Download paper

ISIT 2000,
Sorrento, Italy, June 25-30,2000
Capacity of nearly-decomposable Markovian fading channels
under asymmetric receiver-sender side information
R. Srikant
University of Illinois
e-mail: rsrikantauiuc .edu
Muriel MBdard
MIT
e-mail: m e d a d b i t . edu
Abstract - In modeling wireless channels, slow Define an M x M matrix P as follows: the (i,j) enand fast fades are generally decoupled. We show try of P is given by P,, =
x;)Bk1,
a # j,
that the difference between true capacity and that
k € S , 1€SJ
obtained assuming independence of fast and slow and P,, = 1 P,,. Note that P is also a stochaschannel fades is 0 (elog(e)log(-elog(E))), where e is tic matrix and let p be its stationary probability vector,
the ratio between the average duration of fast and i.e., p = p P . We can interpret P as being the long-term
slow fades.
transition probabilities among macro-states and p , as apOur purpose in this work is to explicitly take into ac- proximating the long-term probability of being in S,, i.e.,
count, in the capacity computation for time-varying fad- pa(.) = p , O ( E )where
,
p , ( e ) is the actual probability of
ing channels, the fact that slow fades and fast fades are being in micro-state i.
not truly decoupled. Decoupling slow fades from fast
Let T ( n ) denote the random variable corresponding
fades has generally been used as a first-order approxi- to the micro-state at time n and define S ( n ) to be ranmation. We consider the case where the sender channel dom variable corresponding to the macro-state at time
side information (SCSI) is a coarse representation of the g. The sample values of T ( n ) is denoted by t(n). Furreceiver channel side information (RCSI). In many cir- ther, let
be the random variable correspondcumstances, RCSI and SCSI are asymmetric, although ing to the signal attenuation at time n. The reckived sigrelated. In particular, when the channel is rapidly vary- nal at time n is given by the random variable Y ( n ) =
ing, providing full feedback from the receiver to the sender
d m X ( n ) W ( n ) ,where X ( n ) is the transmitted
may be onerous and inefficient. Recent work in this area
signal and W ( n )is AWGN with variance u 2 .Our coding
has considered the case where the SCSI is a deterministic
theorem follows.
function of the RCSI [l]. In [l], exact capacity results
are given for the case when the SCSI remains Markov. If Theorem 1 Define
the SCSI and the RCSI can indeed be decoupled, in such
a way that both remain Markov, then the results of [l]
apply directly.
We consider a discrete-time finite-state Markov chan- to ~ ~ l p , ( ~ ) 5
P P
( i, where
)
P is the power constraint
nel (FSMC). The RCSI, which we term the micro states, on the sender.
is a full description of the FSMC. The SCSI, which we
Given R < C and 6 > 0 , we can find an e*(R)and n(6)
term the macro states, is a coarser representation of the such that for all e < e * , there exists a (n/e,2"Ri') code
states: the sender only knows that the current state is whose maximal probability of error is less than 6. (Note
within one of a set of states. The macro states repre- that e* is independent of 6.)
1
sent the long-term behavior of the channel, i.e. the slow
Define Ctrue(e)
= lim - max I ( X " ; { Y " , T " } ) ,
n+o3 72 p ( z " l s n )
fades. Note that fades are possible while we are in the
good macro state and, conversely, energy surges are pos- where p(xkls") = p ( x k l s k ) , for all k 5 n. Then,
sible while we are in the bad macro state. Although the Ctrue(E) = c 0 (elog(€)log(-elog(e))).
model of [l]does not apply, we suspect that, as the spread
Suppose for some R > 0 , we have the fotlowing propbetween the speed of the slow fades and that of the fast erty: for every 6 > 0 , we can find E * ( R )and n ( 6 ) such
fades grows, the results of [l]should become an increas- that for all E < e * , there exists a ( n / e ,2"Ri') code whose
ingly good approximation to the true capacity. Our re- maximal probability of error is less than 6. Then, R < C.
sults support this intuition and quantify the effect of the
spread between the speed of the slow fades and that of
REFERENCES
the fast fades. However, our results also show that convergence is very slow.
[l] G. Caire, S. Shamai, "On the Capacity of Some Channels
with Channel State Information", IEEE rrclnsactions on
We consider a nearly decomposable model ([2]) for our
Information Theory, September 1999, vol. 45, no. 6, pp.
FSMC. Consider a discrete-time Markovian fading pro2007-2020.
cess defined by the stochastic matrix A + E B ,where A is
block-diagonal with M blocks and the ith block (which [2] R.G. Phillips, P.V. Kokotovic, "A Singular Perturbation
Approach to Modeling and Control of Markov Chains",
is also a stochastic matrix) is denoted by A,. We call
IEEE rrclnsactions on Automatic Control, vol. AC-26, no.
the set of fading states associated with the ith block a
5, October 1981, pp. .
macro state and denote it by S,. We assume the RCSI is
the current micro state of the channel whereas the SCSI
is the current macro state. Let d*)
be the stationary
probability vector associated with A , , i.e., x("A, = d').
E,+,
+
d
m
+
+
1-7803-5857-0/00/%10.00 0 2 0 0 0 IEEE.
41 3