文档介绍：Supelec
Random Matrix Theory
for
munications
Merouane´ Debbah
merouane.******@
February, 2008
Presentation
The role of the Cauchy-Stieltjes Transform munications
1
General Multiple Input Multiple Output Model
Model representing multiple-antennas, CDMA, OFDM, ad-works with
cooperation,...
y = W s + n
Received signal MIMO matrix emitted signal AWGN
2
N × 1 N × K K × 1 ∼ N (0, σ IN )
Let
· ¸
s
W = u U , s = 1
x
The goal is to detect s.
2
Communications Notations Used
Receiving vector
• r as ”Received”.
• y as ”Y do we care?”.
Transmitting vector
• s as ”Signal”.
• x as ”Xciting”.
Noise vector
• n as ”Noise”(english).
• b as ”Bruit”(french).
• z as ”Zzzzzzz...”(disturbance)
MIMO matrix
• W when the matrix is non-isometric.
•Θ when the matrix is isometric
• H when we consider multiple-antennas.
3
Shannon Capacity
Mutual information M between input and output:
M(s; (y, W)) = M(s; W) + M(s; y | W)
= M(s; y | W)
= H(y | W) − H(y | s, W)
= H(y | W) − H(n)
The differential entropy of plex Gaussian vector x with covariance Q is given by
log2 det(πeQ).
4
Shannon Capacity
In the case of Gaussian independent entries, since
H 2 H
E(yy ) = σ IN + WQW
H 2
E(nn ) = σ IN
The mutual information per dimension is:
1
CN = (H(y | W) − H(n))
N
³ ´
1 2 H 2
= log det(πe(σ IN + WQW )) − log det(πeσ IN )
N 2 2
µ ¶
1 1 H
= log det(IN + WQW )
N 2 σ2
5
Shannon Capacity
Consider the random variable
µ ¶ XN µ ³ ´¶
1 1 H 1 1 H
CN = log det IN + WW = log 1 + λk WW
N σ2 N σ2
k=1
When N →∞ and K/N →α,
Z µ ¶
1
CN → log 1 + t µ(dt) .
σ2
dCN 2 4 2
→σ−σ Gµ(−σ) .
d 1
σ2
The capacity is strongly related to the Cauchy-Stieltjes transform.
6
Some numerical facts
³ ´
W . zero mean with variance 1 : C = 1 log det I + 1 WWH
N N N N σ2
Mean for variable matrix size at 10dB
Simulations
Theoretical formula
b/s/Hz