Information Theory for Wireless Communications. Lecture 10 Discrete Memoryless Multiple Access Channel (DM-MAC): The Converse Theorem
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1 Information Theory for Wireless Communications. Lecture 0 Discrete Memoryless Multiple Access Channel (DM-MAC: The Converse Theorem Instructor: Dr. Saif Khan Mohammed Scribe: Antonios Pitarokoilis I. THE DISCRETE MEMORYLESS MULTIPLE ACCESS CHANNEL We consider a two-user discrete memoryless multiple access channel (DM-MAC with distribution p(y x,x 2, as shown in Fig.. Users and 2 select their messages, X and X 2, according to the probability density functions p x ( and p x2 ( at rates R and R 2, respectively, while the channel outputs the symbol Y. We define the rate region, R(p x (,p x2 (, as the pair of rates (R,R 2 that satisfy R(p x (,p x2 ( = {(R,R 2 R I(X ;Y X 2,R 2 I(X 2 ;Y X,R +R 2 I(X,X 2 ;Y. ( Further, the capacity region, C DM-MAC is given by the convex hull of the union of all the possible rate regions, R(p x (,p x2 (, over all the possible input distributions p x ( and p x2 (, namely C DM-MAC = Co R(p x (,p x2 (. (2 p x (,p x2 ( We also define the average probablity of error, λ n of a code. Let a code of length n and a set of codewords (W i,w j, i = {,...,2 nr, j = {,...,2 nr 2. The encoders of the two users transmit the codewordsw i and W j. The decoder detects the output, Y, of the DM-MAC channel and decides for the pair of codewords { {Ŵi,Ŵj. Then, the probability of an error even is given byp (Ŵi,Ŵj (W i,w j (W i,w j was sent. The average error probability of this code is given by λ n = i,j { P{(W i,w j P (Ŵi,Ŵj (W i,w j (W i,w j was sent. The channel coding theorem for the DM-MAC can now be stated as follows. Theorem : If there is a sequence of codes ( n, ( 2 nr,2 nr 2, λn such that λn 0, then (R,R 2 C DM-MAC.
2 2 p x ( p(y x,x 2 X Y X 2 p x2 ( Fig.. Two user discrete memoryless multiple access channel m {,...,2 nr C u MC y ndec (ˆm, ˆm 2 m 2 {,...,2 nr 2 C 2 u 2 Fig. 2. Coding for the DM-MAC In the following two important inequalities are given for future reference. The data processing inequality for a system as the one shown in Fig. 2 is given by H(U,U 2 Y n H(U,U 2 (ˆm, ˆm 2. (3 Fano s inequality is given by H(U,U 2 (ˆm, ˆm 2 n(r +R 2 λ n +. (4 For equally-likely codewords
3 3 H(U,U 2 = n(r +R 2 = H(U,U 2 H(U,U 2 Y n +H(U,U 2 Y n I(U,U 2 ;Y n +n(r +R 2 λ n + (5 = H(Y n H(Y n U,U 2 +n(r +R 2 λ n + H(Y i H(Y i Y i,u,u 2 +n(r +R 2 λ n + = = H(Y i H(Y i U,U 2 +n(r +R 2 λ n + (6 (H(Y i H(Y i U,U 2 +n(r +R 2 λ n + I(U,U 2 ;Y i +n(r +R 2 λ n +, (7 where in (5 we made use of the Data Proceessing inequality, (3, and the Fano s inequality (see (4. Eq. (6 follows from the the memoryless property of the channel. Finally, from (7 we get R +R 2 n I(U,U 2 ;Y i +(R +R 2 λ n + n. (8 Further, the rate of user can be bounded as follows. nr = H(U = H(U U 2 = H(U U 2 H(U Y n,u 2 +H(U Y n,u 2 (9 = I(U ;Y n U 2 +H(U Y n,u 2. (0 In (9 we have used the fact that conditioning on an independent random variable does not affect the entropy. By the chain rule for entropy we get H(U,U 2 Y n = H(U 2 Y n +H(U U 2,Y n, and due to the non-negativity of the entropy H(U U 2,Y n H(U,U 2 Y n < n(r +R 2 λ n +
4 = = Therefore, nr < I(U ;Y n U 2 +n(r +R 2 λ n + = I(U ;Y i Y i,u 2 +n(r +R 2 λ n + ( ( H(Yi Y i,u 2 H(Y i U i,u 2i,Y i +n(r +R 2 λ n + (2 (H(Y i U 2 H(Y i U i,u 2i +n(r +R 2 λ n + (3 (H(Y i U 2i H(Y i U i,u 2i +n(r +R 2 λ n + (4 4 R n I(U i ;Y i U 2i +(R +R 2 λ n + n. (5 Similarly, we can prove that R 2 n I(U 2i ;Y i U i +(R +R 2 λ n + n. (6 Definition : For given p X (, p X2 (, define the vector I(p X (, p X2 ( = (I(X ;Y X 2,I(X 2 ;Y X,I(X,X 2 Y R 3 = (I(,I(2,I(3 Based on definition the set C I of rate pairs (R,R 2 can be defined as follows C I = {(R,R 2 R 0,R 2 0,R I(,R 2 I(2,R +R 2 I(3 (7 Further, it can be shown that I(+I(2 I(3 max{i(,i(2. Lemma : Let I,I 2 R 3 withc I, C I2 as in (7. Consider 0 λ and define I λ = λi +( λi 2. Then C I λ = λc I +( λc I2. Theorem 2: The convex hull of the union of the rate regions defined by the individual I vectors, which is C DMAC, is equal to the rate region defined by the convex hull of the I vectors C Co(I, i.e. C DMAC = C Co(I where C Co(I = i Co(I C i.
5 5 II. THE GAUSSIAN MAC CHANNEL We specify the Gaussian MAC Channel as Y = X +X 2 +Z (8 where the power constraints of user and user 2 are EX 2 P and EX 2 2, respectively. Further, we have for the additive noise, Z, that Z N(0,. For fixed input distributions, f X (, f X2 ( the rate region is given by R(f X (,f X2 ( = {(R,R 2 R I(X ;Y X 2,R 2 I(X 2 ;Y X,R +R 2 I(X,X 2 ;Y Then, the capacity region of the Gaussian MAC is given by the convex hull of the union of the rate regions, R(f X (,f X2 (, for every possible choice of the input distributions f X (, f X2 ( that satisfy the power constraints, i.e. We can compute C GMAC = Co f X ( : f X (x x 2 dx P f X2 ( : f X2 (x 2 x 2 2 dx 2 R(f X (,f X2 (. (9 I(X ;Y X 2 = h(y X 2 h(y X,X 2 = h(x +Z 2 log 2(2πe 2 log 2(2πe(P + 2 log 2(2πe = ( 2 log 2 + P, with equality when X N(0,P. Similarly, we can show that I(X 2 ;Y X 2 log 2( + with equality for X 2 N(0,. For the sum rate bound it holds I(X,X 2 ;Y = h(y h(y X,X 2 = h(y 2 log 22πe 2 log 2 ( + P +, (20 with equality when X N(0,P, X 2 N(0,. In (20 the differential entropy of Y, h(y, is bounded by h(y 2 log 22πeVar(Y 2 log 22πeEY 2 = 2 log 22πe ( P + +.
6 6 R log 2 2(+ P R +R 2 = 2 log 2(+ P + 2 log 2(+ R 2 Fig. 3. Capacity Region of a two user Gaussian MAC channel. Finally, the capacity region of the 2-user Gaussian MAC is given by C GMAC = { (R,R 2 R ( 2 log 2 + P, R 2 ( 2 log 2 + P 2, R +R 2 ( 2 log 2 + P +. We note that the capacity region of the 2-user Gaussian MAC is a pentagon and every point in the region can be achieved when transmitters and 2 use the Gaussian codebooks X N(0,P, X 2 N(0,, respectively. The optimal decoding strategy is called successive interference cancelation (SIC and achieves every point in the capacity region. Therefore, SIC achieves also the corner points of the capacity region. ( ( The corner point (R,R 2 = log P, log + 2 2( + is achievable when user is decoded ( first. In this case, the signal of user 2 is treated as interference and the rate R = log P is + achievable. Subsequently, since the message of user is decoded correctly, the receiver can subtract it from the received signal and user 2 can be decoded interference free, achieving the rate R 2 = log 2 2( + σ. ( 2 The other corner point (R,R 2 = log ( P ( σ, log P is achieved at the receiver by + SIC with the reverse decoding order. (2 A. TDMA case In the following, we compare the capacity region of the Gaussian MAC with the achievable rate region, when time division multiple access (TDMA is employed. For the TDMA, we split each channel use of
7 7 R GMAC TDMA log 2 2(+ P α = P P + 2 log 2(+ R 2 Fig. 4. Capacity Region of a two user Gaussian MAC channel vs. the Rate Region of TDMA. T u s into two parts of length αt u s and ( αt u s, respectively, where 0 α. During the first time slot, only the first user is scheduled to transmit with power of P /α and during the second time slot only the second user transmits with power /( α. This way, the users transmit in orthogonal resources and the individual average power constraints are satisfied. Therefore, the rate pair (R,R 2 = (α 2 log 2(+ P α,( α 2 log 2(+ ( α is achievable. In Fig. 4 it is shown that TDMA is strictly suboptimal compared to the GMAC-capacity-achieving strategy of successive interference cancellation. Note that TDMA achieves the sum capacity of GMAC for the choice α = P P +.
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