Chapter 3 OFDM Transmission Over Gaussian Channel_modify Newage

CCU Wireless Access Tech. Lab.

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

2/2

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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.