Communication Over Unknown Channels: A Personal Perspective of Over a Decade Research* Meir Feder Dept. of Electrical Engineering-Systems Tel-Aviv University * The presentation includes joint works with Amos Lapidoth, Neri Merhav, Nadav Shulman, Elona Erez, Ofer Shayevitz and Yuval Lomnitz

The Problem and The Models

Unknown Channels Shannon: A Channel is a known Random Function of the Output y, given the Input x p(y|x) is known Capacity is defined and known Rates up to capacity can be attained reliably Feedback may only help when there is memory in the channel, but is not needed to “learn” the channel Unknown Channels: Possible models Stochastic channels with unknown probability model Individual channels: Arbitrary (unknown) input-output relation Capacity? Communication schemes? Feedback? Follow the success of universal source coding

The Compound Channel First considered by Blackwell, Breiman and Thomasian, 1960 Worst case “capacity”: max min I(X;Y,θ) p(x) θ Can be attained with “universal decoding”: MMI decoder for unknown DMC’s The compound capacity of unknown BSC’s, unknown scalar fading (y = θx + n) and other “common” models is ZERO

The Arbitrary Varying Channel Also considered by Blackwell, Breiman and Thomasian, 1960 With randomization, “worst case” capacity is known:

The Individual Noise Channel May be considered as a special case of the AVC However, since feedback and “variable rate” communication is natural, provides a new point of view Can attain, where unknown. But what does it mean?

The Models Make minimal assumptions about the channel Yet, get beyond worst case “outage” approach: Adapt to the instantaneous specific channel mode. Feedback seems to be a must. Broadcast channel approach?

Outline: The Considered Topics The Receiver Problem: Universal Decoding Universal decoding in the general case, channels with memory The “criterion” Rateless Codes for Universal Communication “Static Broadcast” with Simple Feedback (Ack/Nack) Code generation Universal Communication with Feedback The “Universal Horstein” scheme Individual Channels The setting and the plausible “rateless” solution Lesson Learned Practical: Rateless – incremental redundancy, “universal” decoding Theoretical:

Universal Decoding

The problem

The solution

Composite Hypotheses

A simple fading example Two Codewords: GLRT: New decoder: Uniformly improving the GLRT:

Practical Universal Decoders? Both GLRT and new decoders are exponential in block length Training? Lose rate! Decision Feedback? But apply “weighted” metric

Rateless Codes

A simple “universal” transmission strategy

Universal Prior Look for a prior P attaining:, For binary channels the uniform prior attain all the above: Non binary channels higher loss (uniform, conjecture): Similar result for Gaussian input

Practical Rateless Codes: Use efficiently decodable codes Incremental redundancy “Fountain Codes”? “Raptor Codes”? Rateless codes for Gaussian Channels: “multilevel” construction for incremental redundancy (Erez et al). Recent extension for MIMO channels

Individual Noise Channel with Feedback

The Individual Noise Channel with Feedback SF (‘09): Use “Universal” Horstein’s scheme - Use sequential estimate of the noise empirical probability

Specific scheme outline Randomization is a must Attain “empirical capacity”: Similar performance obtained by using “rateless” codes - Esarwan et al (‘10)

Individual Channels

Short Summary – L&F

The BIG Question: A “True” Universal Communication Scheme

Plausible answer: Reference scheme is constrained to be “finite block” (or FS): For modulo additive “individual” channel optimal performance is where z is the additive individual noise sequence This can be attained universally with feedback (LF submitted to ISIT 2011)

Lesson Learned

Practical Rateless Codes for Universal Communication Feedback is a must Simple Feedback (Ack/Nack) can work – ARQ schemes Incremental redundancy With Practical Universal Decoding Training data is a simple mean for universally decodable codebooks Training and decision based decoding with modified “metric”

Theoretical

Individual channels The problem and possible solution seems to be defined:  A “limited” resource scheme – finite block, finite state, with (or without feedback) can be designed retrospectively after tuned to the channel empirical behavior  The performance of this scheme can be attained universally by the scheme that was not tuned to the empirical measurements – at some “redundancy cost”  The universal scheme will use feedback (probably in the form of rateless coding – i.e. minimal feedback) and randomization  A task to complete!!

THANK YOU!