EE360: Multiuser Wireless Systems and Networks

Lecture 1 OutlineCourse DetailsCourse SyllabusCourse Overview

Future Wireless NetworksMultiuser Channels (Broadcast/MAC

Channels)Spectral Reuse and Cellular SystemsAd-Hoc, Sensor, and Cognitive Radio

NetworksBandwidth Sharing in Multiuser

ChannelsOverview of Multiuser Channel

CapacityCapacity of Broadcast Channels

Course Information*

People Instructor: Andrea Goldsmith,

andrea@ee, Packard 371, 5-6932, OHs: MW after class and by appt.

TA: Mainak Chowdhury, Email: mainakch@stanford.edu OHs: around HWs. Possibly around paper discussions organized via Piazza.

Class Administrator: Pat Oshiro, poshiro@stanford, Packard 365, 3-2681.

*See web or handout for more details

Course InformationNuts and Bolts

Prerequisites: EE359 Course Time and Location: MW 9:30-10:45.

Y2E2 101.

Class Homepage: www.stanford.edu/class/ee360 Contains all required reading, handouts,

announcements, HWs, etc. Class Mailing List: ee360win01314-students

(automatic for on-campus registered students). Guest list: send TA email to sign up

Tentative Grading Policy: 10% Class participation 10% Class presentation 15% Homeworks 15% Paper summaries 50% Project (10% prop, 15% progress report, 25%

final report+poster)

Grade Components Class participation

Read the required reading before lecture/discuss in class

Class presentationPresent a paper related to one of the course

topicsHW 0: Choose 3 possible high-impact

papers, each on a different syllabus topic, by Jan. 13. Include a paragraph for each describing main idea(s), why interesting/high impact

Presentations begin Jan. 22.

HW assignmentsTwo assignments from book or other

problems

Paper summariesTwo 2-4 page summaries of several articlesEach should be on a different topic from the

syllabus

Term project on anything related to multiuser wireless

Analysis, simulation and/or experiment Must contain some original research 2 can collaborate if project merits collaboration

(scope, synergy)

Must set up website for project Will post proposal, progress report, and final report

to website

Project proposal due Jan. 27 at midnight 1-2 page proposal with detailed description of

project plan Comments Feb 3, revised project proposal due Feb

10.

Progress report: due Feb. 24 at midnight

2-3 page report with introduction of problem being investigated, system description, progress to date, statement of remaining work

Poster presentations last week of classes (We 3/13, Th 3/14?)

Final report due March 17 at midnight

Project

See website for details

Tentative Syllabus Weeks 1-2: Multiuser systems

(Chapters 13.4 and 14, additional papers)

Weeks 3-4: Cellular systems (Chapter 15, additional papers)

Weeks 5-6: Ad hoc wireless networks (Chapter 16, additional papers)

Week 7-8: Cognitive radio networks (papers)

Week 8-9: Sensor networks (papers) Weeks 10: Additional Topics. Course

Summary

Future Wireless Networks

Ubiquitous Communication Among People and Devices

Next-generation CellularWireless Internet AccessWireless MultimediaSensor Networks Smart Homes/SpacesAutomated HighwaysIn-Body NetworksAll this and more …

Multiuser Channels:Uplink and Downlink

Downlink (Broadcast Channel or BC): One Transmitter to Many Receivers.

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

R1

R2

R3

x h1(t)x h21(t)

x

h3(t)

x h22(t)

Challenge: How to share the channel among multiuser usersUplink and Downlink typically duplexed in time or frequency

Spectral ReuseDue to its scarcity, spectrum

is reused

BS

In licensed bands

Cellular, Wimax Wifi, BT, UWB,…

and unlicensed bands

Reuse introduces interference

Interference: Friend or Foe?

If treated as noise: Foe

If decodable (MUD): Neither friend nor foe

If exploited via cooperation and cognition: Friend (especially in a network setting)

INPSNR

Increases BERReduces capacity

Cellular Systems Reuse channels to maximize

capacity 1G: Analog systems, large frequency reuse, large cells,

uniform standard 2G: Digital systems, less reuse (1 for CDMA), smaller

cells, multiple standards, evolved to support voice and data (IS-54, IS-95, GSM)

3G: Digital systems, WCDMA competing with GSM evolution.

4G: OFDM/MIMO 5G: ???

BASESTATION MTSO

Rethinking “Cells” in Cellular

Traditional cellular design “interference-limited”MIMO/multiuser detection can remove interferenceCooperating BSs form a MIMO array: what is a cell?Relays change cell shape and boundariesDistributed antennas move BS towards cell boundarySmall cells create a cell within a cell Mobile cooperation via relaying, virtual MIMO, analog

network coding.

SmallCell

Relay

DAS

Coop MIMO

How should cellularsystems be designed?

Will gains in practice bebig or incremental; incapacity or coverage?

ce

Ad-Hoc/Mesh Networks

Outdoor Mesh

Indoor Mesh

Ad-Hoc Networks

Peer-to-peer communications No backbone infrastructure or centralized

control Routing can be multihop. Topology is dynamic. Fully connected with different link SINRs Open questions

Fundamental capacity Optimal routing Resource allocation (power, rate, spectrum,

etc.)

Design Issues Ad-hoc networks provide a flexible

network infrastructure for many emerging applications.

The capacity of such networks is generally unknown.

Transmission, access, and routing strategies for ad-hoc networks are generally ad-hoc.

Crosslayer design critical and very challenging.

Energy constraints impose interesting design tradeoffs for communication and networking.

Cognition Radios Cognitive radios can support new

wireless users in existing crowded spectrumWithout degrading performance of existing

users

Utilize advanced communication and signal processing techniquesCoupled with novel spectrum allocation

policies

Technology could Revolutionize the way spectrum is

allocated worldwide Provide sufficient bandwidth to support

higher quality and higher data rate products and services

Cognitive Radio Paradigms

UnderlayCognitive radios constrained to

cause minimal interference to noncognitive radios

InterweaveCognitive radios find and exploit

spectral holes to avoid interfering with noncognitive radios

OverlayCognitive radios overhear and

enhance noncognitive radio transmissions

KnowledgeandComplexity

Underlay Systems Cognitive radios determine the

interference their transmission causes to noncognitive nodesTransmit if interference below a given

threshold

The interference constraint may be metVia wideband signalling to maintain

interference below the noise floor (spread spectrum or UWB)

Via multiple antennas and beamforming

NCR

IP

NCRCR CR

Interweave Systems Measurements indicate that even

crowded spectrum is not used across all time, space, and frequenciesOriginal motivation for “cognitive” radios

(Mitola’00)

These holes can be used for communicationInterweave CRs periodically monitor

spectrum for holesHole location must be agreed upon between

TX and RXHole is then used for opportunistic

communication with minimal interference to noncognitive users

Overlay SystemsCognitive user has knowledge of

other user’s message and/or encoding strategyUsed to help noncognitive

transmissionUsed to presubtract noncognitive

interferenceRX1

RX2NCR

CR

Wireless Sensor and “Green” Networks

Energy (transmit and processing) is driving constraint Data flows to centralized location (joint compression) Low per-node rates but tens to thousands of nodes Intelligence is in the network rather than in the devices Similar ideas can be used to re-architect systems and

networks to be green

• Smart homes/buildings• Smart structures• Search and rescue• Homeland security• Event detection• Battlefield surveillance

Energy-Constrained Nodes

Each node can only send a finite number of bits.Transmit energy minimized by maximizing

bit timeCircuit energy consumption increases with

bit timeIntroduces a delay versus energy tradeoff

for each bit

Short-range networks must consider transmit, circuit, and processing energy.Sophisticated techniques not necessarily

energy-efficient. Sleep modes save energy but complicate

networking.

Changes everything about the network design:Bit allocation must be optimized across all

protocols.Delay vs. throughput vs. node/network

lifetime tradeoffs.Optimization of node cooperation.

Green” Cellular Networks

Minimize energy at both the mobile and base station viaNew Infrastuctures: cell size, BS placement,

DAS, Picos, relaysNew Protocols: Cell Zooming, Coop MIMO,

RRM, Scheduling, Sleeping, RelayingLow-Power (Green) Radios: Radio Architectures,

Modulation, coding, MIMO

Pico/Femto

Relay

DAS

Coop MIMO

How should cellularsystems be redesignedfor minimum energy?

Research indicates thatsignicant savings is possible

Crosslayer Design in Wireless Networks

ApplicationNetwork

AccessLinkHardwareTradeoffs at all layers of the protocol stack are optimized with respect to

end-to-end performanceThis performance is dictated by the application

Multiuser Channels

Uplink and Downlink

Downlink (Broadcast Channel or BC): One Transmitter to Many Receivers.

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

R1

R2

R3

x h1(t)x h21(t)

x

h3(t)

x h22(t)

Challenge: How to share the channel among multiuser usersUplink and Downlink typically duplexed in time or frequency

RANDOM ACCESS TECHNIQUES

7C29822.038-Cimini-9/97

Random Access Dedicated channels wasteful for data

Use statistical multiplexing

TechniquesALOHA and slotted ALOHACarrier sensing

Collision detection or avoidanceReservation protocols (similar to

deterministic access)

Retransmissions used for corrupted data

Poor throughput and delay characteristics under heavy loading

ALOHA Data is packetized Retransmit

When packets collide Pure ALOHA

send packet whenever data is available a collision occurs for any partial overlap of packets

Slotted ALOHA send packets during predefined timeslots avoids partial overlap of packets

Comments Inefficient for heavily loaded systems Capture effect (packets with high SINR decoded)

improves efficiency

.40

.30

.20

.10

0 0.5 1.0 1.5 2.0 3.0

G (Attempts per Packet TIme)

Thr

ough

put p

er P

acke

t Tim

e)

Pure Aloha

Slotted Aloha

RTS

Channel sensed before transmission To determine if it is occupied

More efficient than ALOHA fewer retransmissions

Carrier sensing is often combined with collision detection in wired networks (e.g. Ethernet)

not possible in a radio environment

Collision avoidance (busy tone, RTS/CTS) can be used

Carrier Sense Techniques

Wired Network

Busy Tone

Wireless Network

CTS

Deterministic Bandwidth Sharing

Frequency Division

Time Division

Code DivisionMultiuser Detection

Space Division (MIMO) Hybrid Schemes

Code Space

Time

Frequency Code Space

Time

Frequency Code Space

Time

Frequency

What is optimal?

Look to Shannon.

Multiple Access SS

Interference between users mitigated by code cross correlation

In downlink, signal and interference have same received power

In uplink, “close” users drown out “far” users (near-far problem)

)()2cos(5.5.)()(5.5.

))(2cos()2cos()()()()2(cos)()()(ˆ

12210

212211

1220

222

111

c

T

cc

cccc

T

cc

fdddttstsdd

dttftftstststftststx

b

b

Multiuser Detection In all CDMA systems and in

TD/FD/CD cellular systems, users interfere with each other.

In most of these systems the interference is treated as noise.Systems become interference-limitedOften uses complex mechanisms to

minimize impact of interference (power control, smart antennas, etc.)

Multiuser detection exploits the fact that the structure of the interference is knownInterference can be detected and

subtracted outBetter have a darn good estimate of the

interference

Ideal Multiuser Detection

Signal 1 Demod

IterativeMultiuserDetection

Signal 2Demod

- =Signal 1

- =

Signal 2

Why Not Ubiquitous Today? Power and A/D Precision

A/D

A/D

A/D

A/D

Multiuser Shannon Capacity

Fundamental Limit on Data Rates

Main drivers of channel capacity Bandwidth and received SINR Channel model (fading, ISI) Channel knowledge and how it is used Number of antennas at TX and RX

Duality connects capacity regions of uplink and downlink

Capacity: The set of simultaneously achievable rates {R1,…,Rn}with arbitrarily small probability of error

R1R2

R3

R1

R2

R3

Broadcast Channel Capacity

Region in AWGN

ModelOne transmitter, two receivers with

spectral noise density n1, n2: n1<n2.Transmitter has average power Pand total

bandwidth B.

Single User Capacity: Maximum achievable rate with asymptotically

small Pe

Set of achievable rates includes (C1,0) and (0,C2), obtained by allocating all resources to one user.

BnPBCi

i 1log

Rate Region: Time Division

Time Division (Constant Power)Fraction of time allocated to each user

is varied

Time Division (Variable Power)Fraction of time and power si allocated

to each user is varied

10;)1(, 2211 CRCRU

.10,)1(

;1log)1(,1log

21

2

22

1

11

ss

ss

P

BnBR

BnBRU

Rate Region: Frequency Division

Frequency DivisionBandwidth Bi and power Si allocated to each

user is varied.

BBBPPP

BnPBR

BnPBR

2121

22

222

11

111

,

;1log,1logU

Equivalent to TD for Bi=iB and Pi=isi.

Superposition Coding

Best user decodes fine pointsWorse user decodes coarse points

Code Division Superposition Coding

Coding strategy allows better user to cancel out interference from worse user.

DS spread spectrum with spreading gain G and cross correlation 12= 21 =G:

By concavity of the log function, G=1 maximizes the rate region.

DS without interference cancellation

PPP

SBnPBR

BnPBR 21

12

22

1

11 ;1log,1logU

PPP

GSGBnP

GBR

GBnP

GBR 21

12

22

1

11 ;

//1log,

/1logU

PPPGPGBn

PGBR

GPGBnP

GBR 21

12

22

21

11 ;

//1log,

//1logU

Broadcast and MAC Fading Channels

Goal: Maximize the rate region {R1,…,Rn}, subject to some minimum rateconstraints, by dynamic allocation of power, rate, and coding/decoding.

WirelessGateway

WiredNetwork

Broadcast: One Transmitter to Many Receivers.

Multiple Access: Many Transmitters to One Receiver.

R1

R2R3

x g1(t)

x g2(t)x g3(t)

Assume transmit power constraint and perfect TX and RX CSI

Fading Capacity Definitions

Ergodic (Shannon) capacity: maximum long-term rates averaged over the fading process.

Shannon capacity applied directly to fading channels.

Delay depends on channel variations. Transmission rate varies with channel quality.

Zero-outage (delay-limited*) capacity: maximum rate that can be maintained in all fading states.

Delay independent of channel variations. Constant transmission rate – much power needed

for deep fading.

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

Constant transmission rate during nonoutage Outage avoids power penalty in deep fades

*Hanly/Tse, IT, 11/98

Two-User Fading Broadcast Channel

+

+X[i]

n1[i]

n2[i]

Y1[i]

Y2[i]

x

x

h1[i]

h2[i]

+

+X[i]

n1[i]=n1[i]/h1[i]

n2[i]=n2[i]/h2[i]

Y1[i]

Y2[i]

At each time i:n={n1[i],n2[i]}

Ergodic Capacity Region*

Capacity region: ,where

The power constraint implies

Superposition coding and successive decoding achieve capacityBest user in each state decoded last

Power and rate adapted using multiuser water-filling: power allocated based on noise levels and user priorities

)()( PFP CPCergodic

}1,][1)(

)(1log)(

1

MjnnnPBn

nPBERC M

iijij

jnj

P

PnPEM

jjn

1

)(

*Li/Goldsmith, IT, 3/01

Zero-Outage Capacity Region*

The set of rate vectors that can be maintained for all channel states under power constraint P

Capacity region defined implicitly relative to power:For a given rate vector R and fading

state n we find the minimum power Pmin(R,n) that supports R.

RCzero(P) if En[Pmin(R,n)] P

)()( PFP CPC Nnzero

MjnnnPBn

nPBRC M

iijij

jj 1,

][1)(

)(1log)(

1

P

*Li and Goldsmith, IT, 3/01

Outage Capacity Region Two different assumptions about outage:

All users turned off simultaneously (common outage Pr)

Users turned off independently (outage probability vector Pr)

Outage capacity region implicitly defined from the minimum outage probability associated with a given rate

Common outage: given (R,n), use threshold policyIf Pmin(R,n)>s* declare an outage,

otherwise assign this power to state n.

Power constraint dictates s* :

Outage probability:

),(min*),(: min nRPEP

snRPn

*),(min:

)(snRPn

npPr

Independent Outage With independent outage cannot use the

threshold approach:Any subset of users can be active in each

fading state.

Power allocation must determine how much power to allocate to each state and which users are on in that state.

Optimal power allocation maximizes the reward for transmitting to a given subset of users for each fading stateReward based on user priorities and outage

probabilities.An iterative technique is used to maximize

this reward.Solution is a generalized threshold-decision

rule.

Minimum-Rate Capacity Region

Combines ergodic and zero-outage capacity:Minimum rate vector maintained in all

fading states.Average rate in excess of the minimum

is maximized.

Delay-constrained data transmitted at the minimum rate at all times.

Channel variation exploited by transmitting other data at the maximum excess average rate.

Minimum Rate Constraints Define minimum rates R* = (R*

1,…,R*M):

These rates must be maintained in all fading states.

For a given channel state n:

R* must be in zero-outage capacity region Allocate excess power to maximize excess

ergodic rate The smaller R*, the bigger the min-rate

capacity region

nRnRnnnPBn

nPBnR jjM

iijij

jj

*

1

)(,][1)(

)(1log)(

Comparison of Capacity Regions

For R* far from Czero boundary, Cmin-rate Cergodic

For R* close to Czero boundary, Cmin-rate CzeroR*

Min-Rate Capacity Region:

Large Deviation in User Channels

Symmetric channel with 40 dB difference in noises in each fading state (user 1 is 40 dB stronger in 1 state, and vice versa).

P = 10 mW, B = 100 KHz

Min-Rate Capacity Region:

Smaller Deviation

Symmetric channel with 20 dB difference in noises in each fading state (user 1 is 20 dB stronger in 1 state, and vice versa).

P = 10 mW, B = 100 KHz

Broadcast Channels with ISI

ISI introduces memory into the channel

The optimal coding strategy decomposes the channel into parallel broadcast channelsSuperposition coding is applied to each

subchannel.

Power must be optimized across subchannels and between users in each subchannel.

Broadcast Channel Model

Both H1 and H2 are finite IR filters of length m.

The w1k and w2k are correlated noise samples.

For 1<k<n, we call this channel the n-block discrete Gaussian broadcast channel (n-DGBC).

The channel capacity region is C=(R1,R2).

w1kH1(w)

H2(w)w2k

xk

y h x wk ii

m

kk i1 11

1

y h x wk ii

m

kk i2 21

2

Circular Channel Model

Define the zero padded filters as:

The n-Block Circular Gaussian Broadcast Channel (n-CGBC) is defined based on circular convolution:

{~} ( ,..., , ,..., )h h hi in

m 1 1 0 0

~ ~(( ))y h x w x h wk i

i

n

k i i kk i1 10

1

1 1 1

~ ~(( ))y h x w x h wk i

i

n

k i i kk i2 20

1

2 2 2

0<k<n

where ((.)) denotes addition modulo n.

Equivalent Channel Model

Taking DFTs of both sides yields

Dividing by H and using additional properties of the DFT yields

~ ~Y H X Wj j j j1 1 1 ~ ~Y H X Wj j j j2 2 2

0<j<n

~

Y X Vj j j1 1

Y X Vj j j2 2

where {V1j} and {V2j} are independent zero-mean Gaussian random variables with s lj l ljn N j n H l2 22 1 2 ( ( / )/| ~ | , , .

0<j<n

Parallel Channel Model

+

+X1

V11

V21

Y11

Y21

+

+Xn

V1n

V2n

Y1n

Y2n

Ni(f)/Hi(f)

f

Channel Decomposition

The n-CGBC thus decomposes to a set of n parallel discrete memoryless degraded broadcast channels with AWGN. Can show that as n goes to infinity, the circular and

original channel have the same capacity region

The capacity region of parallel degraded broadcast

channels was obtained by El-Gamal (1980) Optimal power allocation obtained by Hughes-

Hartogs(’75).

The power constraint on the original channel is converted by Parseval’s theorem to on the equivalent channel.

E x nPii

n

[ ]2

0

1

E X n Pii

n

[( ) ]

0

12 2

Capacity Region of Parallel Set

Achievable Rates (no common information)

Capacity Region For 0<b find {j}, {Pj} to maximize R1+bR2+l SPj. Let (R1

*,R2*)n,b denote the corresponding rate pair.

Cn={(R1*,R2

*)n,b : 0<b }, C=liminfn Cn .1n

PnP

PP

PR

PPP

R

jj

j j

jj

j jjj

jj

j jjj

jj

j j

jj

jjjj

jjjj

2: 2: 2

2

: 1: 11

,10

,)1(

1log5.)1(

1log5.

,)1(

1log5.1log5.

2121

2121

s

s

s

s

ssss

ssss

b

R1

R2

Optimal Power Allocation:

Two Level Water Filling

Capacity vs. Frequency

Capacity Region

Multiple Access Channel

Multiple transmitters Transmitter i sends signal Xi with power Pi

Common receiver with AWGN of power N0B

Received signal:NXYM

ii

1

X1

X2 X3

MAC Capacity Region

Closed convex hull of all (R1,…,RM) s.t.

For all subsets of users, rate sum equals that of 1 superuser with sum of powers from all users

Power Allocation and Decoding OrderEach user has its own power (no power

alloc.)Decoding order depends on desired rate

point

},...,1{,/1log 0 MSBNPBRSi

iSi

i

Two-User RegionSuperposition codingw/ interference canc.

SC w/ IC and timesharing or rate splitting

Frequency division

Time division

C1

C2

Ĉ1

Ĉ2

2,1,1log0

i

BNPBC i

i

,1logˆ,1logˆ10

22

20

11

PBN

PBCPBN

PBC

SC w/out IC

Fading and ISI MAC capacity under fading and ISI

determined using similar techniques as for the BC

In fading, can define ergodic, outage, and minimum rate capacity similar as in BC caseErgodic capacity obtained based on

AWGN MAC given fixed fading, averaged over fading statistics

Outage can be declared as common, or per user

MAC capacity with ISI obtained by converting to equivalent parallel MAC channels over frequency

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

2111211

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 any

scaling

1h

1P

2P 2h

+

+

21 PP

1h

2h+

MAC

BC

The BC from the MAC

0

2121

2121 ),;(),;,(

hhPPChhPPC BCMAC

Blue = Scaled BCRed = MAC 1

2

hh

0

BC in terms of MAC

MAC in terms of BC

PP

MACBC hhPPPChhPC

10

211121 ),;,(),;(

0

2121

2121 ),;(),;,(

hhPPChhPPC 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

PhPhR

221

22

222

22

22

2

121

21

222

121

1

)1log()1log(

)1log()1log(

ss

ss

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 capacity regions can be obtained by duality?Broadcast MIMO

Channels