I W Z M Z . Bangalore. India. October 20-25. 2002
Algorithms for Computing the Capacity of Markov Channels with
Side Information
Shailendra K. Singh
Department of.ECE
Indian Institute of Science
Bangalore, 560012, India
Vinod Sharma
Department of ECE
Indian Institute of Science
Bangalore, 560012, India
shailQpal.ece.iisc.ernet.in
vinodmece.iisc.ernet.in
I. INTRODUCTION
Consider a finite state, irreducible M a r k channel {Sk}. At
time k the channel output Yk depends only on the channel input Xk and the channel state &. Let Uk and Vk he the CSIT
(channel state information at transmitter) and the CSIR (CSI
at &ever) respectively. In this paper we provide algorithms
to compute the capacity of the Markov channel under various assumptions on CSIT and CSIR where the assumption
P(& = s I (I:) = P(Sk I Uk)in (11 is not necessarily satisfied.
Our algorithms are based on the extension of ideas developed
in [3] to the channels with feedback.
11. DISCRETE C HANNEL
In this section we further assume perfect CSIR i.e., V, = S,
and U,,= g(S;-d+l)where g is a deterministic function of
A
%-,,+I =
(Sn-d+l ,...,&). Under these assumptions, we
show that the capacity achieving distribution of X, can he
made a function of the conditional distribution p , of S, giveu
rr; i.e., of
p,(s)
= P(S, = s
I V;).
(1)
converges a.s. i ~ sn + m, for any power allocation policy 7(,).
This limit equals the capacity if y(.) , a function of p , is the
OPA policy with respect to the limiting distribution p of pi.
Theorem 2 With the above assumptions on CSIT and CSIR,
the channel cnpacity of the finite state Markov fading channel
with AWGN noise and avemge power constrain P is
"P
where p is the limiting distribution of pn and (4) wnuerges
0 , s . to (5) for the OPA policy.
Etom the Lagrange multiplier and the Kuhn-Tucker conditions
[2] we get that the optimal power allocation policy y(.) should
.
and y(.) 2 0, for all p for which p ( p ) > 0. Solving (6)
for the minimum X such that the average power constraint
J , y ( p ) d p ( p ) 5 P is satisfied gives the optimal power allom
tion policy.
IV. EXAMPLES
Consider an AWGN channel with the fade values so =
1.0,SI = 0.5, sz = 0.01, s3 = 0.001 and let the average power
constaiut P he 20db. The capacity with perfect CSIT and
CSIR is 1.81 hits. We will take a 1 bit CSIT ( Uk = 0 if
4 E {.OOl,O.Ol} else 1) and perfect CSIR and the delay in
the feedback of the CSI to the transmitter d . Now the assump
and this limit exists a.s. The sup in (2) is taken over all tions in [l]are violated. Then the capacity of this channel with
d = 0 is Co = 1.786 bits and with d = 1 is CI = 1.699 bits.
distributions on the input alphabet.
Theorem 1 With the above mentioned assumptions on CSIT Fig. 1 shows the convergence of (4) for d = 0 and d = 1.
and perfect CSIR, the channel cnpacity of the finite state
Markov channel is
p,, can be computed recursively and { p n } is a regenerative
process which converges in distribution. We also show that
the capacity of the channel is
C
=
1
SUP
W ; Y IS = s,p)p(s)dAp)
(3)
p P(XlP)
where fi is the limiting distribution of p , and (21 converges
a.s. to (31.
111. AWGN C HANNEL
WITH
.
FADING
In this section we consider a Markov fading channel with additive white Gaussian noise (AWGN) with mean zero and variance one. The X, satisfies long term average power constraint
E[X:] 5 P. Here we assume that the receiver has the perfect
knowledge of instantaneous SNR, i.e. V, = SnE(lX,lz I V;]
and the CSIT may be imperfect i.e. U, = g ( x - d + l , Z , )
where g is a deterministic function and { Z , } is an iid noise
sequence. As in the previous section here we show that the
OPA (Optimal Power Allocation Policy) can be made a function of the conditional distribution pn, defined in (1). We
show that
(4)
0-7803-7629-3/02/Sl0.00 @ 2002 IEEE
,
.
,
I
. l . ~
,
,-
I
*.,-
. ..
-
_ .-
-
-
-
Figure 1: Convergence of Mutual Information to the capacity in Gaussian case
REFERENCES
[l] G. Caire and S. Shamai, On the Capacity of Some Channels
with Channel State Information, IEEE 'Pansaction Information
Theory, Vol. 42, 1996, pp. 868 - 886.
[2] T. Cover and J. Thomas, Elements of Information Theory ,
Wiley, New York, 1991.
[3] V. Sharma, S. K. Singh Entmpy mte and Mutual Informotion in
the Regenemtive Setup with Application to Morkov Chonnels,
Proc. IEEE Int. Symp. on Information Theory 2001, Washiug-
.
ton DC, USA June 2429, 2001.
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