Hybrid Codes and the
Point-to-Point Channel
Paul Cuff
Curt Schieler
Source-Channel Coding
f
p(y|x)
g
Correlation between S and Ŝ:
Achieved with separately designed encoder and decoder.
Video Transmission (example)
f
p(y|x)
Systematic Transmission (example)
Transmission
Analog
Digital
Hybrid
Relay
Transmission
Copy-Robust Documents (example)
A document gets printed with redundancy .
Photocopy noise removed by photocopier.
Connection to General Point-to-Point
Channel Setting
f
p(y|x)
g
Digital Watermark (example)
• Media is modified to include extra information
• Scoundrel may try to delete the watermark
– Channel input (X) is modified media
– Ŝ is additional information (digital tag)
Define Empirical Coordination
f
p(y|x)
Empirical distribution:
is achievable if:
g
Separation Method
Channel
Index
Achieves product distributions:
such that
A Better Idea (Hybrid Codes)
Channel
Hybrid Codes
• Digital
– Source is compressed and coded in blocks
• Analog
– Channel input and reconstruction depend letterby-letter on the source and channel output
• Hybrid Codes take advantage of correlation in
network setting (i.e. interference channel)
– [Minero, Lim, Kim – Allerton 2010, ISIT 2011]
Achievable Inner Bound
is achievable if
Source
Channel
Markov Chain
Function (analog encoding)
Function( analog decoding)
Digital Decoding
Binary Example
• Source is Bern(p)
• Binary symmetric channel (Ɛ)
• Require reconstruction to equal channel input
– i.e. X = Ŝ
(systematic transmission)
• Minimize Hamming distortion
If p= .5: D = Ɛ (Optimal)
If p>0 and Ɛ>0 : D > 0 (Suboptimal)
State Amplification
• Channel State is known to the encoder
• Two objectives
No loss of generality
– Transmit a message
– Help decoder estimate state
f
[Kim, Sutivong, Cover – ‘08]
p(y|x,s)
g
[Choudhuri, Kim, Mitra – ‘10, ‘11]
Causal Achievable Region
is achievable iff
Source
Channel
Markov Chain
Function (analog encoding)
Function( analog decoding)
Digital Decoding
Strictly-Causal Achievable Region
is achievable iff
Source
Channel
Markov Chain
Function (analog encoding)
Function( analog decoding)
Digital Decoding