 QAM simulation 64 QAM simulation 64 QAM theorem 256 QAM simulation 256 QAM theorem
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3.3 OFDM System Performance over AWGN Channel
QAM
( )3
2 1sEa
M=
−
( )2
2
0
2| 1 ap c QN
⎡ ⎤⎛ ⎞Ι = −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
( )2 2
0 0
2 2| 1 2 1a ap c Q QN N
⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎢ ⎥ ⎢ ⎥ΙΙ = − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦
( )2
2
0
2| 1 2 ap c QN
⎡ ⎤⎛ ⎞ΙΙΙ = −⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
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( ) ( ) ( ) ( ) ( )2
,2
1 11 4 | 4 2 | 2 |loge M QAMp p c M p c M p c
M M−⎧ ⎫⎡ ⎤= − ⋅ Ι + − ΙΙ + − ΙΙΙ⎨ ⎬⎢ ⎥⎣ ⎦⎩ ⎭
3aa- a- 3a
a
3a
- a
- 3a
na
nb
: I part
: II part
: III part
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3.4 The Signal Constellation of Different Modulation over AWGN Channel
Signal-space diagram for 16-QAMSignal-space diagram for 8-PSK
sE−
2m
3m
4m
5m
6m
7m
8m
Decision boundary
2φ
message point
sE−
sE
d
d
MπMπ 1m
Decision region
1φsE
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3.4 The Signal Constellation of Different Modulation over AWGN Channel
(a) BPSK, SNR=10, (b) BPSK, SNR=20
-2 -1 0 1 2-2
-1
0
1
2BPSK signal constellation with SNR=10
Real part
Imag
e pa
rt(a) (b)
-2 -1 0 1 2-2
-1
0
1
2BPSK signal constellation with SNR=20
Real part
Imag
e pa
rt
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3.4 The Signal Constellation of Different Modulation over AWGN Channel
(c) QPSK, SNR=10, (d) QPSK, SNR=20;
(c) (d)
-2 -1 0 1 2-2
-1
0
1
2QPSK signal constellation with SNR=10
Real part
Imag
e pa
rt
-2 -1 0 1 2-2
-1
0
1
2QPSK signal constellation with SNR=20
Real part
Imag
e pa
rt
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3.4 The Signal Constellation of Different Modulation over AWGN Channel
(e) 8PSK, SNR=10, (f) 8PSK, SNR=20;
(e) (f)
-2 -1 0 1 2-2
-1
0
1
28PSK signal constellation with SNR=10
Real part
Imag
e pa
rt
-2 -1 0 1 2-2
-1
0
1
28PSK signal constellation with SNR=20
Real partIm
age
part
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3.4 The Signal Constellation of Different Modulation over AWGN Channel
(g) 16PSK, SNR=10, (h) 16PSK, SNR=20;
(g) (h)
-2 -1 0 1 2-2
-1
0
1
216PSK signal constellation with SNR=10
Real part
Imag
e pa
rt
-2 -1 0 1 2-2
-1
0
1
216PSK signal constellation with SNR=20
Real partIm
age
part
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3.4 The Signal Constellation of Different Modulation over AWGN Channel
(i) 16QAM, SNR=10, (j) 16QAM, SNR=20
(i) (j)
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
416QAM signal constellation with SNR=10
Real part
Imag
e pa
rt
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
416QAM signal constellation with SNR=20
Real part
Imag
e pa
rt
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References[1] Richard van Nee and Ramjee Prasad, OFDM wireless multimedia communication, Artech House, Boston London, 2000. [2] Ahmad R. S. Bahai and Burton R. Saltzberg, Multi-carrier digital communications - Theory and applications of OFDM, Kluwer Academic / Plenum Publishers ,New York, Boston, Dordrecht, London, Moscow 1999.[3] L. Hanzo, W. Webb and T. Keller, Single- and multi-carrier quadrature amplitude modulation – Principles and applications for personal communications, WLANs and broadcasting, John Wiley & Sons, Ltd, 2000.[4] Zou, W.Y. and Yiyan Wu, “COFDM: An overview,” Broadcasting, IEEE Transactions on, vol. 41, Issue 1, pp. 1 –8, Mar. 1995.[5] Simon Haykin, Communication Systems, John Wiley & Sons, Inc., 3rd edition, 1994.[6] Roger L. Peterson, Rodger E. Ziemer, David E. Borth, Introduction to spread spectrum communications, Prentice Hall International Editions, 1995.