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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