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# The Finite-state channel was introduced as early as 1953 [McMillan'53]. - PowerPoint PPT Presentation

Indecomposable Finite-State Channels with Feedback. Ron Dabora and Andrea Goldsmith. Summary. The Finite-State Channel Model. Definitions. Motivation. The Finite-state channel was introduced as early as 1953 [McMillan'53].

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Presentation Transcript

Ron Dabora and Andrea Goldsmith

Summary

The Finite-State Channel Model

Definitions

Motivation

• The Finite-state channel was introduced as early as 1953 [McMillan'53].

• Memory captured by channel state at end of previous symbol's transmission:

• - Siis the channel state at time i.

• - s0 is the initial channel state.

• p(yi, si|xi, xi-1, yi-1, si-1, s0) = p(yi, si|xi, si-1)

• - Si-1 contains all the history information for time i.

• - ||S|| <

• Example: ISI channel: Si-1 = (Xi-1,Xi-2, … ,Xi-J)

• Cellular Communication, DSL, magnetic recording, partial-response signaling.

• The FSC is defined by: {X x S, p(y,s|x,s’), Y x S }

• The FSC is called indecomposable if the effect of the

• initial state s0 on the state transitions becomes

• negligible as time evolves:

• For all ε>0 there exists N(ε) s.t. for every k > N(ε),

• xk,s0, s’0 it holds that

• |p(sk|xk, s0)-p(sk|xk, s’0)| < ε

• Finite ISI channels are indecomposable (w/o feedback)

• Discrete memoryless PtP channels:

• In recent years there is an increasing interest in time-

• varying channels with memory

• Filters (pulse shape, IF and RF filters)

• AGC, timing, AFC, PLL, equalizer

• FSCs model time-varying channels with memory in the

• discrete channel framework

Indecomposable FSCs with Feedback

Indecomposable FSCs without Feedback

• Notation

• k(n) monotone,

• Result

• Implications

• Can find capacity by optimizing p(xn) for a fixed arbitrary initial state s0

(no need to search over all initial states)

• However

•  Letting the receiver know the initial channel state may improve the rate

• For indecomposable FSCs the maximum achievable rate is the same for all

• initial states [Gallager’68]:

•  Capacity achieving scheme will provide same rate for all initial states s0

• Can find capacity by optimal p(xn) by considering some fixed arbitrary

initial state s0 (no need to search over all initial states)

• Letting the receiver know the initial channel state will not improve the rate

A Remark & a Definition

A channel may be indecomposable without feedback but non-indecomposable with feedback. We call such channels weakly indecomposable

Example:

yi = Q2[xi+ayi-1+ni], a constant, ni is zero-mean unit variance Gaussian RV

 Inhomogeneous Markov chain FSC.

FSCs with and without Feedback

Code for the FSC with Feedback

• An (R, n) code for the FSC with feedback consists

• of

• - Message set: M,

• - Encoder: fi:M × Y i-1X,

• - Decoder: g : Y nM.

• Encoder and decoder do not know the channel

• states

• The maximum average probability of error is

• defined as ,where

• Pe(n)(s0) = Pr(g(Yn) ≠M|s0)

• The capacity for channels with memory is usually given by a

• limiting expression as the blocklength

• Capacity without feedback [Gallager'68]

• Capacity with feedback [Pemuter, Weissman, Goldsmith’08]

• Feedback does not increase the capacity of PtP DMCs

• Feedback increases the capacity of PtP FSCs

• Notation:

?

A Code with Tx-Rx Synchronization Achieves Capacity of the XY-FSC

Conclusions

Rational Transfer Functions Channels

• FSCs model time-varying channels with memory in the

• discrete channel framework

• Indecomposable FSCs without feedback:

• - The effect of the initial state becomes negligible as

• time evolves

• - The capacity-achieving distribution provides the

• same rate for all initial states

• Indecomposable FSCs with feedback

• - The capacity of indecomposable FSCs with feedback

• can be found without searching over all initial

• channel states

• In general, with feedback, knowledge of the initial

• channel state may help achieve higher rates than the

• worst-case rate.

• For some weakly indecomposable channels, codes

• with synchronization can achieve the maximum over

• all initial states

• Capacity of XY-FSCs achieves the maximum over all initial states same as for

• indecomposable FSCs without feedback

• Code construction:

• Let s’0be the optimal initial state. This gives an optimal vector xNx ,yNy.

• For each message generate a codetree with the optimal initial state.

• Generate asequence of length Ls such that the last Nx symbols are the

• optimal vector xNx.

• - Total number of symbols dedicated for synchronization isLsynch=k(n)

• Synchronization: the transmitter starts (re)transmitting the Ls sequence and

• observes the feedback. If at the end of an Ls-sequence the feedback is yNy

• then synchronization is achieved:

• The receiver knows synchronization was achieved when it observes yNy at

• the end of the Ls-sequence.

• The transmitter knows synchronization was achieved as it knows the

• channel outputs through feedback.

• During the synchronization phase, feedback does not affect the transmitted

• sequence.

• Probability of Error: two error events

• Synchronization failed:synchronization was not achieved

• after transmitting all the Lsynch symbols, i.e., the optimal

• yNy was never observed at the end of an Ls-sequence.

• Decoding error: synchronization achieved but the ML

• decoder fails.

• Analysis

• Due to weak indecomposability, when dedicating k(n)

• symbols to synchronization, then taking n large enough the

• probability of failing to synchronize can be made arbitrarily

• small.

• As long as R<CXY-FBthe random coding argument guarantees

• the existence of a code with an arbitrarily small probability

• of error.

• Definition: XY-FSC are FSCs in which the state is a deterministic function of a finite number of the most recent inputs and

• outputs.

• The p.m.f. of XY-FSCs satisfies

• Example:

• Result: XY-FSCs are weakly indecomposable

• Capacity: