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MIMO Channel Capacity• MIMO System Model
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{ }
{ }
)(
powerdtransmittetotal:
VariableGaussian
ddistributeidenticalytindependen:
,0xE
vectortransmit,1:
i
xx
H xx
i
t
Rtr P
P
x x E R
x
n X
=
=
=×
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• Assume the signals transmitted from individual antenna
elements have equal powers of
• Assume that the channel is narrowband and memoryless
and denoted by an complex matrix,
• Assume that the received power for each element is
equal to the total transmitted power
T n P
T n
T
xx I n
P R =
T R nn × H
R
T
niT
n
j
ij nh ,......2,1,
2
1
==
=∑
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• Assume that the channel matrix is known to thereceiver, but not always at the transmitter.
{ }
)(powerreceivedtotalThe
SNRaverage:
antennareceivingeachatpoweraverage:
vectorreceiving,1:
2
22
2
rr
n
H
xxrr
r
r
R
n H
nn
Rtr
I H R H R
n x H r
P
σ
P
P
nr
I nn E R
R
R
=
+=
+=
==
×
==
σ
σ γ
γ
σ
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• MIMO System Capacity Derivation
• The diagonal entries of are the non-negative square
roots of the eigenvalues of matrix
matrixorthognalandnegativenon:
H
−×
==
=
T R
n
H
n
H
H
nn D
I V V
I U U
V DU
T
R
H
H H
H d d
y
y y y H H
iiii
H
of valuesingular:,
reigenvecto:
0,
λ
λ
=
≠=
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• The columns of are the eigenvectors of
• The columns of are the eigenvectors of
U H
H H
V H H H
n X Dr
nU n
X V X
r U r
n X V DU r
H
H
H
H
′+′=′
=′
=′
=′+=
Define
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• For the matrix , the rank is at most
• Let r: the number of nonzero eigenvalues of
T R nn × H
H
Rii
iiii
nr r inr
r in xr
,......,2,1
,......,1
++=′=′
=′+′=′ λ
),min( T R nnm =
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• The equivalent MIMO Channel can be considered as
r uncoupled parallel subchannels
• The channel power gain is equal to the eigenvalue of H
)()(
)()(
)()(
nnnn
xx x x
rr r r
nn
H
nn
xx
H
x x
rr
H
r r
Rtr Rtr
Rtr Rtr
Rtr Rtr
U RU R
V RV R
U RU R
=
=
=
==
=
′′
′′
′′
′′
′′
′′
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• In the equivalent MIMO channel model, the sub-channels are uncoupled and their capacities add up.
)1(log
)1(log
)1(log
21
2
21
2
21
2
σ
λ π
σ
λ
λ
σ
T
ir
i
T
ir
i
T
iri
rir
i
n
P W
n
P W C
n
P P
P
W C
+=
+=
=
+=
=
=
=
∑
∑
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0)(
)det()(
0)(det
00)(
),min(
1=−
−=
=−
≥
<=
≠=−
=
=i
m
i
m
m
T R
H
T R
H
m
T R
Q I P
Q I
nn H H
nn H H Q
y yQ I Define
nnm
λ λ π
λ λ
λ
λ
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)det(log
)det()1(
)det()(
)det()(
22
221
22
1
1
Qn
P I W C
Qn
P I n
P
Q I P n
P n
Q I
T
m
T
m
T
i
r
i
mT i
T
r
i
mi
r
i
σ
σ σ
λ π
σ λ σ π
λ λ λ π
+=
+=+
−−
=−−
−=−
=
=
=
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MIMO Channel Capacity for Adaptive TransmitPower Allocation
• When the channel parameters are known at the
transmitter, the power to various antennas can bedistributed by the “water-filling ” rule.
The power allocated to channel is given byi
r i P i
i ,......,2,1,)(2
=−= +
λ
σ µ
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H
n xx
i
ri
iri
r
i
i
ii
V P P P diag V R
W
P W C
P
P P
T ),......,(
)(1
1log
)1(log
)(
bydeterminedis
)0,max()(
21
2
22
22
2
1
22
=
⎥⎦
⎤
⎢⎣
⎡−+=
+=
−=
=
−=−
+
+
=
+
∑
∑
∑
σ µ λ
σ
σ
σ µ λ
µ
λ
σ
µ λ
σ
µ
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MIMO capacity examples for channel with fixedcoefficients
Example 1. Single Antenna Channel
Hz S bitsW
C dB
P
wC
h H nn RT
/ / 658.6)1001(log,20SNRif
)1(log
11
2
22
=+==
+=
====
σ
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Example 2: A MIMO Channel with UnityChannel Matrix Entries
T Rij n jnih ,......2,1,......1,1 , ===
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Case 1. Coherent combinining
)1(log,1If
)1(log
,,
)(1
1logFrom,
22
2
2
2
2
222
1
2
22
2
σ
σ
λ σ λ σ λ λ
λ
σ µ
σ λ
σ
λ
P nW C h
P hW C
P u P u P
uW C h
T j
j
r
i
i j
+==
⋅+=
=−+=+=
⎢
⎣
⎡−+==
∑
∑∑=
+
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Example 3. A MIMO Channel with Orthogonal
Transmission
Assume nnn T R ==
)1(log
)1(log
)1(detlog
)det(log
22
22
22
22
σ
σ
σ
σ
P nW
P W
P diag W
I n
nP
I W C
I n H H
I n H
n
nn
n
H
n
+=
+=
⎥⎦
⎤⎢⎣
⎡+=
+=
=
=
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Example 4: Receive diversity
)1(log
)det(log
),(
,1
2
2
1
2
2
1
22
,......21
σ
σ
P hW C
h H H
H H n
P I W C
hhh H
n
R
R
T
R
n
i
i
n
i
i
H
H
T
n
T
n
T
∑
∑
=
=
+=
=
⎥⎦
⎤⎢⎣
⎡+=
=
=
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This corresponds to linear maximum ratio combining.
If
{ })max1(log
)1(logmaxC
diversityselectionFor
)1(log
,......,11
2
22
2
22
22
2
i
i
R
Ri
h P
W
h P
W
P nW C
nih
σ
σ
σ
+=
⎭⎬⎫
⎩⎨⎧
+=
+=
==
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Example 1.5 Transmit Diversity
T j
T
j
n
j
j
H
n
R
n jh P W
n
P hW C
h H H
hhh H
n
R
R
,......,11if )1(log
)1(log
),......,,(
1
222
2
2
2
2
1
21
==+=
+=
=
=
=
∑
∑=
σ
σ
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• The above applies to the case when the transmitter
does not know the channel
• For coordinated transmissions,
)1(log,1If
)1(log
,,
)(11logFrom,
22
2
2
2
2
222
1
222
2
σ
σ
λ σ λ σ λ λ λ
σ
µ
σ λ σ
λ
P nW C h
P hW C
P u P u P
uW C h
T j
j
r
i
i j
+==
⋅+=
=−+=+=
⎥⎦⎤⎢⎣
⎡ −+==
∑
∑∑ =
+
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Capacity of MIMO Systems with Random
Channel Coefficients
Assumptions:1. Channel coefficients are perfectly estimated at the
receiver but unknown at the transmitter.
2. are zero mean Gaussian complex randomvariables.
ijh
.2 / 1of variancewitheachvariables,random..Gaussian
meanzerotindependenareimagandReal.3 ,,
d ii
hh ji ji
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[ ]
21
)(
..Gaussiantindependenare,,
1
ondistributiphaseuniformd,distributeRayleighare.4
22
2
21
2
2
2
1
2
,
,
2
2
==
+=
=
−
r
Z
r
ji
ji
r
e
Z
Z P
vr Z Z Z Z Z
h E
h
σ σ
σ
• The antenna spacing is large enough to ensureuncorrelated channel matrix entries.
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Capacity of MIMO Fast and Block Rayleigh
Fading Channels
• For single antenna link, is chi-squared distributed
with two degrees of freedom, denoted by
2h
2
2 χ
⎥⎦
⎤⎢⎣
⎡+=
=
+==−
)1(log
21)(
),0(:,,
2222
22
2
21
2
2
1
1
2
2
2
σ χ
σ
σ χ
σ
P W E C
e y P
N Z Z Z Z y
r
y
r
r
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T R
H T R
H
T
r
T
i
r
i
RT
nn H H
nn H H Q
Qn
P I W E C
n
P W E C
nn H r
≥
<=
⎭⎬
⎫
⎩⎨
⎧
⎥⎦
⎤
⎢⎣
⎡+=
⎥
⎦
⎤⎢
⎣
⎡+=
≤=
∑=
)(detlog
)1(log
),(min)(rank Let
22
2
1
2
σ
σ
λ
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Example: A fast and block fading channel with
receive diversity
∑
∑
=
=
==
=
⎥
⎦
⎤⎢
⎣
⎡+=
=
R
R
R
R
R
R
n
i
in
n
i
in
n
T
n
Z y
h
P
hhh H
2
1
22
2
1
22
2
2
222
21
)1(WlogEC
combiningratiomaximumFor
),......,,(
χ
χ
χ
σ
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⎭⎬⎫
⎩⎨⎧
⎥⎦⎤⎢⎣⎡ +=
Γ
Γ=
−−
)max(1log
combiningselectiveFor
functiongamma:)(
)(2
1
)(
),0(...areWhere
2
22
21
2
2
2
i
y
n
R
nn
r
r i
h P W E C
e yn y P
N d ii Z
r R
R R
σ
σ
σ
σ
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Example: A fast and block fading channel with
transmit diversity
)1(loglim
)1(log
ontransmissiteduncoordinaFor
),......,,(
22
1
22
2
2
222
21
σ
χ
χ σ
P W C
h
n
P W E C
hhh H
T
T
T
T
T
n
n
j
jn
nT
n
+=
=
⎥⎦
⎤
⎢⎣
⎡+=
=
→∞
∑=
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• The system behaves as if the total power is
transmitted over a single unfaded channel.• The transmit diversity is able to remove the effect
of fading for a large number of antennas.
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• For coordinated transmissions, when alltransmitted signals are the same and synchronous.
⎥⎦⎤⎢⎣
⎡ += )1(log 2222 T n
P W E C χ σ
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Example: A MIMO fast and block fadingchannel with transmit-receive diversity
• Assume m=n=nR=nT
• Assume channel parameters are known at the
receiver but not at the transmitter.
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1log
4
11)(log
1lim
log)1log(
411tanh2
2
141
1log
4
11)1(log
1lim
22
4
0 22
2
1
2
2
22
4
0 22
−=
−≥
≥+
++
−+
+−=
−+=
∫
∫
∞→
−
∞→
σ
σ π
σ σ
σ σ
σ π
P
dvv
v P
Wn
C
x x
P P
P
P
dvv
v P
Wn
C
n
n
Q
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• There is a multiplexing gain of n, as there are
n-independent sub-channel, which can be identifiedby their coefficients, perfectly estimated at the
receiver.
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Example: A MIMO fast and block fading channel
with transmit-receive diversity and adaptive
transmitter power allocation
• The average capacity for an ergodic channelcan be obtained by averaging over all
realizations of the cannel coefficients.
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• With adaptive power allocation, the power is
allocated according to water-filling principle, without
adaptive power allocation, the powers from all
antennas are all the same.
• When , there is almost no gain in adaptive
power allocation.
• If , there is a significant potential gain with
power distribution.
• The benefit obtained by adaptive power distribution
is higher for a lower SNR and diminishes at highSNR.
RT nn ≤
RT nn >
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Capacity of MIMO Slow Rayleigh Fading Channels
• is chosen randomly, according to a Rayleighdistribution, at the beginning and held constant for atransmission block, e.g. WLAN.
• Assume that the channel is perfectly estimated at thereceiver and unknown at the transmitter.
• In this system, the capacity is a random variable.
• The complementary cumulative distribution function(ccdf) defines the probability that a specifiedcapacity level is provided, denoted by Pc .
• The outage capacity probability, Pout , specifies theprobability of not achieving a certain level of capacity. Pout=1- Pc .
C i E l f MIMO Sl R l i h
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Capacity Examples for MIMO Slow Rayleigh
Fading Channels
)1(log
,1DiversityTransmit:Example
)1(log
,1
DiversityReceive:Example
)1(log
1
link antennaSingle:Example
2
222
2
222
2222
T
R
n
T
T T R
n
R RT
RT
n
P W C
nnn
P W C
nnn
P W C
nn
χ
σ
χ σ
χ σ
+=
==
+===
+=
==
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Example: Combined Transmit-Receive Diversity
• The upper bound
freedom.of degrees2
withvariablerandomsquaredchiais)(
))(1(logWC
boundlowerThe.Assume
2
2
2
2
n
)1(ni
22
T
T
−
+>
≥
∑−−=
i
i
n T
RT
R n
P
nn
χ
χ
σ
freedom.of degrees2n
withvariablerandomsquaredchiais)(
))(1(log
R
2
2
1
2
222
−
+< ∑=
in
n
i
in
T
R
T
R
n
P W C
χ
χ
σ
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• This case corresponds to a system of uncoupled
parallel transmissions, where each of transmit
antennas is received by a separate set of
receive antennas, so that there is no interference.
T n
Rn
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• In MIMO channels with a large number of antennas,
the capacity grows linearly with the number of antennas.