Optimal Power Control in Decentralized Gaussian Multiple Access Channels

1089-7798 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCOMM.2018.2856240, IEEE Communications Letters 1 Optimal Po wer Control in Decentralized Gaussian Multiple Access Channels Kamal Singh Abstract —W e consider the decentralized power optimization problem for Gaussian fast-fading multiple access channel (MA C) so that the av erage sum-throughput is maximized. In our MA C setup, each transmitter has access to only its own fading coeffi- cient or channel state information (CSI) while the r eceiver has full CSI available at all instants. Unlike centralized MA C (full CSIT MA C) where the optimal powers are known explicitly , the ana- lytical solution for optimal decentralized powers does not seem feasible. In this letter , we specialize alter nating-maximization (AM) method for numerically computing the optimal powers and ergodic capacity of the decentralized MA C for general fading statistics and average power constraints. F or illustration, we apply our AM method to compute the capacity of MA C channels with fading distrib utions such as Rayleigh, Rician etc. I . I N T RO D U C T I O N The multiple access channel (MAC) is a commonly used model to represent communication scenario where multiple senders communicate to a common receiv er, such as the uplink channel of a mobile cellular network. The a vailability of the CSI at the transmitters and receiv er has a significant impact on the achiev able throughput rates of the fading MAC channels. Under full CSI at the receiv er and partial CSI at the trans- mitters, the er godic capacity region of a MA C with additiv e white Gaussian noise (A WGN) and fast fading is completely characterized by the optimal power control schemes [1]. An intuitiv e justification for this property is that in a fast fading scenario, each code word experiences all possible fading real- izations and thus, any rate close to ergodic capacity can be achiev ed by choosing to transmit all code words with the same rate and optimal power strate gies [2]. More precisely , Gaussian codebooks with optimal power control achiev es the ergodic capacity region, see Figure 1. Depending upon the availability of channel state information (CSI) at the transmitters and receiv er, the optimal power control strategy varies. In this letter , we consider a fading MAC where each transmitter knows only its o wn fading coef ficients and the receiver has full CSI. Further, we assume independent fading statistics, with average power constraints not necessary identical, across users. Also, we assume a fast fading model where the channel varies IID (independent and identically distributed) in time. The po wer and er godic capacity problem of the decentral- ized fading Gaussian MA C is a long-standing open problem posed by Shamai and T elatar in [4] suggesting that analyt- ical solution is not feasible. As an alternative, near closed- form lower bounds on ergodic capacity are derived for the identical user 1 MA C channel using a simple heuristic ON- Kamal Singh is with the Department of Electrical Engineering at IIT Bombay , Mumbai, INDIA-400076 (e-mail: {kamalsingh}@ee.iitb .ac.in). 1 Fading distributions and power constraints are identical across users. OFF power scheme which further improve with number of MA C users [4]. In [5], structural properties of the optimals are deriv ed to design suitable power schemes to further raise these near closed-form lower bounds. More recently in [6], tight numerical bounds (upper and lo wer both) to er godic capacity are obtained for the decentralized MAC for identical users setting. This letter is a first attempt to solve numerically the optimal powers and ergodic capacity problem of the decentralized fading MAC for general fading statistics and a verage power constraints. Thus, in our study , the identical users MA C is a special case. Our main contribution is a simple alternate max- imization based numerical algorithm for optimal decentralized powers and capacity where each of the partial maximizations is solved utilizing the monotone structure of the optimals. The organization o f the letter is as follows. Section II details the system model and the optimization problem to be solved. In Section III, the computational algorithm for the optimal po wers based on alternating maximization approach is explained follo wed by proofs of con ver gence and optimal- ity . Numerical results for the decentralized fading MA C are presented in section IV . Section V concludes the letter . I I . S Y S T E M M O D E L Consider a L -user Gaussian fading MA C whose output is giv en by Y = X L i = 1 H i X i + Z , where user- i transmit symbol X i undergoes flat fading denoted by multiplicativ e coefficient H i . The additive noise Z is a normalized A WGN process independent of X i and H i . The fading processes H i are assumed to be independent across users and varies IID in time. In our decentralized model, the fading coefficients H i are known only to the respecti ve transmitters at all instants. The receiv er has access to the full CSI vector ( H 1 , H 2 , · · · , H L ) . W e also assume that the fading distributions are known a priori to all the transmitters and the recei ver . The i -th transmitter, using the av ailable channel state information h i (current realization of H i ), selects transmit power of P i ( h i ) , see Figure 1. For con venience, with a slight abuse of notation, we will use P i to denote the power control of the i -th user . F or a chosen set of po wer schemes denoted by ( P 1 , · · · , P L ) , the average ergodic sum-rate R , E P L i = 1 R i giv en by R ( P 1 , · · · , P L ) = E log  1 + X L i = 1 | H i | 2 P i ( H i )  , (1) is achie ved by employing successive cancellation decoding at the receiv er [3, Chapter 4]. Since H i is known at the respectiv e 1089-7798 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCOMM.2018.2856240, IEEE Communications Letters 2 A WGN coder rate R 1 A WGN coder rate R 2 A WGN coder rate R L W 1 W 2 W L × × × p P 1 ( h 1 ) p P 2 ( h 2 ) p P L ( h L ) × × × X 1 X 2 X L h 1 h 2 h L + Z Y Fig. 1: Po wer control in Decentralized Gaussian f ading MAC. transmitter and receiv er, the sum-rate in (1) depends only on the fading magnitudes. Thus, we can replace | H i | 2 by V i and write P i ( H i ) as P i ( V i ) , 1 ≤ i ≤ L . Our objectiv e is to maximize the sum-rate R over the set of power control schemes P i ( V i ) , 1 ≤ i ≤ L , satisfying po wer constraints associated with the transmitters. Definition 1. The ergodic sum-capacity C s u m is the maximum averag e sum-rate ac hievable [1], i.e. C s u m = max ( P 1 , . .. , P L ) ∈ P E log  1 + X L i = 1 V i P i ( V i )  , (2) wher e the maximization is over the set P defined as collection of all power str ate gies satisfying E P i ( V i ) ≤ P av g i , 1 ≤ i ≤ L . Remark 2. Notice that the average power constr aints are linear and the objective function R is concave in powers P i , 1 ≤ i ≤ L . Furthermor e, R is continuous and has continuous partial derivatives. Also, it can be easily deduced that the set P is a non-empty conve x compact set. W e will use boldface letters to denote vectors. For example, d is a vector with element d i at position- i where d i can be either a scalar or a function. W e also use the notation d D j to represent a vector containing all elements of d excluding element d j . The joint distribution (CDF) is denoted by Ψ , where Ψ i denotes the marginal CDF of f ading of user- i . I I I . O P T I M A L P O W E R C O N T R O L Since (2) is a con ve x program with a strictly feasible point, KKT conditions are necessary and suf ficient condition for the optimal powers. W e obtain the cost function L ,  log * , 1 + L X i = 1 v i P i ( v i ) + - d Ψ ( v ) − L X i = 1 λ i  P i ( v i ) d Ψ i ( v i ) where the constants λ i , 1 ≤ i ≤ L are Lagrange multipliers for each of the po wer constraints. The deri vati ves with respect to the power allocation functions has to be zero for optimality , whenev er non-zero po wer is allocated. Thus,  d Ψ ( v D j ) 1 + v j P j ( v j ) + P L i = 1 , i , j v i P i ( v i ) = λ j v j , 1 ≤ j ≤ L . (3) The analytical solution of (3) for the optimal powers is considered not feasible [4]. Next, we identify a ke y structural property of optimal decentralized po wers that enables the com- putation of optimal po wers and er godic capacity numerically . Theorem 3. The optimal power P ∗ j ( v j ) , whenever non-zer o, must be a monotonically incr easing function of v j , 1 ≤ j ≤ L . Pr oof: W .l.o.g consider the optimal power scheme say P ∗ k of user- k . Using (3), we ha ve v k  d Ψ ( v D k ) 1 + v k P ∗ k ( v k ) + P L i = 1 , i , k v i P ∗ i ( v i ) = λ k , (4) whenev er P ∗ k > 0 . Furthermore, consider any two values of the fading v ariable v k say β > α such that positiv e po wers are allocated. Thus, we ha ve β  d Ψ ( y ) 1 + β P ∗ k ( β ) + y = α  d Ψ ( y ) 1 + α P ∗ k ( α ) + y , i.e.  β 1 + β P ∗ k ( β ) + y − α 1 + α P ∗ k ( α ) + y ! d Ψ ( y ) = 0 . (5) where P L i = 1 , i , k v i P ∗ i ( v i ) is replaced by y for ease of represen- tation. Rewriting the integrand in (5), we get  ( β − α ) ( 1 + y ) + α β ( P ∗ k ( α ) − P ∗ k ( β ) ) ( 1 + α P ∗ k ( α ) + y ) ( 1 + β P ∗ k ( β ) + y ) ! d Ψ ( y ) = 0 · (6) The integrand above is strictly positive for P ∗ k ( β ) ≤ P ∗ k ( α ) , thus violating (6). Therefore, P ∗ k ( β ) > P ∗ k ( α ) . This completes the proof. Remark 4. Though we pr esented pr oof of monotonicity of the optimal P ∗ k assuming that all remaining power s ar e optimal, this is also true for the partially optimal say ˆ P ∗ k for any set of feasible powers for the r emaining users and can be pr oved using r easoning similar to that in the pr oof of the Theor em 3. Let us denote the integral on the LHS in (3) by f j ( v j P j ( v j ) ) . Corollary 5. F or optimal P j ( v j ) , f j ( v j P j ( v j ) ) is monotoni- cally decr easing function of v j . Pr oof: For the optimal case, by Theorem 3, v j P j ( v j ) increases monotonically in v j and the corollary follo ws. A. AM Algorithm W e no w de velop a simple numerical algorithm to solve the joint optimization problem in (2) in terms of partial optimiza- tions using the principle of alternating maximization (AM). As we will later prove, the partial maximization can be solved numerically using the monotone structure of the optimal. The con vergence and the optimality proofs are established in the next sub-section. The alternating maximization (AM) method maximizes R w .r .t. each power scheme sequentially . T o identify this, we de- note the i -th user power by P ( n ) i . The computational algorithm is parameterized in terms of λ i , 1 ≤ i ≤ L . Algorithm Optimal powers for decentralized MA C Initialization: Initialize λ j , 1 ≤ j ≤ L , small step-size δ , approximation error tolerance  . P ( 0 ) j , 1 ≤ j ≤ L denote 1089-7798 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCOMM.2018.2856240, IEEE Communications Letters 3 arbitrarily initialized powers obeying constraints. Set n = 1 . Repeat For j = 1 to L (a) Compute the partial maximization P ( n ) j = arg max P j R ( P ( n ) 1 , · · · , P ( n ) j − 1 , P j , P ( n − 1 ) j + 1 , · · · , P ( n − 1 ) K ) , using the formula: P ( n ) j ( v j ) = 1 v j f − 1 j λ j v j ! · (b) Find ¯ P av g j =  P ( n ) j ( v j ) d Ψ j ( v j ) . (c) If       P av g j − ¯ P av g j  >  , then λ j = λ j + δ ; goto step (a)  P av g j − ¯ P av g j  < −  , then λ j = λ j − δ ; goto step (a) End n = n + 1 Until all the power constraints con verge. Notice that the algorithm outputs power schemes in the order P ( 0 ) 1 , P ( 0 ) 2 , · · · , P ( 0 ) K , P ( 1 ) 1 , P ( 1 ) 2 , · · · , P ( 1 ) K , P ( 2 ) 1 , P ( 2 ) 2 , · · · , P ( 2 ) K , · · · , where each power in the sequence is the partial maximizer in step (a) with the previously av ailable po wers fixed for the remaining users. In the follo wing corollary , we justify the procedure for the partial maximization in step (a). Corollary 6. The partial maximizer P ( n ) j is solved by P ( n ) j ( v j ) = 1 v j f − 1 j λ j v j ! , 1 ≤ j ≤ L , (7) wher e f − 1 j ( · ) is the in verse mapping of the f j ( · ) function. Pr oof: Since for optimal P j ( v j ) , f j ( v j P j ( v j ) ) is strictly decreasing with v j , there is a one-to-one correspondence between the RHS and LHS of (3) for ev ery v j . Hence f − 1 j ( · ) , in verse mapping of f j ( · ) , exists and the corollary follows. Precisely speaking, for every v j , we compare the computed integral f j ( v j P j ( v j ) ) with λ j v j for dif ferent values of P j ( v j ) until the two values agree to the desired accuracy . This is done using bisection method (linear conv ergence, rate 1 / 2 ) to solve (3) for P j ( v j ) . The algorithm repeats until con vergence (step (c)) with stop condition as | P av g j − ¯ P av g j | <  , 1 ≤ j ≤ L . B. Con ver gence and Optimality In general, alternating optimizations need not con ver ge. W e now sho w that AM algorithm al ways con ver ges to the global optimal of (2). The proof follows a set of arguments similar to con vergence and optimality proofs of AM optimization for con vex objectiv e function and conv ex constraints presented in [8, Chapter 9] with appropriate modifications. W e define term “iteration" to indicate updating of all L - user’ s power e xactly once from previously updated L -user’ s powers set. Let P ( n ) : = ( P ( n ) 1 , P ( n ) 2 , · · · , P ( n ) L ) denote the updated all powers set after iteration- n . Lemma 7. Ther e e xists a constant R ∗ such that R ( P ( n ) ) → R ∗ Pr oof: W e describe the proof for L = 2 user case. The extension to higher L is straightforward. For e very run of the algorithm, the ergodic sum-rate improves i.e. R ( P ( n − 1 ) 1 , P ( n − 1 ) 2 ) ≤ R ( P ( n ) 1 , P ( n − 1 ) 2 ) ≤ R ( P ( n ) 1 , P ( n ) 2 ) . In short, R ( P ( n ) ) ≥ R ( P ( n − 1 ) ) holds for all n . Since the rate sequence R ( P ( n ) ) is non-decreasing and bounded from above, it must conv erge i.e. R ( P ( n ) ) → R ∗ for some R ∗ ≤ C s u m . In our main Theorem 9, we sho w that the AM algorithm attains the global optimum irrespecti ve of the chosen starting or initializing conditions. T o wards this end, we define ∆ R ( P ) = R ( P ne x t ) − R ( P ) , where P ne x t is the updated po wers set after an iteration of the AM algorithm using P as previous po wers set. Thus, ∆ R ( P ) is the increment in R ( P ) after an iteration of the algorithm. Corollary 8. If ∆ R ( P ) = 0 for any P ∈ P , then P ne x t = P . Pr oof: The condition ∆ R ( P ) = 0 implies there is no increment in the sum-rate in each of the partial optimizations. This, in turn, implies P ne x t , j = P j , 1 ≤ j ≤ L due to uniqueness of solutions of partial maximizations i.e. each P j is the partial optimizer when remaining powers are fixed. Theorem 9. The AM Algorithm con ver ges to global optimum i.e. R ∗ = C s u m . Pr oof: The proof consists of two parts: (1) showing that if R ( P ( n ) ) < C s u m for an y po wer P ( n ) , then R ( P ( n + 1 ) ) > R ( P ( n ) ) i.e. the algorithm does not get trapped if R ( P ( n ) ) < C s u m , and (2) showing that R ( P ( n ) ) necessarily con ver ges to C s u m . Part (1): If R ( P ) < C s u m for any P ∈ P , then R ( P ne x t ) > R ( P ) or ∆ R ( P ) > 0 . The proof is by contradiction. Consider an y P ∈ P such that R ( P ) < C s u m . Assume ∆ R ( P ) = 0 . Since R ( P ) < C s u m , there exists Q ∈ P such that R ( P ) < R ( Q ) . Consider the direction from P to Q , we see that Q − P = [ Q 1 − P 1 , 0 , 0 , · · · , 0] + [0 , Q 2 − P 2 , 0 , · · · , 0] + · · · + [0 , · · · , 0 , 0 , Q L − P L ] . Rewriting the abov e in unit vector form, we get ~ u = α 1 ~ u 1 + α 2 ~ u 2 + · · · + α L ~ u L , where ~ u is unit vector along Q − P , ~ u 1 along [ Q 1 − P 1 , 0 , 0 , · · · , 0] , ~ u 2 along [0 , Q 2 − P 2 , 0 , · · · , 0] etc. and α i = k Q i − P i k / k Q − P k , 1 ≤ i ≤ L are the scalars. The rate of change of R ( P ) in the direction from P to Q is giv en by ∇ R ( P ) · ~ u = ∇ R ( P ) · ( α 1 ~ u 1 + α 2 ~ u 2 + · · · + α L ~ u L ) , = α 1 ∇ R ( P ) · ~ u 1 + α 2 ∇ R ( P ) · ~ u 2 + · · · + α L ∇ R ( P ) · ~ u L . Consider the direction ~ u 1 : [ Q 1 − P 1 , 0 , 0 , · · · , 0] = [ Q 1 , P 2 , P 3 , · · · , P L ] − [ P 1 , P 2 , P 3 , · · · , P L ] . Corollary 8 implies P 1 maximizes sum-rate R for the fixed set of remaining po wers { P 2 , P 3 , · · · , P L } . Hence ∇ R ( P ) · ~ u 1 = 0 . By similar arguments, ∇ R ( P ) · ~ u i = 0 , 1 ≤ i ≤ L holds. Thus, ∇ R ( P ) · ~ u = X L i = 1 α i ∇ R ( P ) · ~ u i = 0 . (8) 1089-7798 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/LCOMM.2018.2856240, IEEE Communications Letters 4 0 2 4 6 8 10 1 . 5 2 2 . 5 3 3 . 5 4 4 . 5 Po wer P avg (dB) C sum (bits) Centralized Decentralized (a) L = 4 L = 3 L = 2 0 2 4 6 8 10 2 3 4 5 6 Po wer P avg (dB) C sum (bits) (b) Fig. 2: (a) Capacity result for L = 2 non-identical users decentralized MA C with normalized Rayleigh and Rician fading i.e. d Ψ 1 ( h 1 ) = 2 h 1 e − h 2 1 d h 1 , d Ψ 2 ( h 2 ) = 2 h 2 ( K + 1 ) e − h 2 2 ( K + 1 ) − K I 0 ( 2 h 2 √ K ( K + 1 ) ) d h 2 and E [ h 2 1 ] = E [ h 2 2 ] = 1 . Solid curv e is capacity of the centralized MA C [7]. (b) Capacity results for L - identical user s decentralized MA C with all f adings Rayleigh distributed: Dashed curves are the desired capacity of decentralized MA C, dotted curves are near closed-form lower bounds to decentralized MAC capacity obtained in [5] and solid curves represent capacity of centralized MA C [7]. Since sum-rate R is concave, it satisfies R ( Q ) ≤ R ( P ) + ∇ R ( P ) · k Q − P k ~ u , ∀ P ∈ P , ∀ Q ∈ P , which, with (8), implies R ( Q ) ≤ R ( P ) which is a contradic- tion. Therefore, ∆ R ( P ) > 0 . Part (2): Conv ergence to the optimal is not established since the algorithm may produce increments ∆ R ( P ) > 0 arbitrarily small. W e now prove that the sum-rate sequence indeed con verges to C s u m . T o this end, we recall, from Lemma 7, that the sum-rate R ( P ( n ) ) con ver ges to say R ∗ . Thus, for any δ > 0 and for all n sufficiently large, we hav e R ∗ − δ ≤ R ( P ( n ) ) ≤ R ∗ . (9) Let µ = min H P ∆ R ( P ) , where H P = { P ∈ P : R ∗ − δ ≤ R ( P ) ≤ R ∗ } . Recall that R ( P ) is continuous and has continuous partial deri vati ves. Thus, ∆ R ( P ) is also continuous. Since H P is inv erse image of a closed interval under continuous R ( P ) and P is compact, we conclude that the subset H P is also compact. Thus, µ exists. If R ∗ < C s u m , then ∆ R ( P ) > 0 for all power schemes in H P (Part (1)) and hence µ > 0 . Since R ( P ( n ) ) satisfies (9), P ( n ) ∈ P 1 . Therefore, ∆ R ( P ( n ) ) ≥ µ holds for all suf ficiently large n . Since µ > 0 , this implies ∆ R ( P ( n ) ) > 0 for all sufficiently large n suggesting that R ( P ( n ) ) e ventually exceeds R ∗ , which is a contradiction. Therefore, R ( P ( n ) ) → C s u m . L = 3 L = 4 L = 2 0 2 4 6 8 10 12 14 0 1 2 3 4 5 P avg = 0 dB v P ∗ ( v ) Fig. 3: Powers for L - identical users decentralized MAC. I V . N U M E R I C A L R E S U LT S W e demonstrate the utility of the proposed algorithm for Gaussian MA C with independent fadings across users and av erage power constraints assumed identical for simplicity . Figures 2 and 3 illustrate the er godic capacity computed for the non-identical & identical users MA C and the optimal po wers for the identical users MA C respectively . Here, we do not pursue detailed con vergence-rate and complexity analysis of the AM algorithm due to lack of space but empirical estimates suggest that its con ver gence rate slo ws down significantly by sev eral orders of magnitude with number of MA C users. V . C O N C L U S I O N S W e demonstrated, for the first time, the numerical approach based on alternating optimization principle to solv e the decen- tralized powers and ergodic capacity of Gaussian MA C for the general fading statistics and po wer constraints. The proposed algorithm is simple to implement but the computational com- plexity increases with MA C users, thus rendering it useful for MA C with moderate number of users. As future work, we look forward to explore strate gies to impro ve con ver gence rates of our iterativ e AM algorithm. R E F E R E N C E S [1] A. Das and P . Narayan, “Capacities of time-varying multiple-access channels with side information, ” Information Theory , IEEE Tr ansactions on , v ol.48, no.1, pp. 4-25, Jan 2002. [2] G. Caire and S. Shamai, “On the capacity of some channels with channel state information, ” Information Theory , IEEE T ransactions on , vol. 45, pp. 1468-1489, 1998. [3] A. El Gamal and Y .-H. Kim, Network Information Theory , Cambridge Univ ersity Press, 2011. [4] S. Shamai and E. T elatar , “Some information theoretic aspects of decentralized power control in multiple access fading channels, ” in Information Theory and Networking W orkshop, IEEE , June 1999. [5] K. Singh, S. R. B. Pillai, “Decentralized power adaptation in er godic fading multiple access channels, ” in NCC , 2015. [6] N. Mital, K. Singh and S. R. B Pillai, “On the Ergodic Sum-Capacity of Decentralized Multiple Access Channels, ” in IEEE Communications Letters , vol. 20, no. 5, pp. 854-857, May 2016. [7] R. Knopp and P . Humblet, “Information capacity and power control in single-cell multiuser communications," ICC ’95 Seattle , pp. 331-335. [8] R. W . Y eung, Information Theory and Network Coding . Springer Press, 2008.