Capacity Limits of Wireless Channels

with Multiple Antennas:Challenges, Insights, and New

Mathematical Methods

Andrea GoldsmithStanford University

CoAuthors: T. Holliday, S. Jafar, N. Jindal, S. Vishwanath

Princeton-Rutgers Seminar SeriesRutgers University

April 23, 2003

Future Wireless Systems

Nth Generation CellularNth Generation WLANsWireless Entertainment Wireless Ad Hoc NetworksSensor Networks Smart Homes/AppliancesAutomated Cars/FactoriesTelemedicine/LearningAll this and more…

Ubiquitous Communication Among People and Devices

Challenges

The wireless channel is a randomly-varying broadcast medium with limited bandwidth.

Fundamental capacity limits and good protocol designs for wireless networks are open problems.

Hard energy and delay constraints change fundamental design principles

Many applications fail miserably with a “generic” network approach: need for crosslayer design

Outline

Wireless Channel Capacity

Capacity of MIMO ChannelsImperfect channel informationChannel correlations

Multiuser MIMO ChannelsDuality and Dirty Paper Coding

Lyapunov Exponents and Capacity

Wireless Channel Capacity

Fundamental Limit on Data Rates

Main drivers of channel capacity Bandwidth and power Statistics of the channel Channel knowledge and how it is

used Number of antennas at TX and RX

Capacity: The set of simultaneously achievable rates {R1,…,Rn}

R1R2

R3

R1

R2

R3

MIMO Channel Model

x1

x2

x3

y1

y2

y3

h11

h21

h31

h12

h22

h32

h13 h23

h33

),0(~,

2

11

1

1111

INnn

n

x

x

hh

hh

y

ynHxy

mnmnm

n

m

Model applies to any channel described by a matrix (e.g. ISI channels)

n TX antennas m RX antennas

What’s so great about MIMO?

Fantastic capacity gains (Foschini/Gans’96, Telatar’99)Capacity grows linearly with antennas when channel

known perfectly at Tx and Rx

Vector codes (or scalar codes with SIC) optimalAssumptions:

Perfect channel knowledge Spatially uncorrelated fading: Rank

(HTQH)=min(n,m)

)(

1

2

:)(:)1log(max||logmax

QHHRank

iii

PPP

T

PQTrQ

T

iii

pQHHIB

C

What happens when these assumptions are relaxed?

Realistic Assumptions

No transmitter knowledge of HCapacity is much smaller

No receiver knowledge of HCapacity does not increase as the

number of antennas increases (Marzetta/Hochwald’99)

Will the promise of MIMO be realized in practice?

Partial Channel Knowledge

Model channel as H~N(,)Receiver knows channel H perfectlyTransmitter has partial information about H

)|(~

),0(~ 2

HpH

INn

nHxy

Channel

ReceiverTransmitter

x y

,H

Partial Information Models

Channel mean informationMean is measured, Covariance unknown

Channel covariance informationMean unknown, measure covariance

We have developed necessary and sufficient conditions for the optimality of beamformingObtained for both MISO and MIMO channelsOptimal transmission strategy also known

) ,(~ INH

),0(~ NH

Beamforming

Scalar codes with transmit precoding

1x

2x

nx

x Receiver1c

nc

• Transforms the MIMO system into a SISO system.• Greatly simplifies encoding and decoding.• Channel indicates the best direction to beamform

•Need “sufficient” knowledge for optimality

Optimality of Beamforming

Mean Information

Optimality of Beamforming

Covariance Information

No Tx or Rx Knowledge Increasing nT beyond coherence time T in a block fading

channel does not increase capacity (Marzetta/Hochwald’99)Assumes uncorrelated fading.

We have shown that with correlated fading, adding Tx antennas always increases capacitySmall transmit antenna spacing is good!

Impact of spatial correlations on channel capacityPerfect Rx and Tx knowledge: hurts (Boche/Jorswieck’03)Perfect Rx knowledge, no Tx knowledge: hurts (BJ’03)

Perfect Rx knowledge, Tx knows correlation: helpsTX and Rx only know correlation: helps

Gaussian Broadcast and Multiple Access

Channels

Broadcast (BC): One Transmitter to Many Receivers.

Multiple Access (MAC): Many Transmitters to One Receiver.

x h1(t)x h21(t)

x h3(t)

• Transmit power constraint• Perfect Tx and Rx knowledge

x h22(t)

Differences:Shared vs. individual power constraintsNear-far effect in MAC

Similarities:Optimal BC “superposition” coding is also

optimal for MAC (sum of Gaussian codewords)

Both decoders exploit successive decoding and interference cancellation

Comparison of MAC and BC

P

P1

P2

MAC-BC Capacity Regions

MAC capacity region known for many casesConvex optimization problem

BC capacity region typically only known for (parallel) degraded channelsFormulas often not convex

Can we find a connection between the BC and MAC capacity regions?Duality

Dual Broadcast and MAC Channels

x

)(1 nh

x

)(nhM

+

)(1 nz

)(1 nx

)(nxM

)(1 ny)( 1P

)( MP

x

)(1 nh

x

)(nhM

+

)(nzM

)(nyM

+

)(nz

)(ny)(nx)(P

Gaussian BC and MAC with same channel gains and same noise power at each receiver

Broadcast Channel (BC)Multiple-Access Channel (MAC)

The BC from the MAC

Blue = BCRed = MAC

21 hh

P1=1, P2=1

P1=1.5, P2=0.5

P1=0.5, P2=1.5

),;(),;,( 21212121 hhPPChhPPC BCMAC

MAC with sum-power constraint

PP

MACBC hhPPPChhPC

10

211121 ),;,(),;(

Sum-Power MAC

MAC with sum power constraintPower pooled between MAC

transmittersNo transmitter coordination

P

P

MAC BCSame capacity region!

),;(),;,(),;( 210

211121

1

hhPChhPPPChhPC SumMAC

PPMACBC

BC to MAC: Channel Scaling

Scale channel gain by , power by 1/MAC capacity region unaffected by scaling Scaled MAC capacity region is a subset of the scaled BC capacity region for any MAC region inside scaled BC region for anyscaling

1h1P

2P 2h

+

+

21 P

P

1h

2h+

MAC

BC

The BC from the MAC

0

2121

2121 ),;(),;,(

hhPP

ChhPPC BCMAC

Blue = Scaled BCRed = MAC

1

2

h

h

0

BC in terms of MAC

MAC in terms of BC

PP

MACBC hhPPPChhPC

10

211121 ),;,(),;(

0

2121

2121 ),;(),;,(

hhPP

ChhPPC BCMAC

Duality: Constant AWGN Channels

What is the relationship betweenthe optimal transmission strategies?

Equate rates, solve for powers

Opposite decoding order Stronger user (User 1) decoded last in BCWeaker user (User 2) decoded last in MAC

Transmission Strategy

Transformations

BB

BMM

BB

M

MM

RPh

PhPhR

RPh

Ph

PhR

221

22

222

2

222

2

12

121

222

121

1

)1log()1log(

)1log()1log(

Duality Applies to Different

Fading Channel Capacities

Ergodic (Shannon) capacity: maximum rate averaged over all fading states.

Zero-outage capacity: maximum rate that can be maintained in all fading states.

Outage capacity: maximum rate that can be maintained in all nonoutage fading states.

Minimum rate capacity: Minimum rate maintained in all states, maximize average rate in excess of minimum

Explicit transformations between transmission strategies

Duality: Minimum Rate Capacity

BC region known MAC region can only be obtained by duality

Blue = Scaled BCRed = MAC

MAC in terms of BC

What other unknown capacity regions can be obtained by duality?

Dirty Paper Coding (Costa’83)

Dirty Paper Coding

Clean Channel Dirty Channel

Dirty Paper

Coding

Basic premiseIf the interference is known, channel

capacity same as if there is no interference

Accomplished by cleverly distributing the writing (codewords) and coloring their ink

Decoder must know how to read these codewords

Modulo Encoding/Decoding

Received signal Y=X+S, -1X1S known to transmitter, not receiver

Modulo operation removes the interference effectsSet X so that Y[-1,1]=desired message (e.g. 0.5)Receiver demodulates modulo [-1,1]

-1 +3 +5+1-3

…-5 0

S

-1 +10

-1 +10

X

+7-7

…

Broadcast MIMO Channel

111 n x H y 1H

x

1n

222 n x H y 2H

2n

t1 TX antennasr11, r21 RX antennas

)1 t(r

)2 t(r

)IN(0,~n)IN(0,~n21 r2r1

Non-degraded broadcast channel

Perfect CSI at TX and RX

Capacity Results

Non-degraded broadcast channelReceivers not necessarily “better” or

“worse” due to multiple transmit/receive antennas

Capacity region for general case unknown

Pioneering work by Caire/Shamai (Allerton’00): Two TX antennas/two RXs (1 antenna each)Dirty paper coding/lattice precoding*

Computationally very complexMIMO version of the Sato upper bound

*Extended by Yu/Cioffi

Dirty-Paper Coding (DPC)

for MIMO BCCoding scheme:

Choose a codeword for user 1Treat this codeword as interference to user 2Pick signal for User 2 using “pre-coding”

Receiver 2 experiences no interference:

Signal for Receiver 2 interferes with Receiver 1:

Encoding order can be switched

)) log(det(I R 2222THH

) det(I

))( det(Ilog R

121

12111 T

T

HH

HH

Dirty Paper Coding in Cellular

Does DPC achieve capacity?

DPC yields MIMO BC achievable region.We call this the dirty-paper region

Is this region the capacity region?

We use duality, dirty paper coding, and Sato’s upper bound to address this question

MIMO MAC with sum power

MAC with sum power: Transmitters code independentlyShare power

Theorem: Dirty-paper BC region equals the dual sum-power MAC region

PP

MACSumMAC PPPCPC

10

11 ),()(

)()( PCPC SumMAC

DPCBC

P

Transformations: MAC to BC

Show any rate achievable in sum-power MAC also achievable with DPC for BC:

A sum-power MAC strategy for point (R1,…RN) has a given input covariance matrix and encoding order

We find the corresponding PSD covariance matrix and encoding order to achieve (R1,…,RN) with DPC on BC The rank-preserving transform “flips the effective

channel” and reverses the order Side result: beamforming is optimal for BC with 1 Rx

antenna at each mobile

)()( PCPC SumMAC

DPCBC

DPC BC Sum MAC

Transformations: BC to MAC

Show any rate achievable with DPC in BC also achievable in sum-power MAC:

We find transformation between optimal DPC strategy and optimal sum-power MAC strategy “Flip the effective channel” and reverse order

)()( PCPC SumMAC

DPCBC

DPC BC Sum MAC

Computing the Capacity Region

Hard to compute DPC region (Caire/Shamai’00)

“Easy” to compute the MIMO MAC capacity regionObtain DPC region by solving for sum-

power MAC and applying the theoremFast iterative algorithms have been

developedGreatly simplifies calculation of the DPC

region and the associated transmit strategy

)()( PCPC SumMAC

DPCBC

Based on receiver cooperation

BC sum rate capacity Cooperative capacity

Sato Upper Bound on the

BC Capacity Region

+

+

1H

2H

1n

2n

1y

2y

x

|HHΣI|log2

1maxH)(P, T

xsumrateBC

x

C

Joint receiver

The Sato Bound for MIMO BC

Introduce noise correlation between receiversBC capacity region unaffected

Only depends on noise marginals

Tight Bound (Caire/Shamai’00)Cooperative capacity with worst-case noise correlation

Explicit formula for worst-case noise covarianceBy Lagrangian duality, cooperative BC region equals

the sum-rate capacity region of MIMO MAC

|ΣHHΣΣI|log2

1maxinfH)(P, 1/2

zT

x1/2

zsumrateBC

xz

C

Sum-Rate Proof

DPC Achievable

Lagrangian Duality

Obvious

)()( PCPC BCDPCBC

)()( PCPC SumMACMAC

)()( PCPC DPCBC

SumMAC

Duality

Sato Bound

)()( PCPC SumMAC

sumrateCoopBC

)()( PCPC CoopBCBC

)()( PCPC BC

sumrateDPCBC

Compute from MAC

*Same result by Vishwanath/Tse for 1 Rx antenna

MIMO BC Capacity Bounds

Sato Upper Bound

Single User Capacity BoundsDirty Paper Achievable Region

BC Sum Rate Point

Does the DPC region equal the capacity region?

Full Capacity Region

DPC gives us an achievable region

Sato bound only touches at sum-rate point

We need a tighter bound to prove DPC is optimal

A Tighter Upper Bound

Give data of one user to other usersChannel becomes a degraded BCCapacity region for degraded BC knownTight upper bound on original channel capacity

This bound and duality prove that DPC achieves capacity under a Gaussian input restrictionRemains to be shown that Gaussian inputs are optimal

+

+

1H

2H 2n

1y

2y

x 2y

1n

Full Capacity Region Proof

Tight Upper Bound

Worst Case Noise Diagonalizes

Duality

)()( PCPC DSMBCBC

)()( PCPC DPBCMAC

)()( PCPC DPCBCBC

Final Result

Duality

)()( PCPC MACDSMMAC

)()( PCPC DSMMAC

DSMBC

inputsGaussianfor

PCPC BCDPCBC )()(

Compute from MAC

Time-varying Channels

with Memory

Time-varying channels with finite memory induce infinite memory in the channel output.

Capacity for time-varying infinite memory channels is only known in terms of a limit

Closed-form capacity solutions only known in a few casesGilbert/Elliot and Finite State Markov Channels

nn

nXpYXI

nC

n;

1limmax

)(

A New Characterization of Channel Capacity

Capacity using Lyapunov exponents

Similar definitions hold for (Y) and (X;Y)Matrices BYi

and BXiYi depend on input and

channel

)],()()([max)(

YXYXCxp

||...||log1

lim)(21 nXXX

nBBB

nX

where the Lyapunov exponent

for BXi a random matrix whose entries

depend on the input symbol Xi

Lyapunov Exponents and Entropy

Lyapunov exponent equals entropy under certain conditionsEntropy as a product of random matricesConnection between IT and dynamic systems

theory

Still have a limiting expression for entropySample entropy has poor convergence properties

),,(log1

lim)( 1 nn

XXPn

X

),,(log1

lim)( 1 nn

YYPn

Y

)),(,),,((log1

lim),( 11 nnn

YXYXPn

YX

Lyapunov Direction Vector

The vector pn is the “direction” associated with (X) for any .Also defines the conditional channel state

probability

Vector has a number of interesting properties

It is the standard prediction filter in hidden Markov models

Under certain conditions we can use its stationary distribution to directly compute (X) (X)

)|(P||...||

...1

121

21 nn

XXX

XXXn XZ

BBB

BBBp

n

n

Computing Lyapunov Exponents

Define as the stationary distribution of the “direction vector” pnpn

We prove that we can compute these Lyapunov exponents in closed form as

This result is a significant advance in the theory of Lyapunov exponent computation

||]||[log)( , XX pBEX

pn

pn+1

pn+2

Computing Capacity

Closed-form formula for mutual information

We prove continuity of the Lyapunov exponents with respect to input distribution and channelCan thus maximize mutual information

relative to channel input distribution to get capacity

Numerical results for time-varying SISO and MIMO channel capacity have been obtained

We also develop a new CLT and confidence interval methodology for sample entropy

),()()();( YXYXYXI

Sensor Networks

Energy is a driving constraint.Data flows to centralized location.Low per-node rates but up to 100,000 nodes.Data highly correlated in time and space.Nodes can cooperate in transmission and

reception.

Energy-Constrained Network Design

Each node can only send a finite number of bitsTransmit energy per bit minimized by sending each

bit over many dimensions (time/bandwidth product)Delay vs. energy tradeoffs for each bit

Short-range networks must consider both transmit, analog HW, and processing energySophisticated techniques for modulation, coding,

etc., not necessarily energy-efficient Sleep modes save energy but complicate networking

New network design paradigm:Bit allocation must be optimized across all protocolsDelay vs. throughput vs. node/network lifetime

tradeoffsOptimization of node cooperation (coding, MIMO,

etc.)

Results to DateModulation Optimization

Adaptive MQAM vs. MFSK for given delay and rateTakes into account RF hardware/processing tradeoffs

MIMO vs. MISO vs. SISO for constrained energySISO has best performance at short distances

(<100m)

Optimal Adaptation with Delay/Energy Constraints

Minimum Energy Routing

Conclusions Shannon capacity gives fundamental data rate limits

for wireless channels

Many open capacity problems for time-varying multiuser MIMO channels

Duality and dirty paper coding are powerful tools to solve new capacity problems and simplify computation

Lyapunov exponents a powerful new tool for solving capacity problems

Cooperative communications in sensor networks is an interesting new area of research