Comments on the Boundary of the Capacity Region of Multiaccess Fading Channels

1 Comments on the Boundary of the Capacity Re gion of Multiaccess Fading Channels Mohamed Shaqfeh and Norbert Goertz Institute for Digital Communications Joint Research Institute for Signal & Image Processing School of Engineering and Electronics The Uni v ersity of Edinb urgh Mayfield Rd., Edinb urgh EH9 3JL, Scotland, UK Email: { M.Shaqfeh, Norbert.Goertz } @ed.ac.uk Abstract A modification is proposed for the form ula known from the literature that characterizes the boundary of the capacity region of Gaussian multiaccess fading c hannels. The modified version takes into ac count potentially negativ e arguments of the cumulated d ensity fu nction that would affect the accur acy of the numerical ca pacity r esults. Index T erms ergodic capacity region, multiaccess fadin g chann el Nov ember 6, 2018 DRAFT 2 I . I N T R O D U C T I O N The boundary of the capacity region of mu ltiaccess (MA C) fading channels was first characterized in [1] and di scussed in full d etail in [2]. It i s assumed that t he fading processes of all users are independent of each other , are stationary and ha ve continuous probability density functions, f i ( h ) ∀ i , wit h h ≥ 0 the random fading coefficient and i the user index; a total of M users are assumed. The cumulated density function s o f the fading processes are denoted by F i ( h ) . = R h 0 f i ( h ′ ) dh ′ . Note t hat, according to t he standard fading channel m odel with coherent detection, t he support of the channel coef ficients does no t contain n egati ve numbers. The receiver noise is ass umed to be Gaussian wi th the variance σ 2 . I I . B O U N DA RY O F T H E C A P A C I T Y R E G I O N A N D M O D I FI C A T I O N O F T H E S T A N DA R D R E S U L T It was s hown in [2, Theorem 3.16 ] that the boundary of the capacity region of the Gaussian multiaccess channel i s the closure of t he parametricall y defined surfac e ( R ( µ µ µ ) : µ µ µ ∈ ℜ M + , X i µ i = 1 ) (1) where for each i = 1 , ..., M R i ( µ µ µ ) = ∞ Z 0 1 2( σ 2 + z ) ( ∞ Z 2 λ i ( σ 2 + z ) µ i f i ( h ) Y k 6 = i F k  2 λ k h ( σ 2 + z ) 2 λ i ( σ 2 + z ) + ( µ k − µ i ) h | {z } . = x  dh ) dz (2) The vector µ µ µ . = { 0 < µ i ≤ 1 : i = 1 , 2 , ..., M } i s a given “rate award” vector that is specified to pick a desired point on the b oundary of the capacity region. T he vector λ λ λ . = { λ i ∈ ℜ + : i = 1 , 2 , ..., M } is the sol ution o f the equations ∞ Z 0 ( ∞ Z 2 λ i ( σ 2 + z ) µ i 1 h f i ( h ) Y k 6 = i F k  2 λ k h ( σ 2 + z ) 2 λ i ( σ 2 + z ) + ( µ k − µ i ) h | {z } . = x  dh ) dz = ¯ P i for i = 1 , 2 , ..., M , (3) where ¯ P i is the long-term avera ge po wer constrain t of user i . The solutio n of (3) for the vector λ λ λ is unique, and an it erati ve numerical procedure is giv en in [2] to find it. As 0 < µ i ′ ≤ 1 ∀ i ′ , the differ ences µ k − µ i in (2) and (3) can hav e negativ e values and, hence, the arguments of the cumul ated density functions (CDFs) can, depending on the channel coef ficient h , also be negativ e. As the fading coeffic ients can not be negati ve, the CDF is actually Nov ember 6, 2018 DRAFT 3 not defined for such values as they lie out side the support of the random variable. Althoug h it seems natural to assum e the value “zero” in those cases, which might implicitly happen in a implementati on of (2) and (3), this would lead to incorrect results as we show below . T o compensate for this problem, we propose to introduce a modified ar gument in the c umul ated density functions F k ( x ) in the e xpression s i n (2) and (3) as follows: F k ( x ) replac e − − − → F k ([ x ] ∗ ) (4) with x . = 2 λ k h ( σ 2 + z ) 2 λ i ( σ 2 + z ) + ( µ k − µ i ) h (5) and [ x ] ∗ . =    x if x ≥ 0 + ∞ if x < 0 . (6) For negati ve arguments, x , the fun ction [ x ] ∗ takes on the value + ∞ which is inserted i nto a CDF in (4). Hence the value of the CDF for x < 0 i s “1” and not “0”. The j ustification is given in Section III. I I I . E X P L A N A T I O N There is no need to go through the whole deriv ation again to charac terize the capacity boundary surface. W e s tart at the poi nt where we propose a modification, i.e., equati on (18) on page 28 04 of [2]. W e wish t o compute th e rate R i ( µ µ µ ) = Z ∞ 0 1 2( σ 2 + z ) P ( i, z ) dz (7) with P ( i, z ) . = Pr  u i ( z ) > u j ( z ) ∀ j and u i ( z ) > 0  (8) where the mar ginal utiliti es (“rate rev enue minus p owe r cost” [2, p. 2802]) are defined by u i ( z ) . = µ i 2 ( σ 2 + z ) − λ i h i , z ≥ 0 . (9) T o so lve (7) (and also the corresponding problem i n [2, equation (18)] for the vec tor λ λ λ to fulfil the a verage po wer constraint for the user i ) we need to e valuate the probabili ty (8). Nov ember 6, 2018 DRAFT 4 Firstly , it should be noted that the condition u i ( z ) > u j ( z ) ∀ j in (8) (implicitly ) excludes the case j = i because otherwise P ( i, z ) would be “zero” as, trivially , P ( u i ( z ) > u i ( z )) = 0 . Using (9) we can s tate the equivalence u i ( z ) > 0 ⇐ ⇒ h i > 2 λ i ( σ 2 + z ) µ i > 0 . (10) Note that λ i > 0 ∀ i , as λ is a Lagrange mul tiplier th at introduces t he “power p rice” (that can nev er be ne gative) into the optim isation problem that must be s olved to find the capacity region [2]. Using (10), t he probability (8) can now be writt en as P ( i, z ) = Pr  u i ( z ) > 0   u i ( z ) > u j ( z ) ∀ j  · Pr  u i ( z ) > u j ( z ) ∀ j  (11) = Pr  h i > 2 λ i ( σ 2 + z ) µ i   u i ( z ) > u j ( z ) ∀ j  · Pr  u i ( z ) > u j ( z ) ∀ j  (12) = ∞ Z 2 λ i ( σ 2 + z ) µ i f i  h   u i ( z ) > u j ( z ) ∀ j  dh · Pr  u i ( z ) > u j ( z ) ∀ j  (13) = ∞ Z 2 λ i ( σ 2 + z ) µ i f i  h, u i ( z ) > u j ( z ) ∀ j  dh (14) = ∞ Z 2 λ i ( σ 2 + z ) µ i f i ( h ) · Pr  u i ( z ) > u j ( z ) ∀ j   h i = h  dh (15) Since the fading processes of the users are ass umed to be independent, we can write: P ( i, z ) = Z ∞ 2 λ i ( σ 2 + z ) µ i f i ( h ) · Y k 6 = i Pr ( u i ( z ) > u k ( z ) | h i = h ) dh . (16) Now , we need to ev aluate th e probability Pr ( u i ( z ) > u k ( z ) | h i = h ) (17) W e use (9) to rewrite the event u i ( z ) > u k ( z ) and obtain u i ( z ) > u k ( z ) ⇐ ⇒ µ i a − λ i h i > µ k a − λ k h k (18) or , equi valently , h i ( µ k − µ i ) + λ i a aλ k h i < 1 h k (19) Nov ember 6, 2018 DRAFT 5 with the abb re viation a . = 2 ( σ 2 + z ) > 0 and λ i > 0 ∀ i and 0 < µ i ≤ 1 ∀ i . As µ k − µ i can be negati ve, the left-hand side of (19) can be negati ve so we hav e to differe ntiate between two cases: Case A : h i ( µ k − µ i ) + λ i a > 0 ⇐ ⇒  µ k ≥ µ i  or µ k < µ i and h i < λ i a µ i − µ k ! (20) Case B : h i ( µ k − µ i ) + λ i a < 0 ⇐ ⇒ µ k < µ i and h i > λ i a µ i − µ k (21) a) Case A: W ith a = 2( σ 2 + z ) we obt ain from (17 ), (19) and (20) Pr ( u i ( z ) > u k ( z ) | h i = h ) = Pr  h k < 2 λ k h i ( σ 2 + z ) 2 λ i ( σ 2 + z ) + ( µ k − µ i ) h i    h i = h  (22) = Pr  h k < 2 λ k h ( σ 2 + z ) 2 λ i ( σ 2 + z ) + ( µ k − µ i ) h  (23) = F k  2 λ k h ( σ 2 + z ) 2 λ i ( σ 2 + z ) + ( µ k − µ i ) h  (24) with F k ( x ) = R x 0 f k ( h ) dh the cumulated density function of the channel coef ficient k . The solution (24) is t he one orig inally used i n equations (2) and (3) that are taken from [2]. b) Case B: For a n egati ve left-hand si de in (19) we obtain Pr ( u i ( z ) > u k ( z ) | h i = h ) = Pr ( h k > B ) = 1 (25) with B . = 2 λ k h ( σ 2 + z ) 2 λ i ( σ 2 + z ) + ( µ k − µ i ) h < 0 . (26) As h k is a channel coefficient and non-negati ve by definition, t he probabilit y (25) i s simp ly “one”. c) New formulation of the boundary of the capacity re gion: In order to keep th e st ructure of the original solution given in [2] but with the correct ev aluation of the probabilit y in bot h cases A and B, we write the probability Pr ( u i ( z ) > u k ( z ) | h i = h ) = F k " 2 λ k h ( σ 2 + z ) 2 λ i ( σ 2 + z ) + ( µ k − µ i ) h # ∗ ! (27) with the function [ x ] ∗ defined in (6). Wh en we use (27) in (16) and (7) we obtain the corrected solution proposed in Section II. Nov ember 6, 2018 DRAFT 6 I V . A C K N OW L E D G E M E N T S The work reported in this paper has formed part of th e Core 4 Research Program of the V irtual Centre of Excellence in Mobile and Personal Commun ications, Mo bile VCE, www .mobilevce.com, whose funding support, i ncluding t hat of EPSRC , is gratefully acknowl- edged. Full y detailed t echnical reports on this research are av ailable to Industrial Members of Mobile VCE. The authors would also like to thank for the s upport from the Scotti sh Funding Council for th e Join t Research Institute with the Heriot-W att Unive rsity wh ich is a part of the Edinbur gh Research P artnership. R E F E R E N C E S [1] D. Tse and S. Hanly , “Capacity region of the multi -access fading chann el under dynamic resource allocation and polymatroid optimization, ” in Proceed ings IEEE International Sympo sium on Information Theory (ISIT) , 1996, p. 37. [2] D. Tse and S . Hanly , “Multiaccess fading channels – Part I: Polymatroid structure, optimal resource allocation and throughpu t capacities, ” IEEE T ransactions on Information Theory , vo l. 44, no. 7, pp. 2796 –2815, Nov . 1998. Nov ember 6, 2018 DRAFT