On the Burnashev exponent for Markov channels

Transcription

1 On he Burnashev exponen for Markov channels Giacomo Como Dip. di Maem. Poliecnico di Torino, Ialy visiing Yale Universiy, New Haven, CT Serdar Yüksel Elecrical Engineering Yale Universiy, New Haven, CT Sekhar Taikonda Elecrical Engineering Yale Universiy, New Haven, CT Absrac We consider he reliabiliy funcion of Markov channels wih feedback and variable lengh channel codes. We exend Burnashev s [3 classic resul o his case and presen a single leer characerizaion for he reliabiliy funcion. I. INTRODUCTION I is known ha oupu feedback canno increase he capaciy of a discree memoryless channel DMC [7. Feedback, hough, improves is reliabiliy funcion. A classical resul due o Burnashev [3 characerizes he reliabiliy funcion of a DMC wih a simple single-leer formula when one is allowed variable lengh block coding. I is remarkable ha in his case, differenly from when feedback is no available, he reliabiliy funcion is exacly known a any rae below capaciy, and ha i has nonzero slope approaching capaciy. Recenly, he problem of channel coding wih feedback, and Burnashev s resul in paricular, have araced renewed ineres from he researchers; see [8, [9, [5, [. In his paper we address he problem of generalizing Burnashev s resul o channels wih memory. Specifically, we examine he simples case of Markov channels wih perfec sae informaion a he ransmier and receiver and no inersymbolinerference ISI. For his class of channels we are able o exacly characerize, in a single-leer form, he reliabiliy funcion ER = C R C for suiably defined C, C see 3, 4 and Theorem. Hence, Burnashev s exponen exends naurally o he Markov seing. However, he exension is non-rivial as is analysis involves significan echnical challenges. Our proof of he converse is based on an exension of Burnashev s original one [3, following some of he ideas of Berlin e al. [ see also [9. Specifically, we provide wo differen bounds for he error probabiliy, involving respecively he channel capaciy C and is Burnashev coefficien C, corresponding o wo disinc phases which can be recognized in any sequenial ransmission scheme. Similarly o he memoryless case, maringale heory, and in paricular Doob s opional sampling heorem, is widely used joinly wih more sandard informaion heoreic resuls like Fano s and log-sum inequaliies. The use of sandard ergodic heory This work was parially suppored by he Naional Science Foundaion under Gran ECCS for Markov chains coupled wih sopping imes argumens is insead peculiar of he memory case. In order o prove he achievabiliy of he exponen, we propose a simple wo-phase ieraive ransmission scheme based on he one firs considered by Yamamoo and Ioh [0 for DMCs. The performance analysis of his scheme relies on known resuls abou he capaciy of Markov channels, and he error exponen of binary hypohesis ess for irreducible Markov chains. In Secion II we formulae he problem, inroduce noaion, and sae he main resul. In Secion III we prove he converse. In Secion IV we prove achievabiliy by presening a generalizaion of he Yamamoo and Ioh scheme. II. PROBLEM FORMULATION AND MAIN RESULT A. Noaion Given a finie se A, we shall denoe by PA he space of probabiliy measures over A. When µ is in PA and f is in R A, we shall use he noaion µ, f = a A µafa. The se of all infinie A-sequences is denoed by A N, while he se of all finie A-sequences is denoed by A = A n. For a in A N, and s, a s A s+ denoes he resricion of a o he discree inerval [s,. The lengh of an elemen x of A is denoed by Lx, and he empirical frequency funcion is defined as υ A : A PA, [υ A x a = { j n : x j =a} Lx When A is an A-valued random process, we will deal wih finie sopping imes T for A [2. These can be represened as complee prefix-free subses of A, or A-ary rees. Wih his inerpreaion, a T -measurable random variable can hus be hough as a funcion over he se A T of he leaves of his ree. B. Saionary ergodic Markov channels Throughou he paper X, Y, S will respecively denoe inpu, oupu and sae se, all finie. Definiion A saionary Markov channel wih no ISI is described by: a family {P Y x, s PY x X, s S} of probabiliy measures over Y indexed by elemens of X and S; a sochasic marix Π = P S s r s,r over S;.

2 an iniial sae disribuion µ in PS. Throughou he paper we will resric ourselves o ergodic Markov channels, saisfying he following. Assumpion 2 Π is irreducible. We won need any aperiodiciy assumpion. By Perron- Frobenius heorem we have ha Π has a unique invarian measure in PS, whose suppor is all S. Such an ergodic measure will be denoed by µ Π. We will also assume ha λ := min {P Y y x, s x X, y Y, s S} > 0. 2 Assumpion 2 can be relaxed bu we won deal wih he general case here. For each sae s, he family {P Y x, s, x X } describes a DMC: we use he noaion { } C s := max u PX x Xux P Y y x, s P Y y x, slog uzp Y y z, s y Y z X for is capaciy noice ha he opimizing disribuion depends on s, and Cs { := max D PY x, s P Y x, s } x,x X for he Kullback-Leibler divergence beween he pair of is mos disinguishable inpus. Noice ha 2 implies ha C s is finie for every sae s. We will use he compac noaion C = C s [0, log X S, C = C s [0, + S. We define C := µ Π, C. 3 I is known ha Assumpon 2 allows o conclude ha C defined above is acually he capaciy of he Markov channel we are considering when he channel sae is causally known boh a he encoder and a he decoder, wih and wihou oupu feedback. We consider he quaniy C := µ Π, C, 4 which we shall refer o as he Burnashev coefficien of he channel. Noice ha, when he sae space is rivial i.e. S =, C reduces o he usual noion of capaciy of a DMC, while C reduces o he coefficien originally inroduced by Burnashev in [3. C. Causal feedback encoders and sequenial decoders We now inroduce he class of coding schemes we shall consider in his paper. Definiion 3 A causal feedback encoder is he pair of a finie message se and a sequence of maps Φ = W, { φ : W Y S X }. 5 N Wih Def.3, we are implicily assuming ha perfec sae knowledge as well as perfec oupu feedback are causally available a he encoder side. Given a saionary Markov channel and a causal feedback encoder as in Def.3, we will consider a W-valued random variable W describing he message o be ransmied, a sequence X = X N of X -valued r.v.s he channel inpu process, a sequence Y = Y N of Y-valued r.v.s he channel oupu process, and a sequence S = S N S-valued r.v.s he sae process. We consider he ime ordering W, S, X, Y, S 2, X 2, Y 2,..., and assume ha he join disribuion of W, X, Y and S is described by PW = w = / W, PS = s W = µs, PS = x W, S, X, Y = P S s S, PX = x W, S, X, Y = δ {φw,y,s } x, PY = y W, S, X, Y = P Y y S, X. The corresponding expecaion operaor will be denoed by E. We observe ha he absence of ISI ensures ha he sae process S is independen of he ransmied message W. In paricular S forms a saionary Markov chain wih irreducible ransiion probabiliy marix Π. This implies ha he empirical measure process υ S S n converges P-almos surely o he invarian measure µ Π, while saisfying a large deviaions principle see [4 wih convex, good rae funcion I Π θ. Definiion 4 A sequenial decoder for a causal feedback encoder Φ as in 5 is a pair Ψ = T, ψ, where T is a sopping ime for he process S, Y and ψ an W-valued S T, Y T - measurable random variable. Noice ha wih Def.4 we are assuming ha he sae sequence is causally observable a he decoder side. Given a causal feedback encoder Φ as in Def. 3 and a sequenial decoder Ψ as in Def. 4, heir error probabiliy is p e Φ, Ψ := P ψ W. Following Burnashev s approach we shall consider he expeced decoding ime E[T as a measure of he delay of he coding scheme and accordingly define is rae by D. Main resul R := log W /E[T. We are ready o sae our main resul. I is formulaed in an asympoic seing, considering sequences of causal encoders and sequenial decoders wih asympoic average rae below capaciy and vanishing error probabiliy. Theorem 5 For any 0 < R < C:

3 any sequence Φ n, Ψ n of causal encoders sequenial decoder pairs such ha p eφ n, Ψ n = 0, saisfies sup log W n E[T n R, 6 sup E[T n log p e Φ n, Ψ n E B R. 7 2 here exiss a sequence Φ n, Ψ n of encoder-decoder pairs saisfying 6 and such ha E[T n log p e Φ n, Ψ n = E B R. 8 We observe ha Burnashev s original resul [3 for DMCs can be recovered as a paricular case of Theorem 5 when he sae space is rivial. III. AN UPPER BOUND ON THE ERROR EXPONENT The aim of his secion is o skech he proof of Par of Theorem 5. We shall presen a series of parial resuls whose proofs will be given in full deail elsewhere. Our resuls are exensions of hose in [3. In paricular we have followed he approach proposed in [ see also [9,[5 rying o emphasize he emergence of wo disinc ransmission phases, a communicaion one relaed o he channel capaciy, and a binary hypohesis esing one relaed o he Burnashev coefficien. A. A generalized Fano s inequaliy A firs observaion is ha, given a sopping ime T for he process S, Y, he opimal decoder ψ : S Y T W which can be associaed o i is he maximum a poseriori MAP one. Thus, in order o lower bound he error probabiliy we can resric ourselves o MAP sequenial decoders. Given channel oupu and sae observaions up o some ime, we denoe he a poseriori error probabiliy by { p MAP := max P W = w Y, S}. w W Noice ha he quaniy defined above implicily depends on he encoder Φ. Since W is uniformly disribued over he message se W, we have ha p MAP 0 = W / W. Moreover i is no difficul o prove he following recursive lower bound. Lemma 6 Given any causal feedback encoder Φ, for any 0 p MAP + λp MAP P a.s. For every δ > 0 we inroduce he following sopping ime for he process S, Y δ := inf { N : T or p MAP δ}. 9 The following resul can be proved using Fano s inequaliy and Doob s opional sopping heorem. Lemma 7 For any 0 < δ < 2 we have [ δ E C S δ p e Φ, Ψ log W Hδ. 0 δ = B. A lower bound o he error probabiliy of a composie binary hypohesis es We now consider a paricular binary hypohesis esing problem which will arise while proving he main resul. Consider a nonrivial binary pariion of he message se W = W 0 W, W 0 W =, W 0, W, and a sequenial binary hypohesis es Ψ = T, ψ where T is a sopping ime for S, Y and ψ : S Y T {0, } beween he hypoheses {W W 0 } and {W W }. Consider now anoher sooping ime for S, Y, saisfying T a.s. 2 Suppose ha W is a S, Y -measurable random subse of he message se W. The following lower bound o he error probabiliy of he binary es Ψ condiioned on S, Y can be proved using he log-sum inequaliy and Doob s opional sampling heorem. Lemma 8 Le Φ be any causal encoder, and and T sopping imes for he process S, Y saisfying 2. Then, for every S, Y -measurable random message subse W [ T E CS P ψ {W W} S, Y S, Y log Z/4 =+ { } P-a.s., where Z := min i=0, P W W i S, Y. C. Burnashev bound for Markov channels 3 Combining Lemmas 6, 7 and 8, i is possible o prove ha he following. Theorem 9 Given a causal feedback encoder Φ = W,φ and a sequenial decoder Ψ=T,ψ, for every δ > 0 [ C δ [ C E T C S + E CS = = δ + 4 log p e Φ, Ψ + C log W α + β, C where δ is defined by 9, and α := δ + p eφ, Ψ δ, β := log λδ 4 C C Hδ. When he sae space is rivial, he lefhand side of 4 equals C E[T, so ha 4 reduces o Burnashev s inequaliy 4. in [3. For nonrivial sae space, in order o obain he single-leer bound of Theorem 5, i is necessary o consider a counable family of causal encoders Φ k and a corresponding family of sequenial decoders Ψ k saisfying 6.

4 The core idea for proving 7 consiss in inroducing a posiive real sequence δ k, and showing ha boh he random variables k := inf { N T k or p MAP δ k }, and T k k diverge in he probabilisic sense made precise by 6 below. The sequence δ k needs o be properly chosen: we wan i o be asympoically vanishing in order o guaranee ha k diverges, bu no oo fas since oherwise T k k would no diverge. I urns ou ha one possible good choice is δ k := log p eφ. From 2 i follows ha k,ψ k δ p e Φ k, Ψ k k = 0, = 0. 5 δ k The following resul can be proven using Lemma 6. Lemma 0 In he previous seing, for every M in N, we have P k M = 0, P T k k M = 0. 6 We have already noiced ha he irreducibiliy of he sochasic marix Π implies ha he empirical measure process associaed o he sae sequence S converges P-almos surely o he invarian measure µ Π. This fac, ogeher wih 6, allows us o prove he following. Lemma In he previous seing E[ k E k and E[T k k E = C S T k = k + =C, 7 C S = C. 8 By aking he i of boh sides of 4 and subsiuing 7 and 8, we ge 7. IV. AN ASYMPTOTICALLY OPTIMAL SCHEME In order o prove Par 2 of Theorem 5, hus showing he achievabiliy of he Burnashev s exponen E B R, we propose an ieraive ransmission scheme consising in a generalizaion of he one inroduced by Yamamoo and Ioh [0 for DMCs. This scheme consiss of a sequence of epochs. Each epoch is made up of wo disinc ransmission phases, respecively named communicaion and confirmaion phase. In he communicaion phase he message o be sen is encoded in a block code and ransmied over he channel. A he end of his phase he decoder makes a enaive decision abou he message sen based on he observaion of he channel oupus and of he sae sequence. As perfec feedback is available, he resul of his decision is known a he encoder. In he confirmaion phase a binary acknowledge message, confirming he decoder s esimaion if i is correc, or denying i when i is wrong, is sen by he encoder hrough a fixed-lengh repeiion code. The decoder performs a binary hypohesis es in order o decide wheher a deny or a confirmaion message has been sen. If a confirmaion is deeced he ransmission hals, while if a deny is deeced he sysem resars ransmiing he same message wih he same proocol. Again because of perfec feedback availabiliy a he encoder, here are no synchronizaion problems. More precisely we design our scheme as follows. Given a design rae R in 0, C, le us fix an arbirary γ in R C,. For every n in N, consider a message se W n of cardinaliy W n = exp nr and wo blocklenghs ˆn and ñ respecively defined as ˆn = nγ, ñ := n ˆn. A. Fixed-lengh coding for he ransmission phase I is known ha C equals he capaciy of he saionary Markov channel we are considering. This implies ha, since log W n ˆn = R γ < C, here exiss a sequence of causal encoders wih no oupu feedback { ˆφ n : W n S X }, and a sequence of decoders of fixed lengh ˆn { ˆψ n : S ˆn Y ˆn W n } wih error probabiliy asympoically vanishing wih he blocklengh. More precisely, he error probabiliy of he pair ˆΦ n, ˆΨ n goes o zero uniformly wih respec o iniial sae disribuion. Thus, denoing by pn he maximum over all iniial sae disribuions µ of he error probabiliy of he pair ˆφ n, ˆΨ n, we have ha pn = 0. The pair ˆΦ n, ˆΨ n will be used in he firs phase of each epoch of our ieraive ransmission scheme. B. Binary hypohesis es for he confirmaion phase For he second phase we consider a causal repeiion encoder Φ n of lengh ñ, defined by φ n m = x m s for m = 0,, where for every sae s we denoe by x 0 s and x s one of he mos disinguishable inpu pairs for he s-h channel, i.e. such ha C s = DP x 0 s P x s. Suppose ha an acknowledge message m = 0 is sen. Then i is easy o verify ha he pair sequence S, Y ñ= forms a Markov chain wih sae space S Y and ransiion probabiliies P 0 v, y s, z := P S v sp Y y s, x 0 s. Analogously, if a deny message m = has been sen, wih ransiion probabiliies P v, y s, z := P S v sp Y y s, x s. I can be checked ha Assumpion 2 and 2 guaranee ha boh he sochasic marices Π i := P i v, y s, z, i = 0,, are irreducible, wih invarian measures given by Noice ha we can rewrie µ i s, y = µsp Y y x i s, s. C = s S µsd P Y x 0 s, s P Y x 0 s, s = Dµ 0 µ. Using binary hypohesis es resuls generalizing Sein lemma o irreducible Markov chains see [6, [4, i is possible o show ha C is achievable as ype one error exponen, while saisfying he consrain of vanishing ype zero error probabiliy. In fac, for every n in N define he decoder Ψ n : Sñ Yñ {0, }, Ψ n y, s = { if IΠ0 υ S Y s, y > α n 0 if I Π0 υ S Y s, y α n,

5 where I Π0 is he large deviaions rae funcion associaed o he sochasic marix Π 0 [4, and α n is a real posiive sequence saisfying α n = 0, n log α n = 0. Defining p 0 n respecively p n as he maximum over all possible iniial sae disribuions µ of he error probabiliy of he pair Φ n, Ψ n condiioned on he ransmission of a 0 message, i is possible o show ha p 0n = 0, C. Performance of he proposed scheme ñ log p n = C. Once chosen he encoder decoder pairs ˆΦ n, ˆΨ n and Φ n, Ψ n for he communicaion and he confirmaion phase respecively, he ieraive proocol described a he beginning of his secion defines a causal encoder Φ n = W n, φ n and a sequenial decoder Ψ n = T n, ψ n. I can be verified ha he error probabiliy of such a scheme saisfies REFERENCES [ P. Berlin, B. Nakiboglu, B. Rimoldi, E. Telaar, A Simple Derivaion of Burnashev s Reliabiliy funcion, hp://arxiv.org/abs/cs/06045, [2 V. S. Borkar, Probabiliy Theory: An Advanced Course, Springer, 995. [3 M. V. Burnashev, Daa Transmission over a Discree Channel wih Feedback. Random Transmission Time, Prob. of Inform. Transm., vol. 2 4, pp. 030, 976. [4 A. Dembo, O. Zeiouni, Large Deviaions Techniques and Applicaions, 2nd ed., Springer, 998. [5 B. Nakiboglu, R. G. Gallager, Error exponens for variable-lengh block codes wih feedback and cos consrains, preprin, av. a hp://arxiv.org/abs/cs.it/062097, [6 S. Naarajan, Large deviaions, hypoheses esing, and source coding for finie Markov chains, IEEE Trans. Inform. Theory, vol. 3, pp , 985. [7 C. E. Shannon The zero error capaciy of a noisy channel, IEEE Trans. Inform. Theory, vol. 2, pp. 8-9, 956. [8 A. Tchamkeren, I. E. Telaar, On he Universaliy of Burnashev s Error Exponen, IEEE Trans. Inform. Theory, vol.5, pp , [9 A. Tchamkeren, I. E. Telaar, Variable Lengh Coding over an Unknown Channel, IEEE Trans. Inform. Theory, vol.52, pp , [0 H. Yamamoo, K. Ioh, Asympoic performance of a modified Schalkwijk-Barron scheme for channels wih noiseless feedback, IEEE Trans. Inf. Theory, vol. 25, pp , 979. p e Φ n, Ψ n p n, while is decoding ime is dominaed by a scaled geomeric random variable: and PT n > kn p 0 n + pn k, k 0. 9 As a consequence we have ha log W n E[T n = R, log p e Φ n, Ψ n E[T n = C γ. Thus 8 can be deduced from he arbirariness of γ in R C,. We emphasize he fac ha he wo phases in each epoch of he scheme are of fixed lengh, while he number of epochs is possibly variable. Hence, he overall ransmission lengh of he scheme is variable. However, 9 guaranees ha wih high probabiliy he ransmission hals afer he firs epoch, a propery making his scheme appealing for pracical implemenaion, as already noiced in [5. Moreover we observe ha he firs ransmission phase only requires an asympoically vanishing error probabiliy, no necessarily a an exponenial rae. V. CONCLUSIONS AND OPEN PROBLEMS In his paper we have deermined he reliabiliy funcion for Markov channels wih feedback and variable lengh channel codes. In he fuure we will relax he assumpion ha here is no ISI, and ha boh he ransmier and he receiver have access o he sae. In he former case conrolled Markov chain heory will be used. In he laer case oupu feedback will have a dual role: o esimae he sae and o increase he reliabiliy funcion. Exensions o noisy feedback will also be considered.