Capacity Region of Finite State Multiple-Access Channel with Delayed State Information at the Transmitters

0.36 MB·English

by  Uria Basher

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Capacity Region of Finite State
Multiple-Access Channel with Delayed State
Information at the Transmitters
Uria Basher, Avihay Shirazi, and Haim Permuter
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Abstract
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A single-letter characterization is provided for the capacity region of finite-state multiple access channels. The
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J channel state is a Markov process, the transmitters have access to delayed state information, and channel state

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information is available at the receiver. The delays of the channel state information are assumed to be asymmetric
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at the transmitters. We apply the result to obtain the capacity region for a finite-state Gaussian MAC, and for a

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T finite-statemultiple-access fading channel. We derive power control strategies that maximize the capacity region for
I these channels.
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[ Index Terms

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v Capacityregion,Delayedfeedback,Directedinformation,Finite-statechannel,GaussianMultiple-Accesschannel,
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Multiple-Access channel, Multiplexing coding scheme, Successive decoding.
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.
1 I. INTRODUCTION
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Wireless communication is an example of channels where the channel characteristics are time-varying. In a
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v wireless setting, the user’s motion and the changes in the environment, as well as the interference, may lead to
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X temporal changes in the channel quality. Such channel variation models can include fast fading due to multi-path
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and slow fading due to shadowing. In fast fading, the channel state is assumed to be changing for every channel
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use, while in slow fading, the channel is assumed to be constant for each finite block length.
Insuchcommunicationproblems,thechannelstateinformation(CSI)canbetransmittedtothetransmitterseither
explicitly,or throughoutputCSI feedback.Frequently,the CSI feedbackis notinstantaneous;the transmittershave
only delayed information regardingthe state of the channel. The availability of the delayed CSI at the transmitters
willpossiblyincreasethecapacityregion.TheincreaseinthecapacityregionduetoCSIdependsontheCSIdelays
relative to the rate at which the channelis time-varying.When a channelis slowly time-varyingand the delays are
small, CSI may significantly increase the capacity region. However, if the channel is changing rapidly relative to
the CSI delays, the transmitters can no longer adapt to the channel variations. Hence, availability of delayed CSI
may not result in any significant capacity region improvement. Therefore, we are motivated to study the effect of
channel memory and delays on the multiple access channel (MAC) capacity region.
2
Let us now present a brief literature review. We are modeling a time-varying channel as a finite-state Markov
channel(FSMC) [1], [2]. The FSMC is a channelwith a finite numberof states. Duringeach symboltransmission,
the channel’sstate isfixed.The channeltransitionprobabilityfunctionis determinedbythe channelstate. Thetime
variation in the channel characteristics is modeled by the statistics of the underlying state process.
Capacity of memoryless channels, with different cases of state information being available in a causal or non
causal manner at the transmitter and at the receiver, has been studied by Shannon [3] and by Gelfand and Pinsker
[4]. In [5], Goldsmith and Varaiya consider the fading channels with perfect CSI at the transmitter and at the
receiver. They proved that with instantaneous and perfect state information, the transmitter can adapt the data
rates for each channel state to maximize the average transmission rate. Viswanathan [6] loosened this assumption
of perfect instantaneous CSI, and gave a single letter characterization of the capacity of Markov channels with
delayed CSI. Caire and Shamai [7] consider the case that the channel state is identically distributed (i.i.d.), and
the CSI at the transmitter is a deterministic function of the CSI at the receiver. They showed that optimal coding
is particularly simple. Chen and Berger in [8] found the capacity of an FSC with inter-symbol interference (ISI),
where current CSI is available at the transmitter and the receiver. For a comprehensive survey on channel coding
with state information see [9].
TheMACwithstatehasreceivedmuchattentioninrecentyearsduetoitsimportanceinwirelesscommunication
systems.Ontheonehand,completeknowledgeoftheCSIatthetransmittersisanunrealisticassumptioninwireless
communications. On the other hand, it is reasonable to assume that the receiver does possess full knowledge of
the CSI. This practical consideration has motivated the investigationof a MAC where each transmitter is informed
with its own CSI, while the receiver is informed with the full CSI.
Our work is also related to [10], [11], and [12]. In [10] Como and Yu¨ksel found the capacity region of FS-
MAC,wherethechannelstateprocessisi.i.d.,thetransmittershaveaccesstopartial(quantized)CSI,andcomplete
CSI is available at the receiver. In [11] the capacity of general FS-MAC with varying degrees of causal CSI at
the transmitters is characterized in non-single-letter formulas. In [12] the capacity region of the FS-MAC with
feedback that may be an arbitrary time-invariant function of the channel output has been derived. Recent related
work also includes [13], which studies the state-dependent MAC with causal and strictly causal side information
at the transmitters.
In this work,we considerthe capacityregionof a finite state MarkovMultiple-accesschannel(FSM-MAC)with
CSI at the decoder (receiver) and delayed CSI at the encoders (transmitters) with delays d and d as illustrated
1 2
in Fig. 1. The channel probability function at each time instant depends on the state of an underlying finite-state
Markovprocess. The decoder,in addition to the channeloutput,also receivesthe channelstate at each time instant
(perfect CSI). The channel state is fed back to the encoders through a noiseless feedback channel. CSI from the
decoderisreceivedatEncoder1andEncoder2aftertimedelaysof d andd symboldurations,respectively.Each
1 2
encoder, at each time instant, chooses the channel input based on the message to be transmitted and the CSI that
he possesses. A formal description of the system model is presented in Section II. The main result of this paper is
a computable characterization of the capacity region for this channel model.
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M1 Encoder1 X1n Channel
Yn Mˆ1,Mˆ2
Si−d1 p(y|x1,x2,s) Decoder
M2 Encoder2 X2n
S
i
S
i−d2
Fig. 1: FSM-MAC with CSI at the decoder and delayed CSI at the encoders with delays d and d - The state
1 2
process has memoryand is assumed to be FSM. The CSI is fed back to the encodersthrough a noiseless feedback
channel. CSI from the decoder is received at Encoder 1 and Encoder 2 after time delays of d and d symbol
1 2
durations, respectively. We are considering the above problem setting in the cases where d > d , d = d , and
1 2 1 2
d <d =∞.
2 1
The remainder of the paper is organized as follows: In Section II, we concretely describe the communication
model. In Section III, we state our main results, which are the capacity regions for different cases of time delays.
SectionIVprovidestheupperboundonthecapacityregionofFSM-MACwithCSIatthedecoderandasymmetrical
delayed CSI at the encoders. In Section V, we complete the proof of the capacity region, by providing the proof
of the achievability. In Section VI, we provide alternative proof for capacity region. The alternative proof is based
on a multi-letter expression for the capacity region of FS-MAC with time-invariant feedback [12]. In Section VII,
we apply the general results of Section III to obtain the capacity region for a finite-state Gaussian MAC, and
for a finite-state multiple-access fading channel. We derive optimization problems on the power allocation that
maximize the capacity region for these channels. This power allocation would be the optimal power control policy
for maximizing throughput in the presence of delayed CSI. We conclude in Section VIII with a summary of this
work.
II. CHANNEL MODEL AND NOTATION
A. Channel Model
In this paper,we consider the communicationsystem of FSM-MAC with CSI at the decoderand delayed CSI at
the encoders with delays d and d , respectively, as illustrated in Fig. 1. The MAC setting consists of two senders
1 2
andonereceiver.Eachsenderj ∈{1,2}choosesanindexmj uniformlyfromtheset 1,...,2nRj andindependent
(cid:8) (cid:9)
of the other sender. The input to the channelfrom encoder j ∈{1,2} is denoted by {X ,X ,X ,...}, and the
j,1 j,2 j,3
output of the channel is denoted by {Y ,Y ,Y ,...}. We use the notation Vn to denote the sequence (V ,...,V ),
1 2 3 1 n
therefore, Xn, Yn denote the sequences {X ,...,X }, {Y ,...,Y }, respectively. A finite-state Markov channel
j j,1 j,n 1 n
is, ateach time instant,in oneof a finite numberof states S ={s ,s ,...,s }.In each state, the channelis a DMC
1 2 k
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with inputs alphabet X ,X and output alphabet Y. Let the random variables S ,S denote the channel state at
1 2 i i−d
times i and i−d, respectively. Similarly, denote by X ,X , and Y the inputs and the output of the channel
1,i 2,i i
at time i. The channel transition probability function at time i depends on the state S , and the inputs X ,X
i 1,i 2,i
at time i, and is given by P(y |x ,x ,s ). The channel output at any time i is assumed to depend only on the
i 1,i 2,i i
channel inputs and state at time i. Hence
P(y |xi,xi,si)=P(y |x ,x ,s ). (1)
i 1 2 1 i 1,i 2,i i
Thestateprocess{S }isassumedtobeanirreducible,aperiodic,finite-statehomogeneousMarkovchainandhence
i
is ergodic. The state process is independent of the channel inputs and output when conditioned on the previous
states, i.e.,
P(s |si−1,xi−1,xi−1,yi−1)=P(s |s ). (2)
i 1 2 i i−1
Furthermore, we assume that the state process is independent of M and M ,
1 2
n
P(sn,m ,m )=P(sn)P(m )P(m )= P(s |s )P(m )P(m ). (3)
1 2 1 2 i i−1 1 2
Yi=1
Now, let K be the one step state transition probability matrix of the Markov process, and let π be the steady
state probability distribution of the Markov process. The (S ,S ) joint distribution is stationary and is given by
i i−d
π (S =s ,S =s )=π(s )Kd(s ,s ), (4)
d i l i−d j j l j
where Kd(s ,s ) is the (l,j)th elementof the d-step transition probabilitymatrix Kd of the Markovstate process.
l j
For simplicity, let us define S,S˜ as the variables that have the same joint distribution as (S ,S ), i.e.,
1 i i−d1
P(S =sl,S˜1 =sj)=πd1(Si =sl,Si−d1 =sj)=π(sj)Kd1(sl,sj). (5)
Similarly, we define S,S˜ as the variables that have the same joint distribution as (S ,S ).
2 i i−d2
B. Code Description
An (n,2nR1,2nR2,d1,d2) code for FSM-MAC with CSI at the decoder and delayed CSI at the encoders with
delay d and d consists of
1 2
1) Two sets of integers M1 ={1,2,...,2nR1} and M2 ={1,2,...,2nR2}, called the message sets.
2) For each encoder, an encoding function f , j ∈{1,2}, maps the set of messages to channel input words of
j
blocklengthn. Each f worksthrougha sequenceof functionsf thatdependonly onthe message M and
j j,i j
the channel states up to time i−d . For encoder 1 (j =1):
j
f (M ), 1≤i≤d
1,i 1 1
X1,i =  (6)
 f1,i(M1,Si−d1), d1+1≤i≤n 
 
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Similarly for encoder 2 (j =2):
f (M ), 1≤i≤d
2,i 2 2
X2,i =  (7)
 f2,i(M2,Si−d2), d2+1≤i≤n 
 
3) Adecodingfunctionψ thatmapsareceivedsequenceofnchanneloutputsandchannelstatestothemessages
set
ψ :Yn×Sn →M ×M . (8)
1 2
We define the average probability of error for the (n,2nR1,2nR2,d1,d2) code as follows:
1
P(n) = P (sn)Pr{ψ(yn,sn)6=(m ,m )|(m ,m )was sent}. (9)
e 2n(R1+R2) mX1,m2Xsn1 Sn 1 2 1 2
We use standard definitions [14] of achievability and capacity region, namely, a pair rate (R ,R ) is achievable
1 2
for FSM-MAC with CSI at the decoder and delayed CSI at the encoders with delays d and d , if there exists a
1 2
sequence of (n,2nR1,2nR2,d1,d2) codes with Pe(n) → 0 as n goes to infinity. The capacity region is the closure
of the set of achievable (R ,R ) rate pairs.
1 2
III. MAIN RESULTS
Here we present the main results of this paper. Recall, that the joint distributionsof (S,S˜ ), and (S,S˜ ) is given
1 2
in (5). Without loss of generality, let us assume that d ≥d .
1 2
Theorem 1 (Capacity region of FSM-MAC with delayed CSI d ≥d )
1 2
The capacity region of FSM-MAC with CSI at the decoder and asymmetrical delayed CSI at the encoders with
delays d and d as showen in Fig. 1 is given by:
1 2
R <I(X ;Y|X ,S,S˜ ,S˜ ,U),
1 1 2 1 2
 
R= R <I(X ;Y|X ,S,S˜ ,S˜ ,U), , (10)
2 2 1 1 2
P(u|s˜1)P(x1|s˜1[,u)P(x2|s˜1,s˜2,u) R1+R2 <I(X1,X2;Y|S,S˜1,S˜2,U), 
 
where U is an auxiliary random variable with cardinality |U|≤3.
TheproofofTheorem1ispresentedinSectionsIV,andV.InSectionIVweprovetheupperboundofthecapacity
region, and Section V is devoted to the proof of the achievability. The proof of the achievability is based on a
multiplexing coding scheme, and successive decoding. In addition, we provide alternative proof of Theorem 1 in
Section VI. The proof for the cardinality bound of U is presented in Appendix A.
Now, directly from Theorem1 we can derivethe capacity region in the case of d =d . Since d =d we have
1 2 1 2
S˜ =S˜ , hence we denote S˜=S˜ =S˜ . Using Theorem 1 we get,
1 2 1 2
Theorem 2 (Capacity region of FSM-MAC with symmetrical delayed CSI d =d )
1 2
The capacityregionofFSM-MACwith CSI atthe decoderandsymmetricaldelayedCSI atthe encoderswith delay
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d is given by:
R <I(X ;Y|X ,S,S˜,U),
1 1 2
 
R= R <I(X ;Y|X ,S,S˜,U), , (11)
2 2 1
P(u|s˜)P(x1[|s˜,u)P(x2|s˜,u) R1+R2 <I(X1,X2;Y|S,S˜,U), 
 
where U is an auxiliary random variable with cardinality |U|≤3.
Now we consider the case that encoder 1 does not have state information at all, i.e., d =∞.
1
Theorem 3 ( Capacity region of FSM-MAC with delayed CSI only to one encoder)
The capacity region of FSM-MAC with CSI at the decoder and delayed CSI only to one encoder is given by :
R <I(X ;Y|X ,S,S˜,Q),
1 1 2
 
R= R <I(X ;Y|X ,S,S˜,Q), , (12)
2 2 1
P(q)P(x1[|q)P(x2|s˜,q) R1+R2 <I(X1,X2;Y|S,S˜,Q), 
 
where Q is an auxiliary random variable with cardinality |Q|≤3.
The proof of Theorem 3 is quite similar to the proof of Theorem 1; the details are presented in Appendix B.
IV. CONVERSE
In this section we provide the upper bound on the capacity region of MAC with receiver CSI and asymmetrical
delayed CSI feedback, i.e., we give the converse proof for Theorem 1. Without loss of generality let us assume
that d ≥d .
1 2
Proof: Given an achievable rate (R ,R ) we need to show that there exists joint distribution of the form
1 2
P(s,s˜ ,s˜ )P(u|s˜ )P(x |s˜ ,u)P(x |s˜ ,s˜ ,u)P(y|x ,x ,s) such that,
1 2 1 1 1 2 1 2 1 2
R <I(X ;Y|X ,S,S˜ ,S˜ ,U),
1 1 2 1 2
R <I(X ;Y|X ,S,S˜ ,S˜ ,U),
2 2 1 1 2
R +R <I(X ,X ;Y|S,S˜ ,S˜ ,U),
1 2 1 2 1 2
whereU isanauxiliaryrandomvariablewithcardinality|U|≤3.Theproofforthecardinalityboundispresentedin
AppendixA.Since(R1,R2)isanachievablepair-rate,thereexistsacode(n,2nR1,2nR2,d1,d2)withaprobability
of error P(n) arbitrarily small. By Fano’s inequality,
e
H(M ,M |Yn,Sn)≤n(R +R )P(n)+H(P(n)),nε , (13)
1 2 1 2 e e n
and it is clear that ε →0 as P(n) →∞. Then we have
n e
H(M |Yn,Sn)≤H(M ,M |Yn,Sn)≤ε , (14)
1 1 2 n
H(M |Yn,Sn)≤H(M ,M |Yn,Sn)≤ε . (15)
2 1 2 n
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We can now bound the rate R as
1
nR = H(M )
1 1
= H(M )+H(M |Yn,Sn)−H(M |Yn,Sn)
1 1 1
(a)
≤ I(M ;Yn,Sn)+nε
1 n
(=b) I(M ;Yn|Sn)+I(M ;Sn)+nε
1 1 n
(=c) I(M ;Yn|Sn)+nε
1 n
(=d) I(Xn;Yn|Sn)+nε
1 n
= H(Xn|Sn)−H(Xn|Yn,Sn)+nε
1 1 n
(=e) H(Xn|Xn,Sn)−H(Xn|Yn,Sn)+nε
1 2 1 n
(f)
≤ H(Xn|Xn,Sn)−H(Xn|Yn,Xn,Sn)+nε
1 2 1 2 n
= I(Xn;Yn|Xn,Sn)+nε
1 2 n
= H(Yn|Xn,Sn)−H(Yn|Xn,Xn,Sn)+nε
2 1 2 n
n
= H(Y |Yi−1,Xn,Sn)−H(Y |Yi−1,Xn,Xn,Sn)+nε
i 2 i 1 2 n
Xi=1
n
(g)
≤ H(Yi|X2,i,Si,Si−d2,Si−d1,Si−d1−1)−H(Yi|Yi−1,X1n,X2n,Sn)+nεn
Xi=1
n
(=h) H(Yi|X2,i,Si,Si−d2,Si−d1,Si−d1−1)−H(Yi|X1,i,X2,i,Si,Si−d2,Si−d1,S1i−d1−1)+nεn
Xi=1
n
= I(Yi;X1,i|X2,i,Si,Si−d2,Si−d1,S1i−d1−1)+nεn,
Xi=1
where
(a) follows from Fano’s inequality.
(b) follows from chain rule.
(c) follows from the fact that M and Sn are independent.
1
(d) follows from the fact that Xn is a deterministic function of (M ,Sn) and the Markov chain (M ,Sn) −
1 1 1
(Xn,Sn)−Yn.
1
(e) follows from the fact that Xn and M are independent, and the fact that Xn is a deterministic function of
1 2 2
(M ,Sn). Therefore, Xn and Xn are independentgiven Sn .
2 1 2
(f) and (g) follow from the fact that conditioning reduces entropy.
(h) follows fromthe fact that the channeloutputat time i dependsonly on the state S and the the inputsX and
i 1,i
X .
2,i
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Hence, we have
n
1
R ≤ I(Y ;X |X ,S ,S ,S ,Si−d1−1)+ε . (16)
1 n i 1,i 2,i i i−d2 i−d1 1 n
Xi=1
Similarly, we have
n
1
R ≤ I(Y ;X |X ,S ,S ,S ,Si−d1−1)+ε . (17)
2 n i 2,i 1,i i i−d2 i−d1 1 n
Xi=1
To bound the sum of the rates, consider
n(R +R ) = H(M ,M )
1 2 1 2
= H(M ,M )+H(M ,M |Yn,Sn)−H(M ,M |Yn,Sn)
1 2 1 2 1 2
(a)
≤ I(M ,M ;Yn,Sn)+nε
1 2 n
(=b) I(M ,M ;Yn|Sn)+I(M ,M ;Sn)+nε
1 2 1 2 n
(=c) I(M ,M ;Yn|Sn)+nε
1 2 n
(=d) I(Xn,Xn;Yn|Sn)+nε
1 2 n
= H(Yn|Sn)−H(Yn|Xn,Xn,Sn)
1 2
n
= H(Y |Yi−1,Sn)−H(Y |Yi−1,Xn,Xn,Sn)+nε
i i 1 2 n
Xi=1
n
(=e) H(Yi|Yi−1,Sn)−H(Yi|X1,i,X2,i,Si,Si−d2,Si−d1,Si−d1−1)+nεn
Xi=1
n
(f)
≤ H(Yi|Si,Si−d2,Si−d1,Si−d1−1)−H(Yi|X1,i,X2,i,Si,Si−d2,Si−d1,Si−d1−1)+nεn
Xi=1
n
= I(Yi;X1,i,X2,i|Si,Si−d2,Si−d1,Si−d1−1)+nεn,
Xi=1
where
(a) follows from Fano’s inequality.
(b) follows from chain rule.
(c) follows from the fact that M ,M , and Sn are independent.
1 2
(d) follows from the fact that Xn,Xn is a deterministic function of (M ,M ,Sn) and the Markov chain
1 2 1 2
(M ,M ,Sn)−(Xn,Xn,Sn)−Yn.
1 2 1 2
(e) follows from the fact that the channel output at time i depends only on the state S , and the inputs X , and
i 1,i
X .
2,i
(f) follows from the fact that conditioning reduces entropy.
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Hence, we have
n
1
R1+R2 ≤ n I(Yi;X1,i,X2,i|Si,Si−d2,Si−d1,Si−d1−1)+εn. (18)
Xi=1
The expressions in (16), (17), and (18) are the average of the mutual informations calculated at the empirical
distribution in column i of the codebook.We can rewrite these equationswith the new variable Q, where Q=i∈
{1,2,...,n} with probability 1. The equations become
n
n
1
R1 ≤ n I(Yi;X1,i|X2,i,Si,Si−d2,Si−d1,Si−d1−1)+εn
Xi=1
n
1
= n I(YQ;X1,Q|X2,Q,SQ,SQ−d2,SQ−d1,SQ−d1−1,Q=i)+εn
Xi=1
= I(YQ;X1,Q|X2,Q,SQ,SQ−d2,SQ−d1,SQ−d1−1,Q)+εn (19)
Now let us denote X1 ,X1,Q,X2 ,X2,Q,Y ,YQ,S ,SQ,S˜1 ,SQ−d1,S˜2 ,SQ−d2, and U ,(SQ−d1−1,Q).
We have,
R ≤ I(X ;Y|X ,S,S˜ ,S˜ ,U)+ε ,
1 1 2 1 2 n
R ≤ I(X ;Y|X ,S,S˜ ,S˜ ,U)+ε ,
2 2 1 1 2 n
R +R ≤ I(X ,X ;Y|S,S˜ ,S˜ ,U)+ε .
1 2 1 2 1 2 n
To complete the converse proof we need to show the following Markov relations hold:
1) P(u|s,s˜ ,s˜ )=P(u|s˜ ) .
1 2 1
2) P(x |s,s˜ ,s˜ ,u)=P(x |s˜ ,u).
1 1 2 1 1
3) P(x |x ,s,s˜ ,s˜ ,u)=P(x |s˜ ,s˜ ,u).
2 1 1 2 2 1 2
4) P(y|x ,x ,s,s˜ ,s˜ ,u)=P(y|x ,x ,s).
1 2 1 2 1 2
We prove the above using the following claims:
1) follows from the fact that Si−d1−1−Si−d1 −Si−d2 −Si and so is (S1Q−d1−1,Q)−SQ−d1 −SQ−d2 −SQ.
2) follows from the fact that X1,i =f1,i(M1,Si−d1) and that M1 and Sn are independent. Hence
P(x |s ,s ,s ,sq−d1−1,q =i)=P(x |s ,sq−d1−1,q =i).
1,q q q−d1 q−d2 1 1,q q−d1 1
Since this is true for all i,
P(x |s ,s ,s ,sq−d1−1,q)=P(x |s ,sq−d1−1,q).
1,q q q−d1 q−d2 1 1,q q−d1 1
Therefore we have,
P(x |s,s˜ ,s˜ ,u)=P(x |s˜ ,u).
1 1 2 1 1
3) We assume that d ≥ d , since M and (M ,Sn) are independent, and the state process is Markov chain,
1 2 2 1
10
we have
P(m2,si−d2|si,si−d1,si−d2,si−d1,m1) = P(m2,si−d2|si−d1,si−d2,si−d1).
Therefore, we have the Markov chain (M2,Si−d2)−(Si−d1,Si−d2,Si−d1)−(M1,Si,Si−d1). Since X1,i =
f1,i(M1,Si−d1) and X2,i = f2,i(M2,Si−d2) where f1,i,f2,i are deterministic functions, we obtain the
following Markov chain,
X2,i−(M2,Si−d2)−(Si−d1,Si−d2,Si−d1)−(M1,Si,Si−d1)−X1,i. (20)
Which implies,
P(x2,i|x1,i,si,si−d1,si−d2,si−d1−1) = P(x2,i|si−d1,si−d2,si−d1−1).
Since this is true for all i,
P(x |x ,s ,s ,s ,sq−d1−1,q) = P(x |s ,s ,sq−d1−1,q).
2,q 1,q q q−d1 q−d2 1 2,q q−d1 q−d2 1
Therefore we have P(x |x ,s,s˜ ,s˜ ,u)=P(x |s˜ ,s˜ ,u).
2 1 1 2 2 1 2
4) follows from the fact that the channel output at any time i is assumed to depend only on the channel inputs
and state at time i.
Hence, taking the limit as n→∞, P(n) →0, we have the following converse:
e
R ≤ I(X ;Y|X ,S,S˜ ,S˜ ,U),
1 1 2 1 2
R ≤ I(X ;Y|X ,S,S˜ ,S˜ ,U),
2 2 1 1 2
R +R ≤ I(X ,X ;Y|S,S˜ ,S˜ ,U),
1 2 1 2 1 2
for some choice of joint distribution P(s,s˜ ,s˜ )P(u|s˜ )P(x |s˜ ,u)P(x |s˜ ,s˜ ,u)P(y|x ,x ,s) and for some
1 2 1 1 1 2 1 2 1 2
choice of auxiliary random variable U defined on |U|≤3. This completes the proof of the converse.
V. PROOF OFTHEACHIEVABILITY OFTHEOREM 1
In the previous section we proved the converse of the capacity region of Theorem 1. In this section we prove
the achievability part. The main idea of the proof is using multiplexing coding, i.e., multiplexing the input of
the channel at each encoder (the multiplexer is controlled by the delayed CSI), then, using the CSI known at the
decoder, demultiplexing the output at the decoder.
Proof: To prove the achievability of the capacity region, we need to show that for a fix P(x |s˜ )P(x |s˜ ,s˜ )
1 1 2 1 2
and (R ,R ) that satisfy
1 2
R ≤ I(X ;Y|X ,S,S˜ ,S˜ ),
1 1 2 1 2
R ≤ I(X ;Y|X ,S,S˜ ,S˜ ),
2 2 1 1 2