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{Xn = Tn(Un), Tn X L |U|} produces the transmitted symbol Xn according to current feedback Un. The equivalent channel p(y, v|t) is a discrete memoryless channel (DMC) without side information. The capacity can be achieved according to the DMC coding theorem [30]. The codebook is constructed according to p(t(uq)) = p* (t(uq)|uq), where p* ( | ) is the optimal distribution X|U X|U in (3.45). Converse The converse is proved using Fano s lemma [30]. Let the message be W W with a uniform distribution and |W | = 2LNR. The message is related to the channel input XN and the corresponding channel output YN. Let W be 1 1 the detected message at the receiver and de ne the error probability as (LN) Pe = Pr{W W}. The rate R satis es LNR = H (W )
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a = H (W Y1N , V1N ) + I (W ; Y1N , V1N ) b = H (W Y1N , V1N ) + I (W ; Y1N , V1N )
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where (a) follows from the de nition of the mutual information and (b) follows from the independence between W and VN. By Fano s lemma [30], the 1 rst term of (3.46) is upper-bounded as H (W Y1N , V1N ) 1 + Pe(LN ) LNR (3.47)
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In general, Xn and W are related through p(xn|w, un) because of the causal1 ity of the feedback. A key step in the proof of the converse is to introduce an intermediate random variable Tn with conditional distribution p(tn|w, n-1 n-1 u1 ) and assume that Xn is independent of W and U1 given Tn and Un, that n is, p(xn|w, u1 , tn) = p(xn|un, tn). The assumption has no loss of generality because n-1 we can always let Tn = {W, U1 } to restore the original relation. The second term of (3.46) can be upper-bounded as I (W ; Y1N , V1N ) = I (W; Y1N V1N , U1N ) + I (U1N; W V1N ) - I (U1N; W Y1N , V1N )
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I (W; Y1N V1N , U1N )
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= I (W; Yi V1N , U1N , Y1i -1 )
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= H (Yi V1N , U1N , Y1i -1 ) - H (Yi V1N , U1N , Y1i -1 , Ti )
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= I (Ti , Yi Vi , U i )
max p(v )I (T , Y v , u ()
i =1 N h i i i i vi i =1 h i i i i vi
= h(v i ))
max p(v )I (X , Y v , u ()
= h(v i )) (3.49)
where a follows from the mutual information identity (3.41), b follows from N the independence between UN and W given V1 and the nonnegativity of 1 N N N mutual information I(U1 ;W|Y1 ,V1 ), c follows from the chain rule of the mutual information, d follows from the data processing inequality [30] and the i-1 fact that W Ti Yi is a Markov chain given {VN, UN, Y1 }, e follows from 1 1 the fact that the conditioning reduces the entropy, f follows from the fact that Yi is independent from the other variables given {Vi, Ui, Ti}, g follows from Lemma 3.2, h follows from the fact that Ti Xi Yi is a Markov chain given ui, vi, and i follows from Equation (3.45). Combining Equations (3.46), (3.47), and (3.49), we obtain R 1 + Pe(LN ) R + C LN
If P(LN) 0 as N , then the rate R must be less than or equal to C. e Remark 1. If there is a memoryless feedback error, the relation between Un and Vn is an i.i.d. statistical relation pU|V(un|vn). Comparing Equation (3.42) in Lemma 3.2 and the capacity C in (3.45), we observe that the feedback error causes two penalties. The rst penalty is due to the averaging [i.e., SUp(u|v) I(T; Y|u, v) maxuI(T; Y|u, v)], and the second penalty is due to the fact that the receiver is uncertain about U.Therefore, Lemma 3.2 will be useful for the design of the index assignment of the feedback when there is feedback error. Remark 2. Theorem 3.1 is proved with the assumption that the receiver does not know the feedback U. Of course, since the optimal feedback is a deterministic function of V, the receiver knows the feedback implicitly. The capacity is the same even if we start by assuming that the receiver knows the feedback. In this case, the proof of the converse part will start from Equation (3.48). The rest of the proof is the same. For the case of perfect CSIR V = H, the capacity C in (3.45) can be simpli ed to a scalar form as