12- OFDM with Multiple Antennas

Multiple Antenna Systems (MIMO)

TXRX

TN RN

Transmit Antennas

Receive Antennas

RT NN Different paths

Two cases:

1. Array Gain: if all paths are strongly correlated to which other the SNR can be increased by array processing;

2. Diversity Gain: if all paths are uncorrelated, the effect of channel fading can be attenuated by diversity combining

Recall the Chi-Square distribution:

2ny

222

21 ... nxxxy

...,)1,0( diirealNxi

1. Real Case. Let

Then

ny

nyE

2}var{

}{

with

222

1ny

222

21 ||...|||| nxxxy

...,)1,0( diigaussiancomplexCNjbax iii

2. Complex Case. Let

Then

ny

nyE

2

1}var{

}{

with

Receive Diversity:

TXRX

1TN RNTransmit Antennas

Receive Antennas

RN

Different paths

s

1y

RNy

RRR N

S

NN w

w

NsE

h

h

y

y

1

0

11

1h

RNh

Noise PSDEnergy per symbol

Assume we know the channels at the receiver. Then we can decode the signal as

RRR N

iii

N

iiS

N

iii whNshEyhy

1

*0

1

2

1

* ||

signal noise

and the Signal to Nose Ratio

01

2||N

EhSNR S

N

ii

R

Now there are two possibilities:

1. Channels strongly correlated. Assume they are all the same for simplicity

hhhhRN ...21

In the Wireless case the channels are random, therefore is a random variable

RN

iih

1

2||

Then

22

2

1

2 |||| RR

N

ii NhNh

R

and

0

22

0

2

2

1||

N

EN

N

EhNSNR S

RS

R

assuming 1|| 2 hE

0N

ENSNREm SRSNR

02

varN

ENSNR SR

SNR

better on average …

… but with deep fades!

From the properties of the Chi-Square distribution:

Define the coefficient of variation

2

1var

SNR

SNR

m

In this case we say that there is no diversity.

2. Channels Completely Uncorrelated.

22

1

2

2

1||

R

R

N

N

iih

01

2||N

EhSNR S

N

ii

R

Since:

0

222

1

N

ESNR S

NR

with

0N

ENSNRE SR

02

varN

ENSNR SR

RSNR

SNR

Nm 2

1var

Diversity of order RN

0 20 40 60 80 100 120 140 160 180 200-25

-20

-15

-10

-5

0

5

10

15

Example: overall receiver gain with receiver diversity.

1RN

2RN

10RN

Transmitter Diversity

TX RX

TN1RN

Transmit Antennas Receive

Antennas

TN

Different paths

s

y

1h

RNh

wNshN

Ey

TN

ii

T

S0

1

Total energy equally distributed on transmit antennas

Equivalent to one channel, with no benefit.

However there is a gain if we use Space Time Coding (2x1 Alamouti)

Take the case of Transmitter diversity with two antennas

TX RX

][ny

1h

2h

][1 nx

][2 nx

Given two sequences

code them within the two antennas as follows

][],[ 21 nsns

n2 12 n

1s

2s

*2s*1s

1x

2x

1022112]2[ wNshsh

Eny S

20*12

*212

]12[ wNshshE

ny S time

antennas

This can be written as:

*

2

10

2

1*1

*2

21

* 2]12[

]2[

w

wN

s

s

hh

hhE

ny

nyS

To decode, notice that

*

1 1 121 20**

2 2 22 1

[2 ]|| || || ||

[2 1] 2S

z s wy nh h Eh N h

z s wy nh h

Use a Wiener Filter to estimate “s”:

]12[]2[ˆ

]12[]2[ˆ*

1*21

*2

*11

nyhnyhKs

nyhnyhKs

S

S

ENhh

EK

/2||||

/2

02

22

1 with

It is like having two independent channels

1z1s

2s

2||||2

hES

2||||2

hES

2z

0 1|| ||N h w

0

2

2

||||

N

EhSNR S

Apart from the factor ½, it has the same SNR as the receive diversity of order 2.

0 2|| ||N h w

24

22

21

2

2

1|||||||| hhh

TX RX

][1 ny11h

21h

][1 nx

][2 nx ][2 ny

12h

22h

2x2 MIMO with Space Time Coding (2x2 Alamouti)

][

][

][

][

][

][

2

1

2

1

2221

1211

2

1

nw

nw

nx

nx

hh

hh

ny

ny

Same transmitting sequence as in the 2x1 case:

]12[2

]12[

]2[2

]2[

]12[2

]12[

]2[2

]2[

20*122

*2212

202221212

10*112

*2111

102121111

nwNshshE

ny

nwNshshE

ny

nwNshshE

ny

nwNshshE

ny

S

S

S

S

Received sequences:

n2 12 n

1s

2s

*2s*1s

1x

2xantennas

time

][2

]12[

]2[

]12[

]2[

02

1

*21

*22

2221

*11

*12

1211

*2

2

*1

1

nwNs

s

hh

hh

hh

hh

E

ny

ny

ny

ny

S

Write it in matrix form:

Combined as

]12[

]2[

]12[

]2[

*2

2

*1

1

21*2211

*12

22*2112

*11

2

1

ny

ny

ny

ny

hhhh

hhhh

z

z

to obtain

][

2 02

1

*21

*22

2221

*11

*12

1211

21*2211

*12

22*2112

*11

2

1 nwNs

s

hh

hh

hh

hh

E

hhhh

hhhh

z

zS

After simple algebra:

][||||2

|||| 02

12

2

1 nwNhs

sEh

z

zS

with

28

2

1,

22

2

1||||||

jiijhh

diversity 4

This yields an SNR

0

2

2

||||

N

EhSNR S

WiMax Implementation

1h

2h

Base Station

Subscriber Station

Down Link (DL): BS -> SS Transmit Diversity

Uplink (UL): SS->BS Receive Diversity

Down Link: Transmit Diversity

Use Alamouti Space Time Coding:

Error Coding

Data inM-QAM buffer

nX

Block to be transmitted

mX 2

12 mX

STC

IFFT TX

IFFT TX

m2 12 m

mX 2

12 mX

*12 mX

*2mX

Space Time Coding

time

Transmitter:

Error Correction

Data outM-QAM

nXmX 2

12 mX

STD FFT

Receiver:

mY2

12 mY

][][][][][ˆ

][][][][][ˆ

*1212

*212

*1222

*12

kYkHkYkHKkX

kYkHkYkHKkX

mmm

mmm

S

S

ENkHkH

EK

/2|][||][|

/2

02

22

1 with

Space Time Decoding:

For each subcarrier k compute:

2S/PP/S

2

Preamble, Synchronization and Channel Estimation with Transmit Diversity (DL)

The two antennas transmit two preambles at the same time, using different sets of subcarriers

128

12864

CP

k

100 1000

128

128 12864

CP 128

EVEN subcarriers

ODD subcarriers

][1 np

][2 np

++

+

-

n0 319

time frequency

Both preambles have a symmetry:

]128[][

]128[][

22

11

npnp

npnp319,...,128n

][0 nh

][1 nh

][ny

][0 np

][1 np

received signal from the two antennas

Problems:

• time synchronization

• estimation of both channels

Symmetry is preserved even after the channel spreading:

128

12864

CP

128

128 12864

CP 128

][*][ 11 npnh

++

+

-

][*][ 22 npnh

One possibility: use symmetry of the preambles

][ny128z

][*][2][ 222 npnhny

][*][2][ 111 npnhny

0n

64 256 1280 n

64 128

1280 n

64 128

The two preambles can be easily separated

MIMO Channel Simulation

Take the general 2x2 channel

3je

4je

T

T

T

T

1je

2je

Rayleigh

Rayleigh

][1 nx

][2 nx

][1 ny

][2 ny

dBPPP N

N

][

sec][

1

1

10 T

10 R Correlation at the receiver

Correlation at the transmitter