MIMO Channel Capacity

8/4/2019 MIMO Channel Capacity

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1

MIMO Channel Capacity• MIMO System Model



2

{ }

{ }

)(

powerdtransmittetotal:

VariableGaussian

ddistributeidenticalytindependen:

,0xE

vectortransmit,1:

i

xx

H xx

i

t

Rtr P

P

x x E R

x

n X

=

=

=×



3

• 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

==

=∑



4

• 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



5

• 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

=

≠=



6

• 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



7

• 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 =



8



9



10

• 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

=

=

=

==

=

′′

′′

′′

′′

′′

′′



11

• 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

+=

+=

=

+=

=

=

=

∑

∑



12

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



13

)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

+=

+=+

−−

=−−

−=−

=

=

=



14

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



15

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



16

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



17

Example 2: A MIMO Channel with UnityChannel Matrix Entries

T Rij n jnih ,......2,1,......1,1 , ===



18

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

+==

⋅+=

=−+=+=

⎢

⎣

⎡−+==

∑

∑∑=

+



19

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

+=

+=

⎥⎦

⎤⎢⎣

⎡+=

+=

=

=



20

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

∑

∑

=

=

+=

=

⎥⎦

⎤⎢⎣

⎡+=

=

=



21

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

+=

⎭⎬⎫

⎩⎨⎧

+=

+=

==



22

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



23

• 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

+==

⋅+=

=−+=+=

⎥⎦⎤⎢⎣

⎡ −+==

∑

∑∑ =

+



24

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



25

[ ]

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.



26

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



27

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



28

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



29

⎭⎬⎫

⎩⎨⎧

⎥⎦⎤⎢⎣⎡ +=

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



30



31



32

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

+=

=

⎥⎦

⎤

⎢⎣

⎡+=

=

→∞

∑=



33

• 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.



34

• For coordinated transmissions, when alltransmitted signals are the same and synchronous.

⎥⎦⎤⎢⎣

⎡ += )1(log 2222 T n

P W E C χ σ



35

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.



36

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



37

• 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.



38

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.



39



40



41

• 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 >



42

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



43

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

+=

==

+===

+=

==



44

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



45

• 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



46



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48



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• In MIMO channels with a large number of antennas,

the capacity grows linearly with the number of antennas.



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MIMO Channel Capacity