Indecomposable Finite-State Channels with Feedback
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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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The finite state channel was introduced as early as 1953 mcmillan 53

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].

  • 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

    • Correlated fading, multipath, ISI

    • 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:


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