2011 1st semester
MIMO C
Communication
i ti Systems
S t
#5: MIMO Channel Capacity
Kei Sakaguchi
<sakaguchi@mobile ee titech ac jp>
<[email protected]>
May 17, 2011
Schedule (1st half)
Date
Text
Contents
#1
Apr. 12
A-1, B-1
Introduction
#2
Apr 19
Apr.
B5 B
B-5,
B-6
6
Fundamentals of wireless commun.
commun
#3
Apr. 26
B-12
OFDM for wireless broadband
May 3
No class
#4
May 10
B-7
B
7
#5
Nov. 17
A-3, B-10 MIMO channel capacity
#6
Nov. 24
B-2, 3
y 28
May
May 17, 2011
Array signal processing
Spatial channel model
No class
MIMO Commun. Systems (MIMO Channel Capacity)
2
Agenda
 Aim of today
Derive throughput performance of
y
MIMO communication system
 Contents
• SISO, SIMO/MISO channel capacity
• MIMO channel capacity
– Transmit & receive diversity
– Spatial multiplexing, SVD-MIMO
– Water filling power allocation
• Measurement on MIMO channel capacity
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
3
Warming Up
 Question
Calculate Singular Value Decomposition (SVD) of matrix H
z
1
H
0
12 
3 2
x
h1
y
h2
60
 Singular Value Decomposition (SVD)
H  U Σ V H  1 u1 v1H  2 u 2 v 2H    m u m v mH

Σ  diag 1
Singular values:
Singular matrices:
2 
Rt  H H  V Σ U U Σ V  V Λ V
May 17, 2011
H

m  rank H 
VHV  I
UH U  I
R r  HH H  U Σ V H V Σ U H  U Λ U H
H
m
H
H
Λ  diagg1
MIMO Commun. Systems (MIMO Channel Capacity)
2  m 
4
SISO
S
SO System
Sys e
Received signal
y (t )  hs (t )  n(t )
Output
p SNR
2

E[ hs (t ) ]
2
s
2

E[[ n(t ) ]
h
y
h P
2
PDF of output SNR in Rayleigh fading
 
f ( )  exp 

 
1



Channel capacity
A
Average
channel
h
l capacity
it
2


h
P
CSISO    log 2 1  2 

 

May 17, 2011
CSISO     f  CSISO  d
MIMO Commun. Systems (MIMO Channel Capacity)
5
SISO Channel Capacity
Capacity vs.
vs SNR
channel capacityy [bit/s/H
Hz]
10
8
6
4
2
0
0
May 17, 2011
5
10
15
20
SNR [dB]
25
30
MIMO Commun. Systems (MIMO Channel Capacity)
6
SIMO System
y
wr
Received signal vector
h1
y (t )  hs (t )  n(t )
s
Output
p SNR
  max
wr
2
E[ w hs ]
H
r
2
E[ w rH n ]
Mr
  hi
i 1

h2

P
2
y
hM r
2
PDF of OSNR in independent Rayleigh fading
f ( ) 
 
1
M r 1
 

exp
Mr
( M  1)!
 
Channel capacity
A
Average
channel
h
l capacity
it
 Mr 2 P 

CSIMO    log 2 1   hi
2 
 
 i 1
May 17, 2011



CSIMO     f  CSIMO  d
MIMO Commun. Systems (MIMO Channel Capacity)
7
MISO
SO System
Sys e
wt
Received signal
y (t )  hT w t s (t )  n(t )
Output SNR
h1
2
wt
E[ h
s ]
wt
s
T
  max
wt
2
E[ n ]
Mt
  hi
i1
2
P
h2
y
hM t
2
PDF of OSNR in independent Rayleigh fading
 
1
M t 1

exp 
f ( ) 
Mt
( M  1)!
 
Channel capacity
A
Average
channel
h
l capacity
it
 Mt 2 P 

CMISO    log 2 1   hi
2 
 
 i 1
May 17, 2011



CMISO     f  CMISO  d
MIMO Commun. Systems (MIMO Channel Capacity)
8
Chan
nnel caapacityy [bit/
/s/Hz]]
SIMO/MISO Channel Cpacity
SNR vs. capacity
p
y
14
12
SIMO/MISO 1x4/4x1
SIMO/MISO 1x2/2x1
SISO
10
8
6
4
2
0
0
5
10
15
20
25
30
SNR [dB]
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
9
Channnel caapacityy [bit/
/s/Hz]
SIMO/MISO
S
O/ SO Channel
C a e Capacity
Capac y
May 17, 2011
Antenna size vs. capacity
14
12
10
8
6
4
SNR=30[dB]
SNR=20[dB]
[ ]
SNR=10[dB]
2
0
1
2
3
4
5
6
7
8
Number of antennas
MIMO Commun. Systems (MIMO Channel Capacity)
10
MIMO Communication System
– Multi-Input & Multi-Output (MIMO) at the same channel
– Utilization of rank m  min M t , M r  effective channels
(ex. SISO → 1, SIMO → 1)
– Benefits of MIMO = increase of throughput & area coverage
MIMO channel
MIMO transmitter
MIMO receiver
#1
Beamforming
#1
Channel estimation
Spatial multiplexing
Beamforming
Space-time coding
Interference
cancellation
May 17, 2011
# Mｔ
# Mr
MIMO Commun. Systems (MIMO Channel Capacity)
11
Signal Model for MIMO System
Received signal vector
Transmit signal vector
y (t )  Hs(t )  n(t )
MIMO channel matrix
Noise vector
 y1   h11  h1M t   s1   n1 
   

 
       
    
 y M  hM 1  hM M   sM  nM 
r
t 
t 
 r  r
 r
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
12
MIMO
O Diversity
es y
Received signal vector
wt
wr
H
y (t )  Hw t s (t )  n(t )
Output SNR
y
s

2
  max
E[ w rH Hw t s ]
w r ,w t
2
u Hv P
H

 u
2
2
E[ w n ]
H
r
2
P
 2

2
max u Hv
H 
Channel capacity
H
 P 
CMD    log 2 1  2 
  
May 17, 2011
Maximum singular value
u,v
MIMO Commun. Systems (MIMO Channel Capacity)
13
Spatial
p
Multiplexing
p
g
Received signal vector
Mt
y (t )   h i
i 1
si (t )
 n(t )
Mt
Interference
f
cancellation
w ri  H \i
Output SNR
i 
2
si (t )
H
E[ w ri h i
]
Mt
2
E[ w n ]
H
ri
Channel capacity

H \ i  h1 ,  , h i 1 , h i 1  , h M t

H \i H \Hi  EE H

Mt
E  e1 ,  , e M t
i 1
H \i  e M t
CMD     log
g 2 1   i 
May 17, 2011
Null subspace interference cancellation

MIMO Commun. Systems (MIMO Channel Capacity)
14
SVD MIMO
SVD-MIMO
Singular value decomposition
1  max u Hv1
H
1
u1 , v1
H
y
2
Wt
~
H  1 u v  H
H
1 1
Wt
H  U Σ V H  1 u1 v1H  2 u 2 v 2H    m u m v mH
Eigenmodes

  diag 1 , , m

m  min M t , M r 
SVD MIMO
SVD-MIMO
y (t )  WrH HWt s(t )  WrH n(t )
~ (t )
if
 Σs(t )  n
Wt  V
May 17, 2011
Wr  U
MIMO Commun. Systems (MIMO Channel Capacity)
15
SVD MIMO
SVD-MIMO
Singular value decomposition
H  UV H

Tx eigen-beam
mforming
x1
y1
x2
y2
x3
y3
x4
y4
channel
estimation
H
V
feedback channel
May 17, 2011
Rx eigenn-beamforming

S/P
P


CMIMO     log 2 1  2 i 
  m 
i 1
m
  diag 1 , , m
data
MIMO channel capacity
P/S
data
UH
SVD
MIMO Commun. Systems (MIMO Channel Capacity)
16
PDF o
of S
SVD-MIMO
O
PDF of singular
g
values ((Wishart distribution))
1
f (1 ,  , m ) 
K m ,n
K m,n 
 m (m 1)
m (n)m (m)
m
m
i 1
i j
nm
2
exp(
p(


)

(



)

 i j
i
i
n  maxM t , M r 
m  min M t , M r 
Marginal probability
f (1 )     f (1 ,  , m )d2  dm
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
17
PDF o
of S
SVD-MIMO
O (Example)
( a p e)
2x2 MIMO
Mr  2
Mt  2
Joint distribution (Wishart distribution)
f (1 , 2 )  e  1 2 1  2 
2
Marginal probability

f (1 )  2e 2 1  e  1 2  21  12

E1   3.5
E2   0.5
f (2 )  2e 2 2
Cumulative probability
~4
~
~
~
1
~ ~

 1 ~ 2
 2 1
f 1   f (1 )d1  1  e 1  2  e
 1
 
~ ~
f    
0
2
May 17, 2011
~
2
0
f (2 )d2  1  e

~
 2 2

12

~
~2
 2 2  2 2  
MIMO Commun. Systems (MIMO Channel Capacity)
18
CDF of SVD
SVD-MIMO
MIMO
0
Output SNR of SVD-MIMO
cumulaative disstributioon
10
-1
10
-2
10
-20
May 17, 2011
SISO
MIMO ch.
h 4
MIMO ch. 3
MIMO ch. 2
MIMO ch.
h 1
MIMO total
-10
0
10
normalized SNR [dB]
MIMO Commun. Systems (MIMO Channel Capacity)
20
19
MIMO Channel Capacity
Contributions of different eigenmodes
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
20
Chan
nnel caapacityy [bit/ss/Hz]
MIMO Channel Capacity
May 17, 2011
Antenna size vs.
vs capacity
70
SNR=10[dB]
SNR=20[dB]
SNR=30[dB]
60
50
40
30
20
10
0
1
2
3
4
5
6
7
8
MIMO array size
MIMO Commun. Systems (MIMO Channel Capacity)
21
Channnel caapacityy [bit//s/Hz]]
MIMO Channel Capacity
May 17, 2011
SNR vs. capacity
35
30
25
MIMO 4x4
MIMO 2x2
SISO
20
15
10
5
0
0
5
10
15
20
25
30
SNR [dB]
MIMO Commun. Systems (MIMO Channel Capacity)
22
Transmit
a s
Power
o e Op
Optimization
a o
Cost function
Water filling
g
 P 
CMISO     log 2 1  i 2i 
 

i 1
m
m
subject to
P   Pi
i 1
Method of Lagrange multipliers
m

 Pi i  
J   log 2 1  2     P   Pi 
  

i 1
i 1

J
2
log 2 e

0
Pi 

i

Pi
m
O ti l MIMO channel
Optimal
h
l capacity
it

 
Pi     
i 

2
May 17, 2011


 i 
  log 2  2 
  
i 1 
m
CWF

MIMO Commun. Systems (MIMO Channel Capacity)
23
Ergo
odic channel cappacity [bbits/s/H
Hz]
Optimal
p
MIMO Channel Capacity
p
y
35
30
25
MIMO 4x4
MIMO 4x4 wf
MIMO 4x2
MIMO 4x2 wf
MIMO 2x2
MIMO 2x2 wf
SISO
SISO wf
20
15
10
5
0
0
5
10
15
20
25
30
Average SNR per antenna [dB]
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
24
Measurement System
y
ULA
4-ch
AWG
4-ch
transmitter
vATT
ULA
4-ch
receiver
8-ch
DSO
GPIB
position
controller
PC
May 17, 2011
x-y positioner
MIMO Commun. Systems (MIMO Channel Capacity)
25
Photos of Measurement System
y
Rubidium
8-ch DSO
2-ch AWG x2
4-element ULA
Transmitter
a s
e x4
May 17, 2011
PC
C
Receiver
ece ve x 4
MIMO Commun. Systems (MIMO Channel Capacity)
26
Process of Measurement
Transmit frame upload
Position control
Frame data acquisition
y
Channel estimation
Ĥ
sŝ
C h
Coherent
td
detection
t ti
Error count
END?
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
27
Measurement Setup
Center freq.
Array type
Tx. power
Bandwidth
Modulation
Frame
Meas points
Meas.
May 17, 2011
5.2 [GHz]
0 5λ spacing 4
0.5λ
4-element
element sleeve array
0 [dBm/channel] → SNR = 25 [dB]
187.5 [kHz] → 125 [ksps] α=0.5
BPSK,, QPSK,
Q
, 16QAM
Q
512 (31：training、481：data）
256 points (2×2[cm2] step in 30×30 [cm2]））
MIMO Commun. Systems (MIMO Channel Capacity)
28
Measurement Environment
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
29
Transmit Array Antenna (AP)
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
30
Receive Array Antenna (UT)
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
31
RF Transceiver
RF IN
BB OUT
RF OUT
BB IN
Donation from Samsung Yokohama Institute of Technology
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
32
Received Signal of 2x2 MIMO
with QPSK Signaling
Received signal
May 17, 2011
Interference cancellation
MIMO Commun. Systems (MIMO Channel Capacity)
33
Distribution of Channel Capacity
SISO ＠ SNR=10[dB]
May 17, 2011
MIMO ＠ SNR=10[dB]
MIMO Commun. Systems (MIMO Channel Capacity)
34
CDF of Channel Capacity
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
35
Dependency on MIMO Array Size
Channel capacity increases linearly w.r.t. array size in real environment
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
36
Summary
• MIMO channel capacity
– Transmit & receive diversity to enhance reliability
– Multi-stream spatial multiplexing to enhance spectral efficiency
– SVD-MIMO
SVD MIMO & water
t filling
filli power control
t l achieves
hi
b
bestt performance
f
MIMO channel capacity increases linearly with respect to
the number of antenna elements in IID fading
g environment
How about in realistic propagation environment?
Double directional spatial channel model
May 17, 2011
MIMO Commun. Systems (MIMO Channel Capacity)
37