Wireless CommunicationElec 534

Set IVOctober 23, 2007

Behnaam Aazhang

Reading for Set 4

• Tse and Viswanath– Chapters 7,8– Appendices B.6,B.7

• Goldsmith– Chapters 10

Outline

• Channel model• Basics of multiuser systems• Basics of information theory• Information capacity of single antenna single

user channels– AWGN channels– Ergodic fast fading channels– Slow fading channels

• Outage probability• Outage capacity

Outline

• Communication with additional dimensions– Multiple input multiple output (MIMO)

• Achievable rates• Diversity multiplexing tradeoff• Transmission techniques

– User cooperation• Achievable rates• Transmission techniques

Dimension

• Signals for communication– Time period T– Bandwidth W– 2WT natural real dimensions

• Achievable rate per real dimension

)1log(2

12navP

Communication with Additional Dimensions: An Example

• Adding the Q channel– BPSK to QPSK

• Modulated both real and

imaginary signal dimensions• Double the data rate• Same bit error probability

)2

(0N

EQP b

e

Communication with Additional Dimensions

• Larger signal dimension--larger capacity– Linear relation

• Other degrees of freedom (beyond signaling)– Spatial– Cooperation

• Metric to measure impact on– Rate (multiplexing)– Reliability (diversity)

• Same metric for– Feedback– Opportunistic access

Multiplexing Gain

• Additional dimension used to gain in rate• Unit benchmark: capacity of single link AWGN

• Definition of multiplexing gain

Hertzper secondper bit )1log()( SNRSNRC

)log(

)(lim

SNR

SNRCr

SNR

Diversity Gain

• Dimension used to improve reliability• Unit benchmark: single link Rayleigh fading

channel

• Definition of diversity gain

SNRout

1

)log(

))(log(lim

SNR

SNRd out

SNR

Multiple Antennas

• Improve fading and increase data rate• Additional degrees of freedom

– virtual/physical channels– tradeoff between diversity and multiplexing

Transmitter Receiver

Multiple Antennas

• The model

where Tc is the coherence time

cRcTTRcR TMTMMMTM nbHr

Transmitter Receiver

Basic Assumption

• The additive noise is Gaussian

• The average power constraint

)2

,0(Gaussian~ 0RRR MMM I

Nn

avHMM Pbb

TT]}Trace{E[

Matrices

• A channel matrix

• Trace of a square matrix

**

*1

*11

1

111

1

,

RMTMRM

T

RT

RTT

R

TR

hh

hh

H

hh

hh

HM

HMM

MMM

M

MM

M

iiiMM hH

1

]Trace[

Matrices

• The Frobenius norm

• Rank of a matrix = number of linearly independent rows or column

• Full rank if

][Trace][Trace HHHHH HH

F

},min{][Rank TR MMH

},min{][Rank TR MMH

Matrices

• A square matrix is invertible if there is a matrix

• The determinant—a measure of how noninvertible a matrix is!

• A square invertible matrix U is unitary if

IAA 1

IUU H

Matrices• Vector X is rotated and scaled by a matrix A

• A vector X is called the eigenvector of the matrix and lambda is the eigenvalue if

• Then

with unitary and diagonal matrices

Axy

xAx

HUUA

Matrices• The columns of unitary matrix U are

eigenvectors of A• Determinant is the product of all eigenvalues • The diagonal matrix

N

0

01

Matrices

• If H is a non square matrix then

• Unitary U with columns as the left singular vectors and unitary V matrix with columns as the right singular vectors

• The diagonal matrix

HMMMMMM TTTMRMRRTR

VUH

00

00

or

00

0

0

1

1

R

TTR

M

MMM

Matrices

• The singular values of H are square root of eigenvalues of square H

)(eigenvalue

)singular(

iH

MMMM

MMi

RTTR

TR

HH

H

MIMO Channels

• There are channels– Independent if

• Sufficient separation compared to carrier wavelength• Rich scattering

– At transmitter– At receiver

• The number of singular vectors of the channel

• The singular vectors are the additional (spatial) degrees of freedom

RT MM

},min{ RT MM

Channel State Information

• More critical than SISO– CSI at transmitter and received– CSI at receiver– No CSI

• Forward training

• Feedback or reverse training

Fixed MIMO Channel

• A vector/matrix extension of SISO results• Very large coherence time

)log()]det()log[(

)log()|(

)()|(

),|()|(

)|;(

0*

0

0

eNMHQHINe

eNMHrh

nhHrh

HbrhHrh

HbrI

RMMM

RMMM

MMMM

MMMMMMM

MMMM

RR

R

TRR

RTRR

TRTRTRR

TRTR

Exercise

• Show that if X is a complex random vector with covariance matrix Q its differential entropy is largest if it was Gaussian

Solution

• Consider a vector Y with the covariance as X

0log

loglog

loglog)()(

dYf

ff

dYffdYff

dXffdYffXhYh

Y

GaussianY

GaussianYYY

GaussianGaussianYY

Solution

• Since X and Y have the same covariance Q then

dYff

dYQYYf

dXQXXfdXff

GaussianY

Y

GaussianGaussianGaussian

log

][

][log

*

*

Fixed Channel

• The achievable rate

with • Differential entropy maximizer is a complex

Gaussian random vector with some covariance matrix Q

)det(log

)log()]det()log[();(max

0

*

0*

0

N

HQHI

eNMHQHINerbI

RR

RR

R

b

MM

RMMM

p

avMM PbbEbbEQTT

][ and ][ *

Fixed Channel

• Finding optimum input covariance• Singular value decomposition of H

• The equivalent channel

},min{

1

**TR MM

mmmm vuVUH

bVbrUrnbrRTTRR MMMMM

** ~ and ~ with ~~~

Parallel Channels

• At most parallel channels

• Power distribution across parallel channels

},M{M,,,mnbr TRmmmm min21; ~~~

},min{ RT MM

avMM PVQVQbbEbbEQTT

)(tr)(tr][ and ][ **

Parallel Channels

• A few useful notes

)~

1(log

)~

det(log

)det(log

)det(log)det(log

2

0

*

0

**

0

*

0

*

mmmm

MM

MMMMMMMM

MMMMMMMMMM

Q

N

QI

N

QVVI

N

HHQI

N

HQHI

RR

RTTTTR

RR

TRRTTT

TTRR

Parallel Channels

• A note

diagonal is ~

hen equality wwith

)~

1()~

det( 2

0

*

Q

QN

QI mmm

mMM RR

Fixed Channel

• Diagonal entries found via water filling• Achievable rate

with power

)1log();(},min{

1 0

2*

TR MM

m

mm

N

PbrI

mavm

mm PP

NP *

20* with )(

Example

• Consider a 2x3 channel

• The mutual information is maximized at

2/12/16

3/1

3/1

3/1

11

11

11

23

H

2/][ with )6

1log();( *

0

PbbEN

PbrI ji

Example

• Consider a 3x3 channel

• Mutual information is maximized by

100

010

001

H

330 3

with )3

1log(3);( IP

QN

PbrI

Ergodic MIMO Channels

• A new realization on each channel use

• No CSI

• CSIR

• CSITR?

Fast Fading MIMO with CSIR

• Entries of H are independent and each complex Gaussian with zero mean

• If V and U are unitary then distribution of H is the same as UHV*

• The rate

]|;([)|;(

)|;();();,(

hHbrIEHbrI

HbrIbHIbrHITRTR MMMM

MIMO with CSIR

• The achievable rate

since the differential entropy maximizer is a complex Gaussian random vector with some covariance matrix Q

)log()]det()log[();(max 0*

0 eNMHQHINebrI RMMM

p RR

R

b

Fast Fading and CSIR

• Finally,

with

• The scalar power constraint• The capacity achieving signal is circularly

symmetric complex Gaussian (0,Q)

)]det([log);(0N

HQHIErbI

RR MM

][ bbEQ

TT MM

avPbbE ][

MIMO CSIR

• Since Q is non-negative definite Q=UDU*

• Focus on non-negative definite diagonal Q• Further, optimum

)])()(

det([log)]det([log);(0

*

0 N

HUDHUIE

N

HQHIErbI

TT MMIQ

)]det([log);(0NM

HHPIErbI

T

av

Rayleigh Fading MIMO

• CSIR achievable rate

• Complex Gaussian distribution on H• The square matrix W=HH*

– Wishart distribution– Non negative definite– Distribution of eigenvalues

)]det([log);(0NM

HHPIErbI

T

av

Ergodic / Fast Fading

• The channel coherence time is• The channel known at the receiver

• The capacity achieving signal b must be

circularly symmetric complex Gaussian

1cT

)}det({log0

HH

NM

PIEC

T

avMM RR

))/(,0(TT MMTav IMP

Slow Fading MIMO

• A channel realization is valid for the duration of the code (or transmission)

• There is a non zero probability that the channel can not sustain any rate

• Shannon capacity is zero

Slow Fading Channel

• If the coherence time Tc is the block length

• The outage probability with CSIR only

with and

)det(log);(0

*

N

QHHIbrI RTTR

RR

MMMMMM

])det(Pr[loginf),(0

RN

HQHIPR

RR MMQ

avout

avPbbE ][ ][ bbEQ

Slow Fading

• Since

• Diagonal Q is optimum• Conjecture: optimum Q is

])det(Pr[log])det(Pr[log0

*

0

RN

HHUQUIR

N

HQHI

RRRR MMMM

0

0

1

1

1

m

PQ av

opt

Example

• Slow fading SIMO, • Then and

• Scalar

avopt PQ

])1Pr[log(])det(Pr[log0

*

0

RN

HHPR

N

HQHI av

MM RR

1TM

ddistribute is 2* HH

)(),(

)1(

0

1

out

0

R

P

eN

uM

av M

dueu

PR

av

R

R

Example

• Slow fading MISO,• The optimum

• The outage

1RM

Tmmav

opt MmIm

PQ somefor

)(])1Pr[log(

)1(

0

1

0

*

0

m

dueu

RmN

HHP

av

R

P

emN

um

av

Diversity and Multiplexing for MIMO

• The capacity increase with SNR

• The multiplexing gain

)1log(k

SNRkC

)log(

)(lim

SNR

SNRCr

SNR

Diversity versus Multiplexing

• The error measure decreases with SNR increase

• The diversity gain

• Tradeoff between diversity and multiplexing– Simple in single link/antenna fading channels

dSNR

)log(

))(log(lim

SNR

SNRd out

SNR

Coding for Fading Channels

• Coding provides temporal diversity

or

• Degrees of freedom – Redundancy– No increase in data rate

dcSNRgFER

dcSNRgECP )(

M versus D

(0,MRMT)

(min(MR,MT),0)

Multiplexing Gain

Div

ers

ity G

ain