work summary
DESCRIPTION
For a non-ergodic composite channel[1] (e.g. stringent delay constraint and CSIR), Shannon’s coding theorems provide a pessimistic capacity measure by enforcing lossless transmission of the data. - PowerPoint PPT PresentationTRANSCRIPT
Transmission over composite channels with combined source-channel outage:
Reza Mirghaderi and Andrea Goldsmith
Work Summary
STATUS QUO
A subset Vo (with Pr {Vo}≤qs) of source alphabet can be chosen as the outage set. Source outage is declared by the transmitter if any of the symbols in outage set are observed at the source output.
For a source with long distribution tails, source outage avoids allocation of high rate to the symbols on the tail
Source outage allows for a bandwidth expansion inversely proportional to 1-qs.
For a long block-length, the source outage process is stationary and ergodic
The outage indicator process should be transmitted to let the receiver recognize and reconstruct the non-outage part of the block
Source outage can be combined with channel outage
SET UP
The transmitter declares source outage with probability qs
The receiver declares channel outage with probability qc
In non-outage states, the receiver should perfectly decode the source outage indices
The non-outage source symbols should be decoded with distortion less than a given threshold (D)
DESIGN CHALLENGES
How to choose the optimal source outage set (necessary and sufficient conditions)?
What is the optimal source-channel code for this system?
Does the source-channel code separation theorem hold?
SEPARATION THEOREM
Theorem: Given the non-outage sub-source Vqs and channel outage probability qc, the distortion level D is achievable if and only if
where Cqc is the capacity vs. outage, q=1-Pr{Vqs}<qs, and R(Vqs,.) is the distortion rate function for Vqs.
- Proof of the direct part is done by separation of source and channel codes. The source outage indices and the non-outage sub-sequence of the source output are encoded separately and then superimposed and fed into the channel encoder
- For the proof of converse, we consider the strongly typical sequences[3]
•Each reconstruction sequence ṽn, matches with the sent sequence vn in almost (1-qs)n positions (outage indices).
• Empirical distribution [4] P(V*,Ṽ*) between non-outage subsequence of ṽn and vn converges to P(Vqs,Ṽqs). So does the entropy and mutual information. Also, E[d(V*,Ṽ*)]<D
• Use the above to show that for almost all the non-outage blocks (with prob. at least 1-qc), the information density[1], exceeds LHS of the above inequality
HOW IT WORKS
REFERENCES
Definition: The distortion versus (qs, qc)-combined outage is defined as
where the infimum is over all the subsources Vqs with Pr{Vqs}>1-qs and all source-channel encoder-decoder pairs.
- Direct consequence of the separation theorem (q=1-Pr{Vqs})
To design a system with minimum distortion vs. (qs,qc)combined outage
- Find the probability-qs compatible subsource (Pr{Vqs}>1-qs) which solves the above optimization problem (e.g. for Gaussian, choose the best truncated Gaussian)
- Design the source code and channel code separately as described in the proof of separation theorem
- The total fraction of lost symbols is qs+qc-qsqc. We can change source outage and channel outage probabilities to obtain the lowest distortion given a constant probability of losing a data symbol.
For a non-ergodic composite channel[1] (e.g. stringent delay constraint and CSIR), Shannon’s coding theorems provide a pessimistic capacity measure by enforcing lossless transmission of the data.
Declaring channel outage allows for data loss in some channel states in exchange for higher rates in other states.
Channel outage is declared by the receiver side, no data is received in an outage block, the outage process is non-ergodic
Distortion vs (channel) Outage[2] characterizes the transmission of a memory-less data source over a composite channel given a certain channel outage probability
[2] proved the optimality of source and channel code separation to minimize the distortion vs. channel outage.
[2] also proposed end-to-end performance metrics for which separation is not optimal
q2
q2
non-outage set
outage setoutage set Source/Channel Encoder
Information Source
NEW INSIGHTS
The outage process can be generalized to allow for a partial loss in each block of data in exchange for less distortion in the remaining part of the block.
NEW END-TO-END DISTORTION METRIC
Information Source
outage index encoder
non-outage sub-source
encoder
channel encoder
[1] M. Effros, A. Goldsmith and Y. Liang, “Capacity Definitions for General Channels with Receiver Side Information,” submitted to IEEE Transactions on Information Theory, April 2008.
[2] Y. Liang, A. Goldsmith and M. Effros, “Source-Channel Coding and Separation for Generalized Communication Systems,” To be submitted to IEEE Transactions on Information Theory, August 2008.
[3] T. Cover and J. Thomas, Elements of Information Theory. New York: Wiley, 1991.
[4] I. Csiszar and J. Korner. Information Theory: Coding Theorems for Discrete Memoryless Systems. Academic Press, New York, 1981.
Source/Channel Decoder
channel outage