cooperation in wireless networks - wireless systems...
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Cooperation in Wireless Networks
Ivana Maric and Ron Dabora
Stanford
15 September 2008
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 1
Objectives
I The case for cooperation
I Types of cooperation
I Performance measures
I Cooperation schemesI Performance, limitationsI Building blocks
I Small networks, large scale networks
I Fundamental tradeoffs
I Introduce recent results
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 2
Outline
I Introduction
I Relaying strategies
I Conferencing and feedback
I Cooperation in networks with multiple communicating pairs
I Cooperation in fading channels
I Cooperation in large-scale networks
I Summary
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 3
Introduction
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 4
Introduction
I Motivation
I Basic measuresI CapacityI Scaling laws
I Channel modelsI Static channelsI Time-varying channels
I Diversity
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 5
Challenges
I Higher data rates and better coverageI USIIA 2007, RIAA 2006
I Dynamic nature: time-varying channel, users’ mobility,stochastically varying traffic
I Efficient spectrum allocation and coexistence of users
I Security and privacy
I Energy efficiency
I Operating large ad hoc networks
I Guaranteed rate (Quality-of-Service)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 6
Challenges
I Higher data rates and better coverageI USIIA 2007, RIAA 2006
I Dynamic nature: time-varying channel, users’ mobility,stochastically varying traffic
I Efficient spectrum allocation and coexistence of users
I Security and privacy
I Energy efficiency
I Operating large ad hoc networks
I Guaranteed rate (Quality-of-Service)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 6
Traditional Approach: A Network is a Collection of
Point-to-Point Links
I Current wireless networks (cellular networks, Wi-Fi) areviewed as a collection of point-to-point links
I To increase data rates the point-to-point rate is increased
I What happens when this approach is exhausted (tooexpensive, approaching the fundamental limits)?
⇒ Need to find methods to significantly increase data rate forthe same PtP link performance
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 7
Current View: Interference is Harmful
I Wireless networks are inherently broadcastI Any transmission is overheard by neighbouring nodes
T3
R1
R3
R2T2
T1
Interference is extremely harmful for existing wireless networkdesigns
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 8
Addressing the Challenges via Cooperation
I Nodes which are not the source or destination of a givenmessage help communicating the message
I Different types of cooperationI RelayingI Conferencing (iterative decoding)I Feedback
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 9
Addressing the Challenges via Cooperation
I Nodes which are not the source or destination of a givenmessage help communicating the message
I Different types of cooperationI RelayingI Conferencing (iterative decoding)I Feedback
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 9
Addressing the Challenges via Cooperation
I Nodes which are not the source or destination of a givenmessage help communicating the message
I Different types of cooperationI RelayingI Conferencing (iterative decoding)I Feedback
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 9
Addressing the Challenges via Cooperation
I Nodes which are not the source or destination of a givenmessage help communicating the message
I Different types of cooperationI RelayingI Conferencing (iterative decoding)I Feedback
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 9
Addressing the Challenges via Cooperation
I Nodes which are not the source or destination of a givenmessage help communicating the message
I Different types of cooperationI RelayingI Conferencing (iterative decoding)I Feedback
I Future applicationsI Ad-hoc networksI Sensor networks
I Cooperation takes advantage of the broadcast nature of thewireless channel
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 9
Not Just Theoretical
I Dlink High Speed 2.4GHz (802.11g) Wireless RangeExtender
I Under development for the 802.16 (WirelessMAN/WiMAX)I j - multihop relay specificationI m - advanced air interface
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 10
Let’s begin
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 11
Memoryless Point-to-Point Channels
I Gaussian channel
I zi - bandlimited AWGN, i.i.d., E{|zi |2} = N
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 12
Memoryless Point-to-Point Channels
I Gaussian channel
I zi - bandlimited AWGN, i.i.d., E{|zi |2} = N
I Discrete channel: xi ∈ X , yi ∈ Y
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 12
The Memoryless Point-to-Point Channel Model
I A channel is characterized by the conditional distribution ofits output at time i :
p(yi |y i−1, x i ), xi ∈ X and yi ∈ Y,
i = 1, 2, ...
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 13
The Memoryless Point-to-Point Channel Model
I A channel is characterized by the conditional distribution ofits output at time i :
p(yi |y i−1, x i ), xi ∈ X and yi ∈ Y,
i = 1, 2, ...
I p(yi |y i−1, x i ) takes into account all the effects of signalprocessing: time synchronization, frequency synchronization,PLL, equalizer,...
I A channel is called memoryless if p(yi |y i−1, x i ) = p(yi |xi )
X Y p 1
p 2 p 3
q 1 q 2
q 3
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 13
The Memoryless Point-to-Point Channel: BSC
I |X | = 2, |Y| = 2
I xi - BPSK signal
I Decoding takes place after a 2-level quantization at thereceiver with threshold at zero
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 14
The Memoryless Point-to-Point Channel: BSC
I |X | = 2, |Y| = 2
I xi - BPSK signal
I Decoding takes place after a 2-level quantization at thereceiver with threshold at zero
⇔ Binary symmetric channel
X Y p
p
1-p
1-p
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 14
Channel Capacity
X n Y n p(y n |x n ) Encoder Decoder W W ^
I R denotes the information rate in bits/symbol
I In 1948 Claude E. Shannon showed that transmittinginformation over a (memoryless) PtP channel p(y |x) can bedone with an arbitrarily small probability of error as long as
R ≤ maxp(x)
I (X ;Y )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 15
Channel Capacity
X n Y n p(y n |x n ) Encoder Decoder W W ^
I R denotes the information rate in bits/symbol
I In 1948 Claude E. Shannon showed that transmittinginformation over a (memoryless) PtP channel p(y |x) can bedone with an arbitrarily small probability of error as long as
R ≤ maxp(x)
I (X ;Y )
I The capacity achieving code is characterized by thedistribution p(x)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 15
Channel Capacity
X n Y n p(y n |x n ) Encoder Decoder W W ^
I R denotes the information rate in bits/symbol
I In 1948 Claude E. Shannon showed that transmittinginformation over a (memoryless) PtP channel p(y |x) can bedone with an arbitrarily small probability of error as long as
R ≤ maxp(x)
I (X ;Y )
I The capacity achieving code is characterized by thedistribution p(x)
I Average probability of error: P(n)e = Pr(W 6= W )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 15
Channel Capacity
X n Y n p(y n |x n ) Encoder Decoder W W ^
I R denotes the information rate in bits/symbol
I In 1948 Claude E. Shannon showed that transmittinginformation over a (memoryless) PtP channel p(y |x) can bedone with an arbitrarily small probability of error as long as
R ≤ maxp(x)
I (X ;Y )
I The capacity achieving code is characterized by thedistribution p(x)
I Average probability of error: P(n)e = Pr(W 6= W )
I Codebook: Generate 2nR i.i.d. codewordsPr(xn) =
∏n
i=1 pX (xi )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 15
Channel Capacity
X n Y n p(y n |x n ) Encoder Decoder W W ^
I R denotes the information rate in bits/symbol
I In 1948 Claude E. Shannon showed that transmittinginformation over a (memoryless) PtP channel p(y |x) can bedone with an arbitrarily small probability of error as long as
R ≤ maxp(x)
I (X ;Y )
I The capacity achieving code is characterized by thedistribution p(x)
I Average probability of error: P(n)e = Pr(W 6= W )
I Codebook: Generate 2nR i.i.d. codewordsPr(xn) =
∏n
i=1 pX (xi )⇒ ∀ε > 0 we can find n large enough s.t. ∃ at least one
codebook for which P(n)e ≤ ε
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 15
Channel Capacity: Converse
X n Y n p(y n |x n ) Encoder Decoder W W ^
R ≤ maxp(x)
I (X ;Y )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 16
Channel Capacity: Converse
X n Y n p(y n |x n ) Encoder Decoder W W ^
R ≤ maxp(x)
I (X ;Y )
I Conversely, if R > maxp(x) I (X ;Y ) then the averageprobability of error achieved by any code is bounded awayfrom zero for any n
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 16
Channel Capacity: Converse
X n Y n p(y n |x n ) Encoder Decoder W W ^
R ≤ maxp(x)
I (X ;Y )
I Conversely, if R > maxp(x) I (X ;Y ) then the averageprobability of error achieved by any code is bounded awayfrom zero for any n
I Definition: The Capacity of a channel is the maximal ratefor which reliable communication can be achieved
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 16
AWGN Channel Capacity
I Let P be an average power constraint on the channel input:
1
n
n∑
i=1
|xi(w)|2 ≤ P , w ∈ W
I The problem: Find the input distribution p(x) thatmaximizes I(X;Y) subject to average input powerconstraint P
I The solution: X ∼ CN (0,P)I The capacity:
C = log2
(
1 + |g |2 P
N
)
bits/transmission
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 17
Network Capacity
I A network is a collection of K nodes (sources, sinks) anddirected edges (links).
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 18
Network Capacity
I A network is a collection of K nodes (sources, sinks) anddirected edges (links).
I Assume symbol time synchronization of all elements in thenetwork
I Analyze the network throughput for a block of n symbols
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 18
Network Capacity
I A network is a collection of K nodes (sources, sinks) anddirected edges (links).
I Assume symbol time synchronization of all elements in thenetwork
I Analyze the network throughput for a block of n symbols
I Let node k, k = 1, 2, ...,K send message Wk ∈ Wk
I The rate is Rk =log2 |Wk |
n
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 18
Network Capacity
I A network is a collection of K nodes (sources, sinks) anddirected edges (links).
I Assume symbol time synchronization of all elements in thenetwork
I Analyze the network throughput for a block of n symbols
I Let node k, k = 1, 2, ...,K send message Wk ∈ Wk
I The rate is Rk =log2 |Wk |
n
I Let Dk be the set of nodes that decode Wk
I W jk , j ∈ Dk
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 18
Network Capacity
I A network is a collection of K nodes (sources, sinks) anddirected edges (links).
I Assume symbol time synchronization of all elements in thenetwork
I Analyze the network throughput for a block of n symbols
I Let node k, k = 1, 2, ...,K send message Wk ∈ Wk
I The rate is Rk =log2 |Wk |
n
I Let Dk be the set of nodes that decode Wk
I W jk , j ∈ Dk
I The probability of error for network transmission is
P(n)e = Pr
K⋃
k=1
⋃
j∈Dk
{
W jk 6= Wk
}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 18
Network Capacity Region
I The capacity region is the set of all rate vectors
(R1,R2, ...,RK ) such that the probability of error P(n)e can
be made arbitrarily small by taking n large enough
I Why it is important?
1. It is the theoretical upper bound2. Determines optimal communication strategies3. Leads to practical designs
T3
R1
R3
R2T2
T1
R1
R2
I Very hard to find
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 19
Multiuser Networks: The Multiple Access Channel
I The MAC: p(y |x1, x2)I p(x1, x2) = p(x1)p(x2)I Introduced by Shannon in 1961I Capacity known for both discrete and Gaussian channels
I Capacity [Ahlswede’71, Liao’72]I MIMO [Telatar’99]I Fading [Gallager’94, Shamai & Wyner’97, Tse & Hanly’98]
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 20
Multiuser Networks: The Broadcast Channel
I The BC: p(y1, y2|x)
I Introduced by Cover in 1972I Capacity known only for special cases
I Degraded channels [Bergmans’73,74, Gallager’74]I General BC with degraded message sets [Korner & Maron’77]I MIMO BC [Weingarten, Steinberg & Shamai’06]
I Best achievable region due to Marton’79I Best upper bound due to Nair & El-Gamal’07
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 21
Multiuser Networks: The Relay Channel
I The relay channel: p(y , y1|x , x1)
I The most basic form of cooperation
I Introduced by van der Meulen in 1968I Capacity known only for special cases
I Physically degraded channels [Cover & El-Gamal’79]I Gaussian relay channel with SNR → ∞ [Kramer’05]I Stochastically degraded relay channel with deterministic link
[Zhang’88]
I Fundamental schemes introduced by Cover & El-Gamal’79I Decode-and-forwardI Compress-and-forward
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 22
Multiuser Networks: The Interference Channel
I The interference channel: p(y1, y2|x1, x2)
I The building block for multiple-pairs communication
I Introduced by Shannon in 1961I Capacity known only for special cases
I Strong interference [Carleial’75, Sato’81, Costa & El-Gamal’87]I Gaussian IC with very weak interference [Shang, Kramer &
Chen’08, Motahari & Khandani’08]I Cognitive Gaussian ICI No interference
I Best achievable region due to Han & Kobayashi’81
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 23
The Interference Channel: Strong Interference vs. Weak
InterferenceWeak interference
I At least one of the cross-links isworse than the its respectivedirect link
Tx 2
Rx 2
Tx 1 Rx
1
I Decoding W1 at Rx2 reducesthe maximum rate of W1
I No single-letter expression forthe capacity region
I Capacity known for special cases
Strong interference
I The cross-links are better thanthe direct links
Tx 2
Rx 2
Tx 1 Rx
1
I Decoding W1 at Rx2 does notconstrain the maximum rate ofW1
I The capacity achieving schemeis known [Sato’81], [Han andKobayashi’81]
• The rates with weak interference are generally less than therates with (very) strong interference
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 24
The Cut-Set Upper Bound
I A general tool for outer bounding the capacity region of anetwork
I Let V = {1, 2, ...,K} index the network nodes
I Let R ij denote the information rate from node i to node j
I A cut is a partition of V into two sets S and S = V \ STheorem (∼Aref’80)
If the information rates{R ij
}are achievable then there exists
a joint distribution p(x(1), x(2), ..., x(K)
)such that for every cut
(S, S)∑
i∈S,j∈S
R ij ≤ I(
X(S);Y(S)|X(S)
)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 25
Operating Regimes: Half-Duplex vs. Full-Duplex
I We shall compare results for different operating regimes
I In full-duplex the nodes receive and transmit simultaneously
I In half-duplex a node can either receive or transmitI Often encountered in wireless systems in practice
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 26
Time-Varying Channels: Fast vs. Slow Fading
I When the channel is time varying, the received signal isgiven by
yr ,i = htr ,ixt,i + zi
I htr,i is the channel gain between the transmitter and receiverat time i
I htr ,i models a Rayleigh fading channel: htr ,i ∼ CN (0, 1)
I There are three types of Rayleigh fadingI Fast fading: htr,i ∼ CN (0, 1), i.i.d., i = 1, 2, ..., nI block fading: htr,i = htr , i = 1, 2, ..., nI Slow fading: htr,i = htr
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 27
Time-Varying Channels: Fast vs. Slow Fading
I When the channel is time varying, the received signal isgiven by
yr ,i = htr ,ixt,i + zi
I htr,i is the channel gain between the transmitter and receiverat time i
I htr ,i models a Rayleigh fading channel: htr ,i ∼ CN (0, 1)
I There are three types of Rayleigh fadingI Fast fading: htr,i ∼ CN (0, 1), i.i.d., i = 1, 2, ..., nI block fading: htr,i = htr , i = 1, 2, ..., nI Slow fading: htr,i = htr
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 27
Time-Varying Channels: Channel State Information
I CSI is the knowledge a network node has on the channels
I Two types: transmitter CSI (CSIT) and receiver CSI (CSIR)
I Let H denote the random channel state and let the channelbe defined by p(y |x , h).
I There are four possible CSI configurations:
CSIT CSIR Capacity
No No maxp(x) I (X ;Y )
p(y |x) =∑
h p(y |x , h)p(h)No Yes maxp(x) I (X ;Y |H)
Yes No variesYes Yes EH{maxp(x |h) I (X ;Y |h)}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 28
Time-Varying Channels: Outage Capacity
I Shannon’s capacity measure is also called ergodic capacityI Assumes that the channel is information stable (ex. i.i.d.
fading)I Application is delay tolerant
I For slow fading Rayleigh channels, the mutual informationI (X ;Y |h) is a random variable
I depends on the channel realization hI The channel is non-ergodic
I Note that for every R > 0, Pr (I (X ;Y |h) < R) > 0
⇒ The Shannon capacity is zero
I The event {h : I (X ;Y |h) < R} is called outage
I Outage capacity is the maximum rate that can beguaranteed for a given outage probability Pout:
supR
Pr(I (X ;Y |h) < R) ≤ Pout
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 29
Diversity
I Transmitting signals carrying the same information overdifferent paths in time, frequency or space
I Cooperation diversity is achieved when nodes forward toother nodes information
I Enhance desired informationI Facilitate interference cancellation
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 30
Diversity
I Transmitting signals carrying the same information overdifferent paths in time, frequency or space
I Cooperation diversity is achieved when nodes forward toother nodes information
I Enhance desired informationI Facilitate interference cancellation
I Diversity reduces outage probability
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 30
Diversity
I Transmitting signals carrying the same information overdifferent paths in time, frequency or space
I Cooperation diversity is achieved when nodes forward toother nodes information
I Enhance desired informationI Facilitate interference cancellation
I Diversity reduces outage probability
I Transmitter cooperation, receiver cooperation
I When the transmitters cooperate and also the receiverscooperate the system resembles a MIMO system
⇒ Distributed MIMO
I Differences between MIMO and distributed MIMO:I Messages known only at source nodesI Cannot perform antenna power allocationI Nodes may have half-duplex constraints
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 30
Time-Varying Channels: Finite-State Models
I Slow fading, fast fading are extreme cases
I An alternative model for time-varying channels with memoryI Correlated fading, multipathI Filters (pulse shape, IF and RF filters)I AGC, Timing, PLL, equalizer
I The finite-state channel (FSC) was introduced as early as1953 [McMillan’53]
I Time variations are represented by correlated states
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 31
Finite-State Channels
I Memory is captured by the state of the channel at the endof the previous symbol’s transmission
I Si is the channel state at time iI s0 is the initial channel state
p(yi , si |xi , xi−1, y i−1, s i−1, s0) = p(yi , si |xi , si−1)
I Si−1 contains all the history information for time iI S is finite
I ISI channel: Si−1 =(Xi−1,Xi−2, ...,Xi−J )
n i
x i
A/D
y i
ISI Channel
k
h(k)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 32
Analysis of Large Networks: Scaling Laws
I Finding the capacity region of even small networks is a verydifficult task
I Many possibilities for cooperation
I Scaling laws allow us to obtain insights on the performanceof large scale networks.
I Pioneering work of Gupta and Kumar’00
I NotationI f (n) = O(g(n)) ⇔ limn→∞
∣∣∣f (n)g(n)
∣∣∣ < ∞
I f (n) = Θ(g(n)) ⇔ f (n) = O(g(n)) and g(n) = O(f (n))
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 33
Analysis of Large Networks: Scaling Laws
I Definitions:I Assume a network that consists of n nodes that form m
source-destination pairs.I Let dl denote the distance between source and destination
for pair l ≤ mI The transport capacity is defined as
CT = sup(R1,R2,...,Rm) feasible
m∑
l=1
Rldl
I The transport capacityI Provides a single number which summarizes what a network
can deliverI Follows a scaling law such as
CT (n) = Θ(√
n),O(n) bit-meters/second
I Does not provide explicit information on the individual rates
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 34
Some Important Questions
I How to incorporate relaying into the design of a network?I Compare performance of different schemesI Under what conditions capacity is achievedI What are the maximum rate gains we can expect from
adding relays to the network?
I Different aspects of relaying that arise when consideringmultiple communicating pairs
I Do not exist in the classic relay channel
I Understand the fundamental performance tradeoffsassociated with node cooperation
I Analysis of cooperation in large scale networks
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 35
Cooperative Strategies
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 36
In This Section...
I Decode-and-forward
I Compress-and-forward
I Amplify-and-forward
I Capacity upper bound
I Performance comparison
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 37
Cooperation in Wireless Networks
T3
R1
R3
R2T2
T1
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 38
Cooperation in Wireless Networks
T3
R1
R3
R2T2
T1
I Traditional approach: multihop routing
I Many point-to-point links
I Intermediate nodes store and forward packets
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 38
BroadcastI Wireless networks are inherently broadcast
I Any transmission is overheard by neighboring nodes
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 39
BroadcastI Wireless networks are inherently broadcast
I Any transmission is overheard by neighboring nodes
T3
R1
R3
R2T2
T1
I Interference is harmful for current wireless network designsI Cooperative strategies exploit broadcast
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 39
Relay Channel
I Message W ∈ {1, . . . ,M} sent at rate R
I Encoding at the source: X n1 = f1(W )
I At the relay at time i : X2,i = f2,i(Yi−12 ), i = 2, . . . , n
I Decoding: W = g(Y n3 )
I R = log2 M/n
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 40
Decode-and-Forward
I Exploit broadcast transmission at the source
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 41
Decode-and-Forward
I Exploit broadcast transmission at the source
I Source and relay transmit simultaneously
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 41
Decode-and-Forward
I Exploit broadcast transmission at the source
I Source and relay transmit simultaneously
I Messages sent in blocks: w1,w2, . . . ,wb, . . .
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 41
Decode-and-Forward
I Exploit broadcast transmission at the source
I Source and relay transmit simultaneously
I Messages sent in blocks: w1,w2, . . . ,wb, . . .
I Two random codebooks: xn1 , xn
2 generated withp(x2)p(x1|x2)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 41
Superposition Coding
I Random codebooks xn1 , xn
2 generated with p(x2)p(x1|x2)
In block b:
I The source: xn1 (hb(wb−1),wb) block Markov encoding
I The relay: xn2 (hb(wb−1))
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 42
Decode-and-Forward Strategies
I Irregular encoding, successive decodingI Codebooks xn
1 , xn2 have different sizes
I Regular encoding, sliding-window decodingI Decoding over two block
I Regular encoding, backward decodingI Decoding starts from the last received block
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 43
Decode-and-Forward
R ≤ I (X1;Y2|X2)
R ≤ I (X1;Y3|X2) + I (X2;Y3) = I (X1,X2;Y3)
R = maxp(x1,x2)
min{I (X1;Y2|X2), I (X1,X2;Y3)}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 44
Decode-and-Forward in AWGN Channels
I Choose: Gaussian p(x1, x2)I E [|X1|2] = P1, E [|X2|2] = P2, E [X1X
∗2 ] = ρ
√P1P2
I Superposition codebook:
I Gen. symbols: X10 ∼ CN (0, (1 − ρ2)P1), X2 ∼ CN (0,P2)
I In block b: xn10(wb), xn
2 (wb−1)
xn1 (wb−1,wb) = xn
10(wb) + ρ√
P1P2
xn2 (wb−1)
Y2 = h12X1 + Z2
Y3 = h13X1 + h23X2 + Z3
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 45
DF Rate in AWGN Channels
R = maxρ
min
{
log2
(
1 +|h12|2(1 − |ρ|2)P1
N
)
,
log2
(
1 +|h13|2P1
N+
|h23|2P2
N+
2Re{ρh13h∗23}
√P1P2
N
)}
I Signals coherently-combinedI Relay signal perfectly phase-aligned with the source signalI Not practicalI Decoding constraint at the relay can be severeI DF optimal for |h12| → ∞: source and relay act as two
transmit antennasI DF performs well when the relay is close to the source
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 46
Antenna-Clustering Capacity
I Generalizes to multiple relays
I Relays act as a multiple-transmit antenna
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 47
Classic Multihop
R = min
{
T log2(1 +|h12|2P1
TN), T log2(1 +
|h23|2P2
TN)
}
I For α = 2, performs worse then using no relay at all
I Gains for α > 2 and for half-duplex relaysI α-path-loss exponent
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 48
DF in Half-Duplex Relay Channel
I All nodes know a priori when a relay listens/talks
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 49
DF in Half-Duplex Relay Channel
I All nodes know a priori when a relay listens/talks
I Mode modulation: data modulates listen/talk interval
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 49
DF in Half-Duplex Relay Channel
I All nodes know a priori when a relay listens/talks
I Mode modulation: data modulates listen/talk interval
−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 10
0.5
1
1.5
2
2.5
3
3.5
4
4.5 cut−set bound for afixed slot strategy
DF, fixed
DF, random
Pr(M2=L) for DF
Pr(M2=L) for cut−set bound
relay off
d
Rat
e [b
it/us
e]
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 49
Compress-and-Forward
I Relay does not decode the source message
I Relay quantizes Y n2 into quantization codeword Y n
2I By finding a jointly typical yn
2 with received yn2
I Three codebooks: xn1 (wb), y
n2 (sb−1, zb), x
n2 (sb−1)
How does relay operate?
I In block b: knows sb−1, decides on zb thru quantization
I Obtains y2(sb−1, zb)
What does it send?
I Binning: each z randomly assigned to bin s
I In block b + 1 : sends x2(sb) such that zb ∈ sb
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 50
Compress-and-Forward
Destination in block b + 1:
I Decodes sbI Determines zb ∈ sbI Knows y2(sb−1, zb), x2(sb−1)
I Decodes wb using (y2(sb−1, zb), y3,b)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 51
Compress-and-Forward
Destination in block b + 1:
I Decodes sb → RQ ≤ I (X2;Y3)
I Determines zb ∈ sbI Knows y2(sb−1, zb), x2(sb−1)
I Decodes wb using (y2(sb−1, zb), y3,b) R ≤ I (X1; Y2,Y3|X2)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 52
Compress-and-Forward Rate
R = I (X1; Y2,Y3|X2)
subject toI (Y2;Y2|Y3X2) ≤ I (X2;Y3)
for p(x1)p(x2)p(y2|x2, y2)p(y1, y2|x1, x2)
I R is single-user rate when receiver has two antennas
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 53
Compress-and-Forward in AWGN Channels
I Choose: Y2 = Y2 + Z2 Z2 ∼ CN (0, N2)
I For smallest N2 choose: I (Y2; Y2|X2Y3) = I (X2;Y3)
N2 = NP1(|h12|2 + |h13|2) + N
P2|h23|2
R = log2
(
1 +P1|h12|2N + N2
+P1|h13|2
N
)
I Optimal for |h23| → ∞: relay and destination act astwo-receive antenna
I CF performs well when the relay is close to destination
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 54
Antenna-Clustering Capacity
I Generalizes to multiple relays
I Relays act as a multiple-receive antenna
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 55
Antenna-Clustering Capacity
I Two closely spaced clusters: DF and CF
I Achieves optimal scaling behavior
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 56
Amplify-and-Forward
I In discrete channel: X2,i = Y2,i−1, Y ⊆ X
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 57
Amplify-and-Forward
I In discrete channel: X2,i = Y2,i−1, Y ⊆ XI In Gaussian channel: X2,i = aiY2,i−1 i = 1, . . . , n
I ai chosen to satisfy power constraint
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 57
Amplify-and-Forward
I In discrete channel: X2,i = Y2,i−1, Y ⊆ XI In Gaussian channel: X2,i = aiY2,i−1 i = 1, . . . , n
I ai chosen to satisfy power constraint
I At the destination channel with ISI:
Y3,i = h13X1,i + h23X2,i + Z3,i
= h13X1,i + ah12h23X1,i−1 + Z ′3,i
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 57
Amplify-and-Forward
I In discrete channel: X2,i = Y2,i−1, Y ⊆ XI In Gaussian channel: X2,i = aiY2,i−1 i = 1, . . . , n
I ai chosen to satisfy power constraint
I At the destination channel with ISI:
Y3,i = h13X1,i + h23X2,i + Z3,i
= h13X1,i + ah12h23X1,i−1 + Z ′3,i
I Waterfilling optimization of the spectrum of X n1
I Relay should not always transmit with maximum power
I In low-SNR: bursty AF improves performance
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 57
AF Scaling Capacity
I Optimal scaling as number of relays increases
I Requires coherent combining of relay signals
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 58
Cut-Set Bound on Capacity
I Cut: partition of the set of nodes into two sets: (S, S)
I W (S)- set of messages with source in S and sink in SI Choose encoders (inputs): PX .
I Denote as R(PX ,S) set of rates that satisfies:∑
w∈W (S)
Rw ≤ I (XS ;YS |XS) (1)
I Cut-set bound for fixed PX :
R(PX ) =⋂
S
R(PX ,S)
I Cut-set bound:
R =⋃
PX
R(PX )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 59
Cut-Set Bound ExamplesI Point-to-point channel
R =⋃
PX
I (X ;Y )
I Relay Channel
R =⋃
PX1X2
min{I (X1;Y2,Y3|X2), I (X1,X2;Y3)}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 60
Performance Comparison
P1 = P2 = 10, N=1,α = 2
−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 10
1
2
3
4
5
6
upper boundDF
ρ for DF
CF
relay off
AF
d
Ra
te [b
it/u
se]
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 61
Cooperative Strategies: Summary
I DF: when relay is close to source
I CF: when relay is close to destination
I Generalize to multiple relays
I Capacity results are rare
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 62
Conferencing and Feedback
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 63
In This Section...
I Cooperation via conferencing
I FeedbackI Fundamental results for memoryless channelsI Finite-state channels
I Summary
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 64
Conferencing
C 21 C 12
Rx 1
Rx 2
Tx
Y 1 n
Y 2 n
I Conferencing refers to two users interactively helping eachother decode their messages:
I The transmission over the wireless medium is typicallyreceived by users in the vicinity of the target user
I Users have dedicated (orthogonal) links between them, overwhich they communicate
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 65
Multi-Step Conferencing
Tx
W 21
(1)
Rx 1
Rx 2
W 12
(1)
Step 1
W 21
(2)
Rx 1
Rx 2
W 12
(2)
Step 2
W 21
(K)
Rx 1
Rx 2
W 12
(K)
Step K
W 2
W 1 ^
^
I A conference can span several cyclesI At each cycle receivers use more refined knowledge on the
other receiver’s channel outputI Decoding takes place after the last cycleI Admissible conference: the total rates of the conference
messages is less than the capacity of the conference links
1
n
K∑
k=1
log2
∣∣W(k)
ij
∣∣ ≤ Cij , (i , j) ∈
{(1, 2), (2, 1)
}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 66
A Conference: Formal Definition
Tx
W 21
(1)
Rx 1
Rx 2
W 12
(1)
Step 1
W 21
(2)
Rx 1
Rx 2
W 12
(2)
Step 2
W 21
(K)
Rx 1
Rx 2
W 12
(K)
Step K
W 2
W 1 ^
^
I An (C12,C21)-admissible K -cycle conference between Rx1
and Rx2 consists ofI K message sets from node i to node j , (i , j) =
{(1, 2), (2, 1)
}
W(k)ij =
{
1, 2, ..., 2nR(k)ij
}
, k = 1, 2, ..., K .
I K pairs of mapping functions
h(k)12 : Yn
1 ×W(1)21 ×W(2)
21 × · · · ×W(k−1)21 7→ W(k)
12
h(k)21 : Yn
2 ×W(1)12 ×W(2)
12 × · · · ×W(k)12 7→ W(k)
21
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 67
Full Cooperation
C 21 C 12
Rx 1
Rx 2
Tx
Y 1 n
Y 2 n
I Full cooperation: When each receiver sends his channeloutput to the other receiver
I Y n1 becomes available at Rx2
I Y n2 becomes available at Rx1
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 68
Full Cooperation
C 21 C 12
Rx 1
Rx 2
Tx
Y 1 n
Y 2 n
I Full cooperation: When each receiver sends his channeloutput to the other receiver
I Y n1 becomes available at Rx2
I Y n2 becomes available at Rx1
I Full cooperation can be achieved with a single cycle ifI C12 = H(Y1|Y2) and C21 = H(Y2|Y1)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 68
Full Cooperation
C 21 C 12
Rx 1
Rx 2
Tx
Y 1 n
Y 2 n
I Full cooperation: When each receiver sends his channeloutput to the other receiver
I Y n1 becomes available at Rx2
I Y n2 becomes available at Rx1
I Full cooperation can be achieved with a single cycle ifI C12 = H(Y1|Y2) and C21 = H(Y2|Y1)
⇒ In one step Rx2 can send to Rx1 enough information thatwill allow Rx1 to recover Y n
2I Using a scheme by Slepian & Wolf’73I Rate I (X ; Y1, Y2) is achievable at Rx1
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 68
Full Cooperation
C 21 C 12
Rx 1
Rx 2
Tx
Y 1 n
Y 2 n
I Full cooperation: When each receiver sends his channeloutput to the other receiver
I Y n1 becomes available at Rx2
I Y n2 becomes available at Rx1
I Full cooperation can be achieved with a single cycle ifI C12 = H(Y1|Y2) and C21 = H(Y2|Y1)
⇒ In one step Rx2 can send to Rx1 enough information thatwill allow Rx1 to recover Y n
2I Using a scheme by Slepian & Wolf’73I Rate I (X ; Y1, Y2) is achievable at Rx1
I We will focus on results for partial cooperation
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 68
Conferencing: MAC
I The encoders exchange messages prior to transmission
I The capacity region [Willems’83]
R1 ≤ I (X1;Y |X2,U)+C12
R2 ≤ I (X2;Y |X1,U)+C21
R1 + R2 ≤ min{I (X1,X2;Y |U)+C12 + C21, I (X1,X2;Y )
}
for p(u)p(x1|u)p(x2|u)
I This is achieved with a single conference step
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 69
Conferencing: MAC
I The encoders exchange messages prior to transmission
I The capacity region [Willems’83]
R1 ≤ I (X1;Y |X2,U)+C12
R2 ≤ I (X2;Y |X1,U)+C21
R1 + R2 ≤ min{I (X1,X2;Y |U)+C12 + C21, I (X1,X2;Y )
}
for p(u)p(x1|u)p(x2|u)
I This is achieved with a single conference step
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 69
Conferencing: Relay Channel
I Compare two schemes (C = Crd + Cdr ):I Single step (classic relaying, Cdr = 0)I Single cycle with
I Step 1: CF from destination to relayI Step 2: DF from relay to destination
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 70
Conferencing: Broadcast Channel
1I(X;Y )
2I(X;Y )
2I(X;Y )+C
1I(X;Y )
R2
R1
C12
12
I When the channel is physically degraded, a single conferencestep achieves capacity
x
y 1
p(y 1 |x)
y 2
p(y 2 |y 1 )
I It is enough to let the strong receiver assist the weak receiver
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 71
Conferencing: Broadcast Channel
1I(X;Y )
2I(X;Y )
2I(X;Y )+C
1I(X;Y )
R2
R1
C12
12
I When the channel is physically degraded, a single conferencestep achieves capacity
x
y 1
p(y 1 |x)
y 2
p(y 2 |y 1 )
I It is enough to let the strong receiver assist the weak receiver
I For the general BCI It is still an open question whether higher rates can be
achieved with multiple stepsI Can design a K -cycle conference using K − 1 CF cycles and
a final DF step
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 71
Feedback
I PtP channel: the receiver sends back information to thetransmitter
I Allows transmitter to adapt its signal to the channel
Xi = fi (W ,Y i−1)
I Network: the wireless medium is a broadcast mediumI Signals received at nodes in the vicinity of the destination are
correlated with the signal at the destination node
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 72
Feedback in Multiuser Scenarios
I In Multiuser scenarios feedback facilitates both direct andindirect cooperation
I Direct: Feedback sent from the destination receiverI Indirect: Feedback sent from neighbouring receivers
I Consider for example the BCI Feedback from one receiver can increase the rate to both
receivers
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 73
Memoryless Multiuser Scenarios
I Sometimes feedback does not helpI The PtP DMC (p(yn|xn) =
∏n
i=1 p(yi |xi ))I The physically degraded DMBC [El-Gamal’78,81]
x
y 1
p(y 1 |x)
y 2
p(y 2 |y 1 )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 74
Memoryless Multiuser Scenarios
I Sometimes feedback does not helpI The PtP DMC (p(yn|xn) =
∏n
i=1 p(yi |xi ))I The physically degraded DMBC [El-Gamal’78,81]
x
y 1
p(y 1 |x)
y 2
p(y 2 |y 1 )
I Feedback does help in the following scenarios:I The discrete, memoryless MAC [Gaarder & Wolf’75]I The discrete, memoryless relay channel
I Feedback achieves the cut-set bound [Cover & El-Gamal’79]
I The general BC [Ozarow’79]I Including the stochastically degraded channel
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 74
Channels with Memory: Finite-State Channels
I The memory for time i is represented by the state Si−1
I The PtP-FSC:
p(yi , si |xi , xi−1, y i−1, s i−1, s0) = p(yi , si |xi , si−1)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 75
Channels with Memory: Finite-State Channels
I The memory for time i is represented by the state Si−1
I The PtP-FSC:
p(yi , si |xi , xi−1, y i−1, s i−1, s0) = p(yi , si |xi , si−1)
I The FS-MAC:
p(yi , si |x1,i , x2,i , xi−11,1 , x i−1
2,1 , y i−1, s i−1, s0) = p(yi , si |x1,i , x2,i , si−1)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 75
Channels with Memory: Finite-State Channels
I The memory for time i is represented by the state Si−1
I The PtP-FSC:
p(yi , si |xi , xi−1, y i−1, s i−1, s0) = p(yi , si |xi , si−1)
I The FS-MAC:
p(yi , si |x1,i , x2,i , xi−11,1 , x i−1
2,1 , y i−1, s i−1, s0) = p(yi , si |x1,i , x2,i , si−1)
I The FSBC:
p(yi , zi , si |xi , xi−1, y i−1, z i−1, s i−1, s0) = p(yi , zi , si |xi , si−1)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 75
Finite-State Channels
I Can model effects beyond the physical propagation mediumI Filters, loops
I Example: to incorporate the effect of a K -tap equalizer, thestate Si−1 can also be a function of Y i−1
i−K
I Si−1 = f (Y i−1i−K )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 76
Finite-State Channels
I Can model effects beyond the physical propagation mediumI Filters, loops
I Example: to incorporate the effect of a K -tap equalizer, thestate Si−1 can also be a function of Y i−1
i−K
I Si−1 = f (Y i−1i−K )
I Useful for analyzing correlated fading between the twoextremes of fast (i.i.d.) and slow
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 76
Finite-State Channels
I Can model effects beyond the physical propagation mediumI Filters, loops
I Example: to incorporate the effect of a K -tap equalizer, thestate Si−1 can also be a function of Y i−1
i−K
I Si−1 = f (Y i−1i−K )
I Useful for analyzing correlated fading between the twoextremes of fast (i.i.d.) and slow
I Notation: Directed Mutual Information
I (X n → Y n) =
n∑
i=1
I (X i ;Yi |Y i−1)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 76
Finite-State Channels
I Can model effects beyond the physical propagation mediumI Filters, loops
I Example: to incorporate the effect of a K -tap equalizer, thestate Si−1 can also be a function of Y i−1
i−K
I Si−1 = f (Y i−1i−K )
I Useful for analyzing correlated fading between the twoextremes of fast (i.i.d.) and slow
I Notation: Directed Mutual Information
I (X n → Y n) =
n∑
i=1
I (X i ;Yi |Y i−1)︸ ︷︷ ︸
H(X i |Y i−1)−H(X i |Y i )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 76
Finite-State Channels: Capacity of the PtP-FSC
I The capacity of a channel with memory is usually given by alimiting expression as the blocklength n → ∞
I We must verify that the limit exists and is finiteI Otherwise the channel does not support reliable
communication in the Shannon sense
I We assume no CSI
I Capacity without feedback [Gallager’68]
C = limn→∞
maxp(xn)
mins0∈S
1
nI (X n;Y n|s0)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 77
Finite-State Channels: Capacity of the PtP-FSC
I The capacity of a channel with memory is usually given by alimiting expression as the blocklength n → ∞
I We must verify that the limit exists and is finiteI Otherwise the channel does not support reliable
communication in the Shannon sense
I We assume no CSI
I Capacity without feedback [Gallager’68]
C = limn→∞
maxp(xn)
mins0∈S
1
nI (X n;Y n|s0)
I Capacity with feedback [Permuter, Weissman &Goldsmith’08]
CFB = limn→∞
max∏ni=1 p(xi |x i−1,y i−1)
mins0∈S
1
nI (X n → Y n|s0)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 77
Remark
I Capacity without feedback [Gallager’68]
C = limn→∞
maxp(xn)
mins0∈S
1
nI (X n;Y n|s0)
I Capacity with feedback
CFB = limn→∞
max∏ni=1 p(xi |x i−1,y i−1)
mins0∈S
1
nI (X n → Y n|s0)
I Feedback increases the capacity of the PtP-FSC [Permuteret al.’08]
I In contrast to the DMC
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 78
The Finite-State Broadcast Channel with Feedback and
Cooperation
Encoder W
1
W 2
W 0
Broadcast Channel
p(z,y,s|x,s’)
X i
Z i Decoder 2
C
Y i Decoder 1
Z i-1
W 0
W 0 ^
^ ^
W 2 ^
W 1 ^
D A
Z i-1
Y i-1
B D
D
I 8 possible configurations
I Switch C facilitates full cooperation
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 79
The Finite-State Broadcast Channel with Feedback and
Cooperation: Conclusions
I When all switches are openI FSBC without feedback/cooperation
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 80
The Finite-State Broadcast Channel with Feedback and
Cooperation: Conclusions
I When all switches are openI FSBC without feedback/cooperation
I When the FSBC is physically degraded capacity is achievedusing a superposition codebook with memory
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 80
The Finite-State Broadcast Channel with Feedback and
Cooperation: Conclusions
I When all switches are openI FSBC without feedback/cooperation
I When the FSBC is physically degraded capacity is achievedusing a superposition codebook with memory
I Feedback can help the physically degraded FSBCI Although it does not help the physically degraded DMBC
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 80
The Finite-State Broadcast Channel with Feedback and
Cooperation: Conclusions
I When all switches are openI FSBC without feedback/cooperation
I When the FSBC is physically degraded capacity is achievedusing a superposition codebook with memory
I Feedback can help the physically degraded FSBCI Although it does not help the physically degraded DMBC
I When switch C is closed the channel behaves as a physicallydegraded channel
I Capacity is achieved with a superposition codebook for allfeedback configurations
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 80
The Finite-State Broadcast Channel with Feedback and
Cooperation: Conclusions
I When all switches are openI FSBC without feedback/cooperation
I When the FSBC is physically degraded capacity is achievedusing a superposition codebook with memory
I Feedback can help the physically degraded FSBCI Although it does not help the physically degraded DMBC
I When switch C is closed the channel behaves as a physicallydegraded channel
I Capacity is achieved with a superposition codebook for allfeedback configurations
I When switch C is open and feedback comes from one useronly
I Capacity achieved if the channel is physically degraded andthe strong user is sending feedback
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 80
Summary
I Conferencing helps by successively refining the knowledgeeach node has on the received signal at the other node
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 81
Summary
I Conferencing helps by successively refining the knowledgeeach node has on the received signal at the other node
I For the MAC a single cycle achieves capacity
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 81
Summary
I Conferencing helps by successively refining the knowledgeeach node has on the received signal at the other node
I For the MAC a single cycle achieves capacityI For the physically degraded BC a single step achieves
capacity
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 81
Summary
I Conferencing helps by successively refining the knowledgeeach node has on the received signal at the other node
I For the MAC a single cycle achieves capacityI For the physically degraded BC a single step achieves
capacityI For the relay channel:
I When C/g is high ⇒ single CF stepI When C/g is low ⇒ single DF stepI For intermediate values of C/g ⇒ iterative decoding
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 81
Summary
I Conferencing helps by successively refining the knowledgeeach node has on the received signal at the other node
I For the MAC a single cycle achieves capacityI For the physically degraded BC a single step achieves
capacityI For the relay channel:
I When C/g is high ⇒ single CF stepI When C/g is low ⇒ single DF stepI For intermediate values of C/g ⇒ iterative decoding
I Feedback allows the transmitter to adapt its transmissionaccording to the received signal
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 81
Summary
I Conferencing helps by successively refining the knowledgeeach node has on the received signal at the other node
I For the MAC a single cycle achieves capacityI For the physically degraded BC a single step achieves
capacityI For the relay channel:
I When C/g is high ⇒ single CF stepI When C/g is low ⇒ single DF stepI For intermediate values of C/g ⇒ iterative decoding
I Feedback allows the transmitter to adapt its transmissionaccording to the received signal
I Finite-state channels allow modelling of propagation as wellas implementation aspects
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 81
Summary
I In general, feedback is useful in network scenariosI For PtP-DMC feedback does not help
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 82
Summary
I In general, feedback is useful in network scenariosI For PtP-DMC feedback does not help
I When the channel has memory ⇒ feedback is, in general,useful also for PtP links
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 82
Summary
I In general, feedback is useful in network scenariosI For PtP-DMC feedback does not help
I When the channel has memory ⇒ feedback is, in general,useful also for PtP links
I In broadcast scenarios with full cooperationI Capacity achieving schemes can be derived both with and
without feedback
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 82
Summary
I In general, feedback is useful in network scenariosI For PtP-DMC feedback does not help
I When the channel has memory ⇒ feedback is, in general,useful also for PtP links
I In broadcast scenarios with full cooperationI Capacity achieving schemes can be derived both with and
without feedback
I In multiuser scenarios feedback from a neighbouring nodecan help other nodes to communicate
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 82
Summary First Half
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 83
Summary First Half
I Cooperation is an important tool for coping with the designchallenges of future wireless networks
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 84
Summary First Half
I Cooperation is an important tool for coping with the designchallenges of future wireless networks
I Channel modelsI Discrete, AWGNI Fading: fast, block, slow
I Diversity
I Finite-state: correlated time variations
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 84
Summary First Half
I Cooperation is an important tool for coping with the designchallenges of future wireless networks
I Channel modelsI Discrete, AWGNI Fading: fast, block, slow
I Diversity
I Finite-state: correlated time variations
I CSI
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 84
Summary First Half
I Cooperation is an important tool for coping with the designchallenges of future wireless networks
I Channel modelsI Discrete, AWGNI Fading: fast, block, slow
I Diversity
I Finite-state: correlated time variations
I CSI
I Performance metricsI Channel CapacityI Transport capacity
I Scaling laws
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 84
Summary First Half
I RelayingI Decode-and-forwardI Compress-and-forwardI Amplify-and-forward
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 85
Summary First Half
I RelayingI Decode-and-forwardI Compress-and-forwardI Amplify-and-forward
I ConferencingI Interactive decoding
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 85
Summary First Half
I RelayingI Decode-and-forwardI Compress-and-forwardI Amplify-and-forward
I ConferencingI Interactive decoding
I FeedbackI Cooperation between transmitters and receiversI Useful in network scenarios
I Feedback does not have to come from the target receiver
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 85
Networks with Multiple Source-Destination Pairs
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 86
In This Section...
I Differences when relaying for multiple pairs
I Cooperation in interference channel with a relay
I Cooperation in cognitive radio networks
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 87
Relaying
I Relay strategies well developedI decode, compress, amplify -and-forward
I Capture broadcast
I No interferenceI One flow
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 88
Relaying for Multiple Sources?
I The smallest network: interference channel with a relay
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 89
Relaying for Multiple Sources?
I The smallest network: interference channel with a relay
I Simple approach: multihop routing
I Relay time-shares in helping sources
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 89
Multihop
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 90
Multihop
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 90
Multihop
I How can we do better?
I No combining of bits, symbols or packets at the relay
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 90
Generalized Relaying
I Joint encoding and forwarding of multiple data streams
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 91
Network Coding
Butterfly network:
Routing achieves(R1,R2) = (β, 1 − β),for any β ∈ [0, 1]
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 92
Network Coding
Butterfly network:
Routing achieves(R1,R2) = (β, 1 − β),for any β ∈ [0, 1]
Network coding: relaycombines packets. Achieves(R1,R2) = (1, 1)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 92
Joint Encoding of Messages
Network Coding idea:
Generalized relaying:
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 93
Encoding Elements From...
I Relay channel: generalized amplify, quantize, decode-and-forward
I MAC channel: interference cancellation
I Interference channel: rate-splitting
I Broadcast channel: binning, dirty paper coding
I Many encoding strategies can be applied
I Evaluation is difficult
I Goal: Develop strategies that can be applied to largernetworks and can bring gains
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 94
Simple Joint Encoding Strategies: Gaussian Channel
Y3 = h13X1 + h23X2 + Z3
Yj =
3∑
i=1
hijXi + Zj
I Amplify-and-Forward (analog network coding):
X3 = cY3 = c(h13X1 + h23X2 + Z3)
I Decode-and-Forward:
X3 =√
P3(√
cV1(W1) +√
cV2(W2))
I Can outperform time-sharing
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 95
DF with Network Coding
Y1 = h31X3 + Z1
Y2 = h32X3 + Z2
Y3 = h13X1 + h23X2 + Z3
I MAC to relay, BC to sources
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 96
DF with Network Coding
Y1 = h31X3 + Z1
Y2 = h32X3 + Z2
Y3 = h13X1 + h23X2 + Z3
I MAC to relay, BC to sources
Relay broadcasts:
I For R1 = R2 : xn3 (w1 ⊕ w2)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 96
DF with Network Coding
Y1 = h31X3 + Z1
Y2 = h32X3 + Z2
Y3 = h13X1 + h23X2 + Z3
I MAC to relay, BC to sources
Relay broadcasts:
I For R1 = R2 : xn3 (w1 ⊕ w2)
I For R1 ≥ R2 : xn3 (w11,w12 ⊕ w2)
where he splits w1 = (w11,w12) at (R ′1,R2)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 96
DF with Network Coding
Y1 = h31X3 + Z1
Y2 = h32X3 + Z2
Y3 = h13X1 + h23X2 + Z3
I MAC to relay, BC to sources
Relay broadcasts:
I For R1 = R2 : xn3 (w1 ⊕ w2)
I For R1 ≥ R2 : xn3 (w11,w12 ⊕ w2)
where he splits w1 = (w11,w12) at (R ′1,R2)
I AF: x3 = a(h13x1 + h23x2 + z3), under power constraint
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 96
DF with Network Coding
Y1 = h31X3 + Z1
Y2 = h32X3 + Z2
Y3 = h13X1 + h23X2 + Z3
I MAC to relay, BC to sources
Relay broadcasts:
I For R1 = R2 : xn3 (w1 ⊕ w2)
I For R1 ≥ R2 : xn3 (w11,w12 ⊕ w2)
where he splits w1 = (w11,w12) at (R ′1,R2)
I AF: x3 = a(h13x1 + h23x2 + z3), under power constraint
I xn3 (w1,w2)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 96
Differences when Relaying for Multiple Sources
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 97
Differences when Relaying for Multiple Sources
I Joint relaying of multiple data streams
I Interference:
I Sources create interference
I Relaying one message increases interference to other users
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 97
Interference Channel
I No relay
I Capacity region unknown
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 98
Interference Channel
I No relay
I Capacity region unknown
I except...
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 98
In Strong Interference
Gaussian channel:
Y1 = X1 + aX2 + Z1
Y2 = bX1 + X2 + Z2
I Cross-link is ’stronger’ than direct: a, b ≥ 1
I Optimal: jointly decode both messages
I Multiaccess channel to each receiver
I Gains from interference cancellation
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 99
In General: Rate-Splitting
I If interference not strong: unwanted messages cannot bedecoded
I To reduce interference: partial decoding
I An encoder splits message into two messages
I Decoder decodes one unwanted message and cancelsinterference
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 100
Interference Forwarding
I Relay observes signals from both sources
I Relay can use some of its power to forward interference
I Increase interference to cancel it
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 101
Special Case Scenario
I No source1-relay link
I Can forwarding interference W2 help both receivers?
I Increases rate R2 but increases interference at destination 1
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 102
Encoding
I No rate-splitting nor binning
I Block-Markov, regular encoding
I Decoding: sliding-window or backward
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 103
Gaussian Channel
Y1 = X1 + h12X2 + h13X3 + Z1
Y2 = h21X1 + X2 + h23X3 + Z2
Y3 = h32X2 + Z3
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 104
No Relaying
I No relay: strong interference regime
I
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
R1
R2
Rate Regions of Gaussian Channels
without relay
h12 = 1, h221 = 2, h2
23 = 0.15, h232 = 12
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 105
Relaying
I No relay: strong interference regime
I With relay, no interference forwarding
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
R1
R2
Rate Regions of Gaussian Channels
without relay
with relay, h13
=0
h12 = 1, h221 = 2, h2
23 = 0.15, h232 = 12
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 106
Relaying and Interference Forwarding
I No relay: strong interference regime
I With relay, and interference forwarding
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
R1
R2
Rate Regions of Gaussian Channels
without relay
with relay, h13
=0
with relay, h13
=2
h12 = 1, h221 = 2, h2
23 = 0.15, h232 = 12
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 107
Interference Forwarding
Relay can...
I help decoder by interference forwardingI Interference cancelation
I hurt decoder by increasing interferenceI Interference rate becomes too large
Interference forwarding:
I through decode,compress -and-forward
I More general schemes
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 108
Virtual (Distributed) MIMO
I No dedicated relay
I Transmitter cooperation
I Transmitters need knowledge about each other’s messages
I Obtained through:
1. Cooperative strategies2. Dedicated orthogonal links; conferencing3. Feedback4. Cognition
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 109
Gains From Virtual MIMOI Orthogonal links for cooperation
0 10 20 30 40 500.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Exp
ecte
dsu
mra
tes
(bps
)
Cooperation channel gain G (dB)
CMIMO
CBC, CMAC
RTX-RX
RTX
RRX
CNC
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 110
Cognitive Radio Networks
I Motivation: bandwidth gridlock
I Wireless spectrum is crowded
I Licensed band not efficiently used
I Its inefficient use led to spectrum holes
From slides by
B. Brodersen,
BWRC cognitive
radio workshop
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 111
Cognitive Radios
I Co-exist with oblivious users without impacting their service
I Sense the environment
I Use the obtained side information to adaptively transmit
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 112
Interweave (Opportunistic) Approach
From slides by
B. Brodersen,
BWRC cognitive
radio workshop
I Dynamic spectrum access
I Sense the environment
I Transmit in a spectrum hole
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 113
Underlay Approach
I Share the bandwidth; created interference below a threshold
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 114
Cognition and Cooperation
I Why not use obtained information for cooperation?
I In cooperation: a helper needs knowledge about relayedmessage
I Assistance of the source nodeI Listening to the channel
I Cognitive node can obtain similar information throughcognition
I Overlay paradigm: share the band and compensate forinterference by cooperation
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 115
Overlay Approach
I What is the optimal cognitive strategy?
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 116
It All Hinges on...Side Information
I Interweave: users’ activity
I Underlay: channel gains
I Overlay: channel gains, codebooks and (partial) messages
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 117
How Can Side Information be Obtained?
I Interweave: users’ activityI Detection of spectrum holesI Holes common to the transmitter and receiver
I Underlay: channel gains
I If there is a channelreciprocity or feedback
I Overlay: channel gains, codebooks and (partial) messagesI Codebooks: through protocolI Messages via: retransmission; cooperation; decoding
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 118
Cognitive Radio Channel Model
I Two messages: Wk ∈ {1, . . . ,Mk} sent at rates Rk
I Encoding: X n1 = f1(W1,W2), X n
2 = f2(W2)
I Decoding: Wk = gk(Y nk )
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 119
Elements of Cognitive Encoding Strategy
I Opportunistic approach: interference avoidance
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 120
Elements of Cognitive Encoding Strategy
I Opportunistic approach: interference avoidance
I Utilize techniques developed from many canonical models
1. Cooperative strategiesTo increase rate at oblivious receiver
2. Rate-splittingTo allow oblivious decoder to cancel part of interference
3. Precoding against interferenceTo eliminate interference at cognitive receiver
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 120
Cooperation
I To increase rate for the oblivious receiver
I Cognitive radio acts as a relay
X n1 = f1(W1,W2)
I Dedicates some power to transmit the other user’s message
I Increases interference to its own receiver
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 121
Rate-Splitting
I To reduce interference
I Without cognition: interference channel
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 122
Precoding against Interference
I To eliminate interference at cognitive receiver
I Full cognition: MIMO broadcast channel
I Strategy: precoding against interference[Gel’fand and Pinsker, 1979]
I Gaussian channels: Dirty-paper coding (DPC) [Costa, 1981]
I Achieves capacity
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 123
Capacity Results for Gaussian Channels
Y1 = X1 + aX2 + Z1
Y2 = bX1 + X2 + Z2
inteference
������������������������������������
������������������������������������
�������������������������������������������������������
�������������������������������������������������������
1
b1
stronginteference
weak
Wu et.al.
a I Regions for which capacity is known:
I Strong interference, a > b > 1Cooperation achieves capacity
I Weak interference, b ≤ 1Dirty paper coding and cooperationachieve capacity
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 124
Insights
1. Orthogonalizing transmissions is suboptimal
2. Canceling strong interference is beneficial
3. Rate-splitting can be used for partially canceling interference
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 125
Insights
I Side information in cognitive radio networks can be used for:
I Cooperation
I Precoding against interference
I In considered network: cooperation and GP precodingcapacity-achieving in some regimes
I Delay should be considered
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 126
Relaying for Multiple Sources
I Jointly encode messages
I Exploit broadcast
I Relays forward messages and interference
I Create virtual MIMO
joint
jointencodingexploit
broadcast
interferenceforwarding
encoding
f(W1, W3)
W1
W2
W3
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 127
Cooperation in Fading Channels
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 128
In This Section...
I Examples
I Diversity-multiplexing tradeoff for the PtP MIMO channel
I DMT for cooperative systems
I Summary
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 129
Fading Channels
I Rayleigh fading: Many scatterers, no LOSI Communication in dense urban areas
I Rician Fading: Many scatterers with LOSI Satellite communications
0 0.2 0.4 0.6 0.8 1−14
−12
−10
−8
−6
−4
−2
0
2
Time, seconds
Rel
ativ
e P
ower
, dB
, Rel
ativ
e to
RM
S
K = 5
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 130
Channel Model
Consider a point-to-point (PtP) MIMO channel:
H sd
Source (m) Destination (n)
I m transmit antennas
I n receive antennas
I Consider codes with blocklength l
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 131
Channel Model
H sd
Source (m) Destination (n)
I Block-fading model: H is constant for the entire block oflength l .
I
Y =
√
SNR
mHX + Z
X ∈ Cm×l , H ∈ Cn×m, Y ∈ Cn×l , Z ∈ Cn×l
I hi ,j ∼ CN (0, 1), i.i.d.; zi ,j ∼ CN (0, 1), i.i.d.
I Power constraint: 2−Rl∑2Rl
i=1 ||X(i)||2F ≤ ml
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 132
MIMO Systems in Fading Environments: Diversity Gain
I Diversity: Sending the same information through severalpaths.
I Each path is subject to independent fading ⇒ increase thereliability of reception.
I m = 1 ⇒ Pe(SNR).= SNR−n
I Definition: Diversity gain d
d = − limSNR→∞
log Pe(SNR)
log SNR
I For i.i.d. Rayleigh fading the maximal diversity gain is mn
I Probability of error dominated by the outage event
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 133
MIMO Systems in Fading Environments: Multiplexing Gain
I Spatial Multiplexing: Sending independent information overparallel spatial channels.
I Each Tx-Rx pair is fading independently thus creating aparallel channel.
I At high SNR, under i.i.d. Rayleigh fading assumption, the(ergodic) channel capacity is given by
C (SNR) = min{m, n} log SNR + O(1)
I Definition: Multiplexing gain r
r = limSNR→∞
R(SNR)
log SNR
I For i.i.d. Rayleigh fading the maximal multiplexing gain ismin{m, n}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 134
Example: Diversity-Multiplexing Relationship for
Alamouti’s Scheme
I 2 × 2 system
I R = r log SNR
I X =
[x1 −x∗
2
x2 x∗1
]
I Y =√
SNR2 HX + Z
I Using Alamouti’s scheme we arrive at the equivalent channel
yi =
√
SNR||H||2F2
xi + wi
I Pout.= Pr
(
||H||2F ≤ SNR−(1−r)+)
I d(r) = 4(1 − r)+
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 135
Remarks
I The blocklength l is fixed
I Analysis for SNR goes to infinity
I A scheme S is a collection of codes {C(SNR)}, one for eachSNR
I Rate is R(SNR)
I Both data rate and error probability scale with SNR
I Each scheme is characterized by the two parameters (d , r)
I Define for a fixed r the function d∗(r) as
d∗(r) , supS with the same r
d
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 136
Diversity Gain vs. Multiplexing Gain - The Fundamental
Result
I Characterized by Zheng and Tse in 2003.
Theorem (Zheng & Tse’03)
When l ≥ m + n − 1, the optimal tradeoff curve d∗(r) is thepiecewise linear function connecting the points (k, d∗(k)), k =0, 1, ...,min{m, n}, where
d∗(k) = (m − k)(n − k).
I d∗max = mn
I r∗max = min{m, n}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 137
Diversity Gain vs. Multiplexing Gain - The Fundamental
Result
I Figure from Zheng & Tse’03.
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 138
Cooperative Networks
I Cooperation can be used to create a virtual MIMO:I Several single antenna nodes cooperate in sending/receiving
informationI CSI assumptions
I Which cooperation strategy is DMT superior?
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 139
Half Duplex Relay Channel
Relay (1)
Source (1) Destination (1)
H sr
H sd
H rd
I MISO upper bound: d∗(r) = 2(1 − r)
I Orthogonal DF: d∗(r) = 2 − 4r
I Dynamic DF
d∗(r) =
{2(1 − r), , 0 ≤ r ≤ 0.51−rr
, 0.5 ≤ r ≤ 1
I Non-orthogonal AF: d∗(r) = (1 − r) + (1 − 2r)+
I CF with relay CSIT: d∗(r) = 2(1 − r)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 140
Half Duplex Relay Channel
I Figure from Kramer, Maric & Yates’06.
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 141
Full Duplex MIMO Relay Channel: The Optimal DMTRelay (k)
Source (m) Destination (n)
H sr
H sd
H rd
Yr = HsrXs + Zr
Yd = HsdXs + HrdXr + Zd
Theorem (Yuksel & Erkip’07)
The optimal DMT is equal to
d∗(r) = min{dm(n+k)(r), d(m+k)n(r)}.
I The best DMT is achieved by the CF schemeI d∗
max = min{m(n + k), (m + k)n}I r∗max = min{min{m, (n + k)},min{n, (m + k)}}
= min{m, n}Ivana Maric and Ron Dabora Cooperation in Wireless Networks 142
Full Duplex MIMO Relay Channel: DMT Analysis of DFRelay (k)
Source (m) Destination (n)
H sr
H sd
H rd
Theorem (Yuksel & Erkip’07)
The DMT achieved by DF is given by
d∗DF (r) =
min{d(m+k)n(r),
dmn(r) + dmk(r)}, 0 ≤ r ≤ min{m, n, k}dmn(r) ,min{m, n, k} ≤ r ≤ min{m, n}
I If m = 1 or n = 1, DF is optimalI if k < min{m, n} the relay cannot help at high r
I Relay cannot decode if the multiplexing gain is too highI Additional outage event due to decoding at the relay
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 143
Multiple Relays: Non-Clustered
I Single antenna nodes
I Optimal DMT:d∗(r) = d13(r)
I DMT is achieved using DF at both relays
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 144
Multiple Relays: Clustered
I Single antenna nodes
I Clustered nodes: the channel is AWGN (no fading)
I Optimal DMT:
d∗(r) =
{d22(r) , r ≤ 1
0 , r > 1
I DMT is achieved using DF at Relay 1 and CF at Relay 2
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 145
Clustered vs. Non-Clustered
I Figure from Yuksel & Erkip’07.
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 146
DMT of Multi-hop Relaying
I Full duplex: DMT achieved by DF
d∗FD(r) = min
{dmk(r), dkn(r)
}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 147
DMT of Multi-hop Relaying
I Full duplex: DMT achieved by DF
d∗FD(r) = min
{dmk(r), dkn(r)
}
I Half duplex with fixed time allocation (α, 1 − α)
d∗HD-fixed(r) = min
{
dmk
( r
α
)
, dkm
(r
1 − α
)}
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 147
DMT of Multi-hop Relaying
I Full duplex: DMT achieved by DF
d∗FD(r) = min
{dmk(r), dkn(r)
}
I Half duplex with fixed time allocation (α, 1 − α)
d∗HD-fixed(r) = min
{
dmk
( r
α
)
, dkm
(r
1 − α
)}
I Half duplex DDF (m, k, n) = (2, 2, 2)
d∗HD(r) =
8−10r2−r
, r ∈ [0, 1/2)3−4r1−r
, r ∈ (1/2, 2/3)
41−r2−r
, r ∈ (2/3, 1]
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 147
Degrees of Freedom of Cooperative Networks
I The term degrees of freedom is sometimes used instead ofmultiplexing gain.
I The capacity at high SNR:
C (SNR) = min{m, n} logSNR
m+
max{m,n}∑
i=|m−n|+1
E{
log χ22i
}+ o(1)
I For a MIMO channel, when the matrix is full rank, weachieve the maximal degrees of freedom.
I For Rayleigh fading with CSIR the MIMO channel matrix isfull rank
I Cooperation does not increase the DOF
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 148
Degrees of Freedom with Cognitive Cooperation
I When cooperation is based on a cognitive relay node theneach pair can achieve DOF 1
I The sum-rate DOF is 2
⇒ A cognitive relay increases the DOF of the system
I Achieved by instantaneous interference cancellation
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 149
Summary
I DMT is a performance metric for fading channelsI Asymptotically high SNR
I Outage is the dominating error event
I Finite blocklength
I Soft information (CF) is important for achieving themaximum DMT
I With a single relay:I CF achieves the optimal DMTI With full-duplex single antenna nodes: DF also achieves the
optimal DMT
I Clustering improves diversity but does not improve DOFI Source cluster ⇒ use DFI Destination cluster ⇒ use CF
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 150
Scaling Laws
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 151
Large Network Analysis
I Initiated by Gupta and Kumar
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 152
Large Network Analysis
I Initiated by Gupta and Kumar
I n source-destinations: n-dimensional capacity region
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 152
Large Network Analysis
I Initiated by Gupta and Kumar
I n source-destinations: n-dimensional capacity region
I We cannot solve even for n=2!
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 152
Large Network Analysis
I Initiated by Gupta and Kumar
I n source-destinations: n-dimensional capacity region
I We cannot solve even for n=2!
I Assumption:
I Each pair wants to communicate at rate: R(n) bits/s
I Total throughput: T (n) = nR(n)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 152
Large Network Analysis
I Initiated by Gupta and Kumar
I n source-destinations: n-dimensional capacity region
I We cannot solve even for n=2!
I Assumption:
I Each pair wants to communicate at rate: R(n) bits/s
I Total throughput: T (n) = nR(n)
I Max achievable scaling?
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 152
Scaling in Multihop Networks
Upper bound: T (n) ≤ O(√
n)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 153
Scaling in Multihop Networks
Upper bound: T (n) ≤ O(√
n)
I Rate for each pair goes to zero
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 153
Scaling in Multihop Networks
Upper bound: T (n) ≤ O(√
n)
I Rate for each pair goes to zero
I Nearest neighbor communication and spatial reuse optimal
I Number of retransmissions increases
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 153
Scaling in Multihop Networks
Upper bound: T (n) ≤ O(√
n)
I Rate for each pair goes to zero
I Nearest neighbor communication and spatial reuse optimal
I Number of retransmissions increases
I System is interference limited
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 153
Scaling in Multihop Networks
Upper bound: T (n) ≤ O(√
n)
I Rate for each pair goes to zero
I Nearest neighbor communication and spatial reuse optimal
I Number of retransmissions increases
I System is interference limited
Communication scheme:
I Multihop routing
I Treat all unwanted signals as noise
I Achievability proved using percolation theory
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 153
Scaling in Multihop Networks
Upper bound: T (n) ≤ O(√
n)
I Rate for each pair goes to zero
I Nearest neighbor communication and spatial reuse optimal
I Number of retransmissions increases
I System is interference limited
Communication scheme:
I Multihop routing
I Treat all unwanted signals as noise
I Achievability proved using percolation theory
I Can cooperative encoding schemes change the scaling?
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 153
Is Multihop Optimal?
I Dense vs. extended networks
Extended Networks:
For α > 4: multihop is order-optimal
I α-path-loss exponent
I Attenuation is too large for cooperation to help
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 154
Is Multihop Optimal?
I Dense vs. extended networks
Extended Networks:
For α > 4: multihop is order-optimal
I α-path-loss exponent
I Attenuation is too large for cooperation to help
For α ≤ 4?
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 154
Is Multihop Optimal?
I Dense vs. extended networks
Extended Networks:
For α > 4: multihop is order-optimal
I α-path-loss exponent
I Attenuation is too large for cooperation to help
For α ≤ 4?
I 2 ≤ α ≤ 3: n2−α/2
α > 3: n1/2
I For α = 2: linear scaling
I For α > 3: multihop is optimal
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 154
Is Multihop Optimal?
I Dense vs. extended networks
Extended Networks:
For α > 4: multihop is order-optimal
I α-path-loss exponent
I Attenuation is too large for cooperation to help
For α ≤ 4?
I 2 ≤ α ≤ 3: n2−α/2
α > 3: n1/2
I For α = 2: linear scaling
I For α > 3: multihop is optimal
I Dense networks, α ≥ 2: T (n) = O(n1−ε) is achievable
I Arbitrarily close to linear scaling
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 154
Motivation for Cooperative Strategy
I Nearest neighbor communications are not enough
I Linear scaling: in MIMO system
I With n transmit and receive antennas, in high-SNR:n log(SNR)
I We already saw gains from clustering and mimicking MIMO
I This requires cooperation within clusters
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Three-Phase Cooperative Scheme
I Form M-node clusters
I Sources in clustercooperate
I MIMO long-rangetransmissions
I Destinations in clustercooperate
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 156
Within Each Stage
I Transmit cluster: information bits exchanged betweensources
I Receive cluster: quantized observations exchanged betweendestinations
I Spatial reuse: non-adjacent clusters send simultaneously
I TDMA long-range communications between clusters
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 157
Hierarchial Cooperation
I Perform multiple stages of the three-phase scheme
I Each stage improves throughput
I After h stages:T (n) = O(nh/(h+1))
I Choose h s.t.h
h + 1≥ 1 − ε
to obtainT (n) = O(n1−ε)
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 158
Summary
I Lots of progress
I Different behavior for dense vs. extended networks
I Cooperation can change scaling
I For extended networks:
I For α ≥ 3:√
n, multihop is optimal
I For 2 ≤ α ≤ 3: n2−α/2
I Linear scaling only for α = 2
I Dense networks: capacity scales linearly
I Design of practical systems requires more detailed analysis
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 159
Summary and Challenges
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 160
Summary
Relaying for Multiple Sources
I Gains from simple joint encoding strategies
I Canceling strong interference is beneficial
I Relays forward messages and interference
I Gains from virtual MIMO
I Gains from cognitive encoding techniques
Cooperation brings diversity-multiplexing gains
I Compress-and-forward achieves the optimal DMT in singlerelay channel
Cooperative communications can change capacity scaling
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 161
Conventional Network Architecture
I Network protocol layers
I Store-and-forward routing via a sequence of links
I Point-to-point transmissions on the path
I Network layer: decides on the next node, modifies the header
I PHY/Link layer: discards a packet in error
T3
R1
R3
R2T2
T1
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Network Architecture: Cooperative Protocols
Exploit broadcast
I Nodes collect erroneous packets
I A link is not necessarily point-to-point
Allow for encoding at the nodes
I Relaying, joint encoding of messages, network coding
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 163
Network Architecture: Cooperative Protocols
I Decoder: soft combining of packets
I Protocol: provide for relaying and routing
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 164
Network Architecture: Cooperative Protocols
I Decoder: soft combining of packets
I Protocol: provide for relaying and routing
I For example:
I Routing on the network layer
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 164
Network Architecture: Cooperative Protocols
I Decoder: soft combining of packets
I Protocol: provide for relaying and routing
I For example:
I Routing on the network layer
I Sequence of cooperative links
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 165
Demonstrated Gains from Cooperative Communications
For small networks
I Rates in relay channel
I Diversity-multiplexing gains
I Rate regions for multiple sources
For large networks
I Scaling law O(√
n) → O(n)
−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 10
1
2
3
4
5
6
upper boundDF
ρ for DF
CF
relay off
AF
d
Rat
e [b
it/u
se]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
multiplexing gain r
div
ers
ity g
ain
d dynamic DF
CF andMISO upper bound
nonorthogonalAF
orthogonal AF andorthogonal DF
no cooperation
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
R1
R2
Rate Regions of Gaussian Channels
without relay
with relay, h13
=0
with relay, h13
=2
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 166
Challenges
I Capacity results for canonical models
I Many encoding possibilities at the relays
I Practical cooperative schemes
I Cooperative protocols
joint
jointencodingexploit
broadcast
interferenceforwarding
encoding
f(W1, W3)
W1
W2
W3
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 167
Bibliography
I T. Cover and A. El Gamal, ”Capacity Theorems for the RelayChannel” IEEE Trans. Inf. Th., vol. 25, no. 5, Sept. 1979.
I A. B. Carleial, ”Multiple- Access Channels with DifferentGeneralized Feedback Signals” IEEE Trans. Inf. Th., vol. 28, no.6, Nov. 1982.
I F.M.J. Willems, ”Informationtheoretical Results for the DiscreteMemoryless Multiple Access Channels”. Ph.D. dissertation, Oct1982.
I G. Kramer. M. Gastpar and P. Gupta, ”Cooperative Strategiesand Capacity Theorems for Relay Networks” IEEE Trans. Inf. Th.,vol. 51, no. 9, Sept. 2005.
I G. Kramer, I. Maric and R. Yates, ”CooperativeCommunications,” , Foundations and Trends in Networking,Hanover, MA: NOW Publishers Inc., vol. 1, no. 3-4, 2006
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 168
Bibliography
I L. Zheng and D. Tse, ”Diversity and Multiplexing: A FundamentalTradeoff in Multiple Antenna Channels” IEEE Trans. Inf. Th.,Vol. 49(5), May 2003.
I M. Yuksel and E. Erkip. Multi-antenna Cooperative WirelessSystems: A Diversity-multiplexing Tradeoff Perspective, IEEETrans. Inf. Th., Special Issue on Models, Theory, and Codes forRelaying and Cooperation in Communication Networks, vol. 53,no. 10, pp. 3371-3393, October 2007.
I F. Xue and P.R. Kumar, ”Scaling Laws for Ad Hoc WirelessNetworks: An Information Theoretic Approach” NOW Publishers,2006, and references therein.
I A. Ozgur, O. Leveque and D. Tse, ”Hierarchical Cooperationachieves Optimal Capacity Scaling in Ad Hoc Networks” IEEETrans. Inf. Th., vol. 53, no. 10, Oct. 2007.
Ivana Maric and Ron Dabora Cooperation in Wireless Networks 169