Lecture 1: The Multiple Access Channel. Copyright G. Caire 12

Transcription

1 Lecture 1: The Multiple Access Channel Copyright G. Caire 12

2 Outline Two-user MAC. The Gaussian case. The K-user case. Polymatroid structure and resource allocation problems. Copyright G. Caire 13

3 Two-user MAC W 1 Tx1 X 1 P Y { W c 1, W c 2 } Y X1,X 2 Rx W 2 Tx2 X 2 Copyright G. Caire 14

4 The networking approach to the MAC The networking approach to the MAC is to avoid multiuser interference in the presence of users random access. Examples: Aloha, CSMA, , Packet-Reservation Multiple-Access, TDMA/FDMA. Another line of thought focused on signal design in order to make interference similar to noise. Examples: CDMA, UWB. Information theory shows that neither of the above approaches is generally optimal (although it can be optimal or near-optimal in some cases). Copyright G. Caire 15

5 Definitions Let {X 1 X 2,P Y X1,X 2, Y} denote a memoryless stationary MAC. A (2 nr 1, 2 nr 2,n) MAC code is defined by the two message sets M 1 = {1,...,2 nr 1 }, M 2 = {1,...,2 nr 2 } Two encoding functions f 1 : M 1! X n 1, f 2 : M 2! X n 2 such that x 1 (m 1 )=f 1 (m 1 ) and x 2 (m 2 )=f 2 (m 2 ). A decoding function g : Y n! M 1 M 2 Copyright G. Caire 16

6 Capacity region The average probability of error is defined as P e (f 1,f 2,g)=P ( M c 1, M c 2 ) 6= (M 1,M 2 ), where M 1,M 2 are independent and uniformly distributed over M 1 and M 2, respectively, and where X n 1 = x 1 (M 1 ),X n 2 = x 2 (M 2 ), ( c M 1, c M 2 )=g(y n ). The MAC capacity region C is the closure of the set of all pairs (R 1,R 2 ) for which there exists a sequence of codes (f (n) 1,f (n) 2,g (n) ) with rates (R 1,R 2 ) and P e (f (n) 1,f (n) 2,g (n) )! 0 as n!1. Copyright G. Caire 17

7 Simple inner and outer bounds Time-sharing inner bound (TDMA): let C 1 = max I(X 1 ; Y X 2 = x 2 ), C 2 = max I(X 2 ; Y X 1 = x 1 ) P X1,x 2 2X 2 P X2,x 1 2X 1 then, the following region is achievable R 1 C 1 + R 2 C 2 apple 1 Sum-rate outer bound: R 1 + R 2 apple where (X 1,X 2 ) P X1 (x 1 )P X2 (x 2 ). max I(X 1,X 2 ; Y ) P X1,P X2 Copyright G. Caire 18

8 How the capacity region looks like? d on the capacity region of any DM-MAC R 2 C 12 Time-division inner bound C 2 Outer bound? R 1 C 1 Notice: from a time-sharing argument we have that C must be a convex region. Copyright G. Caire 19

9 Capacity region of the 2-user MAC Theorem 1. MAC Capacity region: The MAC capacity region is the convex closure of the rates satisfying R 1 apple I(X 1 ; Y X 2,Q), R 2 apple I(X 2 ; Y X 1,Q), R 1 + R 2 apple I(X 1,X 2 ; Y Q) for some P Q P X1 QP X2 Q. } apple apple R 2 I(X 2 ; Y X 1 ) I(X 2 ; Y ) I(X 1 ; Y ) I(X 1 ; Y X 2 ) R 1 Copyright G. Caire 20

10 Proof: Achievability Fix P Q, P X1 Q and P X2 Q. We shall show that any point in the interior of R(X 1,X 2,Q) (pentagon) is achievable. Codebook generation: generate a typical q with i.i.d. components P Q, and independently two codebooks {x 1 (m 1 ):m 1 2 [1 : 2 nr 1]} and {x 2 (m 2 ): m 2 2 [1 : 2 nr 2]} with independent entries P X1 Q( q i ) and P X2 Q( q i ), respectively. Encoding: to send message m 1, encoder 1 transmits x 1 (m 1 ). encoder 2 sends x 2 (m 2 ). Similarly, Decoding: Declare bm 1, bm 2 if this is the unique index pair such that (x 1 ( bm 1 ), x 2 ( bm 2 ), y) 2 T (n) (X 1,X 2,Y q), otherwise declare error. Copyright G. Caire 21

11 By symmetry we have P e = P(E M 1 =1,M 2 = 1) where E = {g(y n ) 6= (1, 1)}. When analyzing the random coding ensemble average error probability, (Q n,x n 1 (m 1 ),X n 2 (m 2 ),Y n ) are random vectors. We consider the possible joint conditional probability distributions of (X n 1 (m 1 ),X n 2 (m 2 ),Y n ) given Q n : m 1 m 2 joint pmf 1 1 P X1 QP X2 QP Y X1,X 2 * 1 P X1 QP X2 QP Y X2,Q 1 * P X1 QP X2 QP Y X1,Q * * P X1 QP X2 QP Y Q Copyright G. Caire 22

12 The error event E is contained in the union of the following events: E1 c = {(Q n,x1 n (1),X2 n (1),Y n ) /2 T (n) (Q, X 1,X 2,Y)} E 2 = {(Q n,x n 1 (m 1 ),X n 2 (1),Y n ) 2 T (n) (Q, X 1,X 2,Y) for some m 1 6=1} E 3 = {(Q n,x n 1 (1),X n 2 (m 2 ),Y n ) 2 T (n) (Q, X 1,X 2,Y) for some m 2 6=1} E 4 = {(Q n,x1 n (m 1 ),X2 n (m 2 ),Y n ) 2 T (n) (Q, X 1,X 2,Y) for some m 1 6=1,m 2 6=1} Copyright G. Caire 23

13 We bound each term: 1. By LLN P(E1 M c 1 =1,M 2 = 1)! P(E 2 M 1 =1,M 2 = 1) apple 2 n(i(x 1;Y X 2,Q) R 1 ( )). 3. P(E 3 M 1 =1,M 2 = 1) apple 2 n(i(x 2;Y X 1,Q) R 2 ( )). 4. P(E 4 M 1 =1,M 2 = 1) apple 2 n(i(x 1,X 2 ;Y Q) R 1 R 2 ( )). The result follows from the union bound. Copyright G. Caire 24

14 Proof: Converse Assume that there is a sequence of (R 1,R 2,n)-codes such that P e (n) n!1.! 0 as The n-letter multivariate joint distribution is given by ny (M 1,M 2,X1 n,x2 n,y n ) 2 n(r 1+R 2 ) P Y X1,X 2 (y i x 1i,x 2i ) 1{X n 1 = x 1 (m 1 )} 1{X n 2 = x 2 (m 2 )} Copyright G. Caire 25

15 Sum rate: from Fano inequality we have: H(M 1,M 2 Y n ) apple 1+n(R 1 + R 2 )P (n) e apple n n H(M 1 Y n,m 2 ) apple H(M 1,M 2 Y n ) apple n n H(M 2 Y n,m 1 ) apple H(M 1,M 2 Y n ) apple n n Therefore: n(r 1 + R 2 ) = H(M 1,M 2 ) H(M 1,M 2 Y n )+H(M 1,M 2 Y n ) apple I(M 1,M 2 ; Y n )+n n nr 1 = H(M 1 M 2 ) H(M 1 Y n,m 2 )+H(M 1 Y n,m 2 ) apple I(M 1 ; Y n M 2 )+n n nr 2 = H(M 2 M 1 ) H(M 2 Y n,m 1 )+H(M 2 Y n,m 1 ) apple I(M 2 ; Y n M 1 )+n n Copyright G. Caire 26

16 Sum rate inequality n(r 1 + R 2 ) apple I(M 1,M 2 ; Y n )+n n nx = I(M 1,M 2 ; Y i Y i 1 )+n n = = = nx I(M 1,M 2,X 1i,X 2i ; Y i Y i 1 )+n n nx I(X 1i,X 2i ; Y i Y i 1 )+I(M 1,M 2 ; Y i Y i 1,X 1i,X 2i )+n n nx I(X 1i,X 2i ; Y i )+n n Copyright G. Caire 27

17 Individual rate inequalities nr 1 apple I(M 1 ; Y n M 2 )+n n nx = I(M 1 ; Y i Y i 1,M 2 )+n n = apple = = = nx I(M 1 ; Y i Y i nx I(M 1,M 2,Y i nx I(X 1i,M 1,M 2,Y i nx 1,M 2,X 2i )+n n 1 ; Y i X 2i )+n n 1 ; Y i X 2i )+n n I(X 1i ; Y i X 2i )+I(M 1,M 2,Y i 1 ; Y i X 1i,X 2i ) nx I(X 1i ; Y i X 2i )+n n + n n Copyright G. Caire 28

18 Multiletter rate inequalities We arrive at R 1 apple 1 n R 2 apple 1 n R 1 + R 2 apple 1 n nx I(X 1i ; Y i X 2i )+ n nx I(X 2i ; Y i X 1i )+ n nx I(X 1i,X 2i ; Y i )+ n Notice that these mutual informations depend on the marginal distributions at time i induced by the joint distribution induced by the specific sequence of codes that we have supposed to exist. Copyright G. Caire 29

19 From multiletter to single-letter Time-sharing argument: define a new random variance Q Uniform[1 : n] independent of (X n 1,X n 2,Y n ). We can write nx R 1 apple 1 n I(X 1i ; Y i X 2i )+ n = I(X 1Q ; Y Q X 2Q,Q)+ n Now we identify X 1 = X 1Q,X 2 = X 2Q,Y = Y Q and, letting n!1, we obtain R 1 apple I(X 1 ; Y X 2,Q) R 2 apple I(X 2 ; Y X 1,Q) R 1 + R 2 apple I(X 1,X 2 ; Y Q) for some (Q, X 1,X 2 ) P Q P X1 QP X2 Q. Copyright G. Caire 30

20 The Gaussian MAC The Gaussian MAC is described by Y = g 1 X 1 + g 2 X 2 + Z Real case: X 1, X 2, Y = R, g 1,g 2 2 R, Z N (0,N 0 /2). Complex circularly symmetric case: X 1, X 2, Y = C, g 1,g 2 2 C, Z CN(0,N 0 ). The input constraints are given by 1 n nx x k,i (m k ) 2 apple E sk, 8 m k 2 [1 : 2 nr k], k =1, 2 Copyright G. Caire 31

21 For the real case, we define S 1 =2g1E 2 s1 /N 0 and S 2 =2g2E 2 s2 /N 0, and the function C(S) = 1 2 log(1 + S). For the complex case, we define S 1 = g 1 2 E s1 /N 0 and S 2 = g 2 2 E s2 /N 0, and the function C(S) = log(1 + S). Choosing X 1,X 2 Gaussian and independent we find the region R 1 apple C(S 1 ) R 2 apple C(S 2 ) R 1 + R 2 apple C(S 1 + S 2 ) It is easy to see that this is in fact the capacity region (no convex hull needed since it is achieved by a single input distribution). Copyright G. Caire 32

22 The Wyner-Cover pentagon apple R 2 C(S 2 ) C(S 2 /(1 + S 1 )) C(S 1 /(1 + S 2 )) C(S 1 ) R 1 Copyright G. Caire 33

23 Comparison with multiaccess techniques R 2 High SNR R 2 Low SNR C(S 2 ) TDMA C(S 2 ) S2 C S 1 +1 S2 C S 1 +1 S1 C S 2 +1 C(S 1 ) Treating other codeword as noise R 1 S1 C S 2 +1 C(S 1 ) R 1 Copyright G. Caire 34

24 Generalization to K users Theorem 2. K-user MAC capacity region. The capacity region of the K-user MAC with common message is given by the convex closure of the set of rates satisfying X R k apple I(X K ; Y X K c,q) k2k for all K [1 : K] (we use the notation X K = {X k : k 2 K}), for some P Q Q K k=1 P X k Q. Copyright G. Caire 35

25 K-user Gaussian MAC The Gaussian MAC with K users is described by Y = KX g k X k + Z k=1 with 1 P n n x k,i(m k ) 2 apple E sk, 8 k =1,...,K and all messages m k. Letting S k =2gk 2E sk/n 0 (real case), or S k = g k 2 E sk /N 0 (complex case), we have: ( X C(S 1,...,S K )= R k apple C X! ) S k 8 K {1,...,K} k2k k2k Copyright G. Caire 36

26 Resource allocation on the Gaussian MAC In resource allocation problems we care about the transmit power (not receive SNR): let k = S k / g k 2, such that X R k apple C X! g k 2 k k2k k2k We are interested in solving problems of the following form: maximize subject to KX w k R k, k=1 R 2 C KX k=1 k apple Copyright G. Caire 37

27 Polymatroid structure of C Definition 1. Sub-modular rank function. Let K =[1: K] and let f :2 [1:K]! R + denote a set function with the following properties: 1. f(;) =0(normalized). 2. f(k) f(k 0 ) if K K 0 (non-decreasing). 3. f(k)+f(k 0 ) f(k [ K 0 )+f(k \ K 0 ) (submodular). Definition 2. Polymatroid. The polyhedron defined by the inequalities x 2 R K +, and X x k apple f(k), 8 K [1 : K] k2k where f :2 [1:K]! R + is a submodular rank function, is called a polymatroid. Copyright G. Caire 38

28 Copyright G. Caire 39

29 Polymatroids and linear programming Suppose that we are interested in the problem of maximizing the weighted rate-sum: KX maximize w k R k, subject to R 2 C k=1 This is a linear program with K! relevant constraints, corresponding to all possible decoding orders. Using the fact that C is a polymatroid, we immediately have the optimal solution: Let denote the permutation of {1,...,K} that sorts the weights in increasing order: w 1 apple w 2 apple apple w K Copyright G. Caire 40

30 Then, the weighted rate-sum is maximized by the vertex R, obtained by decoding in the order, i.e., 1! 2!! K. Namely, we have R sum (w) = =! KX g k 2 w k C k 1+ P K j=k+1 g j 2 j 0 1 KX KX (w k w k 1 g j 2 A j k=1 k=1 j=k where w 0 = w 0 =0. Notice that the solution is a sum of concave functions of 1,..., K, which can be further maximized subject to the constraint P K k=1 k apple, if required. Copyright G. Caire 41

31 End of Lecture 1 Copyright G. Caire 42