Open Access

Distributed algorithms for sum rate maximization in multi-cell downlink OFDMA with opportunistic DF relaying

EURASIP Journal on Wireless Communications and Networking20142014:154

https://doi.org/10.1186/1687-1499-2014-154

Received: 12 March 2014

Accepted: 11 September 2014

Published: 23 September 2014

Abstract

This paper considers a multi-cell orthogonal frequency division multiple access (OFDMA) downlink system with several decode-and-forward (DF) relay stations (RSs) aiding the base station (BS) transmissions. The problem considered is the maximization of the system sum rate with a total power constraint in each cell. An iterative semi-distributed resource allocation (RA) algorithm is first proposed to optimize mode selection (decision whether relaying should be used or not and which relay), subcarrier assignment (MSSA), and power allocation (PA), alternatively. During the MSSA stage, the problem is decoupled into subproblems which can be solved distributively in linear time. During the PA stage, an algorithm based on single condensation and Lagrange duality (SCLD) is designed to optimize PA with the tentative MSSA results. The convergence of the SCLD-based RA algorithm is theoretically guaranteed and an local optimum is reached after convergence. To solve the formulated problem autonomously, a modified iterative water-filling (IWF) algorithm is further proposed. Specifically, each cell autonomously optimizes its own sum rate with the estimated power values of the received interferences from the other cells. An optimum algorithm is proposed to solve the local RA problem in each cell. Through numerical experiments, the convergence of the two proposed algorithms as well as their benefits compared with a centralized algorithm (CA) are illustrated.

Keywords

OFDMA Opportunistic DF relaying Distributed resource allocation Cochannel interference mitigation

1 Introduction

High data rate and ubiquitous coverage are strongly required in the next-generation wireless communication networks. To achieve this goal, the emerging technology incorporating orthogonal frequency division multiple access (OFDMA) with relaying is receiving a lot of interest from both academia and industry. In particular, the OFDMA technology is able to combat frequency-selective multi-path fading and flexible in applying dynamic radio resource allocation (RA) for performance improvement. Moreover, the relaying technology has the attractive feature of coverage extension and data rate improvement [1].

Concerning relay-aided OFDM(A) transmissions, [2] and [3] have proposed two types of relaying, namely amplify-and-forward (AF) and decode-and-forward (DF) relays. With DF/AF relaying, symbol transmission is carried out in two time slots (TSs). During the first TS, the source broadcasts symbols on all subcarriers with the relay keeping quiet. During the second TS, except for the relay, the source might also transmit symbols on subcarriers unused by the relay. Adopting the DF relaying, [423] have investigated intensively the RA problems for downlink OFDMA.

Considering the case when the source to destination (S-D) link is unavailable, [47] have proposed algorithms for RA in the DF relay-aided OFDM systems. Considering the case when the S-D link is available, [823] have studied RA in systems with opportunistic relaying (sometimes termed as selection relaying). To start with, [813] have studied a low spectrum efficiency (LSE) protocol, when only the relay transmits symbols during the second TS. Specifically, each subcarrier can choose either the relay-aided mode or the direct mode for data transmission. In the relay-aided mode, a symbol is first transmitted from the source to the relay during the first TS, which is then forwarded from the relay to the destination during the second TS. While in the direct mode, the symbol is transmitted directly to the targeted destination during the first TS without the help of the relay. Note that, with the LSE protocol, subcarriers unused by the relay during the second TS are actually wasted. To address this issue, [1422] have proposed and studied improved high spectrum efficiency (HSE) protocols which allow new symbols to be transmitted on the subcarriers unused by the relay during the second TS.

Note that all these papers consider RA in single-cell situations and model the cochannel interference (CCI) as additive background noise. This is reasonable only when the frequency reuse factor 1/W is low. Here, W is the number of cells in a cluster which cannot use the same frequencies for transmission. However, due to its ability to achieve higher system capacity, aggressive frequency reuse is recommended in the next generation cellular systems [24]. When the frequency reuse factor is high, the CCI becomes a key factor affecting the system performance and thus cannot be ignored [25, 26].

Considering the CCI, system models in multi-cell DF relay-aided OFDMA become quite interesting and challenging. With multi-cell DF relay-aided OFDMA systems, [27] and [28] have discussed RA algorithms when the powers are uniformly allocated to all stations. For multi-cell OFDMA systems without DF relaying, several RA algorithms have been proposed in [2932]. However, these methods cannot be extended directly to solve RA problems jointly optimizing transmission mode selection, subcarrier assignment (MSSA), as well as power allocation (PA) in multi-cell OFDMA systems with opportunistic DF relaying. Concerning the opportunistic DF protocol in multi-cell relayed OFDMA systems, [25] and [26] have recently proposed joint RA schemes to maximize the sum rate over all cells and the weighted sum of each cell min-rate, respectively. However, the proposed centralized algorithms (CAs) seem to be quite heavy to implement in practice.

Compared with the above existing works, the contributions of this paper are as follows:

  • We propose an iterative semi-distributed RA algorithm to optimize the MSSA and the PA alternatively with the sum rate keeping increasing. For the MSSA stage, the optimization problem is decoupled into subproblems which can be solved distributively in linear time. For the PA stage, the algorithm based on single condensation and Lagrange duality (SCLD) is designed to optimize PA iteratively with an analytical solution at each iteration. As will be illustrated through numerical results, the proposed algorithm converges quite fast and is more practical compared to the CA of [25].

  • We further propose a modified iterative water-filling (IWF) algorithm to solve the formulated RA problem autonomously. Specifically, each cell autonomously optimizes its own sum rate with the estimated power values of the received interferences from the other cells. An optimum algorithm is proposed to solve the local RA problem in each cell. As will be illustrated, the convergence of the modified IWF algorithm is always observed in simulations, although it seems intractable to derive the conditions of convergence theoretically. Through numerical results, the modified IWF algorithm provides a good trade-off between complexity and performance. Thus, it is recommended for practical implementation.

The rest of this paper is organized as follows: First, the system model and the problem formulation for the considered system are presented in the next section. The proposed SCLD-based RA algorithm and the modified IWF algorithm are described in ‘Section 3’ and ‘Section 4’, respectively. The convergence of the proposed algorithms and their benefits compared with a CA are illustrated by numerical experiments in ‘Section 5’. Finally, the conclusions are drawn in ‘Section 6’.

2 System model and problem formulation

2.1 System model

We consider a cellular OFDMA system with N cells coordinated by a central controller for message passing among cells. In each cell, downlink transmission is carried out from a base station (BS) to U mobile stations (MSs) with the help of J relay stations (RSs) which are assumed to be of the DF type. For each link, the channel is assumed to be frequency selective and transformed into K parallel subchannels by using OFDM with sufficiently long cyclic prefix. The data transmission is carried out in two TSs. During the first TS, a symbol is first broadcast by the BS at a subcarrier k, which is in either relay-aided mode or direct mode. Both a selected RS and a targeted MS receive this symbol. If the relay-aided mode is used, the RS decodes the received symbol and forwards it to the targeted MS over the subcarrier k with the BS keeping quiet on this subcarrier during the second TS. The MS only decodes the symbol received during the second TS. If the direct mode is used, the targeted MS decodes the symbol received during the first TS. Also, another symbol is broadcast by the BS at the same subcarrier during the second TS, which is received and only decoded by the targeted MS.

With OFDMA, each subcarrier is allocated to only one MS in each cell. Throughout this paper, we assume the timing and carrier synchronization is perfect and there are no errors during the SR transmission at the first TS. We further assume that the central controller can obtain/distribute messages from/to all nodes. Moreover, we assume that the coherence time of each link is sufficiently long for implementing the RA. Note that by assuming the above idealities, an upper bound on the system performances is obtained.

The data transmission procedure in every cell is identical. Thus, we only analyze the downlink data transmission inside one selected cell n, which is impaired by cochannel interference from the other cells. Specifically in cell n, data transmissions are carried out either in relay-aided mode or direct mode, as will be elaborated in the following section. Note that, as will be used, a variable with the indice t 1/t 2 means that this variable corresponds to the transmission at TS1/TS2.

Let us first see more closely the relay-aided data transmission. As illustrated in Figure 1a, the BS s n first produces a symbol P s n , t 1 k x s n , t 1 k at subcarrier k during the first TS, while the transmitter of the selected RS r jn remains idle. Here, x s n , t 1 k denotes the normalized symbol (meaning E | x s n , t 1 k | 2 = 1 ) transmitted by s n at subcarrier k during the first TS, and P s n , t 1 k denotes the corresponding transmit power. Simultaneously in an interfering cell n, a symbol P s n , t 1 k x s n , t 1 k is also produced from the interfering BS s n at the same subcarrier. Here again E | x s n , t 1 k | 2 = 1 . Note that, instead of using an additional integer variable to indicate whether s n transmits data on subcarrier k or not, we use P s n , t 1 k to do it. Specifically, P s n , t 1 k > 0 means that s n uses subcarrier k for data transmission during the first TS, and P s n , t 1 k = 0 means that s n transmits nothing at the subcarrier k during the first TS. This choice is motivated to simplify the system sum rate expression and facilitate the algorithm design. Specifically, when using an integer variable to indicate whether a subcarrier k in a cell n is used or not, the rate expression would contain integer variables in both the denominators and the numerators, which makes it a non-linear function of the integer variables. When using the corresponding power value to do it, the rate expression contains integer variables only in the numerators, which makes it a linear function of the integer variables. At the end of the first TS, the signal received by r jn for subcarrier k can be expressed as
y r jn k = P s n , t 1 k h s n , r jn k x s n , t 1 k + v r jn k + n = 1 , n n N P s n , t 1 k h s n , r jn k x s n , t 1 k
(1)
Figure 1

Illustration of a relay-aided transmission in cell n with cochannel interference from cell n . The solid lines and the dashed lines represent communication links and interfering links, respectively. (a) At subcarrier k and the first TS. (b) At subcarrier l and the second TS.

where v r jn k denotes the additive white Gaussian noise (AWGN) at subcarrier k and r jn during the first TS. h s n , r jn k denotes the channel frequency response (CFR) for subcarrier k from s n to r jn .

During the second TS, as illustrated in Figure 1b, the selected RS r jn re-encodes the decoded symbol and forwards P r jn k x r jn k at the same subcarrier k. Here, x r jn k = x s n , t 1 k . The BS s n transmits nothing on this subcarrier, meaning that P s n , t 2 k = 0 . Here, P r jn k and P s n , t 2 k denote the transmit power allocated to r jn and s n , respectively at subcarrier k during the second TS. At the same time, in an interfering cell n, r j n and s n also transmit P r j n k x r j n k and P s n , t 2 k x s n , t 2 k at subcarrier k, where at most only one power value out of P r j n k x r j n k and P s n , t 2 k x s n , t 2 k can be non-zero. More specifically, when subcarrier k of cell n chooses the direct mode for data transmission, P r j n k = 0 . Otherwise, P s n , t 2 k = 0 . Here again E | x r j n k | 2 = 1 and E | x s n , t 2 k | 2 = 1 . At the end of the second TS, the signal received by the targeted MS d un at subcarrier k can be expressed as
y d un , t 2 k = P r jn k h r jn , d un k x r jn k + v d un , t 2 k + n = 1 , n n N P s n , t 2 k h s n , d un k x s n , t 2 k + n = 1 , n n N j = 1 J P r j n k h r j n , d un k x r j n k
(2)

where v d un , t 2 k denotes the AWGN corrupting d un at subcarrier k during the second TS, h r j n , d un k denotes the CFR for subcarrier k from r j n to d un . h s n , d un k denotes the CFR of subcarrier k from s n to d un .

We assume v r jn k , v d un , t 2 k are independent zero-mean circular Gaussian random variables with the same variance σ2. After some mathematical calculations, the signal-to-interference-plus-noise ratio (SINR) associated with decoding x s n , t 1 k from y r jn k at r jn during the first TS is expressed by
Γ r jn k = P s n , t 1 k G s n , r jn k f r jn k
(3)

where f r jn k = σ 2 + n , n n P s n , t 1 k G s n , r jn k denotes the sum power of the AWGN and the interference received by r jn at subcarrier k during the first TS. G s n , r jn k = | h s n , r jn k | 2 denotes the channel gain of subcarrier k from s n to r jn .

Also, the SINR associated with decoding x r jn k , which equals x s n , t 1 k , from y d un , r jn k at d un during the second TS is expressed by
Γ d un , t 2 k = P r jn k G r jn , d un k f d un , t 2 k
(4)

where f d un , t 2 k = σ 2 + n = 1 , n n N P s n , t 2 k G s n , d un k + n = 1 , n n N j = 1 J P r j n k G r j n , d un k denotes the sum power of the AWGN and the interference received by d un at subcarrier k during the second TS. G r jn , d un k = | h r jn , d un k | 2 denotes the channel gain of subcarrier k from r jn to d un . G s n , d un k = | h s n , d un k | 2 denotes the channel gain of subcarrier k from s n to d un .

The maximum achievable rate for the subcarrier k when allocated to d un is given by [3]
R ujn , 2 k = min ln 1 + Γ r jn k , ln 1 + Γ d un , t 2 k
(5)

in nats/two-TSs.

Let us describe further the direct data transmission in cell n. During the first TS, s n broadcasts P s n , t 1 k x s n , t 1 k at subcarrier k. The targeted MS d un receives signals from all BSs. Finally, the signal received by d un at subcarrier k can be expressed as
z d un , t 1 k = P s n , t 1 k h s n , d un k x s n , t 1 k + v d un , t 1 k + n = 1 , n n N P s n , t 1 k h s n , d un k x s n , t 1 k
(6)

where v d un , t 1 k denotes the AWGN corrupting d un at subcarrier k during the first TS.

During the second time slot, another symbol P s n , t 2 k x s n , t 2 k is broadcast by s n at subcarrier k and received by the same MS d un . The received signal can be expressed as:
z d un , t 2 k = P s n , t 2 k h s n , d un k x s n , t 2 k + v d un , t 2 k + n = 1 , n n N P s n , t 2 k h s n , d un k x s n , t 2 k + n = 1 , n n N j = 1 J P r j n k h r j n , d un k x r j n k .
(7)
Thus, the achievable sum rate for subcarrier k during two TSs is given by
R un , 1 k = ln 1 + Υ d un , t 1 k + ln 1 + Υ d un , t 2 k
(8)
in nats/two-TSs, where
Υ d un , t 1 k = P s n , t 1 k G s n , d un k f d un , t 1 k
(9)
denotes the SINR associated with the decoding of x s n , t 1 k from y d un , t 1 k at d un during the first TS. f d un , t 1 k = σ 2 + n = 1 , n n N P s n , t 1 k G s n , d un k denotes the sum power of the AWGN and the interference received by d un at subcarrier k during the first TS.
Υ d un , t 2 k = P s n , t 2 k G s n , d un k f d un , t 2 k
(10)

denotes the SINR associated with the decoding of x s n , t 2 k from y d un , t 2 k at d un during the second TS.

2.2 RA problem formulation

In order to formulate the RA problem, we now introduce binary variables a un k and b ujn k to describe the mode selection and subcarrier assignment in both TSs. To be more specific, a un k = 1 indicates that subcarrier k is allocated for data transmission to d un in direct mode during two TSs. b ujn k = 1 indicates that subcarrier k is allocated for data transmission to d un aided by r jn .

We consider maximizing the sum rate of all MSs in all cells under per cell total power constraints. The optimization variables are the transmission mode for each subcarrier, the subcarrier assignments and the power allocations at the BSs and the RSs. According to the system model, the considered RA problem can be formulated as follows:
max P , A , B R P , A , B s.t. C 1 : u = 1 U a un k + u = 1 U j = 1 J b ujn k 1 , n , k , C 2 : a un k { 0 , 1 } , b ujn k { 0 , 1 } , u , j , n , k , C 3 : k = 1 K P s n , t 1 k + k = 1 K P s n , t 2 k + j = 1 J k = 1 K P r jn k P t , n , n , C 4 : P s n , t 1 k 0 , P s n , t 2 k 0 , P r jn k 0 , j , n , k , C 5 : P s n , t 1 k 1 u = 1 U a un k + u = 1 U j = 1 J b ujn k = 0 , n , k , C 6 : P s n , t 2 k 1 u = 1 U a un k = 0 , n , k , C 7 : P r jn k 1 u = 1 U b ujn k = 0 , j , n , k ,
(11)
where
R P , A , B = n = 1 N u = 1 U k = 1 K a un k R un , 1 k P n k + n = 1 N u = 1 U j = 1 J k = 1 K b ujn k R ujn , 2 k P n k ,
(12)

P=[[P1]T,…,[P N ]T]T, P n = P n 1 T , , P n K T T , P n k = P s n , t 1 k , P s n , t 2 k , P r 1 n k , , P r Jn k T , A = a un k , B = b ujn k , Pt,n denotes the available total power in cell n.

Here, C 1 and C 2 ensure that each subcarrier k during the first TS can select only one mode (direct/relay-aided) to transmit data towards only one MS d un . If the relay-aided mode is selected, the data transmission can be helped by only one RS r jn . Moreover, C 3 and C 4 ensure that the consumed sum power for each cell is less than its available sum power. This type of power constraints gives an upper bound of the system performance. In practice, each node (BS, RS) in each cell will have an individual power constraint. Finally, C 5, C 6, and C 7 guarantee that no data is transmitted on an unused subcarrier and subcarrier k is used by only one node (either the BS or the RS) in each cell during the second TS.

3 SCLD-based coordinate ascent RA algorithm

3.1 The overall RA algorithm

In order to solve problem (11), an iterative coordinate ascent approach is adopted. Each iteration is carried out in two stages: the MSSA stage and the PA stage. We introduce integer m to indicate the iteration number and add superscript m to variables obtained at the end of iteration m. We now propose the overall RA algorithm, as depicted in Algorithm 1.

Specifically in the proposed RA algorithm, we first set m=0 and initialize the power P0 either by the uniform power allocation (UPA) or by using the optimal algorithm, as proposed in [16] when CCI is ignored. Each iteration consists of the MSSA stage and the PA stage. During the MSSA stage of iteration m, we set P=Pm−1 and decouple problem (11) into nk integer linear programs (ILPs), which can easily be solved in linear time, as will be shown in ‘Section 3.2’. After optimization, {A m ,B m } will be obtained. Finally, we have R(Pm−1,Am−1,Bm−1)≤R(Pm−1,A m ,B m ).

During the PA stage of iteration m, we set A=A m , B=B m and solve problem (11) using the SCLD PA algorithm proposed in ‘Section 3.3’. The output delivered is denoted as P m . As will be shown in ‘Section 3.3’, the SCLD PA algorithm approximates the original non-convex problem into a series of convex problems. The solutions of the approximated convex problems converge to a local optimum satisfying the Karush-Kuhn-Tucker (KKT) conditions of the non-convex problem. After convergence, the power vector P m will be the output. Finally, we have R(Pm−1,A m ,B m )≤R(P m ,A m ,B m ).

Considering the proposed RA algorithm, we now have the following:
R P m 1 , A m 1 , B m 1 E 1 : R P m 1 , A m , B m E 2 : R P m , A m , B m ,

where E 1 and E 2 are due to the MSSA stage and the PA stage, respectively. This means that Algorithm 1 yields non-decreasing sum rates along with iterations. Moreover, due to the total power constraint in each cell, the optimum sum rate is upper bounded and the sum rate values will not increase indefinitely along with iterations. This means that the iterations will eventually converge [31]. Algorithm 1 will stop when the sum rate increase is below a prescribed value ε1 or when m reaches a prescribed value M.

At the end of iteration m, a local optimum of the formulated problem is obtained at the PA stage, which is then improved at the MSSA stage of the next iteration. After that, a better local optimum can be calculated at the PA stage of iteration m+1. Finally, a good local optimum can be obtained after convergence. Note that each stage is carried out with analytical equations. Thus, the proposed RA algorithm has a lower-complexity compared to the centralized algorithm (CA) of [25]. As will be elaborated in ‘Section 3.2’ and ‘Section 3.3’, the proposed MSSA algorithm can be implemented distributively in each cell and the SCLD PA algorithm can be implemented semi-distributively in each cell. Thus, the proposed iterative RA algorithm can be implemented semi-distributively in each cell with the help of a central controller (CC) exchanging messages among cells.

3.2 MSSA optimization

In this subsection, the MSSA stage for iteration m is considered. After setting P to P m , problem (11) can be rewritten as follows:
max A , B R P m , A , B s.t. C 1 , C 2 , C 5 , C 6 , C 7 .
(13)
Note that each subcarrier in each cell has independent constraints. Therefore, problem (13) can be decoupled into NK subproblems. Specifically, subproblem i, which corresponds to subcarrier k0 in cell n0 is formulated as the following:
max A i , B i R i P m , A i , B i s.t. C 1 : u a un 0 k 0 + u j b ujn 0 k 0 1 , C 2 : a un 0 k 0 { 0 , 1 } , b ujn 0 k 0 { 0 , 1 } , u , j , C 5 : P s n 0 , t 1 k 0 , m 1 1 u = 1 U a un 0 k 0 + u = 1 U j = 1 J b ujn 0 k 0 = 0 , C 6 : P s n 0 , t 2 k 0 , m 1 1 u = 1 U a un 0 k 0 = 0 , C 7 : P r jn 0 k 0 , m 1 1 u = 1 U b ujn 0 k 0 = 0 , j ,
(14)
where A i = a un k | n = n 0 , k = k 0 , B i = b ujn k | n = n 0 , k = k 0 and R i are given by
R i = u a un 0 k 0 R un 0 , 1 k 0 P m + u j b ujn 0 k 0 R ujn 0 , 2 k 0 P m .
As will be explained later, problem (14) has the same optimum solution as the following problem, where the constraints C 5C 7 are removed.
max A i , B i R i P m 1 , A i , B i s.t. C 1 , C 2 .
(15)

Specifically, n0,k0, when P s n 0 , t 1 k 0 , m 1 > 0 , we certainly have R un 0 , 1 k 0 > 0 , u . Therefore at the optimum of problem (15), we have u = 1 U a un 0 k 0 + u = 1 U j = 1 J b ujn 0 k 0 = 1 , meaning that this subcarrier k0 should be used for data transmission. Otherwise, we can still improve R i by randomly selecting one variable of the set a un k | k = k 0 , n = n 0 and setting it to 1. Thus in this case, we have P s n 0 , t 1 k 0 , m 1 1 u = 1 U a un 0 k 0 + u = 1 U j = 1 J b ujn 0 k 0 = 0 . Also, when P s n 0 , t 1 k 0 , m 1 = 0 , we certainly have P s n 0 , t 1 k 0 , m 1 1 u = 1 U a un 0 k 0 + u = 1 U j = 1 J b ujn 0 k 0 = 0 . Thus at the optimum of problem (15), the constraint C 6.5 is fulfilled. Similarly, at the optimum of problem (15), the constraints C 6.6 and C 6.7 are fulfilled. This means that the optimum solution of problem (15) is also feasible for problem (14). Note that problem (14) is the same as problem (15) except that its feasible set is a subset of that of problem (15). Thus, the optimum solution of problem (15) is also the optimum solution of problem (14).

According to the expression of R i and the constraints (C 1,C 2), this subproblem (15) is further denoted as
max A i , B i u = 1 U a un 0 k 0 R un 0 , 1 k 0 P m 1 + u = 1 U j = 1 J b ujn 0 k 0 R ujn 0 , 2 k 0 P m 1 , C 1 : u a un 0 k 0 + u j b ujn 0 k 0 1 , C 2 : a un 0 k 0 { 0 , 1 } , b ujn 0 k 0 { 0 , 1 } , u , j.

It is obvious that (15) can easily be solved by searching the optimal mode, MS, and RS which provide the maximum rate for subcarrier k0 in cell n0 with P=Pm−1.

3.3 PA optimization

In this subsection, the PA stage for iteration m is considered. After setting the indicators to {A m ,B m }, the transmission modes are fixed in all cells. We denote by S un ( d ) the set of subcarriers allocated to MS d un in direct mode and S ujn ( r ) the set of subcarriers allocated to MS d un and RS r jn in relay-aided mode. Then, the objective function of problem (11) can be rewritten as R(P,A m ,B m ), given by the following:
n k S un ( d ) log 1 + Υ d un , t 1 k + log 1 + Υ d un , t 2 k + k S ujn ( r ) log min 1 + Γ r jn k , 1 + Γ d un , t 2 k .

Note that R is a non-convex function due to the presence of interfering power terms in the denominators of Υ d un , t 1 k , Υ d un , t 2 k , Γ r jn k , and Γ d un , t 2 k . To solve problem (11), we first replace it with a minimization problem that is then solved by using the SCLD PA algorithm.

The equivalent minimization problem is obtained in two steps. Problem (11) is first converted into an equivalent one given by
min P R P , A m , B m s.t. C 3 C 7 ,
(16)
where −R(P,A m ,B m ) is given by
n k S un ( d ) log 1 1 + Υ d un , t 1 k + log 1 1 + Υ d un , t 2 k k S ujn ( r ) log max 1 1 + Γ r jn k , 1 1 + Γ d un , t 2 k ,
(17)
1 1 + Υ d un , t 1 k = f d un , t 1 k f d un , t 1 k + P s n , t 1 k G s n , d un k ,
(18)
1 1 + Υ d un , t 2 k = f d un , t 2 k f d un , t 2 k + P s n , t 2 k G s n , d un k ,
(19)
1 1 + Γ d un , t 2 k = f d un , t 2 k f d un , t 2 k + P r jn k G r jn , d un k ,
(20)
1 1 + Γ r jn k = f r jn k f r jn k + P s n , t 1 k G s n , r jn k .
(21)
After that, another equivalent formulation is obtained by introducing slack variables Ψ r = { Ψ r nk , n , k S ujn ( r ) } . Then, problem (16) can be formulated as follows:
min P , Ψ r n k S un ( d ) log f d un , t 1 k g d un , t 1 k f d un , t 2 k g s n d un , t 2 k + k S ujn ( r ) Ψ r nk s.t. C 3 C 7 , C 8 : log f r jn k g r jn k Ψ r nk 0 , n , k S ujn ( r ) , C 9 : log f d un , t 2 k g r jn d un , t 2 k Ψ r nk 0 , n , k S ujn ( r ) ,
(22)

where g d un , t 1 k = f d un , t 1 k + P s n , t 1 k G s n , d un k , g s n d un , t 2 k = f d un , t 2 k + P s n , t 2 k G s n , d un k , g r jn d un , t 2 k = f d un , t 2 k + P r jn k G r jn , d un k , and g r jn k = f r jn k + P s n , t 1 k G s n , r jn k .

Problem (22) is still non-convex. In order to solve it, we now propose the SCLD PA algorithm. To derive it, two steps are involved. During the first step, a convex approximation of problem (22) is constructed. We first use the method of Lemma 1 in [33] to condense all denominator posynomials g d un , t 1 k , g s n d un , t 2 k , g r jn d un , t 2 k , g r jn k into monomials ĝ d un , t 1 k , ĝ s n d un , t 2 k , ĝ r jn d un , t 2 k , ĝ r jn k using tentative PA results P 0 = P s n , t 1 , 0 k , P s n , t 2 , 0 k , P r jn , 0 k . Functions h 1 nk = log f r jn k g r jn k Ψ r nk , h 2 nk = log f d un , t 2 k g r jn d un , t 2 k Ψ r nk , and h 3 nk = n k S un ( d ) log f d un , t 1 k g d un , t 1 k f d un , t 2 k g s n d un , t 2 k + k S ujn ( r ) Ψ r nk are approximated by ĥ 1 nk = log f r jn k ĝ r jn k Ψ r nk , ĥ 2 nk = log f d un , t 2 k ĝ r jn d un , t 2 k Ψ r nk , and ĥ 3 nk = n k S un ( d ) log f d un , t 1 k ĝ d un , t 1 k f d un , t 2 k ĝ s n d un , t 2 k + k S ujn ( r ) Ψ r nk , respectively. Then, by introducing the logarithmic change of the variables (e.g., x ~ = log x ), the approximated problem is formulated as the following:
minimize P ~ , Ψ ~ r n R n P ~ , Ψ ~ r s.t. C 3 , C 5 C 7 , C 4 : 1 P T n k e P ~ r jn k + e P ~ s n , t 1 k + e P ~ s n , t 2 k 1 , n , C 8 : log f r jn k e P ~ ĝ r jn k e P ~ e Ψ ~ r nk 0 , n , k S ujn ( r ) , C 9 : log f d un , t 2 k e P ~ ĝ r jn d un , t 2 k e P ~ e Ψ ~ r nk 0 , n , k S ujn ( r ) ,
(23)
where R n P ~ , Ψ ~ r is given by
k S un ( d ) log f d un , t 1 k e P ~ ĝ d un , t 1 k e P ~ f d un , t 2 k e P ~ ĝ s n d un , t 2 k e P ~ + k S ujn ( r ) log e Ψ ~ r nk .

P ~ denotes the vector of logarithmic power variables, and Ψ ~ r denotes the vector of logarithmic slack variables Ψ ~ r nk .

Note that the method we used in this step belongs to the class of successive convex approximation methods [34]. After some mathematical calculations, it is easy to verify that all the approximations of problem (22) satisfy the three conditions proposed in [34] for the convergence of the successive approximation method. By denoting Θ={P,Ψ r }, the three conditions are listed as follows:
  1. 1.
    Bounding condition: Θ,
    h i n , k ( Θ ) h ~ i n , k ( Θ ) , i = 1 , 2 , 3 .
     
  2. 2.
    Tightness condition: At the beginning of iteration t,
    h i n , k Θ m , t 1 = h ~ i n , k Θ m , t 1 , i = 1 , 2 , 3 .
     
  3. 3.
    Differential condition: At the beginning of iteration t, θΘ,
    h i n , k Θ m , t 1 ∂θ = h ~ i n , k Θ m , t 1 ∂θ , i = 1 , 2 , 3 .
     

During the second step, the Lagrange dual method is applied to solve the approximated convex problem. Let us denote the Lagrange multipliers of C 4, C 5, and C 6 by λ={λ n }, μ = { μ nk } k S ujn ( r ) , and ν = { ν nk } k S ujn ( r ) , respectively. Then, the dual problem of (23) is written as max λ , μ , ν ( min P ~ , Ψ ~ r L ( λ , μ , ν , P ~ , Ψ ~ r ) ) . Here, L ( λ , μ , ν , P ~ , Ψ ~ r ) denotes the Lagrange of problem (16).

After taking the derivative of L with relation to (w.r.t.) Ψ ~ r nk and setting it to 0, we obtain μ nk + ν nk = 1 , n , k S ujn ( r ) . By applying it, the dual variables are updated with the subgradient method as the following:
λ n ( i + 1 ) = λ n ( i ) + δ λ , n ( i ) 1 P T n k P r jn k , ( i ) + P s n k , ( i ) + P s n , t 2 k , ( i ) 1 ,
(24)
μ nk ( i + 1 ) = μ nk ( i ) + δ μ , nk ( i ) log f r jn k P ( i ) ĝ r jn d un , t 2 k P ( i ) ĝ r jn k P ( i ) f d un , t 2 k P ( i ) ,
(25)
ν nk ( i + 1 ) = 1 μ nk ( i + 1 ) .
(26)

Here, δ λ , n ( i ) = α λ , n / i and δ μ , nk ( i ) = α μ , nk / i are the step sizes chosen for λ and μ, respectively. We assume that each receiver in a cell n can estimate the received power from each of the interfering stations (either the BS or a RS of an interfering cell). Thus, f r jn k P ( i ) , f d un , t 2 k P ( i ) , ĝ r jn d un , t 2 k P ( i ) , and ĝ r jn k P ( i ) can be calculated locally in the cell n.

We further take the derivative of L w.r.t. P ~ s n , t 1 k , P ~ r jn k , and P ~ s n , t 2 k , and force them to 0. With the logarithmic power variables transformed back to the original P-space, fixed-point equations for P s n , t 1 k , ( i ) , n , k , P r jn k , ( i ) , n , k S ujn ( r ) , and P s n , t 2 k , ( i ) , n , k S un ( d ) can be formulated as
h 4 nk λ n ( i ) , μ ( i ) , P ( i ) = X 1 nk ( P 0 ) + X 2 nk μ ( i ) , P 0 λ n ( i ) P T n + Y 1 nk P ( i ) + Y 2 nk μ ( i ) , P ( i ) ,
(27)
h 5 nk λ n ( i ) , ν ( i ) , P ( i ) = X 3 nk P 0 + X 4 nk ν ( i ) , P 0 λ n ( i ) P T n + Y 3 nk P ( i ) + Y 4 nk ν ( i ) , P ( i ) ,
(28)
and
h 6 nk λ n ( i ) , ν ( i ) , P ( i ) = X 5 nk P 0 + X 6 nk ν ( i ) , P 0 λ n ( i ) P T n + Y 5 nk P ( i ) + Y 6 nk ν ( i ) , P ( i ) ,
(29)
respectively. Here, μ ( i ) = μ nk ( i ) , ν ( i ) = ν nk ( i ) ,
X 1 nk ( P 0 ) = n , k S u n ( d ) P s n , 0 k G s n , d u n k g d u n , t 1 k P 0 , X 2 nk μ ( i ) , P 0 = n , k S u j n ( r ) μ n k ( i ) P s n , 0 k G s n , r j n k g r j n k P 0 , X 3 nk ( P 0 ) = n , k S u n ( d ) P r jn , 0 k G r jn , d u n k g d u n , t 2 k P 0 , X 4 nk ν ( i ) , P 0 = n , k S u j n ( r ) ν n k ( i ) P r jn , 0 k G r jn , d u n k g d u n , t 2 k P 0 , X 5 nk ( P 0 ) = n , k S u n ( d ) P s 1 n , 0 k G s n , d u n k g d u n , t 2 k P 0 , X 6 nk ν ( i ) , P 0 = n , k S u j n ( r ) ν n k ( i ) P s 1 n , 0 k G s n , d u n k g d u n , t 2 k P 0 , Y 1 nk P ( i ) = n n , k S u n ( d ) G s n , d u n k f d u n , t 1 k P ( i ) , Y 2 nk μ ( i ) , P ( i ) = n n , k S u j n ( r ) μ n k ( i ) G s n , r j n k f r j n k P ( i ) , Y 3 nk P ( i ) = n n , k S u n ( d ) G r jn , d u n k f d u n , t 2 k P ( i ) , Y 4 nk ν ( i ) , P ( i ) = n n , k S u j n ( r ) ν n k ( i ) G r jn , d u n k f d u n , t 2 k P ( i ) , Y 5 nk P ( i ) = n n , k S u n ( d ) G s n , d u n k f d u n , t 2 k P ( i ) , Y 6 nk ν ( i ) , P ( i ) = n n , k S u j n ( r ) ν n k ( i ) G s n , d u n k f d u n , t 2 k P ( i ) .

We now consider the following theorem.

Theorem 1

H ( P ) = h 4 nk ( P ) , h 5 nk ( P ) , h 6 nk ( P ) satisfies the three properties of the standard interference function proposed in [35].

Proof

Taking equations h 4 nk for example, we denote the right-hand-side functions as
h 4 nk ( P ) = X 1 nk ( P 0 ) + X 2 nk ( μ , P 0 ) λ n P T n + Y 1 nk ( P ) + Y 2 nk ( μ , P ) , Y 1 nk ( P ) = n n , k S u n ( d ) G s n , d u n k σ 2 + I d u n , t 1 k ( P ) , Y 2 nk ( μ , P ) = n n , k S u j n ( r ) μ n k G s n , r j n k σ 2 + I r j n k ( P ) .
Here, the values of λ n ( i ) , μ n k ( i ) , X 1 nk ( P 0 ) , and X 2 nk ( μ ( i ) , P 0 ) are all constant and positive. Therefore, it is obvious that h 4 nk ( P ) > 0 . Also, both I d u n , t 1 k ( P ) and I r j n k ( P ) are linear functions of power variables. Thus, it is obvious that h 4 nk ( P ) is a monotonic increasing function of P, meaning that h 4 nk ( P ) > h 4 nk ( P ) , P > P . Here, the inequality should be understood as element-wise. Note for all α>1, we have
α h 4 nk ( P ) = X 1 nk ( P 0 ) + X 2 nk ( μ , P 0 ) λ n α P T n + Y 1 nk ( P ) α + Y 2 nk ( μ , P ) α > X 1 nk ( P 0 ) + X 2 nk ( μ , P 0 ) λ n P T n + Y ̌ 1 nk ( α , P ) + Y ̌ 2 nk ( α , μ , P ) ,
where
Y ̌ 1 nk ( α , P ) = n n , k S u n ( d ) G s n , d u n k σ 2 + α I d u n , t 1 k ( P ) , Y ̌ 2 nk ( α , μ , P ) = n n , k S u j n ( r ) μ n k G s n , r j n k σ 2 + α I r j n k ( P ) .
According to the expression of I d un , t 1 k ( P ) and I r jn k ( P ) , it is obvious that α I d un , t 1 k ( P ) = I d un , t 1 k ( α P ) and α I r jn k ( P ) = I r jn k ( α P ) . Thus, we have
α h 4 nk ( P ) > h 4 nk ( α P ) , α > 1 , n , k.

Similarly, we have h 5 nk ( P ) > 0 , α h 5 nk ( P ) > h 5 nk ( α P ) , α > 1 , n , k S ujn ( r ) , and h 6 nk ( P ) > 0 , α h 6 nk ( P ) > h 6 nk ( α P ) , α > 1 , n , k S un ( d ) . Both h 5 nk ( P ) and h 6 nk ( P ) are monotonic increasing functions of P.

Note that each element of H(P) satisfies the three properties (positivity, monotonicity, scalability) of the standard interference function defined in [35]. Thus, H(P) is a standard interference function.

Thanks to the Theorems 1 and 2 in [35], the standard interference function H(P) has a unique fixed point solution, which can be found iteratively as follows:
P s n , t 1 k , ( i , i + 1 ) = h 4 nk λ n ( i ) , μ ( i ) , P ( i , i ) , n , k , i ,
(30)
P r jn k , ( i , i + 1 ) = h 5 nk λ n ( i ) , ν ( i ) , P ( i , i ) , n , k S ujn ( r ) , i ,
(31)
P s n , t 2 k , ( i , i + 1 ) = h 6 nk λ n ( i ) , ν ( i ) , P ( i , i ) , n , k S un ( d ) , i.
(32)
Let us denote
X nk , n = P s n , 0 k G s n , d u n k g d u n , t 1 k ( P 0 ) , μ n k ( i ) P s n , 0 k G s n , r j n k g r j n k ( P 0 ) , ν n k ( i ) P r jn , 0 k G r jn , d u n k g d u n , t 2 k ( P 0 ) , P r jn , 0 k G r jn , d u n k g d u n , t 2 k ( P 0 ) , ν n k ( i ) P s 1 n , 0 k G s n , d u n k g d u n , t 2 k ( P 0 ) , P s 1 n , 0 k G s n , d u n k g d u n , t 2 k ( P 0 ) ,
Y nk , n = 1 f d u n , t 1 k ( P ( i ) ) , μ n k ( i ) f r j n k ( P ( i ) ) , ν n k ( i ) f d u n , t 2 k ( P ( i ) ) , G r jn , d u n k f d u n , t 2 k ( P ( i ) ) , ν n k ( i ) G s n , d u n k f d u n , t 2 k ( P ( i ) ) , G s n , d u n k f d u n , t 2 k ( P ( i ) ) .
As we assume that each receiver in each cell can estimate the received power from each of the interfering stations (either the BS or a RS of an interfering cell), both X nk , n and Y nk , n can be calculated locally in each interfering cell n for a considered cell n. We further assume that the central controller (CC) has access to the estimates of all the cross gains G s n , d u n k , G s n , r j n k , G r jn , d u n