1. Transmission over composite channels with combined source-channel outage: Reza Mirghaderi and Andrea Goldsmith Work Summary STATUS QUO A subset Vo (with Pr {V o }≤q s ) 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-q s. 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 q s The receiver declares channel outage with probability q c 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 V qs and channel outage probability q c, the distortion level D is achievable if and only if where C qc is the capacity vs. outage, q=1-Pr{V qs }<q s, and R(V qs,.) is the distortion rate function for V qs. - 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 ṽ n, matches with the sent sequence v n in almost (1-q s )n positions (outage indices). Empirical distribution [4] P(V*, Ṽ *) between non-outage subsequence of ṽ n and v n converges to P(V qs, Ṽ qs ). 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-q c ), the information density[1], exceeds LHS of the above inequality HOW IT WORKS REFERENCES Definition: The distortion versus (q s, q c )-combined outage is defined as where the infimum is over all the subsources V qs with Pr{V qs }>1-q s and all source-channel encoder-decoder pairs. - Direct consequence of the separation theorem (q=1-Pr{V qs }) To design a system with minimum distortion vs. (q s,q c )combined outage - Find the probability-qs compatible subsource (Pr{V qs }>1-q s ) 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 q s +q c -q s q c. 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 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 [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.