EE360: Multiuser Wireless Systems and Networks
Lecture 1 Outline
Course Details
 Course Syllabus
 Course Overview

 Future Wireless Networks
 Multiuser Channels (Broadcast/MAC Channels)
 Spectral Reuse and Cellular Systems
 Ad-Hoc, Sensor, and Cognitive Radio Networks
Bandwidth Sharing in Multiuser Channels
 Overview of Multiuser Channel Capacity
 Capacity 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 Information
Nuts and Bolts



Prerequisites: EE359
Course Time and Location: MW 9:30-10:45. Y2E2 101.
Class Homepage: www.stanford.edu/class/ee360


Class Mailing List: ee360win01314-students (automatic for oncampus registered students).


Contains all required reading, handouts, announcements, HWs, etc.
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


Class presentation




Present a paper related to one of the course topics
HW 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 assignments


Read the required reading before lecture/discuss in class
Two assignments from book or other problems
Paper summaries


Two 2-4 page summaries of several articles
Each should be on a different topic from the syllabus
Project


Term project on anything related to multiuser wireless
Analysis, simulation and/or experiment



Must set up website for project



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


Will post proposal, progress report, and final report to website
Project proposal due Jan. 27 at midnight



Must contain some original research
2 can collaborate if project merits collaboration (scope, synergy)
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
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 Cellular
Wireless Internet Access
Wireless Multimedia
Sensor Networks
Smart Homes/Spaces
Automated Highways
In-Body Networks
All this and more …
Multiuser Channels:
Uplink and Downlink
Uplink (Multiple Access
Channel or MAC):
Many Transmitters
to One Receiver.
Downlink (Broadcast
Channel or BC):
One Transmitter
to Many Receivers.
R3
x
h3(t)
x
h22(t)
x
x
h1(t)
h21(t)
R2
R1
Challenge: How to share the channel among multiuser users
Uplink and Downlink typically duplexed in time or frequency
Spectral Reuse
Due to its scarcity, spectrum is reused
In licensed bands
and unlicensed bands
BS
Cellular, Wimax
Wifi, BT, UWB,…
Reuse introduces interference
Interference: Friend or Foe?

If treated as noise: Foe
P
SNR 
NI
Increases BER
Reduces capacity

If decodable (MUD): Neither friend nor foe

If exploited via cooperation and cognition:
Friend (especially in a network setting)
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: ???
BASE
STATION
MTSO
Rethinking “Cells” in Cellular
Small
Cell
Coop
MIMO
How should cellular
systems be designed?
Relay
DAS

Will gains in practice be
big or incremental; in
capacity or coverage?
Traditional cellular design “interference-limited”
 MIMO/multiuser detection can remove interference
 Cooperating BSs form a MIMO array: what is a cell?
 Relays change cell shape and boundaries
 Distributed antennas move BS towards cell boundary
 Small cells create a cell within a cell
 Mobile cooperation via relaying, virtual MIMO, analog network
coding.
Ad-Hoc/Mesh Networks
Outdoor Mesh
ce
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 spectrum


Utilize advanced communication and signal
processing techniques


Without degrading performance of existing users
Coupled 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

Underlay
 Cognitive
radios constrained to cause minimal
interference to noncognitive radios

Interweave
 Cognitive
radios find and exploit spectral holes
to avoid interfering with noncognitive radios

Overlay
 Cognitive
radios overhear and enhance
noncognitive radio transmissions
Knowledge
and
Complexity
Underlay Systems

Cognitive radios determine the interference their
transmission causes to noncognitive nodes

Transmit if interference below a given threshold
IP
NCR
NCR

CR
CR
The interference constraint may be met


Via wideband signalling to maintain interference
below the noise floor (spread spectrum or UWB)
Via multiple antennas and beamforming
Interweave Systems

Measurements indicate that even crowded spectrum
is not used across all time, space, and frequencies


Original motivation for “cognitive” radios (Mitola’00)
These holes can be used for communication



Interweave CRs periodically monitor spectrum for holes
Hole location must be agreed upon between TX and RX
Hole is then used for opportunistic communication with
minimal interference to noncognitive users
Overlay Systems

Cognitive user has knowledge of other
user’s message and/or encoding strategy
 Used
to help noncognitive transmission
 Used to presubtract noncognitive interference
CR
NCR
RX1
RX2
Wireless Sensor and “Green” Networks
•
•
•
•
•
•





Smart homes/buildings
Smart structures
Search and rescue
Homeland security
Event detection
Battlefield surveillance
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
Energy-Constrained Nodes

Each node can only send a finite number of bits.




Short-range networks must consider transmit,
circuit, and processing energy.



Transmit energy minimized by maximizing bit time
Circuit energy consumption increases with bit time
Introduces a delay versus energy tradeoff for each bit
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
Pico/Femto
Coop
MIMO
Relay
DAS
 Minimize



How should cellular
systems be redesigned
for minimum energy?
Research indicates that
signicant savings is possible
energy at both the mobile and base station via
New Infrastuctures: cell size, BS placement, DAS, Picos, relays
New Protocols: Cell Zooming, Coop MIMO, RRM,
Scheduling, Sleeping, Relaying
Low-Power (Green) Radios: Radio Architectures, Modulation,
coding, MIMO
Crosslayer Design in
Wireless Networks

Application

Network

Access

Link

Hardware
Tradeoffs at all layers of the protocol stack are
optimized with respect to end-to-end performance
This performance is dictated by the application
Multiuser Channels
Uplink and Downlink
Uplink (Multiple Access
Channel or MAC):
Many Transmitters
to One Receiver.
Downlink (Broadcast
Channel or BC):
One Transmitter
to Many Receivers.
R3
x
h3(t)
x
h22(t)
x
x
h1(t)
h21(t)
R2
R1
Challenge: How to share the channel among multiuser users
Uplink and Downlink typically duplexed in time or frequency
RANDOM ACCESS TECHNIQUES
Random Access

Dedicated channels wasteful for data


Use statistical multiplexing
Techniques



ALOHA and slotted ALOHA
Carrier sensing
 Collision detection or avoidance
Reservation protocols (similar to deterministic access)

Retransmissions used for corrupted data

Poor throughput and delay characteristics under
heavy loading
7C29822.038-Cimini-9/97

Data is packetized

Retransmit


Pure ALOHA



Slotted Aloha
.40
.30
.20
.10
Pure Aloha
0
0.5
1.0
1.5
2.0
G (Attempts per Packet TIme)
send packet whenever data is available
a collision occurs for any partial overlap of packets
Slotted ALOHA



When packets collide
Throughput per
Packet Time)
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
3.0
Carrier Sense Techniques

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
CTS
RTS
Busy Tone
Wired Network

Wireless Network
Collision avoidance (busy tone, RTS/CTS) can be used
Deterministic Bandwidth Sharing

Code Space
Frequency Division
Time

Time Division
Code Space
Frequency
Time

Code Division



Frequency
Multiuser Detection
Space Division (MIMO)
Hybrid Schemes
Code Space
Time
Frequency
What is optimal? Look to Shannon.
Multiple Access SS

Interference between users mitigated by code
cross correlation
Tb
xˆ (t )   1s1 (t ) sc21 (t ) cos 2 (2f c t )   2 s2 (t   ) sc 2 (t   ) sc1 (t ) cos( 2f c t ) cos( 2f c (t   )) dt
0
Tb
 .51d1  .5 2 d 2  sc1 (t ) sc 2 (t )dt  .5d1  .5d 2 cos( 2f c ) 12 ( )
0


In downlink, signal and interference have
same received power
In uplink, “close” users drown out “far” users
(near-far problem)


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-limited
Often 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 known


Interference can be detected and subtracted out
Better have a darn good estimate of the interference
Ideal Multiuser Detection
-
Signal 1
=
A/D
A/D
A/D
Signal 1
Demod
A/D
Iterative
Multiuser
Detection
Signal 2
Signal 2
Demod
-
=
Why Not Ubiquitous Today? Power and A/D Precision
Multiuser Shannon Capacity
Fundamental Limit on Data Rates
Capacity: The set of simultaneously achievable rates {R1,…,Rn}
with arbitrarily small probability of error
R3
R2
R1

R3
R1
Main drivers of channel capacity





R2
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
Broadcast Channel Capacity
Region in AWGN

Model
 One 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

P 
Ci  B log 1 

 ni B 

Set of achievable rates includes (C1,0) and (0,C2), obtained
by allocating all resources to one user.
Rate Region: Time Division

Time Division (Constant Power)

Fraction of time  allocated to each user is varied
U R
1

 C1 , R2  (1   )C 2 ;0    1
Time Division (Variable Power)

Fraction of time  and power si allocated to each user
is varied
 
 s1 
 s2  
U R1  B log 1 
, R2  (1   ) B log 1 
 ;
 
 n1 B 
 n2 B   


s 1  (1   )s 2  P,
0    1.


Rate Region: Frequency Division

Frequency Division

Bandwidth Bi and power Si allocated to each user is varied.
 



 
P
P
1
2
R1  B1 log 1 
;
U
, R2  B2 log 1 



n1 B1 
n2 B2   
 








P1  P2  P, B1  B2  B


Equivalent to TD for Bi=iB and Pi=isi.
Superposition Coding
Best user decodes fine points
Worse user decodes coarse points
Code Division

Superposition Coding

Coding strategy allows better user to cancel out interference from
worse user.






P1 
P2




U
R

B
log
1

,
R

B
log
1

;
P

P

P


1
2
1
2





n1 B 
n2 B  S1  









DS spread spectrum with spreading gain G and cross
correlation 12= 21 =G:







B
P1
B
P2

; P1  P2  P 
R

log
1

,
R

log
1

U

1
2





G
n1 B / G 
G
n2 B / G  S1 / G  










By concavity of the log function, G=1 maximizes the rate region.
DS without interference cancellation







B
P1
B
P2



R

log
1

,
R

log
1

;
P

P

P
U

1
2
1
2





G
n1 B / G  P2 / G 
G
n2 B / G  P1 / G  







Broadcast and MAC
Fading Channels
Broadcast:
One Transmitter
to Many Receivers.
Multiple Access:
Many Transmitters
to One Receiver.
Wireless
Gateway
x
x
g2(t)
Wired
Network
x
g3(t)
g1(t)
R2
R3
R1
Goal: Maximize the rate region {R1,…,Rn}, subject to some minimum rate
constraints, by dynamic allocation of power, rate, and coding/decoding.
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.




Zero-outage (delay-limited*) capacity: maximum rate
that can be maintained in all fading states.



Shannon capacity applied directly to fading channels.
Delay depends on channel variations.
Transmission rate varies with channel quality.
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
h1[i]
X[i]
n1[i]
x
+
Y1[i]
x
+
Y2[i]
h2[i]
n2[i]
At each time i:
n={n1[i],n2[i]}
n1[i]=n1[i]/h1[i]
X[i]
+
Y1[i]
+
Y2[i]
n2[i]=n2[i]/h2[i]
Ergodic Capacity

*
Region
Capacity region: Cergodic( P )  P F C (P ) ,where






P
(
n
)
j
,
C (P )  R j  E n  B log 1 
M



n
B

P
(
n
)
1
[
n

n
]


j
i
j
i 

i

1



1  j  M}
M

The power constraint implies En  Pj (n)  P
j 1

Superposition coding and successive decoding
achieve capacity


Best user in each state decoded last
Power and rate adapted using multiuser water-filling:
power allocated based on noise levels and user priorities
*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
C zero ( P )   P F  nN C (P )






P
(
n
)

j
,
C (P )   R j  B log 1 
M



 n j B   Pi (n)1[n j  ni ] 

i 1






1 j  M


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.
 RCzero(P) if En[Pmin(R,n)]  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 policy
 If Pmin(R,n)>s* declare an outage, otherwise assign this
power to state n.


min
Power constraint dictates s* : P  E n:P min ( R ,n ) s* P ( R, n)

Outage probability: Pr

n:P min ( R ,n ) s*
p ( n)

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 state
 Reward 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:




P
(
n
)
j
,
R j (n)  B log 1 
M


 n j B   Pi (n)1[n j  ni ] 
i 1



R j (n)  R*j 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
Comparison of Capacity Regions

For R* far from Czero boundary, Cmin-rate Cergodic

For R* close to Czero boundary, Cmin-rate CzeroR*
Min-Rate Capacity Region:
Large Deviation in User Channels
P = 10 mW,
B = 100 KHz
Symmetric channel with 40 dB difference in noises in each fading state
(user 1 is 40 dB stronger in 1 state, and vice versa).
Min-Rate Capacity Region:
Smaller Deviation
P = 10 mW,
B = 100 KHz
Symmetric channel with 20 dB difference in noises in each fading state
(user 1 is 20 dB stronger in 1 state, and vice versa).
Broadcast Channels with ISI

ISI introduces memory into the channel

The optimal coding strategy decomposes the
channel into parallel broadcast channels


Superposition coding is applied to each subchannel.
Power must be optimized across subchannels
and between users in each subchannel.
Broadcast Channel Model
w1k
xk
H1(w)
w2k
H2(w)




m
y1k   h1i xk i w1k
i 1
m
y2 k   h2i xk i w2 k
i 1
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).
Circular Channel Model

Define the zero padded filters as:
~ n
{hi }i 1  (h1 ,..., hm ,0,...,0)

The n-Block Circular Gaussian Broadcast Channel
(n-CGBC) is defined based on circular convolution:
n 1
~
~
y1k   h1i x(( k i )) w1k  xi  h1i  w1k
i 0
n 1
~
~
y2 k   h2i x(( k i )) w2 k  xi  h2i  w2 k
i 0
where ((.)) denotes addition modulo n.
0<k<n
Equivalent Channel Model

Taking DFTs of both sides yields
~
~
Y1 j  H1 j X j  W1 j
~
~
Y2 j  H2 j X j  W2 j

0<j<n
~
Dividing by H and using additional properties
of the DFT yields
Y1j  X j  V1j
Y2j  X j  V2j
0<j<n
where {V1j} and {V2j} are independent zero-mean Gaussian
~
random variables with s 2lj  n( Nl (2j / n)/|H lj |2 , l  1,2.
Parallel Channel Model
V11
X1
+
Y11
+
Y21
V21
V1n
Xn
f
+
Y1n
+
Y2n
V2n
Ni(f)/Hi(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).
n 1

The power constraint  E[ xi2 ]  nP on the original channel is
n 1
i 0
converted by Parseval’s theorem to  E[( X i) 2 ]  n 2 P on the
i 0
equivalent channel.
Capacity Region of Parallel Set

Achievable Rates (no common information)
  j Pj 


 j Pj



,
R1  .5  log 1 
 .5  log 1 
 (1   ) P  s 
s 1 j 
j:s 1 j s 2 j
j:s 1 j s 2 j
j
j
1j 



 (1   j ) Pj 
(1   j ) Pj 
  .5  log 1 
,
R2  .5  log 1 





P

s
s
j:s 1 j s 2 j
j
:
s

s
j
j
2
j
2
j
1
j
2
j




2
0   j  1,  Pj  n P 

Capacity Region
 For 0<b find {j}, {Pj} to maximize R1+bR2+l SPj.
(R1*,R2*)n,b

Let
denote the corresponding rate pair.

Cn={(R1*,R2*)n,b : 0<b  }, C=liminfn n1 Cn .
R2
b
R1
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:
M
Y   Xi  N
i 1
X1
X2
X3
MAC Capacity Region

Closed convex hull of all (R1,…,RM) s.t.


Ri  B log 1   Pi / N 0 B ,

iS
 iS



S  {1,..., M }
For all subsets of users, rate sum equals that of 1
superuser with sum of powers from all users
Power Allocation and Decoding Order


Each user has its own power (no power alloc.)
Decoding order depends on desired rate point
Two-User Region
Superposition coding
w/ interference canc.
C2
Time division
SC w/ IC and time
sharing or rate splitting
Ĉ2
Frequency division
SC w/out IC

Pi 
Ci  B log 1 
, i  1,2
 N0 B 


P1
ˆ
C1  B log 1 
,
N 0 B  P2 

Ĉ1
C1


P2
ˆ
C2  B log 1 
,
N 0 B  P1 

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 case



Ergodic 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
Comparison of MAC and BC

Differences:



Shared vs. individual power constraints
Near-far effect in MAC
Similarities:


P
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


BC capacity region typically only known for
(parallel) degraded channels


Convex optimization problem
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  s
h22 P2M
s2
 log( 1 
)  log( 1 
Opposite decoding order


h12 P1B
s2
h22 P2B
h22 P1B  s
)  R1B
B
)

R
2
2
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 capacity regions can be obtained by duality?
Broadcast MIMO Channels
