On the capacity of the general trapdoor channel with feedback

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1 On the caacity of the general tradoor channel with feedback Jui Wu and Achilleas Anastasooulos Electrical Engineering and Comuter Science Deartment University of Michigan Ann Arbor, MI, {juiwu, Abstract The tradoor channel is a binary inut/outut/state channel with state changing deterministically as the modulo- sum of the current inut, outut and state. At each state, it behaves as one of two Z channels, each with crossover robability 1/. Permuter et al. formulated the roblem of finding the caacity of the tradoor channel with feedback as a stochastic control roblem. By solving the corresonding Bellman fixed-oint equation, they showed that the caacity equals C = log In this aer, we consider a generalization of the tradoor channel, whereby at each state the corresonding Z channel has crossover robability. We characterize the caacity of this roblem as the solution of a Bellman fixed-oint equation corresonding to a Markov decision rocess (MDP). Numerical solution of this fixed-oint equation reveals an unexected behavior, that is, for a range of crossover robabilities, the caacity seems to be constant. To the authors knowledge, this is the first time that such behaviour is observed for the Shannon caacity of any channel. Our main contribution is to formalize and rove this observation. In articular, by exlicitly solving the Bellman equation, we show the existence of an interval [0.5, ] over which the caacity remains constant. I. INTRODUCTION Shannon [1] showed that feedback information does not increase the caacity of discrete memoryless channels (DMCs). However, the situation changes in channels with memory. There exists a rich literature on the feedback caacity of channels with memory. Viswanathan [] derived the caacity of a finite-state channel (FSC) with receiver channel state information (CSI) and delayed outut feedback in the absence of intersymbol interference (ISI). Chen and Berger [3] derived the feedback caacity of a FSC with ISI where current CSI is available at both the transmitter and the receiver. Yang et al [4] used a stochastic control methodology to derive the caacity of finite-state machine channels. Permuter et al. [5] derived a single-letter exression for the caacity of unifilar FSCs, which is a family of FSCs with deterministic evolution of the channel state. Later Tatikonda and Mitter [6] derived the caacity of finite-state Markov channels. In all the above works, the caacity is characterized as an otimization of a single-letter exression. However, this otimization roblem is rarely solved analytically. A notable excetion is [5] where the caacity of the tradoor channel is analytically derived. The tradoor channel is a secial case of unifilar FSCs with binary inut, outut and channel state alhabets. For each state, the tradoor channel behaves as a Z channel with crossover robability 0.5. The authors of [5] formulated the caacity as the otimal exected reward of a Markov decision rocess (MDP) and roved it to be equal to C = 1+ 5 by exlicitly solving the Bellman fixed oint equation. In this aer, we consider a generalization of the tradoor channel where we allow the crossover robability of each Z channel to be instead of 0.5. We characterize the caacity of the channel though a Bellman fixed-oint equation. Numerical evaluation of the caacity for different values of reveals a surrising result: the caacity seems to remain constant over a range of in the neighborhood of 0.5. To the authors knowledge this is the first time such a henomenon is observed for any kind of channel (with/without memory, with/without feedback, single-/multi-user, etc). To conclusively rove this conjecture, we exlicitly solve the corresonding Bellman fixed-oint equation. The roof follows the general technique of [5] for the secial tradoor channel. In articular, we define a initial function V 0 and the iterates V k+1 = T V k through an aroriate oerator T. Subsequently we show existence of a limit V of {V k } k 0 which satisfies the Bellman equation. As exected, several technical stes are more involved comared to the secial case treated in [5]. These include the definition of the initial function V 0, as well as the roof of Lemma, where it is shown that arts of functions V k remain stationarity for all k. The remaining of this aer is organized as follows. In section II, we describe the channel model for the general tradoor channel and formulate the caacity as the otimal exected reward for an MDP. In section III, we solve the corresonding Bellman fixed oint equation for a range of the crossover robabilities and show that over this range the caacity is constant. Concluding remarks are discussed in section IV. II. CHANNEL MODEL AND PRELIMINARIES Consider a family of unifilar channels with inuts X t X, outut Y t Y and state S t S at time t. The channel conditional robability is P (Y t, S t+1 X t, Y t 1, S t ) = Q(Y t X t, S t )δ h(st,x t,y t)(s t+1 ), (1) for a given stochastic kernel Q X S P(Y) and a deterministic kernel h : S X Y S, where P(Y) denotes the sace of all robability measure on Y.

2 Permuter et al. [5] showed that the caacity of the unifilar finite-state channel with noiseless outut feedback can be exressed as C = 1 su lim inf {(x t s t,y t 1 )} N t 1 N N I(X t, S t ; Y t Y t 1 ).() t=1 Moreover, the caacity can be viewed as the otimal exected average reward of a Markov decision rocess (MDP) with state B t 1 P(S), action U t : S P(X ) and instantaneous reward g : P(S) (S P(X )) R at time t defined by B t 1 (s) = P (S t = s Y t 1 ) s S (3a) U t (x s) = P (X t = x S t = s, Y t 1 ) g(b, u) = I(X t, S t ; Y t B t 1 = b, U t = u) = x,s,y [ Q(y x, s)u(x s)b(s) x X, s S(3b) Q(y x, s) ] log x, s Q(y x, s)u( x s)b( s) b P(S), u S P(X ). (3c) Through Bayes law, the current state B t can be recursively udated through x,s B t (s) = δ h(s,x,y t)(s)q(y t x, s )U t (x s )B t 1 (s ) x,s Q(Y t x, s )U t (x s )B t 1 (s ) s S. (4) Let ψ : (P(S) Y (S P(X ))) P(S) denote the udating formula, i.e., B t = ψ(b t 1, Y t, U t ). The general tradoor channel with arameter (denoted GTC()) is a secial family of unifilar FSCs with X = Y = S = {0, 1}, stochastic kernel Q defined in Tabel I, and deterministic kernel h defined as h(s, x, y) = s x y, (5) where denotes modulo- addition. We can interret Q as s t x t y t Q(y t x t, s t) TABLE I: The kernel Q of the general tradoor channel with crossover robability. the kernel of two Z channels (see Fig. 1) with crossover robability for different states. Since the inut and channel state alhabets are binary, the quantity B t 1 (0) is equivalent to B t 1, and similarly, (U t (0 0), U t (0 1)) is equivalent to U t. Therefore, with a slight abuse of notation, from now on we will consider the equivalent MDP with state B t 1 = B t 1 (0) [0, 1], action U t = Fig. 1: The Z channels that comose the GTC(). (U t,0, U t,1 ) = (U t (0 0), U t (0 1)) [0, 1] and instantaneous reward g at time t defined as g(b, u 0, u 1 ) = H(bu 0 + b(1 u 0 ) + (1 )u 1 (1 b)) H()[b(1 u 0 ) + (1 b)u 1 ], (6a) where H( ) is the binary entroy function. The Bellman fixed oint equation that defines the otimal inut distribution and caacity is given by C+V (b) = su { H(y 0 (b, u 0, u 1 )) H()[b(1 u 0 ) + (1 b)u 1 ] +y 0 (b, u 0, u 1 )V (φ 0 (b, u 0, u 1 )) } +y 1 (b, u 0, u 1 )V (φ 1 (b, u 0, u 1 )) where y 0, y 1, φ 0, φ 1 : [0, 1] 3 [0, 1] are defined as b [0, 1], (7) y 0 (b, u 0, u 1 ) = bu 0 + b(1 u 0 ) + (1 )u 1 (1 b)(8a) y 1 (b, u 0, u 1 ) = 1 y 0 (b, u 0, u 1 ) bu 0 φ 0 (b, u 0, u 1 ) = y 0 (b, u 0, u 1 ) (8b) (8c) φ 1 (b, u 0, u 1 ) = (1 )(1 u 0)b + u 1 (1 b). (8d) y 1 (b, u 0, u 1 ) Permuter et al. [5] showed that for the secial case of GTC(0.5) the caacity is C = log 1+ 5 by roving the existence of a function V that together with C satisfies the Bellman fixed oint equation (7). III. THE CAPACITY OF THE GENERAL TRAPDOOR CHANNEL Numerical evaluation of the Bellman fixed oint equation in (7) results in caacity values shown in Fig.. With the otimal inut distribution obtained numerically, the steadystate distributions π(b) of B t can be evaluated through the equation π(b) = 1 [ Q(y x, 1)P X SB (x 1, b )(1 b ) y,x 0 +Q(y x, 0)P X SB (x 0, b )b ] π(b )db b [0, 1].(9) Fig. 3 deicts the suort of π( ) for differnet values of. In articular, for any, the marks reresent the oints where π( ) is ositive. Based on Fig., several comments on the caacity of GTC() can be made. First, GTC(0) is simly a noiseless channel and therefore achieves a rate of 1. Similarly, in

3 Caacity Suuort of π( ) C = log Crossover robability, Fig. : Caacity for different values of Crossover robability, Fig. 3: Suort of the steady-state distribution π( ) of B t for different values of. GTC(1), Y t = S t for all t, which results in Y t+1 = X t. In other words, GTC(1) is a noiseless channel with unit-delayed outut and therefore it also achieves rate 1. A surrising finding from Fig. is that for a range of values of above = 0.5 the caacity seems to remain constant. To the authors knowledge this is the first time such a henomenon is observed for any kind of channel (with/without memory, with/without feedback, single-/multiuser, etc). Further investigation of the steady-state distribution of B t for this rage of values of reveals (see Fig. 3) that the suort of the steady-state distribution π( ) is finite, and in articular, it consists of exactly four values (as reorted in [5] for the secial case = 1/). It is natural to conjecture that the caacity of GTC() is constant for all in the flat region. This is exactly what we set out to rove in the remaining of this section. The sketch of the roof is as follows. We first define a secial function V 0 : [0, 1] R and the iterates V k+1 = T V k k 0 for an aroriately defined maing T (deending on C). We then show that the resulting sequence {V k } k 0 is monotonically nonincreasing, bounded, and therefore has a limit V. Moreover, the convergence is uniform and consequently V satisfies the fixed-oint equation V = T V, due to continuity of T. This fixed-oint equation is exactly the Bellman equation in (7), which roves that C is the caacity. Define T : ([0, 1] R) ([0, 1] R) by (T v)(b) = su { H(y 0 (b, u 0, u 1 )) H()[b(1 u 0 ) + (1 b)u 1 ] +y 0 (b, u 0, u 1 )v(φ 0 (b, u 0, u 1 )) } +y 1 (b, u 0, u 1 )v(φ 1 (b, u 0, u 1 )) C Define two constants m, M by M = v [0, 1] R. (10) ( 1 ) 1 C+1 /(3 5) + ( 1 (11a) ) 1 m = 1 φ 0 (M, 1, 3 5 ). (11b) Finally, define V 0 : [0, 1] R as the maximal concave function that satisfies { H(b) Cb, b [m, M] V 0 (b) =.(1) H(b) C(1 b), b [1 M, 1 m] To be more recise, for b [0, m], V 0 is the tangent line of the function H(b) bc at b = m. Similarly, for b [M, 0.5], V 0 is the tangent line of the function H(b) bc at b = M. Now the shae of V 0 is exlicitly described for b [0, 0.5]. The shae of V 0 in [0.5, 1] is the mirror image of that of V 0 in [0, 0.5]. Fig. 4 is the grah of V 0 for = 0.6. The bold solid lines are values defined in (1) and the thin solid lines are the tangent lines which make V 0 the maximal concave function subject to (1). We begin by roving the reservation of continuity and concavity under the maing T. Since V 0 is continuous and concave by construction, this roerty of T will in turn establish the continuity and the concavity of {V k } k 0. Lemma 1. If v : [0, 1] R is continuous and concave, then T v is continuous and concave. Proof. Since y 0 ( ), y 1 ( ), y 0 ( )v(φ 0 ( )), and y 1 ( )v(φ 1 ( )) are continuous w.r.t. (b, u 0, u 1 ) on the closed region [0, 1] 3 and thus uniformly continuous, T v is continuous on [0, 1]. To show the concavity of T v, we introduce new variables w = bu 0 and z = (1 b)u 1 and define h by h(b, w, z) = H(ỹ 0 (b, w, z)) H()(b w + z) +ỹ 0 (b, w, z)v( φ 0 (b, w, z)) +ỹ 1 (b, w, z)v( φ 1 (b, w, z)), (13)

4 V 0 (b) H(b) Cb H(b) C(1 b) b Fig. 4: Function V 0 ( ) for = 0.6. where ỹ 0, ỹ 1, φ0, φ1 : [0, 1] 3 [0, 1] are defined as ỹ 0 (b, w, z) = b + (1 )w + (1 )z (14a) ỹ 1 (b, w, z) = 1 ỹ 0 (b, w, z) (14b) w φ 0 (b, w, z) = (14c) ỹ 0 (b, w, z) (1 )(b w) + z φ 1 (b, w, z) =. (14d) ỹ 1 (b, w, z) With these definitions, T v can be rewritten as (T v)(b) = su h(b, w, z) C b [0, 1](15) w [0,b],z [0,1 b] We will first show that h is concave in (b, w, z) and then show the concavity of T v based on that of h. Consider the first term of h, which is H(ỹ 0 (b, w, z)). Since ỹ 0 is affine in (b, w, z) and H is concave, H(ỹ 0 (b, w, z)) is concave in (b, w, z). The second term, H()(b w + z), is affine in (b, w, z). Recall the concavity of a function v imlies that tv( r t ) is concave in (r, t). Since ỹ 0(b, w, z) is affine in (b, w, z), the third term ỹ 0 (b, w, z)v( φ w 0 (b, w, z)) = ỹ 0 (b, w, z)v( ỹ 0 (b, w, z) ) (16) is concave in (b, w, z). Similarly, the fourth term, ỹ 1 (b, w, z)v( φ 1 (b, w, z)), is concave in (b, w, z). Subsequently, h is concave in (b, w, z). Now we are ready to rove the concavity of T v. Given any θ, b 1, b [0, 1], (T v)(θb 1 + θb ) = su h(θb 1 + θb, w, z) C w [0,θb 1+ θb ] z [0,1 (θb 1+ θb )] = su (a) w 1 [0,b 1],z 1 [0,1 b 1] w [0,b ],z [0,1 b ] su w 1 [0,b 1],z 1 [0,1 b 1] w [0,b ],z [0,1 b ] = su θh(b 1, w 1, z 1 ) θc w 1 [0,b 1] z 1 [0,1 b 1] + su w [0,b ] z [0,1 b ] h(θb 1 + θb, θw 1 + θw, θz 1 + θz ) C θh(b 1, w 1, z 1 ) + θh(b, w, z ) C θh(b, w, z ) θc = θ(t v)(b 1 ) + θ(t v)(b ), (17) where (a) is due to the concavity of v. Define as the ositive solution to the equation = ( 1 ) 1. (18) reresents the uer bound on the flat region in Fig. 1, a fact that will be roved later. In the next lemma, we rove the stationarity of {V k } k 0 in [m, M] [1 M, 1 m]. Lemma. For any [0.5, ], the values of functions {V k } k 0 in [m, M] [1 M, 1 m] are stationary. In other words, V k (b) = V 0 (b) b [m, M] [1 M, 1 m]. (19) Proof. It is sufficient to rove that V 1 (b) = V 0 (b) [m, M]. Define f by b f(b, u 0, u 1 ) = H(y 0 (b, u 0, u 1 )) H()[b(1 u 0 ) + (1 b)u 1 ] +y 0 (b, u 0, u 1 )V 0 (φ 0 (b, u 0, u 1 )) +y 1 (b, u 0, u 1 )V 0 (φ 1 (b, u 0, u 1 )). (0) Consider the otimization roblem for a fixed b [m, M] su f(b, u 0, u 1 ). (1) Since f is concave in (u 0, u 1 ) due to the concavity of V 0 and the inequality constraints are convex, the KKT conditions are necessary and sufficient for otimality. In the following we will show that (u 0, u 1) = (1, 3 5 ) satisfy the KKT f conditions, i.e., u 0 (b, u 0, u 1) 0 and f u 1 (b, u 0, u 1) = 0. For b = m, we have φ 0 (m, 1, 3 5 ) (a) = φ 0 (1 φ 0 (M, 1, 3 5 ), 1, 3 5 ) (b) = 1 M, () where (a) is due to the definition of m in (11) and (b) is due to the following roerty of φ 0 φ 0 (1 φ 0 (b, 1, u ), 1, u ) = 1 b b [0, 1], u [0, 1]. (3)

5 Similarly, for b = M we have form the definition of m in (11) that φ 0 (M, 1, 3 5 ) = 1 m. Since φ 0 (b, 1, 3 5 ) is increasing w.r.t. b, we conclude that φ 0 (b, 1, 3 5 ) [1 M, 1 m] for all b [m, M]. On the other hand, φ 1 (b, 1, 3 5, which is an 1+ 5 ) = + increasing function of and constant w.r.t. b. By observing that = m = 0.5 (4a) + + = M = (1 ) 1, (4b) we deduce that φ 1 (b, 1, 3 5 ) [m, M] whenever [0.5, ]. The above arguments result in that b [m, M], φ 0 (b, 1, 3 5 ) and φ 1 (b, 1, 3 5 ) are in [m, M] [1 M, 1 m] and therefore the closed-form exression in (1) can be used in calculating the values artial derivatives of f w.r.t. to u 0 and u 1 at (b, 1, 3 5 ) resectively. In articular, we have f (b, 1, 3 5 ) u 0 = b[( 1) log 1 + log 1 b b (a) M 1 b = b log( ) 1 M b + C + log 3 5 ] 0 (5a) f (b, 1, 3 5 ) = 0, (5b) u 1 where (a) is due to the definition of M in (11) and (5a) is due to b M. Therfore, the KKT conditions are satisfied, so V 1 (b) = (T V 0 )(b) = su f(b, u 0, u 1 ) C = f(b, 1, 3 5 ) C = H(b) bc = V 0 (b) b [m, M]. (6) Due to symmetry, the stationarity also holds for b [1 M, 1 m]. Reeated alication of this argument shows that V k+1 (b) = V k (b) for all b [m, M] [1 M, 1 m] and k 1. In anticiation of Lemma 3, we now define an order on [0, 1] R as follows. For any functions v, v : [0, 1] R, we say that v is greater or equal to v, denoted by v v, if and only if v(b) v (b) b [0, 1]. (7) From the definition of T, we can easily verify that T v T v if v v. (8) Lemma 3. There exists a function V : [0, 1] R such that lim V k = V, (9) k where convergence is interreted in the su-norm. Proof. Since V 0 is concave and satisfies (1), by Lemma 1 and Lemma, V k is concave and satisfies (1) for all k 0. Furthermore, V 0 : [0, 1] R is the maximal concave function that satisfies (1). This results in V k = T k (V 0 ) (a) T k (V 1 ) = V k+1 k 0, (30) where (a) is due to (8). Therefore {V k } k 0 is monotonically nonincreasing, and subsequently {V k } k 0 has a ointwise limit V such that V (b) lim k V k(b). (31) To rove convergence of {V k } k 0 to V in the su-norm, it is equivalent to rove that {V k (b)} k 0 converges to V (b) uniformly on [0, 1]. By the concavity of V k k 0, V is concave and thus is continuous on (0, 1). Let Ṽ be the continuous extension of V from (0, 1) to [0, 1]. Ṽ V due to the concavity of V. On the other hand, since {V k } k 0 is monotonically nonincreasing, V k V and thus V k Ṽ by the continuity of Ṽ. Taking k in the above inequality we have V Ṽ and thus V = Ṽ. Since {V k} k 0 is a sequence of nonincreasing continuous functions and V is continuous, {V k } k 0 converges to V uniformly by Dini s theorem [7]. With the convergence of {V k } k 0 to V in the su-norm, we are ready to resent our main result. Proosition 1. For any [0.5, ], the caacity of the GTC() is equal to C. Proof. By Lemma 3, {V k } k 0 converges to V in the sunorm. Therefore we have V = lim V k+1 = lim T V (a) k = T ( lim V k) = T V, (3) k k k where (a) is due to that T is su-norm continuous and the uniform convergence of {V k } k 0 to V. The equation T V = V imlies that (C, V ) satisfies the Bellman fixed oint equation (7) and thus C is the caacity. IV. CONCLUSION In this aer, we considered a generalization of the tradoor channel and showed that for a secific region of crossover robabilities, the channel caacity is constant. This is a surrising and unique henomenon, the consequences of which we have not yet fully understood. An obvious research question that has not been addressed in this aer is whether it is ossible to find a closed-form exression for the caacity for values of outside the interval [0.5, ]. This seems to be a much harder roblem since for this range of values we do not have a guess for the caacity exression as a function of and therefore the techniques used in this work cannot be readily extended.

6 REFERENCES [1] C. E. Shannon, The zero error caacity of a noisy channel, Information Theory, IRE Transactions on, vol., no. 3,. 8 19, [] H. Viswanathan, Caacity of markov channels with receiver csi and delayed feedback, Information Theory, IEEE Transactions on, vol. 45, no., , [3] J. Chen and T. Berger, The caacity of finite-state markov channels with feedback, IEEE Transactions on Information Theory, vol. 51, no. 3, , 005. [4] S. Yang, A. Kavčić, and S. Tatikonda, Feedback caacity of finite-state machine channels, Information Theory, IEEE Transactions on, vol. 51, no. 3, , 005. [5] H. Permuter, P. Cuff, B. Van Roy, and T. Weissman, Caacity of the tradoor channel with feedback, IEEE Transactions on Information Theory, vol. 54, no. 7, 008. [6] S. Tatikonda and S. Mitter, The caacity of channels with feedback, Information Theory, IEEE Transactions on, vol. 55, no. 1, , 009. [7] K. A. Ross, Elementary analysis. Sringer, 1980.

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