The Finite-state channel was introduced as early as 1953 [McMillan'53]. Memory captured by channel state at end of previous symbol's transmission: - S i is the channel state at time i. - s 0 is the initial channel state. p(y i, s i |x i, x i-1, y i-1, s i-1, s 0 ) = p(y i, s i |x i, s i-1 ) - S i-1 contains all the history information for time i. - || S || < Example: ISI channel: S i-1 = (X i-1,X i-2, ,X i-J ) Cellular Communication, DSL, magnetic recording, partial-response signaling. Indecomposable FSCs with Feedback Indecomposable Finite-State Channels with Feedback Ron Dabora and Andrea Goldsmith 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 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 s 0 on the state transitions becomes negligible as time evolves: For all >0 there exists N() s.t. for every k > N(), x k,s 0, s 0 it holds that |p(s k |x k, s 0 )-p(s k |x k, s 0 )| < Finite ISI channels are indecomposable (w/o feedback) Code for the FSC with Feedback ? The Finite-State Channel Model 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 Motivation Definitions FSCs with and without Feedback Conclusions Rational Transfer Functions Channels An (R, n) code for the FSC with feedback consists of - Message set: M, - Encoder: f i : M Y i-1 X, - Decoder: g : Y n M. -Encoder and decoder do not know the channel states The maximum average probability of error is defined as,where P e (n) (s 0 ) = Pr(g(Y n ) M|s 0 ) 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, Goldsmith08] Feedback does not increase the capacity of PtP DMCs Feedback increases the capacity of PtP FSCs Notation: For indecomposable FSCs the maximum achievable rate is the same for all initial states [Gallager68]: Capacity achieving scheme will provide same rate for all initial states s 0 Can find capacity by optimal p(x n ) by considering some fixed arbitrary initial state s 0 (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: y i = Q 2 [x i +ay i-1 +n i ], a constant, n i is zero-mean unit variance Gaussian RV Inhomogeneous Markov chain FSC. Notation k(n) monotone, Result Implications Can find capacity by optimizing p(x n ) for a fixed arbitrary initial state s 0 (no need to search over all initial states) However Letting the receiver know the initial channel state may improve the rate Summary 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: Capacity of XY-FSCs achieves the maximum over all initial states same as for indecomposable FSCs without feedback Code construction: Let s 0 be the optimal initial state. This gives an optimal vector x Nx,y Ny. For each message generate a codetree with the optimal initial state. Generate a sequence of length L s such that the last Nx symbols are the optimal vector x Nx. - Total number of symbols dedicated for synchronization is L synch =k(n) Synchronization: the transmitter starts (re)transmitting the L s sequence and observes the feedback. If at the end of an L s -sequence the feedback is y Ny then synchronization is achieved: The receiver knows synchronization was achieved when it observes y Ny at the end of the L s -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. Indecomposable FSCs without Feedback A Code with Tx-Rx Synchronization Achieves Capacity of the XY-FSC Probability of Error: two error events Synchronization failed: synchronization was not achieved after transmitting all the L synch symbols, i.e., the optimal y Ny was never observed at the end of an L s -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