Capacity Limits of Wireless Channels
with Multiple Antennas:
Challenges, Insights, and New Mathematical Methods
Andrea Goldsmith
Stanford University
CoAuthors: T. Holliday, S. Jafar, N. Jindal, S. Vishwanath
Princeton-Rutgers Seminar Series
Rutgers University
April 23, 2003
Future Wireless Systems
Ubiquitous Communication Among People and Devices
Nth Generation Cellular
Nth Generation WLANs
Wireless Entertainment
Wireless Ad Hoc Networks
Sensor Networks
Smart Homes/Appliances
Automated Cars/Factories
Telemedicine/Learning
All this and more…
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 Channels
 Imperfect channel information
 Channel correlations

Multiuser MIMO Channels
 Duality

and Dirty Paper Coding
Lyapunov Exponents and Capacity
Wireless Channel Capacity
Fundamental Limit on Data Rates
Capacity: The set of simultaneously achievable rates {R1,…,Rn}
R3
R1

R2
R3
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
R2
R1
MIMO Channel Model
n TX antennas
m RX antennas
h11
x1
h31
h22
x2
h32
h13
x3
h12
h21
y1
y2
h23
h33
 y1   h11  h1n   x1   n1 
               ,
  
   
 ym  hm1  hmn   xn  nm 
y3
y  Hx  n
n ~ N (0,  2 I )
Model applies to any channel described by a matrix (e.g. ISI channels)
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
C
 max log | I  H T QH | max
Pi : Pi  P
B Q:Tr (Q ) P
i


Rank ( H T QH )
2
log(
1

p


i i )
i 1
Vector codes (or scalar codes with SIC) optimal
Assumptions:


Perfect channel knowledge
Spatially uncorrelated fading: Rank (HTQH)=min(n,m)
What happens when these assumptions are relaxed?
Realistic Assumptions

No transmitter knowledge of H
 Capacity

is much smaller
No receiver knowledge of H
 Capacity
does not increase as the number of
antennas increases (Marzetta/Hochwald’99)

Will the promise of MIMO be realized in
practice?
Partial Channel Knowledge
H ,q
Channel
x
Transmitter
y  Hx  n
n ~ N (0,  2 I )
y
Receiver
H ~ p( H | q )
q



Model channel as H~N(m,S)
Receiver knows channel H perfectly
Transmitter has partial information q about H
Partial Information Models

Channel mean information

Mean is measured, Covariance unknown
H ~ N ( m , I)

Channel covariance information

Mean unknown, measure covariance
H ~ N (0, S)

We have developed necessary and sufficient
conditions for the optimality of beamforming


Obtained for both MISO and MIMO channels
Optimal transmission strategy also known
Beamforming

Scalar codes with transmit precoding
c1
x1
x2
cn
xn
x
Receiver
• 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)


We have shown that with correlated fading, adding Tx
antennas always increases capacity


Assumes uncorrelated fading.
Small transmit antenna spacing is good!
Impact of spatial correlations on channel capacity
 Perfect Rx and Tx knowledge: hurts (Boche/Jorswieck’03)
 Perfect Rx knowledge, no Tx knowledge: hurts (BJ’03)
 Perfect Rx knowledge, Tx knows correlation: helps
 TX and Rx only know correlation: helps
Gaussian Broadcast and
Multiple Access Channels
Broadcast (BC):
One Transmitter
to Many Receivers.
• Transmit power constraint
• Perfect Tx and Rx knowledge
Multiple Access (MAC):
Many Transmitters
to One Receiver.
x
x
x
h1(t)
x
h22(t)
h21(t)
h3(t)
Comparison of MAC and BC

Differences:
 Shared vs. individual power
 Near-far effect in MAC

P
Similarities:
constraints
P1
P2
 Optimal
BC “superposition” coding is also
optimal for MAC (sum of Gaussian codewords)
 Both
decoders exploit successive decoding and
interference cancellation
MAC-BC Capacity Regions

MAC capacity region known for many cases
 Convex

optimization problem
BC capacity region typically only known for
(parallel) degraded channels
 Formulas

often not convex
Can we find a connection between the BC
and MAC capacity regions?
Duality
Dual Broadcast and MAC Channels
Gaussian BC and MAC with same channel gains
and same noise power at each receiver
z1 (n)
h1 (n)
x
+
h1 (n)
y1 (n)
x1 (n)
x
z (n)
( P1 )
x(n)
(P )
h M (n)
x
+
zM (n)
+
y (n)
h M (n)
yM (n)
xM (n)
x
( PM )
Broadcast Channel (BC)
Multiple-Access Channel (MAC)
The BC from the MAC
C MAC ( P1 , P2 ; h1 , h2 )  C BC ( P1  P2 ; h1 , h2 )
h1  h2
P1=0.5, P2=1.5
P1=1, P2=1
Blue = BC
Red = MAC
P1=1.5, P2=0.5
MAC with sum-power constraint
C BC ( P; h1 , h2 ) 

0 P1  P
C MAC ( P1 , P  P1 ; h1 , h2 )
Sum-Power MAC
CBC ( P; h1 , h2 ) 
Sum
C
(
P
,
P

P
;
h
,
h
)

C
 MAC 1
1 1
2
MAC ( P; h1 , h2 )
0 P1  P

MAC with sum power constraint
 Power pooled between MAC transmitters
 No transmitter coordination
Same capacity region!
MAC
P
P
BC
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 any scaling
P1

 h1
MAC
+
P2
h2
P1
 P2

BC
 h1
h2
+
+
The BC from the MAC
  
Blue = Scaled BC
Red = MAC

h2
h1
  0
C MAC ( P1 , P2 ; h1 , h2 )   C BC (
 0
P1

 P2 ;  h1 , h2 )
Duality: Constant AWGN Channels

BC in terms of MAC
C BC ( P; h1 , h2 ) 


0 P1  P
C MAC ( P1 , P  P1 ; h1 , h2 )
MAC in terms of BC
P1
C MAC ( P1 , P2 ; h1 , h2 )   C BC (  P2 ; h1 , h2 )
 0

What is the relationship between
the optimal transmission strategies?
Transmission Strategy
Transformations

Equate rates, solve for powers
R1M  log( 1 
R2M  log( 1 

h12 P1M
)
M
2
h2 P2  
h22 P2M
2
 log( 1 
)  log( 1 
h12 P1B
2
h22 P2B
h22 P1B  
)  R1B
B
)

R
2
2
Opposite decoding order
 Stronger
user (User 1) decoded last in BC
 Weaker user (User 2) decoded last in MAC
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
MAC in terms of BC
Blue = Scaled BC
Red = MAC
BC region known
 MAC region can only be obtained by duality

What other unknown capacity regions
can be obtained by duality?
Dirty Paper Coding (Costa’83)

Basic premise
 If
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
Dirty
Paper
Coding
Clean Channel
Dirty
Paper
Coding
Dirty Channel
Modulo Encoding/Decoding
Received signal Y=X+S, -1X1


S known to transmitter, not receiver
Modulo operation removes the interference effects



Set X so that Y[-1,1]=desired message (e.g. 0.5)
Receiver demodulates modulo [-1,1]
-1
0
+1
…
…
-7
-5
-3
-1
0
+1
+3
S
+5
X
-1
0
+1
+7
Broadcast MIMO Channel
(r1  t )
n1
y1  H1x  n1
H1
x
(r2  t )
H2
t1 TX antennas
r11, r21 RX antennas
n2
Perfect CSI at TX and RX
y2  H2 x  n 2
n1 ~ N(0, I r1 ) n 2 ~ N(0, I r2 )
Non-degraded broadcast channel
Capacity Results

Non-degraded broadcast channel
 Receivers 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 complex
 MIMO
version of the Sato upper bound
*Extended
by Yu/Cioffi
Dirty-Paper Coding (DPC)
for MIMO BC

Coding scheme:



Choose a codeword for user 1
Treat this codeword as interference to user 2
Pick signal for User 2 using “pre-coding”

Receiver 2 experiences no interference:

R 2  log(det(I  H 2S 2 H 2T ))
Signal for Receiver 2 interferes with Receiver 1:
 det(I  H1 (S1  S 2 ) H1T ) 

R1  log 
T

det(I

H
S
H
)
1 2 1



Encoding order can be switched
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
 Share power
Sum
MAC
C
( P) 
code independently
P

CMAC ( P1 , P  P1 )
0 P1  P

Theorem: Dirty-paper BC region equals the
dual sum-power MAC region
C
DPC
BC
( P)  C
Sum
MAC
( P)
Transformations: MAC to BC

Show any rate achievable in sum-power MAC also
achievable with DPC for BC:
DPC BC
Sum MAC
DPC
Sum
C BC
( P)  CMAC
( P)


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
Transformations: BC to MAC

Show any rate achievable with DPC in BC also
achievable in sum-power MAC:
DPC
Sum
C BC
( P)  CMAC
( P)
 We
DPC BC
Sum MAC
find transformation between optimal DPC
strategy and optimal sum-power MAC strategy
 “Flip the effective channel” and reverse order
Computing the Capacity Region
C
DPC
BC
( P)  C
Sum
MAC
( P)

Hard to compute DPC region (Caire/Shamai’00)

“Easy” to compute the MIMO MAC
capacity region
 Obtain
DPC region by solving for sum-power
MAC and applying the theorem
 Fast iterative algorithms have been developed
 Greatly simplifies calculation of the DPC region
and the associated transmit strategy
Sato Upper Bound on the
BC Capacity Region
 Based on receiver cooperation
n1
x
H1
+
H2
n2
+
y1
Joint receiver
y2
 BC sum rate capacity  Cooperative capacity
sumrate
CBC
(P, H)
max 1

log | I  HΣ x H T |
Sx 2
The Sato Bound for MIMO BC

Introduce noise correlation between receivers

BC capacity region unaffected


Only depends on noise marginals
Tight Bound (Caire/Shamai’00)

Cooperative capacity with worst-case noise correlation
sumrate
CBC
(P, H)


inf max 1

log | I  Σ z1/2HΣ x H T Σ z1/2 |
Sz Sx 2
Explicit formula for worst-case noise covariance
By Lagrangian duality, cooperative BC region
equals the sum-rate capacity region of MIMO MAC
Sum-Rate Proof
DPC Achievable
C
DPC
BC
Duality
Sum
DPC
CMAC
( P)  C BC
( P)
( P)  C BC ( P )
sumrate
DPC
CBC
( P)  C BC ( P)
C BC ( P )  C
Coop
BC
CMAC ( P)  C
(P)
C
( P)
Obvious
Sato Bound
*Same result by
Vishwanath/Tse
for 1 Rx antenna
Sum
MAC
Coop
BC
sumrate
( P)  C
Sum
MAC
( P)
Lagrangian Duality
Compute from MAC
MIMO BC Capacity Bounds
Single User Capacity Bounds
Dirty Paper Achievable Region
BC Sum Rate Point
Sato Upper Bound
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
n1
H1
x

y1
y2
H2
n2
y2
+
Give data of one user to other users




+
Channel becomes a degraded BC
Capacity region for degraded BC known
Tight upper bound on original channel capacity
This bound and duality prove that DPC achieves
capacity under a Gaussian input restriction

Remains to be shown that Gaussian inputs are optimal
Full Capacity Region Proof
Tight Upper Bound
C BC ( P)  C
DSM
BC
( P)
C
DPC
BC
Final Result
DPC
C BC ( P)  C BC
( P)
( P)  CBC ( P)
for Gaussian inputs
C
DSM
BC
( P)  C
DSM
MAC
( P)
DP
CMAC ( P)  C BC
( P)
Duality
Duality
Compute from MAC
DSM
CMAC
( P)  CMAC ( P)
Worst Case Noise Diagonalizes
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

1
C  maxn lim I X n ; Y n
p ( X ) n  n


Closed-form capacity solutions only known in a
few cases

Gilbert/Elliot and Finite State Markov Channels
A New Characterization of
Channel Capacity

Capacity using Lyapunov exponents
C  max[ ( X )   (Y )   ( X , Y )]
p( x )
where the Lyapunov exponent
1
 ( X )  lim log || B X1 B X 2 ...B X n ||
n  n
for BXi a random matrix whose entries
depend on the input symbol Xi

Similar definitions hold for (Y) and (X;Y)

Matrices BYi and BXiYi depend on input and channel
Lyapunov Exponents
and Entropy

Lyapunov exponent equals entropy under
certain conditions


Entropy as a product of random matrices
Connection between IT and dynamic systems theory
1
 ( X )  lim log P ( X 1 ,, X n )
n  n

1
 (Y )  lim log P(Y1 ,, Yn )
n  n
1
 ( X , Y )  lim log P(( X 1 , Y1 ), , ( X n , Yn ))
n  n
Still have a limiting expression for entropy

Sample entropy has poor convergence properties
Lyapunov Direction Vector

The vector pn is the “direction” associated with
(X) for any m.

Also defines the conditional channel state probability
mBX BX ...BX
n
pn 
 P( Z n1 | X )
|| mBX BX ...BX ||1
1
1

2
2
n
n
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)
Computing Lyapunov Exponents

Define p as the stationary distribution of the
“direction vector” pnpn
pn+2
p
pn
pn+1

We prove that we can compute these Lyapunov
exponents in closed form as
 ( X )  Ep , X [log || pBX ||]

This result is a significant advance in the theory
of Lyapunov exponent computation
Computing Capacity



Closed-form formula for mutual information
I ( X ; Y )   ( X )   (Y )   ( X , Y )
We prove continuity of the Lyapunov exponents
with respect to input distribution and channel
 Can 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
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 bits



Short-range networks must consider both transmit,
analog HW, and processing energy



Transmit energy per bit minimized by sending each bit
over many dimensions (time/bandwidth product)
Delay vs. energy tradeoffs for each bit
Sophisticated 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 protocols
Delay vs. throughput vs. node/network lifetime tradeoffs
Optimization of node cooperation (coding, MIMO, etc.)
Results to Date

Modulation Optimization



Adaptive MQAM vs. MFSK for given delay and rate
Takes into account RF hardware/processing tradeoffs
MIMO vs. MISO vs. SISO for constrained energy

SISO 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