Refinement of the outer bound of capacity region in Gaussian multiple access channel with feedback

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1 International Symposium on Information Theory and its Applications, ISITA2004 Parma, Italy, October 0 3, 2004 Refinement of the outer bound of capacity region in Gaussian multiple access channel with feedback Kenjiro YANAGI and Takashi HIRAYAMA Department of Applied Science, Faculty of Engineering, Yamaguchi University, Ube, , JAPAN yanagi@yamaguchi-uacjp Graduate School of Science and Engineering, Yamaguchi University, Ube, , JAPAN (Current) Hitachi Government and Public Corporation System Engineering, Ltd hirayama-t@gphitachicojp Abstract (S j (X j ),S 2j (X j,y ),,S nj (X j,y n )), The capacity region of Gaussian multiple access channel without feedback was already exactly obtained But it is not yet obtained in the feedback case We give the α outer bound of capacity region in Gaussian multiple access channel with noiseless and no-delayed feedback in general which is an extension of the result given by Ordentlich INTRODUCTION The m user Gaussian multiple access channel with feedback is defined by m Y i = S ij + Z i, i =, 2, j= For j m, S j = {S ij ; i =, 2, } is a stochastic process representing input signal transmitted by the sender j, where S ij is input signal at the time i sent by the sender j, Y = {Y i ; i, 2, } is a stochastic process representing output signal, and Z = {Z i ; i, 2, } is a non-degenerate, zeromean Gaussian process representing noise, respectively Though S,S 2,,S m,z are mutually independent in the non-feedback case, we have a different aspect in the feedback case Let X,X 2,,X m denote the input messages, where each X j is uniformly distributed in (, 2,, 2 n ) and is independent of the other messages Since we have feedback, the input signal S ij of sender j at time i is a function of the message X j and the past values of the putput signal Y i = (Y,Y 2,,Y i ) For block length n we must specify a ((2 nr, 2 nr 2,, 2 nr m ),n) code with codewords Sij n (X j,y n )= X j {, 2,, 2 nrj },j =, 2,,m In addition, we require that the codewords satisfy the expected power constraints E[ n n Sij(X 2 j,y i )] P j,j =, 2,,m, i= where the expectation is taken over all possible noise sequences The capacity of the non-white Gaussian channel without feedback was first characterized by Keilers [4] In the m user case the capacity region for n uses of the channel is the set of all rate vectors (R,R 2,,R m ) satisfying log K(n) S j + K (n), for every subset Γ of the senders {, 2,,m}, for some n n covariance matrices K (n) S j of the vectors S j = (S j,s 2j,,S nj ), satisfying the power constraint n Tr[K(n) S j ] P j, j m Keilers [4] used a sequential water filling procedure to obtain the extreme points of the convex hull of the capacity region On the other hand, the capacity region of m-user non-white Gaussian multiple access channel with feedback was given by Pombra and Cover [6] After then Ordentlich [5] gave the more refined version in the following log i S i j Γ S j+, ()

2 where, Γ are subsets of {, 2,,m} such that Γ and ={, 2,,m} The power constraint is given by n Tr[K(n) S j ] P j, j m (2) We denote the feedback capacity region by C n,f B (P,P 2,,P m ) Also we denote the nonfeedback capacity region by C n (P,P 2,,P m ) The total capacity C n,f B (P,P 2,P m ) is defined as m the maximum of j= under the power constraint (2) For the convenient use we introduce the notation C n,f B (P,P 2,,P Γ ) which is Γ-total capacity defined by the maximum of, where Γ {, 2,,m} 2 ORDENTLICH S RESULT Pombra and Cover [6] obtained the result about total capacity Proposition C n (P,P 2,,P m ) C n,f B (P,P 2,,P m ) 2 C n (P,P 2,,P m ) On the other hand, Ordentlich [5] obtained the following result Proposition 2 Proof of Proposition 2 C n (P,P 2,,P m ) C n,f B (P,P 2,,P m ) 2C n (P,P 2,,P m ) We put R( ) = and S( ) = S j For Γ, we give the expectation of R( ) and then we obtain R( ) 2 2 R(Γ) Since 2 Γ Γ Γ K (n) S( ) S( Γ)+Z = K (n) S i + K (n) Z, i Γ we give the expectation of () for Γ and then we obtain the right hand side in the following Then 2 Γ Γ log (n) K S( ) S( log Γ)+ 2 Γ Γ K(n) S( ) S( Γ)+ log K(n) S j + K (n) C n,f B (P,P 2,,P m ) 2C n (P,P 2,,P m ) qed 3 UPPER BOUND OF CAPACITY IN GAUSSIAN CHANNEL WITH FEEDBACK It is well known that tha capacity of Gaussian channel without feedback is given by the foolowing Proposition 3 C n (P ) k i= log np + r + r r k kr i, where 0 <r r 2 r n are eigenvalues of K (n) Z, and k( n) is tha maximal integer satisfying np + r + r r k >kr k On the other hand, since the capacity C n,f B (P )of Gaussian channel with feedback is not given in the exact form, it is useful to obtain the smaller outer bound in possible In particular Cover and Pombra [2] gave the following remarkable result Proposition 4 C n (P ) C n,f B (P ) 2C n (P ), C n (P ) C n,f B (P ) C n (P )+ log 2 2 After then Chen and Yanagi [] gave the more refined following upper bound Proposition 5 The following (), (2) holds () For any α>0 and for any P > 0 we have the followings; C n (P ) C n,f B (P ) ( + α )C n(αp )

3 (2) We put G(P, α) = ( + α )C n(αp ) Lemma For any α>0 and any P>0 C n,f B (P, P,,P) Then the following (a), (b) hold (a) There exists P > 0 such that ( + α ) C n (αp, αp,,αp) G(P, ) = min α>0 G(P,α) (b) For any P P there exists α>0 such that G(P, α) <G(P, ) Proposition 6 The following (), (2) hold () For any α>0 and for any P > 0 we have the followings; (2) We put C n (P ) C n,f B (P ) C n (αp )+ 2 log( + α ) F (P, α) =C n (αp )+ 2 log( + α ) Then the following (a), (b) hold (a) There exists P > 0 such that F (P, ) = min α>0 F (P,α) (b) For any P P, there exists α>0 such that F (P, α) <F(P, ) 4 α-outer BOUND OF CAPACITY RE- GION We obtain the α-outer bound which is the extension of Proposition 2 given by Ordentlich [5] -total capacity is given by the following C n,f B (P,P 2,,P ) = max R( ), where Γ satisfies R( ) Also the folowings are satisfied (n) K S( ) S( log Γ)+ n Tr[K(n) S j ] P j, j m Lemma 2 For the non-white additive noise multi-access channel with feedback, a rate vector (R,R 2,,R m ) is achievable only if there exists a feedback code such that: K log Sj L j Γ Sj+Z for all subsets, Γ( Γ {, 2,,m}) and all lower triangular matrices L Proof of Lemma In () we put = Γ Then we have log K S j+ ( + K ) log Sj+Z α/(+α) α α/(+α) ( + K ) log S j+α/(+α) /(+α) α We put K =K S j+z and K2 =K α S j Z Since K α S j, ( + α ) log K α/(+α) K2 /(+α) ( + α ) log α +α K+ +α K2 ( + αk ) log Z j α (3) If the same power is assumed for any j Γ, then (3) is rewritten as follows ( + α ) log Γ αk X + Γ ( Γ )αk XX 2 + K,

4 We put D = Γ αk X + K Z Then we have Γ αk X + Γ ( Γ )αk X X 2 + K = D + Γ ( Γ )αk X X 2 = D /2 (I + Γ ( Γ )αd /2 K X X 2 D /2 )D /2 = D I + Γ ( Γ )αd /2 K X X 2 D /2 We put DKD = D /2 K X X 2 D /2 Thus ( + α ) log Γ αk X + K + ( + ) log I + Γ ( Γ )αdkd α ( + α ) log Γ αk X + K + 2 ( + α ) log{ tr[i + Γ ( Γ )αdkd]} n ( + α ) log Γ αk X + K + 2 ( + α ) log{+ Γ ( Γ )α tr[dkd]} n We put P Γ P j Then we have C n,f B ( P,, P ) ( + α ) C n (α P,,α P )+T, (4) where T 2 ( + α ) log{+ Γ ( Γ )α tr[dkd]} n In order to consider the second inequality, we take α> 0 and Γ such that = Γ α +α Then Γ = Γ +α The number of these sets have to be chosen to be integers Then α has to satisfy the following α i = i Γ i, 0 i Γ By Lemma 2 we put L = αi Then we have K log S j j Γ αs j+ log Γ αk X + K Z Γ αk XX 2 And we put D = Γ αk X + K Z Thus log Γ αk X + K + log I Γ αd /2 K X X 2 D /2 log Γ αk X + K + 2 log{ n tr[i Γ αd /2 K X X 2 D /2 ]} log Γ αk X + K + Γ α log{ 2 n tr[d /2 K X X 2 D /2 ]} We set the right side by B Then B We take the addition about all subsets satisfying the size = Γ α +α We denote the number of those subsets by N Then we have N N B We put the left hand side by N Γ Then We remark that ( Γ N Γ = Then Hence we have N N Γ B ) ( Γ, N = N Γ N =+ α ) ( + α ) log Γ αk X + K + 2 ( + Γ α ) log{i α n tr[d /2 K X X 2 D /2 ]} ( + α ) log Γ αk X + K + T 2, (5) where T 2 2 ( + Γ α ) log{ α n tr[dkd]}

5 By (4), (5), C n,f B ( P,, P ) (+ α ) C n (α P,,α P )+min{t,t 2 } Since sgnt = sgnt 2, min{t,t 2 } 0 Thus C n,f B ( P,, P ) ( + α ) C n (α P,,α P ) Now we have the following theorem Theorem For any and any α>0 C n,f B (P,P 2,,P ) ( + α ) C n (αp,αp 2,,αP ) Proof of Theorem By Lemma we have for any P>0 qed C n,f B (P,P,,P) ( + α ) C n (αp,αp,,αp) Since C n,f B (P,P,,P ) is symmetric function of P,P 2,,P and C n,f B (,, ) is concave function, we obtain the following C n,f B (P,P 2,,P ) = C n,f B (P,P 2,,P ) where P j,, C n,f B ( = C n,f B ( P, P,, P ) ( + α ) C n (α P,,α P ) = (+ α ) C n (αp,,αp ), Theorem 2 We put P P j P j ) Ḡ (P, α) = ( + α ) C n (αp,,αp) Then we have the following (a), (b) (a) There exists P > 0 such that Ḡ (P, ) = min Ḡ (P,α) α>0 qed (b) For any P P, there exists α>0 such that Ḡ (P, α) < Ḡ (P, ) Proof Since we can prove it by the same way as Proposition 5, we omit the proof qed Since Theorem 2 holds for any subset {, 2,,m}, α-outer bound can be constructed in the following If P P j, then we choose α And if P P j = P, then we choose α such that Ḡ ( P,α) < Ḡ ( P,) Then we can give the α-outer region which is a strict extension of Ordentrich s outer region Remark The proof of Theorem is given in the case of i α = α i = i, 0 i In order to complete the proof, we have to show the result in the case including all positive rational numbers We consider any rational number α = q p We can choose two integers K and L, with L between 0 and K, such that α = L K L Indeed, take L = p and K = p + q Furthermore we may assume that K> If this is not the case, p and q users can be scaled by the same constant Now consider a set of K users, composed of the original users and K virtual users The virtual users have power constraints P j, where j ranges from + to K We now apply the user bound on the maximum sum rate for these K users and the parameter α, and we obtain C n,f B ( P, P 2,, P K ) (+ α ) C n (α P,α P 2,,α P K ) Now taking the limit as P j 0, for all + j K, we obtain the corresponding upper bound for the users References [] HWChen and KYanagi, Refinements of the halfbit and factor-of-two bounds for capacity in Gaussian channels with feedback, IEEE Trans IT, 45(999),

6 [2] TMCover and SPombra, Gaussian feedback capacity, IEEE Trans IT, 35(989), [3] TMCover and JAThomas, Elements of Information Theory, John Wiley and Sons, Inc, 99 [4] CWKeilers, The capacity of the spectral Gaussian multiple-access channel, PhD, Thesis, Dept Elect Eng, Stanford, CA, May 976 [5] EOrdentlich, On the factor of two bound for Gaussian multiple access channels with feedback, IEEE Trans IT, 42(996), [6] SPombra and TMCover, Non white Gaussian multiple access channels with feedback, IEEE Trans IT, 40(994), [7] JAThomas, Feedback can at most double Gaussian multiple access channel capacity, IEEE Trans IT, 33(987), 7-76 [8] KYanagi, Outer bound of capacity region in m- user non-white Gaussian multiple access chahnnelwith feedback, Proc Sym Applied Functional Analysis, 995, [9] KYanagi and TWani-ishi, On the outer bound of capacity region in Gaussian multiple access channel with feedback, Proc SITA, 995, [0] KYanagi and THirayama, On the α outer bound of capacity region in Gaussian multiple access channel with feedback, Proc SITA, 2003, [] KYanagi, JWYu and I-FChao, On some inequalities for capacity of mixed Gaussian channels with feedback, Arch Inequalities Appl, 2(2004), 3-24