joint source-channel coding over broadcast channels with
TRANSCRIPT
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Joint Source-Channel Codingover Broadcast Channels
with Side Information at the Receivers
Ertem TuncelUniversity of California, Riverside
November 30, 2011
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Sensor networks
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Special case: Many-to-one communication
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Special case: One-to-many communication
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Communication model
Broadcast Channel
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Equivalent communication model
Channel 2
Channel K
Channel 1
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Start simple: Point-to-point communication
Channel
Case with memoryless sources and channels studied for 40 years.
Known as the Slepian-Wolf problem when P[Xn 6= Xn]→ 0.
Known as the Wyner-Ziv problem when P[d(Xn, Xn) > D]→ 0.Both are separable problems, i.e., source and channel codes canbe designed separately without compromising optimality.
R < κC where κ = mn .
Source coding in both problems utilizes the tool known asbinning.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Start simple: Point-to-point communication
Channel
Case with memoryless sources and channels studied for 40 years.
Known as the Slepian-Wolf problem when P[Xn 6= Xn]→ 0.
Known as the Wyner-Ziv problem when P[d(Xn, Xn) > D]→ 0.Both are separable problems, i.e., source and channel codes canbe designed separately without compromising optimality.
R < κC where κ = mn .
Source coding in both problems utilizes the tool known asbinning.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Start simple: Point-to-point communication
Channel
Case with memoryless sources and channels studied for 40 years.
Known as the Slepian-Wolf problem when P[Xn 6= Xn]→ 0.
Known as the Wyner-Ziv problem when P[d(Xn, Xn) > D]→ 0.Both are separable problems, i.e., source and channel codes canbe designed separately without compromising optimality.
R < κC where κ = mn .
Source coding in both problems utilizes the tool known asbinning.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Start simple: Point-to-point communication
Channel
Case with memoryless sources and channels studied for 40 years.
Known as the Slepian-Wolf problem when P[Xn 6= Xn]→ 0.
Known as the Wyner-Ziv problem when P[d(Xn, Xn) > D]→ 0.
Both are separable problems, i.e., source and channel codes canbe designed separately without compromising optimality.
R < κC where κ = mn .
Source coding in both problems utilizes the tool known asbinning.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Start simple: Point-to-point communication
Channel
Case with memoryless sources and channels studied for 40 years.
Known as the Slepian-Wolf problem when P[Xn 6= Xn]→ 0.
Known as the Wyner-Ziv problem when P[d(Xn, Xn) > D]→ 0.Both are separable problems, i.e., source and channel codes canbe designed separately without compromising optimality.
R < κC where κ = mn .
Source coding in both problems utilizes the tool known asbinning.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Start simple: Point-to-point communication
Channel
Case with memoryless sources and channels studied for 40 years.
Known as the Slepian-Wolf problem when P[Xn 6= Xn]→ 0.
Known as the Wyner-Ziv problem when P[d(Xn, Xn) > D]→ 0.Both are separable problems, i.e., source and channel codes canbe designed separately without compromising optimality.
R < κC where κ = mn .
Source coding in both problems utilizes the tool known asbinning.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Start simple: Point-to-point communication
Channel
Case with memoryless sources and channels studied for 40 years.
Known as the Slepian-Wolf problem when P[Xn 6= Xn]→ 0.
Known as the Wyner-Ziv problem when P[d(Xn, Xn) > D]→ 0.Both are separable problems, i.e., source and channel codes canbe designed separately without compromising optimality.
R < κC where κ = mn .
Source coding in both problems utilizes the tool known asbinning.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
How binning works
When there is no side information, no need for binning:
SOURCE SPACE
CHANNEL SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
How binning works
When there is side information at the receiver, assign multiplesource words the same channel word:
SOURCE SPACE
CHANNEL SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
How binning works
Decode by first decoding the channel word. Trace the decodedchannel word back to the source space.
SOURCE SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
How binning works
You have both the side information Yn and possible Xn’s. If Xn
and Yn are known to be correlated, which one would you choose?
SOURCE SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
How binning works
SOURCE SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Random binning (hats off to Slepian-Wolf)
Typical set
Randomly assign source vectors to bins such that there are≈ 2n[I(X;Y)−ε] elements in each bin.
Sufficiently few in each bin to decode Xn using typicality.
Even if the sender knew Yn, source coding rate could not belower than H(X|Y).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Random binning (hats off to Slepian-Wolf)
Typical set
Randomly assign source vectors to bins such that there are≈ 2n[I(X;Y)−ε] elements in each bin.
Sufficiently few in each bin to decode Xn using typicality.
Even if the sender knew Yn, source coding rate could not belower than H(X|Y).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Random binning (hats off to Slepian-Wolf)
Typical set
Randomly assign source vectors to bins such that there are≈ 2n[I(X;Y)−ε] elements in each bin.
Sufficiently few in each bin to decode Xn using typicality.
Even if the sender knew Yn, source coding rate could not belower than H(X|Y).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Wyner-Ziv extension
If distortion is OK, first quantize and then bin.
Create a codebook of Zn’s of size ≈ 2n[I(X;Z)+ε].Randomly assign these codevectors into bins such that there are≈ 2n[I(Y;Z)−ε] elements in each bin.Sufficiently few in each bin to decode Zn using typicality.To ensure that the correct Zn satisfies typicality, we needY − X − Z.Once Zn is decoded, use it with Yn through a single-letterfunction Xi = φ(Yi,Zi).
The minimum source coding rate within distortion D thenbecomes
RWZ(D) = minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Y)− I(Y;Z)
= minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Z|Y)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Wyner-Ziv extension
If distortion is OK, first quantize and then bin.Create a codebook of Zn’s of size ≈ 2n[I(X;Z)+ε].
Randomly assign these codevectors into bins such that there are≈ 2n[I(Y;Z)−ε] elements in each bin.Sufficiently few in each bin to decode Zn using typicality.To ensure that the correct Zn satisfies typicality, we needY − X − Z.Once Zn is decoded, use it with Yn through a single-letterfunction Xi = φ(Yi,Zi).
The minimum source coding rate within distortion D thenbecomes
RWZ(D) = minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Y)− I(Y;Z)
= minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Z|Y)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Wyner-Ziv extension
If distortion is OK, first quantize and then bin.Create a codebook of Zn’s of size ≈ 2n[I(X;Z)+ε].Randomly assign these codevectors into bins such that there are≈ 2n[I(Y;Z)−ε] elements in each bin.
Sufficiently few in each bin to decode Zn using typicality.To ensure that the correct Zn satisfies typicality, we needY − X − Z.Once Zn is decoded, use it with Yn through a single-letterfunction Xi = φ(Yi,Zi).
The minimum source coding rate within distortion D thenbecomes
RWZ(D) = minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Y)− I(Y;Z)
= minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Z|Y)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Wyner-Ziv extension
If distortion is OK, first quantize and then bin.Create a codebook of Zn’s of size ≈ 2n[I(X;Z)+ε].Randomly assign these codevectors into bins such that there are≈ 2n[I(Y;Z)−ε] elements in each bin.Sufficiently few in each bin to decode Zn using typicality.
To ensure that the correct Zn satisfies typicality, we needY − X − Z.Once Zn is decoded, use it with Yn through a single-letterfunction Xi = φ(Yi,Zi).
The minimum source coding rate within distortion D thenbecomes
RWZ(D) = minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Y)− I(Y;Z)
= minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Z|Y)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Wyner-Ziv extension
If distortion is OK, first quantize and then bin.Create a codebook of Zn’s of size ≈ 2n[I(X;Z)+ε].Randomly assign these codevectors into bins such that there are≈ 2n[I(Y;Z)−ε] elements in each bin.Sufficiently few in each bin to decode Zn using typicality.To ensure that the correct Zn satisfies typicality, we needY − X − Z.
Once Zn is decoded, use it with Yn through a single-letterfunction Xi = φ(Yi,Zi).
The minimum source coding rate within distortion D thenbecomes
RWZ(D) = minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Y)− I(Y;Z)
= minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Z|Y)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Wyner-Ziv extension
If distortion is OK, first quantize and then bin.Create a codebook of Zn’s of size ≈ 2n[I(X;Z)+ε].Randomly assign these codevectors into bins such that there are≈ 2n[I(Y;Z)−ε] elements in each bin.Sufficiently few in each bin to decode Zn using typicality.To ensure that the correct Zn satisfies typicality, we needY − X − Z.Once Zn is decoded, use it with Yn through a single-letterfunction Xi = φ(Yi,Zi).
The minimum source coding rate within distortion D thenbecomes
RWZ(D) = minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Y)− I(Y;Z)
= minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Z|Y)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Wyner-Ziv extension
If distortion is OK, first quantize and then bin.Create a codebook of Zn’s of size ≈ 2n[I(X;Z)+ε].Randomly assign these codevectors into bins such that there are≈ 2n[I(Y;Z)−ε] elements in each bin.Sufficiently few in each bin to decode Zn using typicality.To ensure that the correct Zn satisfies typicality, we needY − X − Z.Once Zn is decoded, use it with Yn through a single-letterfunction Xi = φ(Yi,Zi).
The minimum source coding rate within distortion D thenbecomes
RWZ(D) = minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Y)− I(Y;Z)
= minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Z|Y)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Wyner-Ziv extension
If distortion is OK, first quantize and then bin.Create a codebook of Zn’s of size ≈ 2n[I(X;Z)+ε].Randomly assign these codevectors into bins such that there are≈ 2n[I(Y;Z)−ε] elements in each bin.Sufficiently few in each bin to decode Zn using typicality.To ensure that the correct Zn satisfies typicality, we needY − X − Z.Once Zn is decoded, use it with Yn through a single-letterfunction Xi = φ(Yi,Zi).
The minimum source coding rate within distortion D thenbecomes
RWZ(D) = minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Y)− I(Y;Z)
= minZ,φ: Y−X−Z, E{d(X,φ(Y,Z))}≤D
I(X;Z|Y)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Back to broadcast channels (2 receivers)
The most general message sending scenario:
Channel 1
Channel 2
Encoder
Decoder 1
Decoder 2
Sending degraded messages:
Channel 1
Channel 2
Encoder
Decoder 1
Decoder 2
Sometimes, the latter is dictated by the structure of the channel(e.g., binary symmetric channels and Gaussian channels).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Back to broadcast channels (2 receivers)
The most general message sending scenario:
Channel 1
Channel 2
Encoder
Decoder 1
Decoder 2
Sending degraded messages:
Channel 1
Channel 2
Encoder
Decoder 1
Decoder 2
Sometimes, the latter is dictated by the structure of the channel(e.g., binary symmetric channels and Gaussian channels).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Back to broadcast channels (2 receivers)
The most general message sending scenario:
Channel 1
Channel 2
Encoder
Decoder 1
Decoder 2
Sending degraded messages:
Channel 1
Channel 2
Encoder
Decoder 1
Decoder 2
Sometimes, the latter is dictated by the structure of the channel(e.g., binary symmetric channels and Gaussian channels).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Back to broadcast channels (2 receivers)
In our problem, the degraded messages would be nested bins:
SOURCE SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Back to broadcast channels (2 receivers)
The weak channel can decode only the coarse bin:
SOURCE SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Back to broadcast channels (2 receivers)
The strong channel can decode the finer bin:
SOURCE SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A simple joint source-channel coding scheme
Xn Xnk
Y nk
V mk
xn(i) um(i)
i.i.d. ∼ PX i.i.d. ∼ PU
2n[H(X)+ε]
sourcevectors
Decoder
Probability of error
Pke ≤ 2n[H(X)−I(X;Yk)−κI(U;Vk)+ε] → 0
DecodingFind the unique i s.t.
(um(i),Vmk ) jointly typical
(xn(i), Ynk ) jointly typical
Output Xn = xn(i).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A simple joint source-channel coding scheme
Xn Xnk
Y nk
V mk
xn(i) um(i)
i.i.d. ∼ PX i.i.d. ∼ PU
2n[H(X)+ε]
sourcevectors
Decoder
Probability of error
Pke ≤ 2n[H(X)−I(X;Yk)−κI(U;Vk)+ε] → 0
DecodingFind the unique i s.t.
(um(i),Vmk ) jointly typical
(xn(i), Ynk ) jointly typical
Output Xn = xn(i).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A simple joint source-channel coding scheme
Xn Xnk
Y nk
V mk
xn(i) um(i)
i.i.d. ∼ PX i.i.d. ∼ PU
2n[H(X)+ε]
sourcevectors
Decoder
Probability of error
Pke ≤ 2n[H(X)−I(X;Yk)−κI(U;Vk)+ε] → 0
DecodingFind the unique i s.t.
(um(i),Vmk ) jointly typical
(xn(i), Ynk ) jointly typical
Output Xn = xn(i).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Achievable channel uses per source symbol
Theorem (Tuncel 2006)Asymptotically lossless transmission is possible for κ = m
n iff∃U s.t. H(X|Yk) ≤ κI(U;Vk) for k = 1, . . . ,K.
R1
R2
κC1
κC2
H(X|Y1)
H(X|Y2)
CorollaryIf ∃U maximizing all I(U;Vk)
simultaneously, then we can use all channelsin full capacity for the purpose of SWcoding!!!
Examples
Gaussian channelsBinary symmetric channels
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Achievable channel uses per source symbol
Theorem (Tuncel 2006)Asymptotically lossless transmission is possible for κ = m
n iff∃U s.t. H(X|Yk) ≤ κI(U;Vk) for k = 1, . . . ,K.
R1
R2
κC1
κC2
H(X|Y1)
H(X|Y2)
CorollaryIf ∃U maximizing all I(U;Vk)
simultaneously, then we can use all channelsin full capacity for the purpose of SWcoding!!!
Examples
Gaussian channelsBinary symmetric channels
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Achievable channel uses per source symbol
Theorem (Tuncel 2006)Asymptotically lossless transmission is possible for κ = m
n iff∃U s.t. H(X|Yk) ≤ κI(U;Vk) for k = 1, . . . ,K.
R1
R2
κC1
κC2
H(X|Y1)
H(X|Y2)
CorollaryIf ∃U maximizing all I(U;Vk)
simultaneously, then we can use all channelsin full capacity for the purpose of SWcoding!!!
Examples
Gaussian channelsBinary symmetric channels
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Achievable channel uses per source symbol
Theorem (Tuncel 2006)Asymptotically lossless transmission is possible for κ = m
n iff∃U s.t. H(X|Yk) ≤ κI(U;Vk) for k = 1, . . . ,K.
R1
R2
κC1
κC2
H(X|Y1)
H(X|Y2)
CorollaryIf ∃U maximizing all I(U;Vk)
simultaneously, then we can use all channelsin full capacity for the purpose of SWcoding!!!
Examples
Gaussian channelsBinary symmetric channels
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtual binning
There is no deliberate binning. However, in effect, channelperforms virtual binning.
SOURCE SPACE CHANNEL SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtual binning
When the channel is strong:
SOURCE SPACE CHANNEL SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtual binning
When the channel is weak:
SOURCE SPACE CHANNEL SPACE
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
Operational separation: Encoding and decoding can be brokeninto source and channel coding modes that operate somewhatseparately:
Source encoder finds the index on the source codebook.Channel encoder maps this index onto a channel word.Channel decoder creates a list of possible channel words, orequivalently, a list of source codebook indices (a virtual bin).Source decoder uses the side information to identify the sourceword in the list.
As a result, source and channel random variables do not interactat all:
H(X|Yk) ≤ κI(U;Vk)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
Operational separation: Encoding and decoding can be brokeninto source and channel coding modes that operate somewhatseparately:
Source encoder finds the index on the source codebook.
Channel encoder maps this index onto a channel word.Channel decoder creates a list of possible channel words, orequivalently, a list of source codebook indices (a virtual bin).Source decoder uses the side information to identify the sourceword in the list.
As a result, source and channel random variables do not interactat all:
H(X|Yk) ≤ κI(U;Vk)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
Operational separation: Encoding and decoding can be brokeninto source and channel coding modes that operate somewhatseparately:
Source encoder finds the index on the source codebook.Channel encoder maps this index onto a channel word.
Channel decoder creates a list of possible channel words, orequivalently, a list of source codebook indices (a virtual bin).Source decoder uses the side information to identify the sourceword in the list.
As a result, source and channel random variables do not interactat all:
H(X|Yk) ≤ κI(U;Vk)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
Operational separation: Encoding and decoding can be brokeninto source and channel coding modes that operate somewhatseparately:
Source encoder finds the index on the source codebook.Channel encoder maps this index onto a channel word.Channel decoder creates a list of possible channel words, orequivalently, a list of source codebook indices (a virtual bin).
Source decoder uses the side information to identify the sourceword in the list.
As a result, source and channel random variables do not interactat all:
H(X|Yk) ≤ κI(U;Vk)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
Operational separation: Encoding and decoding can be brokeninto source and channel coding modes that operate somewhatseparately:
Source encoder finds the index on the source codebook.Channel encoder maps this index onto a channel word.Channel decoder creates a list of possible channel words, orequivalently, a list of source codebook indices (a virtual bin).Source decoder uses the side information to identify the sourceword in the list.
As a result, source and channel random variables do not interactat all:
H(X|Yk) ≤ κI(U;Vk)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
Operational separation: Encoding and decoding can be brokeninto source and channel coding modes that operate somewhatseparately:
Source encoder finds the index on the source codebook.Channel encoder maps this index onto a channel word.Channel decoder creates a list of possible channel words, orequivalently, a list of source codebook indices (a virtual bin).Source decoder uses the side information to identify the sourceword in the list.
As a result, source and channel random variables do not interactat all:
H(X|Yk) ≤ κI(U;Vk)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
But is this the same as traditional separation?
Through nested binning, a traditional source coder can achieve afirst layer rate of H(X|Y1) and a total rate of H(X|Y2), assumingthat the latter is larger.
But the channel capacity region is not the same as union of all[I(U;V1), I(U;V2)] pairs.
Consider the binary symmetric channel
U
!"
#"
V2 V1
R
!
C
!
R2
!
R1
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
But is this the same as traditional separation?
Through nested binning, a traditional source coder can achieve afirst layer rate of H(X|Y1) and a total rate of H(X|Y2), assumingthat the latter is larger.
But the channel capacity region is not the same as union of all[I(U;V1), I(U;V2)] pairs.
Consider the binary symmetric channel
U
!"
#"
V2 V1
R
!
C
!
R2
!
R1
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
But is this the same as traditional separation?
Through nested binning, a traditional source coder can achieve afirst layer rate of H(X|Y1) and a total rate of H(X|Y2), assumingthat the latter is larger.
But the channel capacity region is not the same as union of all[I(U;V1), I(U;V2)] pairs.
Consider the binary symmetric channel
U
!"
#"
V2 V1
R
!
C
!
R2
!
R1
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
But is this the same as traditional separation?
Through nested binning, a traditional source coder can achieve afirst layer rate of H(X|Y1) and a total rate of H(X|Y2), assumingthat the latter is larger.
But the channel capacity region is not the same as union of all[I(U;V1), I(U;V2)] pairs.
Consider the binary symmetric channel
U
!"
#"
V2 V1
R
!
C
!
R2
!
R1
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Virtues of virtual binning
But is this the same as traditional separation?
Through nested binning, a traditional source coder can achieve afirst layer rate of H(X|Y1) and a total rate of H(X|Y2), assumingthat the latter is larger.
But the channel capacity region is not the same as union of all[I(U;V1), I(U;V2)] pairs.
Consider the binary symmetric channel
U
!"
#"
V2 V1
R
!
C
!
R2
!
R1
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
WZ extensions: The common description scheme (CDS)
Xn
Xnk
Ynk
Vmk
zn(i) u
m(i)
i.i.d. ∼ PZ i.i.d. ∼ PU
2n[I(X ;Z)+ε]
codevectors
Decoder
Analysis
Let Z be s.t. (Y1, . . . , YK)− X − Z and E{dk(X, φk(Z, Yk))} ≤ Dk for some φk.
Pke = Pr[dk(Xn, Xn
k ) > Dk] ≤ 2n[I(X;Z)−I(Yk;Z)−κI(U;Vk)+ε]
Thus, it suffices to have I(X; Z|Yk) ≤ κI(U;Vk) for Pke → 0.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
WZ extensions: The common description scheme (CDS)
Xn
Xnk
Ynk
Vmk
zn(i) u
m(i)
i.i.d. ∼ PZ i.i.d. ∼ PU
2n[I(X ;Z)+ε]
codevectors
Decoder
Analysis
Let Z be s.t. (Y1, . . . , YK)− X − Z and E{dk(X, φk(Z, Yk))} ≤ Dk for some φk.
Pke = Pr[dk(Xn, Xn
k ) > Dk] ≤ 2n[I(X;Z)−I(Yk;Z)−κI(U;Vk)+ε]
Thus, it suffices to have I(X; Z|Yk) ≤ κI(U;Vk) for Pke → 0.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
WZ extensions: The common description scheme (CDS)
Xn
Xnk
Ynk
Vmk
zn(i) u
m(i)
i.i.d. ∼ PZ i.i.d. ∼ PU
2n[I(X ;Z)+ε]
codevectors
Decoder
Analysis
Let Z be s.t. (Y1, . . . , YK)− X − Z and E{dk(X, φk(Z, Yk))} ≤ Dk for some φk.
Pke = Pr[dk(Xn, Xn
k ) > Dk] ≤ 2n[I(X;Z)−I(Yk;Z)−κI(U;Vk)+ε]
Thus, it suffices to have I(X; Z|Yk) ≤ κI(U;Vk) for Pke → 0.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
WZ extensions: The common description scheme (CDS)
Xn
Xnk
Ynk
Vmk
zn(i) u
m(i)
i.i.d. ∼ PZ i.i.d. ∼ PU
2n[I(X ;Z)+ε]
codevectors
Decoder
Analysis
Let Z be s.t. (Y1, . . . , YK)− X − Z and E{dk(X, φk(Z, Yk))} ≤ Dk for some φk.
Pke = Pr[dk(Xn, Xn
k ) > Dk] ≤ 2n[I(X;Z)−I(Yk;Z)−κI(U;Vk)+ε]
Thus, it suffices to have I(X; Z|Yk) ≤ κI(U;Vk) for Pke → 0.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
WZ extensions: The common description scheme (CDS)
Xn
Xnk
Ynk
Vmk
zn(i) u
m(i)
i.i.d. ∼ PZ i.i.d. ∼ PU
2n[I(X ;Z)+ε]
codevectors
Decoder
Analysis
Let Z be s.t. (Y1, . . . , YK)− X − Z and E{dk(X, φk(Z, Yk))} ≤ Dk for some φk.
Pke = Pr[dk(Xn, Xn
k ) > Dk] ≤ 2n[I(X;Z)−I(Yk;Z)−κI(U;Vk)+ε]
Thus, it suffices to have I(X; Z|Yk) ≤ κI(U;Vk) for Pke → 0.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
CDS with dirty paper coding (DPC-CDS)
V |V21
VK
pS,U
S m
nX mUf ( )
1( )g
^1X n
nY 1
1V m
g ( )2
nX 2^
2Y n
2mV
K^X n
nYK
KV m K( )g
.
.
. .
.
.
...
Theorem
(D1, . . . ,DK) is achievable if there exist
(Y1, . . . , YK)− X − Z and E{dk(X, φk(Z, Yk))} ≤ Dk for some φk
T − (U, S)− (V1, . . . ,VK)
such that I(X; Z|Yk) ≤ κ[I(T;Vk)− I(T; S)].
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
CDS with dirty paper coding (DPC-CDS)
V |V21
VK
pS,U
S m
nX mUf ( )
1( )g
^1X n
nY 1
1V m
g ( )2
nX 2^
2Y n
2mV
K^X n
nYK
KV m K( )g
.
.
. .
.
.
...
Theorem
(D1, . . . ,DK) is achievable if there exist
(Y1, . . . , YK)− X − Z and E{dk(X, φk(Z, Yk))} ≤ Dk for some φk
T − (U, S)− (V1, . . . ,VK)
such that I(X; Z|Yk) ≤ κ[I(T;Vk)− I(T; S)].
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Proof sketch
i.i.d. PZ i.i.d. PT
Xn
zn(i) tm(j|i)
codevectors
codevectors
2n[I(X;Z)+ε]
2m[I(S;T )+ε]
Um is chosen such that (Sm, tm(j|i),Um) is jointly typical
Decoder finds unique (i, j) such that (zn(i),Ynk ) and (tm(j|i),Vm
k ) areboth jointly typical
Pke ≤ 2n[I(X;Z)+κI(T;S)−I(Yk;Z)−κI(T;Vk)+ε]
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Proof sketch
i.i.d. PZ i.i.d. PT
Xn
zn(i) tm(j|i)
codevectors
codevectors
2n[I(X;Z)+ε]
2m[I(S;T )+ε]
Um is chosen such that (Sm, tm(j|i),Um) is jointly typical
Decoder finds unique (i, j) such that (zn(i),Ynk ) and (tm(j|i),Vm
k ) areboth jointly typical
Pke ≤ 2n[I(X;Z)+κI(T;S)−I(Yk;Z)−κI(T;Vk)+ε]
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Proof sketch
i.i.d. PZ i.i.d. PT
Xn
zn(i) tm(j|i)
codevectors
codevectors
2n[I(X;Z)+ε]
2m[I(S;T )+ε]
Um is chosen such that (Sm, tm(j|i),Um) is jointly typical
Decoder finds unique (i, j) such that (zn(i),Ynk ) and (tm(j|i),Vm
k ) areboth jointly typical
Pke ≤ 2n[I(X;Z)+κI(T;S)−I(Yk;Z)−κI(T;Vk)+ε]
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Proof sketch
i.i.d. PZ i.i.d. PT
Xn
zn(i) tm(j|i)
codevectors
codevectors
2n[I(X;Z)+ε]
2m[I(S;T )+ε]
Um is chosen such that (Sm, tm(j|i),Um) is jointly typical
Decoder finds unique (i, j) such that (zn(i),Ynk ) and (tm(j|i),Vm
k ) areboth jointly typical
Pke ≤ 2n[I(X;Z)+κI(T;S)−I(Yk;Z)−κI(T;Vk)+ε]
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Gaussian Problem
From now on, we exclusively work with
κ = 1 (i.e., m = n)K = 2 recieversGaussian sources and channelsSquare error distortion measure d(X, X) = (X − X)2
Average power constraint E{U2} ≤ P
Compare the performances of
Uncoded (analog) transmissionSeparate codingCDSLayered coding based on DPC-CDS.Hybrid digital/analog (HDA) schemes.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Gaussian Problem
From now on, we exclusively work withκ = 1 (i.e., m = n)K = 2 recieversGaussian sources and channelsSquare error distortion measure d(X, X) = (X − X)2
Average power constraint E{U2} ≤ P
Compare the performances of
Uncoded (analog) transmissionSeparate codingCDSLayered coding based on DPC-CDS.Hybrid digital/analog (HDA) schemes.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Gaussian Problem
From now on, we exclusively work withκ = 1 (i.e., m = n)K = 2 recieversGaussian sources and channelsSquare error distortion measure d(X, X) = (X − X)2
Average power constraint E{U2} ≤ P
Compare the performances ofUncoded (analog) transmissionSeparate coding
CDSLayered coding based on DPC-CDS.Hybrid digital/analog (HDA) schemes.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Gaussian Problem
From now on, we exclusively work withκ = 1 (i.e., m = n)K = 2 recieversGaussian sources and channelsSquare error distortion measure d(X, X) = (X − X)2
Average power constraint E{U2} ≤ P
Compare the performances ofUncoded (analog) transmissionSeparate codingCDS
Layered coding based on DPC-CDS.Hybrid digital/analog (HDA) schemes.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Gaussian Problem
From now on, we exclusively work withκ = 1 (i.e., m = n)K = 2 recieversGaussian sources and channelsSquare error distortion measure d(X, X) = (X − X)2
Average power constraint E{U2} ≤ P
Compare the performances ofUncoded (analog) transmissionSeparate codingCDSLayered coding based on DPC-CDS.
Hybrid digital/analog (HDA) schemes.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The Gaussian Problem
From now on, we exclusively work withκ = 1 (i.e., m = n)K = 2 recieversGaussian sources and channelsSquare error distortion measure d(X, X) = (X − X)2
Average power constraint E{U2} ≤ P
Compare the performances ofUncoded (analog) transmissionSeparate codingCDSLayered coding based on DPC-CDS.Hybrid digital/analog (HDA) schemes.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Uncoded transmission and separate coding
Yk = ρkX + Nk with σ2X = σ2
Yk= 1
Vk = U + Wk with σ2U = P.
Uncoded transmission
U =√
PX.
Dk =σ2
Nkσ2
Wkσ2
Wk+σ2
NkP ≥ Dp2p
k (P) =σ2
Nkσ2
Wkσ2
Wk+P
Separate coding
Since both the channel and the side information is degraded,two-layered source and channel codes are optimal
Achievable (D1,D2) is completely known (Steinberg and Merhav2004, Tian and Diggavi 2007)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Uncoded transmission and separate coding
Yk = ρkX + Nk with σ2X = σ2
Yk= 1
Vk = U + Wk with σ2U = P.
Uncoded transmission
U =√
PX.
Dk =σ2
Nkσ2
Wkσ2
Wk+σ2
NkP ≥ Dp2p
k (P) =σ2
Nkσ2
Wkσ2
Wk+P
Separate coding
Since both the channel and the side information is degraded,two-layered source and channel codes are optimal
Achievable (D1,D2) is completely known (Steinberg and Merhav2004, Tian and Diggavi 2007)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Uncoded transmission and separate coding
Yk = ρkX + Nk with σ2X = σ2
Yk= 1
Vk = U + Wk with σ2U = P.
Uncoded transmission
U =√
PX.
Dk =σ2
Nkσ2
Wkσ2
Wk+σ2
NkP ≥ Dp2p
k (P) =σ2
Nkσ2
Wkσ2
Wk+P
Separate coding
Since both the channel and the side information is degraded,two-layered source and channel codes are optimal
Achievable (D1,D2) is completely known (Steinberg and Merhav2004, Tian and Diggavi 2007)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Uncoded transmission and separate coding
Yk = ρkX + Nk with σ2X = σ2
Yk= 1
Vk = U + Wk with σ2U = P.
Uncoded transmission
U =√
PX.
Dk =σ2
Nkσ2
Wkσ2
Wk+σ2
NkP ≥ Dp2p
k (P) =σ2
Nkσ2
Wkσ2
Wk+P
Separate coding
Since both the channel and the side information is degraded,two-layered source and channel codes are optimal
Achievable (D1,D2) is completely known (Steinberg and Merhav2004, Tian and Diggavi 2007)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Uncoded transmission and separate coding
Yk = ρkX + Nk with σ2X = σ2
Yk= 1
Vk = U + Wk with σ2U = P.
Uncoded transmission
U =√
PX.
Dk =σ2
Nkσ2
Wkσ2
Wk+σ2
NkP ≥ Dp2p
k (P) =σ2
Nkσ2
Wkσ2
Wk+P
Separate coding
Since both the channel and the side information is degraded,two-layered source and channel codes are optimal
Achievable (D1,D2) is completely known (Steinberg and Merhav2004, Tian and Diggavi 2007)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance of CDS
Distortion performance
Dk =1
1σ2
Nk
+ Pmax
{σ2
N1σ2
W1,σ2
N2σ2
W2
}
When σ2W1σ2
N1= σ2
W2σ2
N2, this reduces to
Dk =σ2
Nkσ2
Wk
σ2Wk
+ P= Dp2p
k (P) k = 1, 2
Otherwise, whoever has the larger σ2Wkσ2
Nkachieves Dp2p
k (P).
We refer to σ2Wkσ2
Nkas the combined channel/side information
quality.
Once again, the better channel can afford having the worse sideinformation and vice versa.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance of CDS
Distortion performance
Dk =1
1σ2
Nk
+ Pmax
{σ2
N1σ2
W1,σ2
N2σ2
W2
}
When σ2W1σ2
N1= σ2
W2σ2
N2, this reduces to
Dk =σ2
Nkσ2
Wk
σ2Wk
+ P= Dp2p
k (P) k = 1, 2
Otherwise, whoever has the larger σ2Wkσ2
Nkachieves Dp2p
k (P).
We refer to σ2Wkσ2
Nkas the combined channel/side information
quality.
Once again, the better channel can afford having the worse sideinformation and vice versa.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance of CDS
Distortion performance
Dk =1
1σ2
Nk
+ Pmax
{σ2
N1σ2
W1,σ2
N2σ2
W2
}When σ2
W1σ2
N1= σ2
W2σ2
N2, this reduces to
Dk =σ2
Nkσ2
Wk
σ2Wk
+ P= Dp2p
k (P) k = 1, 2
Otherwise, whoever has the larger σ2Wkσ2
Nkachieves Dp2p
k (P).
We refer to σ2Wkσ2
Nkas the combined channel/side information
quality.
Once again, the better channel can afford having the worse sideinformation and vice versa.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance of CDS
Distortion performance
Dk =1
1σ2
Nk
+ Pmax
{σ2
N1σ2
W1,σ2
N2σ2
W2
}When σ2
W1σ2
N1= σ2
W2σ2
N2, this reduces to
Dk =σ2
Nkσ2
Wk
σ2Wk
+ P= Dp2p
k (P) k = 1, 2
Otherwise, whoever has the larger σ2Wkσ2
Nkachieves Dp2p
k (P).
We refer to σ2Wkσ2
Nkas the combined channel/side information
quality.
Once again, the better channel can afford having the worse sideinformation and vice versa.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance of CDS
Distortion performance
Dk =1
1σ2
Nk
+ Pmax
{σ2
N1σ2
W1,σ2
N2σ2
W2
}When σ2
W1σ2
N1= σ2
W2σ2
N2, this reduces to
Dk =σ2
Nkσ2
Wk
σ2Wk
+ P= Dp2p
k (P) k = 1, 2
Otherwise, whoever has the larger σ2Wkσ2
Nkachieves Dp2p
k (P).
We refer to σ2Wkσ2
Nkas the combined channel/side information
quality.
Once again, the better channel can afford having the worse sideinformation and vice versa.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance of CDS
Distortion performance
Dk =1
1σ2
Nk
+ Pmax
{σ2
N1σ2
W1,σ2
N2σ2
W2
}When σ2
W1σ2
N1= σ2
W2σ2
N2, this reduces to
Dk =σ2
Nkσ2
Wk
σ2Wk
+ P= Dp2p
k (P) k = 1, 2
Otherwise, whoever has the larger σ2Wkσ2
Nkachieves Dp2p
k (P).
We refer to σ2Wkσ2
Nkas the combined channel/side information
quality.
Once again, the better channel can afford having the worse sideinformation and vice versa.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The common receiver (the one with larger σ2Wkσ2
Nk)
sees Unr as interference
cannot decode either Unc or Un
rdecodes Zn
c and Tn, but throws away the latteroutputs Xn
c using φc(Zc,Yc).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The common receiver (the one with larger σ2Wkσ2
Nk)
sees Unr as interference
cannot decode either Unc or Un
rdecodes Zn
c and Tn, but throws away the latteroutputs Xn
c using φc(Zc,Yc).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The common receiver (the one with larger σ2Wkσ2
Nk)
sees Unr as interference
cannot decode either Unc or Un
rdecodes Zn
c and Tn, but throws away the latteroutputs Xn
c using φc(Zc,Yc).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The common receiver (the one with larger σ2Wkσ2
Nk)
sees Unr as interference
cannot decode either Unc or Un
rdecodes Zn
c and Tn, but throws away the latter
outputs Xnc using φc(Zc,Yc).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The common receiver (the one with larger σ2Wkσ2
Nk)
sees Unr as interference
cannot decode either Unc or Un
rdecodes Zn
c and Tn, but throws away the latteroutputs Xn
c using φc(Zc,Yc).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The refinement receiver (the one with smaller σ2Wkσ2
Nk)
decodes Znc and Tn as before, but keeps Tn
sees Unc as additional noise and Tn as an additional channel output
decodes Unr (and hence the bin index)
decodes Znr and outputs Xn
r using φr(Zr,Yr).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The refinement receiver (the one with smaller σ2Wkσ2
Nk)
decodes Znc and Tn as before, but keeps Tn
sees Unc as additional noise and Tn as an additional channel output
decodes Unr (and hence the bin index)
decodes Znr and outputs Xn
r using φr(Zr,Yr).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The refinement receiver (the one with smaller σ2Wkσ2
Nk)
decodes Znc and Tn as before, but keeps Tn
sees Unc as additional noise and Tn as an additional channel output
decodes Unr (and hence the bin index)
decodes Znr and outputs Xn
r using φr(Zr,Yr).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The refinement receiver (the one with smaller σ2Wkσ2
Nk)
decodes Znc and Tn as before, but keeps Tn
sees Unc as additional noise and Tn as an additional channel output
decodes Unr (and hence the bin index)
decodes Znr and outputs Xn
r using φr(Zr,Yr).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The layered description scheme (LDS)
Consider the following layered encoding scheme (Nayak,Tuncel, and Gunduz 2010):
Xn
Qc Σ−
+
Qr
Znc
Znr BINNING
CHANNELENCODER
Unr
DPC-CDSUn
c
Σ
+
+
T n
Un
The refinement receiver (the one with smaller σ2Wkσ2
Nk)
decodes Znc and Tn as before, but keeps Tn
sees Unc as additional noise and Tn as an additional channel output
decodes Unr (and hence the bin index)
decodes Znr and outputs Xn
r using φr(Zr,Yr).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance comparisons
σ2W1σ2
N1= σ2
W2σ2
N2
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.2
0.25
0.3
0.35
0.4
D1
D2
N1 = 0.9,W1 = 0.4,N2 = 0.4,W2 = 0.9
ConverseUncodedCDSLDSSeparate
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance comparisons
σ2W1σ2
N1> σ2
W2σ2
N2, σ2
W1> σ2
W2, and σ2
N1< σ2
N2
0.2 0.25 0.3 0.35 0.40.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
D1
D2
N1 = 0.4,W1 = 0.9,N2 = 0.6,W2 = 0.4
ConverseUncodedCDSLDSSeparate
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance comparisons
σ2W1σ2
N1> σ2
W2σ2
N2, σ2
W1> σ2
W2, and σ2
N1> σ2
N2
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
D1
D2
N1 = 0.9,W1 = 0.4,N2 = 0.4,W2 = 0.1
ConverseUncodedCDSLDSSeparate
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Performance comparisons
σ2N1
= σ2N2
= 1
0.5 0.6 0.7 0.8 0.9 1
0.3
0.35
0.4
0.45
0.5
D1
D2
N1 = 1,W1 = 0.9,N2 = 1,W2 = 0.4
ConverseUncodedCDSLDSSeparate
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The HDA-WZ scheme
T n
Encoder
Σ
T n
Decoder
EstimatorXn Un
W n
V n Tn Xn
Y n
Y n
V n
Wilson, Narayanan, and Caire 2010.
The same codebook is used for source coding and channelcoding.
Minimum distortion is achieved when I(X;T)→ I(T;V,Y).
D =σ2
Nσ2W
P+σ2W= Dp2p(P)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The HDA-WZ scheme
T n
Encoder
Σ
T n
Decoder
EstimatorXn Un
W n
V n Tn Xn
Y n
Y n
V n
Wilson, Narayanan, and Caire 2010.
The same codebook is used for source coding and channelcoding.
Minimum distortion is achieved when I(X;T)→ I(T;V,Y).
D =σ2
Nσ2W
P+σ2W= Dp2p(P)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The HDA-WZ scheme
T n
Encoder
Σ
T n
Decoder
EstimatorXn Un
W n
V n Tn Xn
Y n
Y n
V n
Wilson, Narayanan, and Caire 2010.
The same codebook is used for source coding and channelcoding.
Minimum distortion is achieved when I(X;T)→ I(T;V,Y).
D =σ2
Nσ2W
P+σ2W= Dp2p(P)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
The HDA-WZ scheme
T n
Encoder
Σ
T n
Decoder
EstimatorXn Un
W n
V n Tn Xn
Y n
Y n
V n
Wilson, Narayanan, and Caire 2010.
The same codebook is used for source coding and channelcoding.
Minimum distortion is achieved when I(X;T)→ I(T;V,Y).
D =σ2
Nσ2W
P+σ2W= Dp2p(P)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
HDA-WZ scheme over broadcast channels
T n
Encoder
Σ
T n
Decoder
Estimator
Xn
Un
W n1
V n1
Tn Xn1
Y n1
Y n1
ΣUn
W n2
V n2
T n
Decoder
Y n2
EstimatorTn Xn
2
Y n2
V n1
V n2
Similar to the CDS, HDA-WZ achieves Dk = Dp2pk (P)
simultaneously for k = 1, 2 if
σ2N1(P + σ2
W1) = σ2
N2(P + σ2
W2)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
HDA-WZ scheme over broadcast channels
T n
Encoder
Σ
T n
Decoder
Estimator
Xn
Un
W n1
V n1
Tn Xn1
Y n1
Y n1
ΣUn
W n2
V n2
T n
Decoder
Y n2
EstimatorTn Xn
2
Y n2
V n1
V n2
Similar to the CDS, HDA-WZ achieves Dk = Dp2pk (P)
simultaneously for k = 1, 2 if
σ2N1(P + σ2
W1) = σ2
N2(P + σ2
W2)
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Head-to-head: HDA-WZ vs CDS
Fix σ2W1
, σ2W2
and assume σ2W1> σ2
W2
Conditions for Dk = Dp2pk (P) for k = 1, 2:
σ2N1
σ2N2
σ2N
2
=σ2N
1
1
1 Uncoded
CD
S
HDA-W
Z
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Head-to-head: HDA-WZ vs CDS
Fix σ2W1
, σ2W2
and assume σ2W1> σ2
W2
Conditions for Dk = Dp2pk (P) for k = 1, 2:
σ2N1
σ2N2
σ2N
2
=σ2N
1
1
1 UncodedC
DS
HDA-W
Z
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A new point-to-point scheme
Σ++DPC-CDS
EncoderXn
Σ++
Wn
Und DPC-CDS
DecoderEstimator
Xn
V n
Y n
Zn
Σ+
Zn−
EnHDA-WZEncoder
Unh
HDA-WZDecoder
Tnh
Tnd
Gao and Tuncel 2011.
Dirty-paper codeword Td = γUnh + Un
d
P is split into νP and (1− ν)P for DPC-CDS and HDA-WZstreams
For fixed ν, γ = γCosta =νP
νP+σ2W
maximizes the capacitybetween Un
d and Vn.
Turns out that we do not have to use γ = γCosta to achieveD = Dp2p(P).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A new point-to-point scheme
Σ++DPC-CDS
EncoderXn
Σ++
Wn
Und DPC-CDS
DecoderEstimator
Xn
V n
Y n
Zn
Σ+
Zn−
EnHDA-WZEncoder
Unh
HDA-WZDecoder
Tnh
Tnd
Gao and Tuncel 2011.
Dirty-paper codeword Td = γUnh + Un
d
P is split into νP and (1− ν)P for DPC-CDS and HDA-WZstreams
For fixed ν, γ = γCosta =νP
νP+σ2W
maximizes the capacitybetween Un
d and Vn.
Turns out that we do not have to use γ = γCosta to achieveD = Dp2p(P).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A new point-to-point scheme
Σ++DPC-CDS
EncoderXn
Σ++
Wn
Und DPC-CDS
DecoderEstimator
Xn
V n
Y n
Zn
Σ+
Zn−
EnHDA-WZEncoder
Unh
HDA-WZDecoder
Tnh
Tnd
Gao and Tuncel 2011.
Dirty-paper codeword Td = γUnh + Un
d
P is split into νP and (1− ν)P for DPC-CDS and HDA-WZstreams
For fixed ν, γ = γCosta =νP
νP+σ2W
maximizes the capacitybetween Un
d and Vn.
Turns out that we do not have to use γ = γCosta to achieveD = Dp2p(P).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A new point-to-point scheme
Σ++DPC-CDS
EncoderXn
Σ++
Wn
Und DPC-CDS
DecoderEstimator
Xn
V n
Y n
Zn
Σ+
Zn−
EnHDA-WZEncoder
Unh
HDA-WZDecoder
Tnh
Tnd
Gao and Tuncel 2011.
Dirty-paper codeword Td = γUnh + Un
d
P is split into νP and (1− ν)P for DPC-CDS and HDA-WZstreams
For fixed ν, γ = γCosta =νP
νP+σ2W
maximizes the capacitybetween Un
d and Vn.
Turns out that we do not have to use γ = γCosta to achieveD = Dp2p(P).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A new point-to-point scheme
Σ++DPC-CDS
EncoderXn
Σ++
Wn
Und DPC-CDS
DecoderEstimator
Xn
V n
Y n
Zn
Σ+
Zn−
EnHDA-WZEncoder
Unh
HDA-WZDecoder
Tnh
Tnd
Gao and Tuncel 2011.
Dirty-paper codeword Td = γUnh + Un
d
P is split into νP and (1− ν)P for DPC-CDS and HDA-WZstreams
For fixed ν, γ = γCosta =νP
νP+σ2W
maximizes the capacitybetween Un
d and Vn.
Turns out that we do not have to use γ = γCosta to achieveD = Dp2p(P).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A new degree of freedom
For any power allocation, instead of only Costa’s γ, a range of γ leadsto the optimal distortion D = Dp2p(P).
00
1Feasible region vs. γCosta
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A new WZBC scheme
Constructed with the same idea as in the point-to-point scheme:
Σ++DPC-CDS
EncoderXn
Σ++
Wn1
Und
DPC-CDSDecoder
EstimatorXn
1
V n1
Y n1
ZnΣ+
Zn−
EnHDA-WZEncoder
Unh
HDA-WZDecoder
Tnh
Tnd
Σ++
Wn2
DPC-CDSDecoder
EstimatorXn
2
V n2
Y n2
Zn
HDA-WZDecoder
Tnh
Tnd
We explore the freedom in (ν, γ) to see if we can achieveDk = Dp2p
k (P) simultaneously for k = 1, 2.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
A new WZBC scheme
Constructed with the same idea as in the point-to-point scheme:
Σ++DPC-CDS
EncoderXn
Σ++
Wn1
Und
DPC-CDSDecoder
EstimatorXn
1
V n1
Y n1
ZnΣ+
Zn−
EnHDA-WZEncoder
Unh
HDA-WZDecoder
Tnh
Tnd
Σ++
Wn2
DPC-CDSDecoder
EstimatorXn
2
V n2
Y n2
Zn
HDA-WZDecoder
Tnh
Tnd
We explore the freedom in (ν, γ) to see if we can achieveDk = Dp2p
k (P) simultaneously for k = 1, 2.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Main result
Theorem: Dk =σ2
Nkσ2
WkP+σ2
Wk
= Dp2pk (P) is achieved whenever
P + σ2W1
P + σ2W2
σ2N1≤ σ2
N2≤σ2
W1
σ2W2
σ2N1
σ2N1
σ2N2
σ2N
2
=σ2N
1
1
1 Uncoded
CD
S
HDA-W
Z
Lemma: The combination of DPC-CDS and HDA-WZ cannever perform worse than LDS.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Main result
Theorem: Dk =σ2
Nkσ2
WkP+σ2
Wk
= Dp2pk (P) is achieved whenever
P + σ2W1
P + σ2W2
σ2N1≤ σ2
N2≤σ2
W1
σ2W2
σ2N1
σ2N1
σ2N2
σ2N
2
=σ2N
1
1
1 Uncoded
CD
S
HDA-W
Z
Lemma: The combination of DPC-CDS and HDA-WZ cannever perform worse than LDS.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Main result
Theorem: Dk =σ2
Nkσ2
WkP+σ2
Wk
= Dp2pk (P) is achieved whenever
P + σ2W1
P + σ2W2
σ2N1≤ σ2
N2≤σ2
W1
σ2W2
σ2N1
σ2N1
σ2N2
σ2N
2
=σ2N
1
1
1 Uncoded
CD
S
HDA-W
Z
Lemma: The combination of DPC-CDS and HDA-WZ cannever perform worse than LDS.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Summary
An unexpectedly simple method based on virtual binning turnsout to be optimal for lossless coding with side information overbroadcast channels.
When generalized to lossy coding for Gaussian sources andchannels, it can still be optimal if the combined quality σ2
Nσ2W is
constant among receivers.
If not, then you can send refinement information to whoever hasless σ2
Nσ2W .
At worst case, the performance of the digital layered schemeLDS coincides with that of separate coding. It is always better ifthe weaker channel has better side information.
When combined with a hybrid digital/analog method, ourmethod actually achieves the optimum Dp2p
k (P) for both channelsin a certain region in the parameter space of (σ2
W1, σ2
W2, σ2
N1, σ2
N2).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Summary
An unexpectedly simple method based on virtual binning turnsout to be optimal for lossless coding with side information overbroadcast channels.
When generalized to lossy coding for Gaussian sources andchannels, it can still be optimal if the combined quality σ2
Nσ2W is
constant among receivers.
If not, then you can send refinement information to whoever hasless σ2
Nσ2W .
At worst case, the performance of the digital layered schemeLDS coincides with that of separate coding. It is always better ifthe weaker channel has better side information.
When combined with a hybrid digital/analog method, ourmethod actually achieves the optimum Dp2p
k (P) for both channelsin a certain region in the parameter space of (σ2
W1, σ2
W2, σ2
N1, σ2
N2).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Summary
An unexpectedly simple method based on virtual binning turnsout to be optimal for lossless coding with side information overbroadcast channels.
When generalized to lossy coding for Gaussian sources andchannels, it can still be optimal if the combined quality σ2
Nσ2W is
constant among receivers.
If not, then you can send refinement information to whoever hasless σ2
Nσ2W .
At worst case, the performance of the digital layered schemeLDS coincides with that of separate coding. It is always better ifthe weaker channel has better side information.
When combined with a hybrid digital/analog method, ourmethod actually achieves the optimum Dp2p
k (P) for both channelsin a certain region in the parameter space of (σ2
W1, σ2
W2, σ2
N1, σ2
N2).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Summary
An unexpectedly simple method based on virtual binning turnsout to be optimal for lossless coding with side information overbroadcast channels.
When generalized to lossy coding for Gaussian sources andchannels, it can still be optimal if the combined quality σ2
Nσ2W is
constant among receivers.
If not, then you can send refinement information to whoever hasless σ2
Nσ2W .
At worst case, the performance of the digital layered schemeLDS coincides with that of separate coding. It is always better ifthe weaker channel has better side information.
When combined with a hybrid digital/analog method, ourmethod actually achieves the optimum Dp2p
k (P) for both channelsin a certain region in the parameter space of (σ2
W1, σ2
W2, σ2
N1, σ2
N2).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Summary
An unexpectedly simple method based on virtual binning turnsout to be optimal for lossless coding with side information overbroadcast channels.
When generalized to lossy coding for Gaussian sources andchannels, it can still be optimal if the combined quality σ2
Nσ2W is
constant among receivers.
If not, then you can send refinement information to whoever hasless σ2
Nσ2W .
At worst case, the performance of the digital layered schemeLDS coincides with that of separate coding. It is always better ifthe weaker channel has better side information.
When combined with a hybrid digital/analog method, ourmethod actually achieves the optimum Dp2p
k (P) for both channelsin a certain region in the parameter space of (σ2
W1, σ2
W2, σ2
N1, σ2
N2).
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Not included in this talk
We have an even more general HDA scheme than thecombination of DPC-CDS and HDA-WZ. We send an additionalanalog signal, thereby performing in the worst case as good asuncoded transmission.
We are also working on schemes that transmit blocks of size 2n,and treat them as n-length blocks of 2-D vectors. This mightcreate even more freedom in point-to-point transmission.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Not included in this talk
We have an even more general HDA scheme than thecombination of DPC-CDS and HDA-WZ. We send an additionalanalog signal, thereby performing in the worst case as good asuncoded transmission.
We are also working on schemes that transmit blocks of size 2n,and treat them as n-length blocks of 2-D vectors. This mightcreate even more freedom in point-to-point transmission.
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Collaborators
Deniz Gunduz Jayanth Nayak Yang Gao
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Collaborators
Deniz Gunduz
Jayanth Nayak Yang Gao
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Collaborators
Deniz Gunduz Jayanth Nayak
Yang Gao
Introduction and Background Virtual Binning Digital WZBC Schemes HDA Schemes Summary and Conclusions
Collaborators
Deniz Gunduz Jayanth Nayak Yang Gao