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].

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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].
- 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: