Communication Over Unknown Channels:

A Personal Perspective of Over a Decade Research*

Meir Feder

Dept. of Electrical Engineering-SystemsTel-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

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

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!