The document discusses the capacity regions and duality of multiple input multiple output (MIMO) Gaussian channels, focusing on transmit precoding and receiver shaping. It introduces concepts like multiplexing gain, MIMO channel capacity, and the duality between multiple access and broadcast channels. Various mathematical formulations for optimal power allocation and capacity calculation are presented.

1. Mul$user
MIMO
Gaussian
Channels:
Capacity
Regions
and
Duality
Shris$
Pradhan
University
of
Bri.sh
Columbia
December
2011

2. Mul$ple
Input
Mul$ple
Output
(MIMO)
y = Hx + n
MIMO
Model
Transmit
Precoding
and
Receiver
Shaping
Singular
Value
Decomposi$on
(SVD):
H =UΣV H
y = Σx + n
Mul$plexing
gain
=
Rank
of
H
=
RH
Power
Noise
1 2 3
Subchannel
Water
Level
MIMO
Channel
Capacity
C = max
p(x)
I(X;Y) = max
p(x)
[H(Y)− H(Y | X)]
C = max
Pi:ΣiPi≤P
log2 1+
Piγi
P
#
$
%
&
'
(
i
∑ γ = SNR
Pi
P
=
1
γ0
−
1
γi
0
γi ≥γo
γi <γ0
#
$
%%
&
%
%
C = log
γi
γ0
!
"
#
$
%
&
i:γi≥γ0
∑

3. Mul$user
MIMO
MAC
and
BC
Broadcast
(Downlink)
Channel
Capacity
0 1 2 3 4 5 6 7 8 9
0
1
2
3
4
5
6
7
8
9
R1 [bps/Hz]
R2[bps/Hz]
BC
Mul$ple
Access
(Uplink)
Channel
Capacity
0 1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
R1 (bps)
R2(bps)
Capacity Region of MAC
Duality
of
MAC
and
BC
CBC (P;g1,g2 ) = CMAC (P1,P − P1;g1,g2 )
0≤P1≤P

CMAC (P1,P2;g1,g2 ) = CBC
P1
α
+ P2;αg1,g2
!
"
#
$
%
&
α>0

R1 = log I + H1
H
Q1H1
R2 = log I + H1
H
Q1H1 + H2
H
Q2H2 − R1
CDPC (P, H) = Co R(π,Σi )
π,Σi

"
#
$$
%
&
''