WELCOME
Chen Chen
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Simulation of MIMO Capacity Limits
Professor:
Patric Östergård
Supervisor:
Kalle Ruttik
Communications Labortory
2
Agenda
1.
Introduction to Multiple-In Multiple-Out(MIMO)
2.
MIMO Multiple Access Channel(MAC)
3.
Water-filling algorithm(WF)
4.
MIMO Broadcast Channel(BC)
5.
Zero-forcing method(ZF)
6.
Simulation results
7.
Conclusion
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What is MIMO
Y j  H ij X i  n j
Input vector:
X i  [ x1 , x2 ,...xNt ]T
Output vector:
Y j  [ y1 , y2 ,... yNr ]T
Noise vector :
n j  [n1 , n2
nN ]T
r
2
Hij is the channel gain from Txi to Rxj with E | hij |   1
 h1,1 h1, Nt 


H ij     

 hN ,1 hN , N 
r
t 
 r
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MIMO MAC (uplink)
MAC is a channel which two
(or more) senders send
information to a common
receiver
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Water-filling algorithm
 n2  En   ; n2  
En  0; n2  
The optimal strategy is to ‘pour
energy’ (allocate energy on each
channel).
In channels with lower effective
noise level, more energy will be
allocated.
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Iterative water filling algorithm
Initialize Qi = 0, i = 1 …K.
repeat;
for j = 1 to K;
Q 
'
z
k

j 1, j i
H j Q j H Tj  Qz
1
Qi  arg max log | H i QH iT  Qz' |
Q
2
end;
until the desired accuracy is reached
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MIMO MAC capacity
Single-user water filling
K-user Water-filling
K
1
1
1
1
H
max
imize

log
| H i Qi H iH  Qz |  log | Qz |
max imize log | HQH  Qz |  log | Qz |

2
2
2
2
i 1
subjecttotr (Qi )  Pi ,i  1,..., K
subjecttotr (Q )  P,
Q  0.
Qi  0,i  1,..., K
When we apply the water filling
Qi=Q.
1
Qi  arg max log | H i QH iT  Qz' |
Q
2
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MIMO MAC capacity region
The capacity region of the MAC is the closure of the set
of achievable rate pairs (R1, R2).
R1  I ( X 1 ; Y | X 2 ),
R2  I ( X 2 ; Y | X 1 ),
R1  R2  I ( X 1 , X 2 ; Y ).
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MAC sum capacity region (WF)
The sum rate converges to the sum
capacity.
(Q1……. Qk) converges to an optimal
set of input covariance matrices.
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MIMO BC (downlink)
Single transmitter
for all users
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Zero-forcing method
K
x j =  H jM i d i + n j  H jM s d s + n j
i=1
= H j M jd j  H j M ' j d ' j  n j ,
signal
interferecnce
noise
To find out the optimal transmit vector,
such that all multi-user interference is
zero, the optimal solution is to force
HjMj = 0, for i≠ j,
so that user j does not interfere with
any other users.
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BC capacity region for 2 users
The capacity region of a BC depends only on the
Conditional distributions of p( y1 | x)and p( y2 | x).
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BC sum capacity
CBD  max log 2 | I 

2 
n
2
|
 1







 k 
1. Use water filling on the diagonal elements of  to determine
the optimal power loading matrix  under power constraint P.
2. Use water-filling on the diagonal elements of  j to calculate
the power loading matrix  jthat satisfies the power constraint Pj
corresponding to rate Rj. (power control)
3. Let mj be the number of spatial dimensions used to transmit to
user j, The number of sub-channels allocated to each user must be
a constant when K = Nt/ mj ,  m j  Nrj (known sub-channel)
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Examples of simulation results
Ergodic capacity with different correlations (single user)
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Ergodic capacity (single user)
4 different set correlations magnitude coefficient
Ergodic capacity
Tx = Rx = 3
Correlation
(0, 0)
(0, 0.2)
(0.2, 0.95)
(0.95, 0.95)
Max(SNR=20)
16.37
16.26
11.68
8.07
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MIMO MAC sum capacity (2 users)
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MIMO MAC sum capacity (2 users)
3
2
1
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MIMO MAC sum capacity (2 users)
Ergodic capacity
Max(SNR=20)
Tx = Rx = 3
sum capacity
user 1
user2
21.32
16.31
16.34
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MIMO MAC sum capacity (3 users)
Tx = Rx= 5
SNR=20
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MIMO MAC capacity (3 users)
Ergodic capacity
Max(SNR=20)
Tx = Rx = 3
sum capacity
user 1
user2
user3
23.45
16.60
16.19
16.16
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MIMO MAC capacity (WF)(2 users)
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MIMO MAC capacity (WF) (2 users)
Ergodic capacity
Tx = Rx = 3
With water filling
sum capacity
user 1
user2
Max(SNR=20)
21.96
16.36
16.45
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MIMO MAC capacity (WF) (3 users)
Tx= Rx =4
SNR=20
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MIMO MAC capacity (WF) (3 users)
Ergodic capacity
Max
4Tx X 4Rx, SNR=20
sum capacity
user 1
user2
user3
39.60
27.57
27.58
27.78
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BC sum capacity
Tx=4;
Rx=2;
SNR=20;
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BC sum capacity: with Power Control
Tx=4;
Rx=2;
SNR=20;
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BC sum capacity: Coordinated Tx-Rx
Tx=4;
Rx=2;
SNR=20;
mj =2
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BC sum capacity
BC sum capacity
Max(SNR=20)
Tx=4, Rx=2, mj= 2
sum capacity
user 1
user2
27.03
15.67
14.81
28.18
16.68
17.03
31.79
16.61
18.43
With Power Control
Max (SNR=20)
Known sub-channel
Max (SNR=20)
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Conclusion
MIMO capacity:
1. It depends on H, the larger rank and eigen values of H, the
more MIMO capacity will be.
2. If we understood better the knowledge of Tx and Rx, we can
get higher channel capacity. With power control, the capacity
will also be increased.
3. When water-filling is applied: the capacity will be incresaing
significantly.
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Main references
1. T. M. Cover, “Elements if information theory”, 1991.
2. W. Yu, “Iterative water-filling for Gaussian vector multiple
access channels”, 2004.
3. Quentin H.Spencer, “Zero-forcing methods for downlink
spatial multiplexing”, 2004.
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THANK YOU!
Any questions?
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