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Sum Rate of Gaussian Multiterminal Source
Coding
Pramod Viswanath
University of Illinois, Urbana-Champaign
March 19, 2003
Gaussian Multiterminal Source Coding
ENCODER 2
ENCODER 1
ENCODER K
E
DECOD
R
yK = htKn
n2
n1
nN
n =n
y2 = ht2n
y1 = ht1n
mK
m2
m1
Sum of rates of encoders R and distortion metric d (n, n)
Quadratic Gaussian CEO Problem
ENCODER K
ENCODER 1
ENCODER 2
ER
DOCED
y1 = n + z1
n n
yK = n + zK
x2 = n + z2
mK
m2
m1
Sum of rates of encoders R and distortion metric d (n, n)
Result: Quadratic Gaussian CEO
• quadratic distortion metric d (n, n) = (n− n)2
• For large number of encoders K,
R(D) =1
2log+
(σ2
n
D
)+
σ2z
2σ2n
(σ2
n
D− 1
)+
– Second term is loss w.r.t. cooperating encoders
Outline
• Problem Formulation:
Tradeoff between sum rate R and distortion
(metric d (n, n)).
• Main Result:
Characterize a class of distortion metrics for which no loss
in sum rate compared with encoder cooperation
– A multiple antenna test channel
Random Binning of Slepian-Wolf
∆
O
∆
××
×O
O
O
×
∆∆
∆
×
××
∆
∆
∆
O
O
O
y1
y2
×
∆
×
O
O
×
• Rate is number of quantizers
Encoding in Slepian-Wolf
∆
O
∆
××
×O
O
O
×
∆∆
∆
×
××
∆
∆
∆
O
O
O
y1
y2
×
∆
×
O
O
×
• Quantizer closest to realization
Decoding in Slepian-Wolf
• Decoder knows joint distribution of y1, y2
• It is given the two quantizer numbers from the encoders
• Picks the pair of points in the quantizers which best matches
the joint distribution
– For jointly Gaussian y1, y2 nearest neighbor type test
• R1 + R2 = H (y1, y2) is sufficient for zero distortion
Deterministic Broadcast Channel
y2 = g(x)
x
y1 = f(x)
• Pick distribution on x such that y1, y2 have desired joint dis-
tribtion (Cover 98)
Slepian-Wolf code for Broadcast Channel
• Encoding: implement Slepian-Wolf decoder
– given two messages, find the appropriate pair y1, y2 in the
two quantizers
– transmit x that generates this pair y1, y2.
Slepian-Wolf code for Broadcast Channel
• Encoding: implement Slepian-Wolf decoder
– given two messages, find the appropriate pair y1, y2 in the
two quantizers
– transmit x that generates this pair y1, y2.
• Decoding: implement Slepian-Wolf encoder
– quantize y1, y2 to nearest point
– messages are the quantizer numbers
Lossy Slepian-Wolf Source Coding
∆
O
∆
××
×O
O
O
×
∆∆
∆
×
××
∆
∆
∆
O
O
O
u1
u2
×
∆
×
O
O
×
• Approximate y1, y2 by u1, u2
Lossy Slepian-Wolf Source Coding
• Encoding: Find ui that matches source yi, separately for
each i
– For jointly Gaussian r.v. s, nearest neighbor calculation
– Each encoder sends quantizer number containing the u
picked
Lossy Slepian-Wolf Source Coding
• Encoding: Find ui that matches source yi, separately foreach i
– For jointly Gaussian r.v. s, nearest neighbor calculation
– Each encoder sends quantizer number containing the u
picked
• Decoding: Reconstruct the u’s picked by the encoders
– reconstruction based on joint distribution of u’s
– Previously ui = yi were correlated
– Here u’s are independently picked
Lossy Slepian-Wolf
• We require
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p [ui|yi]
Lossy Slepian-Wolf
• We require
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p [ui|yi]
• Generate n1, . . . , nK deterministically from reconstructed u’s.
• Need sum rate
Rsum = I (u1, . . . , uK; y1, . . . , yK) .
• Distortion equal to
E [d (n, n)] .
Marton Coding for Broadcast Channel
Dec1
Dec2
w ∼ N (0, I)
Hty
u1
u2
xu1, u2
Enc
• Reversed encoding and decoding operations
• Sum rate I (u1, . . . , uK; y1, . . . , yK) .
• No use for the Markov property
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p [ui|yi]
Achievable Rates: Costa Precoding
m1 u1
w
xHt
m2 u2
m1
m2
• Users’ data modulated onto spatial signatures u1,u2
Stage 1: Costa Precoding
Dec1
m1 u1
w
xHt
m2 u2
m1
• Encoding for user 1 treating signal from user 2 as known
interference at transmitter
Stage 2
Dec2
m1 u1
w
xHt
m2 u2
m2
• Encode user 2 treating signal for user 1 as noise
Adaptation to Lossy Slepian-Wolf
DecHt
z ∼ N (0, Kz)
xu1, u2
Ency
u1
u2
• Joint distribution of u’s and y’s depends on noise z
• Performance independent of correlation in z
Noise Coloring
• Fix particular Costa coding scheme - fixes u’s and x.
• Idea:
Choose z such that
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p [ui|yi]
and (Kz)ii = 1
• Then can adapt to Lossy Multiterminal Source Coding
Markov Condition and Broadcast Channel
• The Markov condition
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p [ui|yi]
of independent interest in the broadcast channel
Markov Condition and Broadcast Channel
• The Markov condition
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p [ui|yi]
of independent interest in the broadcast channel
•
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p
[ui|y1, . . . , yK, u1, u2, . . . , ui−1
]
Markov Condition and Broadcast Channel
• The Markov condition
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p [ui|yi]
of independent interest in the broadcast channel
•
p [u1, . . . , uK|y1, . . . , yK] = ΠKi=1p
[ui|y1, . . . , yK, u1, u2, . . . , ui−1
]
• Equivalent to: given u1, . . . , ui−1
ui −→ yi −→ y1, . . . , yi−1, yi+1, . . . , yK
Implication
DecHt
z ∼ N (0, Kz)
xu1, u2
Ency
u1
u2
• Need only y1 to decode u1
• Given u1, need only y2 to decode u2
Performance of Costa scheme equals that when receivers
cooperate
Markov Condition and Noise Covariance
• The sum capacity is also achieved by such a scheme
(CS 01, YC 01, VT 02, VJG 02)
• For every Costa scheme, there is a choice of Kz such that
Markov condition holds (Yu and Cioffi, 01)
Sum Capacity
z ∼ N (0, Kz)ReceiversCooperating
Ht
H
Ht
H
Sato
Enc
w
wBroadcast
DecEnc
DecEnc
w
E[xtKzx
] ≤ P
ReciprocityReciprocity
x1, x2 independent
Multiple Access Cooperating Transmitters
Sum Capacity
z ∼ N (0, Kz)ReceiversCooperating
Ht
H
Ht
H
Sato
Enc
w
wBroadcast
DecEnc
DecEnc
w
E[xtKzx
] ≤ P
Reciprocity
Convex Duality
Reciprocity
x1, x2 independent
Multiple Access Cooperating Transmitters
Gaussian Multiterminal Source Coding
ENCODER 2
ENCODER 1
ENCODER K
E
DECOD
R
yK = htKn
n2
n1
nN
n =n
y2 = ht2n
y1 = ht1n
mK
m2
m1
H = [h1, . . . ,hK]
Sum of rates of encoders R and distortion metric d (n, n)
Main Result
• Distortion metric
d (n; n) =1
N(n− n)t
(I + Hdiag {p1, . . . , pK}Ht
)(n− n)
– Here p1, . . . , pK - powers of users in reciprocal MAC
• Rate distortion function
R(D) = Sum rate of MAC−N logD
= Sum rate of Broadcast Channel−N logD
= logdet(I + HDHt
)−N logD
Bells and Whistles
• For quadratic distortion metric
d (n; n) =1
N(n− n)t (n− n)
set of H can be characterized
• Analogy with CEO problem:
For large number of encoders and random H
characterization of R(D) almost surely
Discussion
• A “connection” made between coding schemes for multiter-
minal source and channel coding
Discussion
• A “connection” made between coding schemes for multiter-
minal source and channel coding
• Connection somewhat superficial
– relation between source coding and broadcast channel
through a common random coding argument (PR 02, CC
02)
– relation between source coding and multiple access chan-
nel through a change of variable (VT 02, JVG 01)
Discussion
• A “connection” made between coding schemes for multiter-minal source and channel coding
• Connection somewhat superficial
– relation between source coding and broadcast channelthrough a common random coding argument
– relation between source coding and multiple access chan-nel through a change of variable
• Connection is suggestive
– a codebook level duality