The capacity of finite-State Markov Channels With feedback

Abstract

We consider a class of finite-state Markov channels with feedback. We first introduce a simplified equivalent channel model, and then construct the optimal stationary and nonstationary input processes that maximize the long-term directed mutual information. Furthermore, we give a sufficient condition under which the channel's Shannon capacity can be achieved by a stationary input process. The corresponding converse coding theorem and direct coding theorem are proved.

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SUBMITTED TO IEEE TRANSACTION ON INFORM THEORY, OCT. 20031
The Capacity of Finite-State Markov Channels
with Feedback
Jun Chen, Student Member, IEEE, Toby Berger†, Fellow, IEEE
Abstract
We consider a class of finite-state Markov channels with feedback. After introducing a troduce
a simplified equivalent channel model, we construct the optimal stationary and nonstationary input
processes that maximize the long-term directed mutual information. Furthermore, we give a sufficient
condition under which the channel’s Shannon capacity can be achieved by a stationary input process.
The corresponding converse coding theorem and direct coding theorem are proved.
Index Terms—Channel capacity, Markov channel, feedback, typicality.
I. INTRODUCTION
We study the capacity of a feedback channel whose state process can be affected by its input
and whose state information is available at both the transmitter and the receiver. Our channel
model is illustrated in Fig. 1. Were it not for the feedback, our channel would belong to the
family of finite-state channels. The FSC literature is vast; see, for example, [1]–[3]. In [4], Verdu
and Han gave a general capacity formula for channels without feedback. If in our model the
state process were not affected by the input, i.e., p(sk+1|xk,sk) = p(sk+1|sk), the model would
reduce to a special case in the general framework of [5] and [6].
The feedback channel coding problem goes back to early work by Shannon [7], Dobrushin
[8] and Wolfowitz [9]. Tatikonda [10] introduced a model of feedback channels which can be
viewed as a generalization of the formulation in [4], derived a general formula for the capacity of
†Corresponding author.
J. Chen and T. Berger are with the School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853
USA (email: jc353@cornell.edu, berger@ece.cornell.edu). This work was supported in part by NSF Grant No. CCR-9980616,
ARO P-40116-PH-MUR, DAAD 19-99-1-0215.
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2SUBMITTED TO IEEE TRANSACTION ON INFORM THEORY, OCT. 2003
Encoder
W
Message
Channel
p(vk|xk,sk)
Xk
Decoder
W
Vk
Estimate
of Message
∧
State
p(sk+1|xk,sk)
Sk+1
Sk
Sk
∆
Vk−1
Fig. 1 Our Model
channels in this class, and used dynamic programming to compute the optimal input distribution.
We show that channels descirbed by our model, which is called the Markov channel in [10],
possess under certain conditions a relatively simple capacity formula and that the corresponding
optimal input distribution can be computed with markedly less complexity than in the general
case.
It’s easy to see that if we let Yk= (Vk,Sk+1) and Q(yk|xk,yk−1) = p(vk|xk,sk)p(sk+1|xk,sk),
then Fig. 1 can be simplified to the model shown in Fig. 2. Therefore, we henceforth consider
only this simplified channel model. The model in Fig. 2 was perhaps first introduced in [11],
Encoder
W
Message
Channel
Q(yk|xk,yk−1)
Xk
Decoder
W
Yk
Estimate
of Message
∧
∆

Yk−1
Fig. 2 An Equivalent Model
[12]. Ying and Berger [13] analyzed the capacity of this channel model when the output is
binary. In this paper we will give a more general treatment.
The rest of this paper is divided into eight sections. In Section II we introduce several basic
notations and definitions. In Section III we prove the converse channel coding theorem for
our model, which provides an upper bound on the achievable rate of information transmission
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CHEN AND BERGER3
through the channel. Then we give a recursive formula to calculate the maximal directed mutual
information in Section IV. In Section V we analyze the optimum stationary input distribution that
maximizes the long-term directed mutual information. We generalize in Section VI to analyze
the optimum not-necessarily-stationary input. A sufficient condition under which the optimum
stationary input is actually optimum among all the input distributions is given in Section VII. We
prove the direct channel coding theorem and suggest a coding scheme in Section VIII. Finally,
several directions to extend our results are discussed in Section IX which serves as a conclusion.
II. PRELIMINARIES
A. Notation
We assume throughout the paper that the channel input and output alphabets both are finite .
Without loss of generality, we let x ∈ {1,2,··· ,Nx} and y ∈ {1,2,··· ,Ny}.
B. Code Description
An (n,M,ε,y0) feedback code for our channel consists of
1) An encoding function f that maps the set of messages W = {1,··· ,M} to channel input
words of blocklength n through a sequence of functions {fy0,k}n
the message W and the channel outputs up to time k − 1, i.e.,
k=1that depend only on
Xk= fy0,k(W,Yk−1
1
).
(1)
Although it might seem to be more general to let
Xk=˜fy0,k(W,Xk−1
1
,Yk−1
1
),
this actually is equivalent to (1), as shown by the following argument:
X1 =
˜fy0,1(W) = fy0,1(W),
X2 =
˜fy0,2(W,X1,Y1) =˜fy0,2(W,˜fy0,1(W),Y1) = fy0,2(W,Y1).
It follows easily by induction that Xk=˜fy0,k(W,Xk−1
of full generality.
1
,Yk−1
1
) = fy0,k(W,Yk−1
1
), so (1) is
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4SUBMITTED TO IEEE TRANSACTION ON INFORM THEORY, OCT. 2003
2) A decoding function gy0that maps a received sequence of n channel outputs to the message
set gy0: Yn
M
?
Note: The encoding function fy0and decoding function gy0both depend on the initial
channel state y0.
1→ W such that the average probability of decoding error satisfies
Pe?
1
M
w=1
P(ˆ W ?= w|W = w,Y0= y0) ≤ ε, whereˆ W = gy0(Yn
1).
(2)
Definition 1: Ry0is an ε-achievable rate given the initial state y0if for every δ > 0 there
exists, for all sufficiently large n, an (n,M,ε,y0) code such that
achievable if it is ε-achievable for all ε > 0. The supremum of all achievable rates Ry0is defined
as the feedback capacity Cfd
1
nlogM ≥ Ry0− δ. Ry0is
y0given the initial state y0.
III. CONVERSE CHANNEL CODING THEOREM
This section is devoted to the proof of the converse channel coding theorem.
Theorem 1(converse channel coding theorem): Given the initial state y0, information trans-
mission with an arbitrary small expected frequency of errors is not possible if R > limsup
n→∞
Cy0,n
n.
Here
Cy0,n= max
1)∈P∗(Xn
p(Xn
1)[I(X1;Y1|Y0= y0) +
n
?
k=2
I(Xk;Yk|Yk−1)],
and P∗(Xn
functions that satisfy
1) is the set of input distributions on Xn
1which consists of all the probability mass
p(Xk|Xk−1
1
,Yk−1
0
) = p(Xk|Yk−1), k = 1,2,··· ,n.
Proof:
In the proof we implicitly assume that P(Y0= y0) = 1 and thus use Y0instead of y0.
Let W be the message random variable. For any (n,M,ε,y0) code, by Fano’s inequality
H(W|Yn
0) ≤ h(Pe) + PelogM.
(3)
Since
H(W|Yn
0) = H(W) − I(W;Yn
0) = logM − I(W;Yn
0),
we have
(1 − Pe)logM ≤ h(Pe) + I(W;Yn
0),
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CHEN AND BERGER5
which we rewrite as
1
nlogM ≤h(Pe) + I(W;Yn
0)
n(1 − Pe)
.
(4)
As n → ∞, Pe→ 0. Hence the channel capacity
Cfd
y0
= limsup
n→∞
≤ limsup
n→∞
1
nlogM
max
p(Xn
1)
1
nI(W;Yn
0).
(5)
We have
I(W;Yn
0)=H(Yn
n
?
n
?
n
?
n
?
0) − H(Yn
[H(Yk|Yk−1
0|W)
) − H(Yk|Yk−1
=
k=1
0
0
,W)]
≤
k=1
[H(Yk|Yk−1) − H(Yk|Yk−1
0
,Xk,W)]
(a)
=
k=1
[H(Yk|Yk−1) − H(Yk|Yk−1,Xk)]
=
k=1
I(Xk;Yk|Yk−1),
(6)
where (a) holds because, when conditioned on the input Xkand the previous output Yk−1(i.e.,
the current channel state), the channel output Ykbecomes independent of both the message W
and the earlier outputs Yk−2
0
.
n ?
information was introduced by Massey [14] who attributes it to Marko [15]. See [10] for a
detailed discussion of this concept. It has been shown in [13], that the maximum directed mutual
We call
k=1I(Xk;Yk|Yk−1) the directed mutual information. The concept of directed mutual
information for our channel model is attained inside P∗(Xn
from restricting p(Xn
1); i.e., no loss of generality results
1) to P∗(Xn
1) when maximizing the directed information.
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