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Feasibility of SINR guarantees for downlink transmissions in relay-enabled OFDMA networks
Automatica 48 (2012) 1818–1824
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Brief paper
Feasibility of SINR guarantees for downlink transmissions in relay-enabled OFDMA networks✩ Vishwesh V. Kulkarni a,1 , Joo Ghee Lim b , Sanjay K. Jha c a
Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455, United States
b
School of Electrical and Electronic Engineering, Singapore Polytechnic, Singapore 139651, Singapore
c
School of Computer Science and Engineering, University of New South Wales, Sydney, NSW 2033, Australia
article
info
Article history: Received 25 October 2010 Received in revised form 16 November 2011 Accepted 4 February 2012 Available online 18 June 2012 Keywords: OFDMA Downlink transmission SINR feasibility M-matrices
abstract We examine the feasibility of the signal-to-interference-and-noise-ratio (SINR) guarantees for downlink transmissions in relay-enhanced OFDMA networks that feature stationary users. The constraints are as follows: (i) the SINR of every user exceeds a certain threshold and (ii) the transmit power for each transmission is less than a certain threshold. We first derive a set of necessary and sufficient feasibility conditions for the specific case in which a user i served by the relay station shares at most one subchannel with a user j served by the base station. These conditions are a function of the target SINR values and the channel gains, and derived using a property of an M-matrix. If the geographical positions of the base station, the relay station, and the users are fixed, then our result determines the feasible SINR values for the users. Conversely, if these entities are not geographically fixed but the target SINR values are fixed, our result determines the geographical locations for these entities that ensure that the target SINR values are realized. We then extend these results to the case of networks featuring multiple base stations and multiple relays. Our conditions for checking the feasibility can be easily implemented in practice. Published by Elsevier Ltd
1. Introduction Orthogonal frequency division multiple-access (OFDMA) technology is expected to play a significant part in the next generation wireless networks such as WiMAX, 3GPP-LTE, and IEEE 802.22 WRAN (see Alavi, Zhou, & Cheng, 2009 and Kivanc, Li, & Liu, 2003). In OFDMA, the base station assigns sets of orthogonal subcarriers, i.e., subchannels, to the users. OFDMA can utilize frequency and multi-user diversities in order to maximize the system capacity. To further improve the system performance, relay stations are often introduced (see Soldani & Dixit, 2008): the examples include WiMAX (802.16j) and the 3GPP LTE-Advanced standards (see Yang, Hu, Xu, & Mao, 2009). The relays are wireless stations that (i) receive the data from a base station and transmit it to a set of users, and (ii) receive the data from a set of users and transmit it to
✩ This research was supported by the ARC Discovery Grant DP1096353. The material in this paper was partially presented at the 50th IEEE Conference on Decision and Control (CDC 2011), December 12–15, 2012, Orlando, Florida, USA. This paper was recommended for publication in revised form by Associate Editor Huijun Gao, under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (V.V. Kulkarni), [email protected] (J.G. Lim), [email protected] (S.K. Jha). 1 Tel.: +1 612 625 3300; fax: +1 612 625 4583.
0005-1098/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.automatica.2012.05.048
their designated base station (see Fig. 1). Now, if the SINR requirements of a set of users cannot be satisfied by a base station, could those be satisfied by adding a finite number of relays and, if so, where should the relays be placed? We investigate this problem in the context of downlink transmit power allocation in a relayenabled OFDMA network in which the number of users exceeds the number of subcarriers. To solve this problem, we build on the framework developed in Alpcan, Fan, Basar, Arcak, and Wen (2008) to characterize the maximum number of uplink transmissions that a base station can support in a multi-cell code division multiple-access (CDMA) network. Let the γ i denote the target signal-to-interferenceand-noise-ratio (SINR) values of user i and let L denote the CDMA spreading gain. Then, the base station can support the M uplink transmissions from M users simultaneously if i=1 γ i /(γ i + L) < 1 (see Alpcan et al., 2008, Lemma 2). We consider a similar convex optimization problem of minimizing the sum downlink transmit power subject to the constraint that the SINR exceeds the desired threshold for all users. In OFDMA networks, the CDMA spreading effect is absent, so L = 1. Hence, using the same logic, it trivially follows that, in an OFDMA network, a base station – unaided by a relay – may satisfactorily serve only as many users as there are subchannels available to it. In this paper, we make use of certain properties of M-matrices to establish the required feasibility conditions and extend these results to the general case of networks featuring multiple base stations and multiple relays.
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transmit powers pi obey the constraint pi ∈ [0, pmax ] for some predefined pmax ; we assume that pmax is sufficiently large. The channel gain from the base station to the ith user is denoted as hiB and the channel gain from the relay to the ith user is denoted as hiR . At times, we shall refer to these channel gains as his(i) , i.e., the channel gain for the transmission of the sender s(i) of user i to user i—the sender s(i) can be either the base station B or the relay station R. Both hiB and hiR are constrained to lie between 0 and 1. The received signal at the ith M1 user is xi = hiB pi + hφ(i)R pφ(i) . Likewise, the received signal at the ith M2 user is xi = hiR pi + hφ(i)B pφ(i) . Thus, the SINR for the ith M1 user is given as hiB pi
γi = Fig. 1. Two scenarios of a relay-enabled OFDMA network. The squares denote either the base station or a relay whereas the circles denote users. The relay R1 is used to serve users beyond the coverage range of the base station B while the relay R2 is used to serve the users within the coverage range of the base station who, otherwise, cannot be served due to data rate (or SINR) requirements. The dotted circles denote the coverage ranges of B, R1, and R2.
hφ(i)R pφ(i) + σi2
,
and the SINR for the ith M2 user is given as hiR pi
γi =
hφ(i)B pφ(i) + σi2
,
The paper is organized as follows. In Section 2, we introduce the notation and describe the single-cell network featuring a base station and a relay. In Section 3, the optimization problem is stated for this network. In Section 4, the SINR feasibility conditions are established for this network using certain properties of the M-matrices. In Section 5, these results are generalized to establish the SINR feasibility conditions for networks featuring multiple base stations and multiple relays. Simulation results for a single-cell network featuring one base station, one relay, and three users are presented in Section 6. The paper is concluded in Section 8 after a brief discussion in Section 7.
where σi2 is the background noise on the subchannel assigned to the ith user. We refer to this network as NM . To better illustrate our notation, let us consider the case of an OFDMA wireless network having two subchannels, say f1 and f2 , and a single base station enhanced by a relay station. Let there be three users, indexed 1, 2, 3. Thus, in this example, N = 2 and M = 3. Let us refer to this network as N3 . Suppose the base station assigns the subchannels f1 and f2 to users 1 and 2, respectively, and does not serve the third user, who is served by the relay station instead. Suppose the relay station assigns the subchannel f2 to user 3. Thus, user 1 experiences no interference whereas the users 2 and 3 interfere with each other. The SINRs for the users are as follows:
2. OFDMA system description
γ1 =
We assume that the OFDMA network differentiates between the uplink and downlink traffics, either in time (e.g., using time-domain duplexing) or frequency (e.g., using frequencydomain duplexing). In addition, we assume that the downlink transmissions from a base station to the relays (i.e., the so-called relay zones) are separated from the transmissions from the base station or relays to the users (i.e., the so-called access zones), again in time or frequency. This is commonly used, e.g. in WiMAX relay operations (see Genc, Murphy, Yu, & Murphy, 2008), so that users do not require any modifications to support relay operations. Thus, the downlink transmission progresses as the following repeating sequence of two time-slots: (i) in the first time-slot, the base station transmits the data to the relay station, and (ii) in the next time-slot, the base station and the relay transmit the data to their users (see Table 1). The OFDMA wireless network comprises a base station ‘‘B’’, a relay station ‘‘R’’, N subchannels and M users, indexed 1, 2, . . . , M, with 2N > M > N. Without loss of generality, we assume that M1 = N users receive the data directly from the base station; let us refer to this set of users as M1 . The remaining M2 = M − N users receive the data from the relay station; let us refer to this set of users as M2 . The base station assigns one orthogonal channel each to the M1 users while the relay station assigns one orthogonal channel each to the M2 users. Thus, an M1 user may experience interference from at most one M2 user and vice versa. So, given a user i, let φ(i) denote the user interfering with it. Let M denote the union of M1 and M2 . Without loss of generality, we index the users in M such that the first N users are from M1 . Let us denote the transmission power of the base station for the ith user as piB and that of the relay station for the ith user as piR . With a slight abuse of notation, we will, at times, refer to piB and piR as pi . All
h1B p1
σ
2 1
,
γ2 =
h2B p2 h2R p3 + σ
2 2
,
γ3 =
h3R p3 h3B p2 + σ32
.
3. Problem formulation Let M denote the number of users in the OFDMA networks. Then, the optimization problem that we wish to solve is as follows: min
M
p
Ci (pi )
s.t. γi ≥ γ i , 0 ≤ pi ≤ pmax ∀i,
(1)
i=1
.
where Ci (·) is a convex continuously differentiable function, p = [p1 p2 . . . pM ]T is the vector of the downlink transmit powers and γ i is the target SINR for user i. Let the channel gain for transmission to user i from the sender (base station or relay) of its interfering user φ(i) be denoted as his(φ(i)) . Let the channel gain for transmission to user φ(i) from the sender (base station or relay) of its interfering user i be denoted as hφ(i)s(i) . Let
.
Ai,j =
his(i) −his(j) γ¯j
if i = j; if j = φ(i).
(2)
.
b = [γ 1 σ12 γ 2 σ22 . . . γ M σM2 ]T ,
. Ω = {p ∈ RM : Ap ≥ b, pi ∈ [0, pmax ] ∀i}.
(3) (4)
For example, for the case of N3 , we have
. A=
h1B 0 0
0 h2B −h3B γ 3
0
−h2R γ 2 ; h3R
. b = [γ 1 σ12 γ 2 σ22 γ 3 σ32 ]T .
For the general case of an NM ℓ network in which at most ℓ > 1 users may interfere with a user i, φ(i) is not a single number but,
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Table 1 Notation. Symbol Meaning The set of users served by the base station The set of users served by the relay φ(i) The set of users interfering with user i s(i) Sender (B or R) of the downlink transmission to user i his(i) The channel gain for transmission to user i from its sender s(i) (which is either B or R) his(φ(i)) The channel gain for transmission to user i from the sender s(φ(i)) of φ(i) (which may be either B or R) hφ(i)s(i) The channel gain for transmission to user φ(i) from the sender s(i) of i (which may be either B or R) γi The target SINR for user i pi Transmit power for downlink transmission to user i diag(xi ) Diagonal matrix with xi as its elements NM A relay enabled OFDMA network with one base station and M users s.t. a user served by B may share at most one subchannel with a user served by the relay R NM ℓ A relay enabled OFDMA network with one base station and M users s.t. a user served by B may share up to ℓ subchannels with a user served by the relay R A(ij) (i, j)th element of the matrix A Z Set of non-negative integers
M1 M2
rather, a vector of indices of users that cause interference to user i. In this case, let
.
Ai,j =
his(i) −his(j) γ¯j
if i = j; if j ∈ φ(i).
(5)
. b = [γ 1 σ12 γ 2 σ22 . . . γ M σM2 ]T .
(6)
. Ω = {p ∈ RM : Ap ≥ b, pi ∈ [0, pmax ] ∀i}.
(7)
Now, the optimization problem given by (1) can be recast as M
min p
Ci (pi )
subject to p ∈ Ω .
(8)
i=1
Theorem 1 (SINR Guarantees in NM ). Consider the network NM . The optimization problem given by (2)–(4) and (8) is feasible if and only if his(φ(i)) his(i)
Proof. The objective function i=1 Ci (pi ) is clearly strictly convex in its arguments due to the strict convexity of Ci with respect to pi for all i. We now establish the convexity and compactness of the constraint set Ω . Let rowi (X ) denote the ith row of the matrix X . First, note that Ω is bounded by definition since pi ∈ [0, pmax ]. In addition, it is closed since it consists of the intersection of halfspaces, namely, rowi (A)p ≥ bi i = 1, 2, . . . , M, which include their respective separating hyperplanes, and the closed set pi ∈ [0, pmax ]. Thus, the constraint set Ω is compact. Finally, following a similar argument it is easy to see that it is also convex, since it is the intersection of convex sets. Hence the proof.
M
4. An upper bound on simultaneous downlink transmissions in NM and NM ℓ To determine the largest target SINR values that can be supported by a relay-enhanced base station in downlink transmissions in N3 , we now extend the technique used in Alpcan et al. (2008) to prove its Lemma 3.2 (also see Kulkarni, Kothare, & Safonov, 2010). We first note that a matrix C is said to be an M-matrix if it can be decomposed as C = kI − B where k is a positive real number, I is the identity matrix of suitable size, and B is a matrix comprising non-negative elements such that the spectral radius of B is strictly less than k (see Berman & Plemmons, 1994 and Horn & Johnson, 1991 for details on M-matrices).
(9)
for every user i that shares a subchannel with some other user φ(i). Furthermore, if (9) holds, then every p satisfying Ap ≥ b satisfies p > 0. Proof. To prove the result, we use the approach used in Alpcan et al. (2008) to prove Alpcan et al. (2008, Lemma 3.2). Let us rewrite . . Ap ≥ b as Ax ≥ b where x = [x1 x2 xM ]T , xi = his(i) pi (i = 1, 2, . . . , M ), and
For the sake of completeness, we now re-state Alpcan et al. (2008, Lemma 3.1) in the setting of our optimization problem. Lemma 1. Let Ω be non-empty. Then, the system problem given by (2)–(4) and (8) is a strictly convex optimization problem with a convex, compact constraint set, and hence admits a unique global minimum.
γi < 1
A(i,j)
1 h is(φ(i)) . γi = − h is(i) 0
if j = i; if j = φ(i);
(10)
else.
Now, if the condition (9) holds, then A is a nonsingular M-matrix. Note that b is componentwise positive. Hence, it follows from the result (Berman & Plemmons, 1994, Chapter 6, N39) that Ax ≥ 0 implies x ≥ 0, i.e., p ≥ 0 since the channel gains his(i) ∈ (0, 1]. This proves that if the condition (9) holds, then every p satisfying Ap ≥ b also satisfies p ≥ 0, whence Ω is non-empty, i.e., the given optimization problem is feasible. To prove that if the condition (9) does not hold then Ω is empty, let us use a contradiction. Suppose, on the contrary, that the condition (9) does not hold but Ω is non-empty. Thus, by assumption, A is not an M-matrix and there exists x ≥ 0 such that Ax ≥ b. Since each element of b is strictly positive, it follows that x and Ax are componentwise strictly positive. Hence, by Berman and Plemmons (1994, Chapter 6, I39), A is an M-matrix, which is a contradiction. Hence the proof. Remark 1. We assume that the base station and the relay have sufficiently large transmit power budgets. This assumption is important because the sum of the downlink transmit power at either the base station or the relay is constrained to be less than that threshold. If the threshold is too low, the solution p described in Theorem 1, which disregards the upper limit on transmit powers, might not be implementable in practice. In Kulkarni et al. (2010), sufficient conditions under which such a p may still lead to a stable closed-loop system are described. Next, we extend Theorem 1 to the general case of a NM ℓ network in which a user served by the base station may share up to ℓ subchannels with a user served by the relay. Let the vector of
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Table 2 Simulation parameters. Basestn (300, 300)
Relay 1 (593, 300)
Relay 2 (7, 300)
(365, 300)
User 2
Coords
User 1
(690, 350)
User 3 (670, 100)
(10, 150)
Case 1 Case 2 Case 3 Case 4
Used Used Used Used
Used Used Used Used
Unused Used Used Used
γ1 γ1 γ1 γ1
γ2 γ2 γ2 γ2
γ3 γ3 γ3 γ3
γ4 γ4 γ4 γ4
channel gains for transmission to user i from the sender (base station or relay) of its interfering users φ(i) be denoted as his(φ(i)) . Let the vector of channel gains for transmission to user φ(i) from the sender (base station or relay) of its interfering users i be denoted as hφ(i)s(i) . Note that a user i may experience interference from at most ℓ users. Then, the feasibility of SINR guarantees in NM ℓ is characterized as follows. Theorem 2 (SINR Guarantees in NM ℓ ). Consider the network NM ℓ . The optimization problem given by (5)–(8) is feasible if and only if
his(j) j∈φ(i)
his(i)
γi < 1
(11)
for every user i. Furthermore, if (11) holds, then every p satisfying Ap ≥ b satisfies p > 0. Proof. Since no two users served by the base station share the same subchannel and since no two users served by the relay R share the same subchannel, we can view the M-user system as a collection of mutually exclusive subsystems in which at most ℓ users may interfere with each other. Then, the arguments used to prove Theorem 1 can be easily adopted to prove the necessity and sufficiency of (11) for the feasibility of the optimization problem.
We now generalize the results derived in Section 4 to determine the SINR feasibility conditions for networks featuring multiple base stations, multiple relays, and possibly multiple (shared) subchannels for downlink transmission to each user. In this general case, the interference for the downlink transmission to user i on any given subchannel is caused by other base stations and other relay stations that use the same subchannel. Suppose the OFDMA network has L base stations. Let Γi denote the set of users in the ith cell. Let κ denote the number of users in the entire network. Let the users be indexed 1 through κ . For every user i, let φ(i) denote the set of interfering users—these users receive downlink transmissions either from one of the base stations or from one of the relays on the set, say, Ξ (i) of subchannels assigned to it. Then, the optimization problem of interest given by (1) is now generalized as follows: L
p
Ci (pi )
s.t. γi ≥ γ i , 0 ≤ pi ≤ pmax ∀i,
j=1 i∈Γj
.
where Ci (·) is a convex continuously differentiable function, p = [p1 p2 . . . pκ ]T is the vector of the downlink transmit powers and γ i is the target SINR for user i. This optimization problem can be rewritten as min p
L
Ci (pi ) s.t. Ap ≥ b, p ≥ 0,
(12)
j=1 i∈Γj
.
= 10 = 10 = 10 = 18
= 10 = 10 = 18 = 18
= 10 = 10 = 10 = 18
.
can rewrite the condition Ap ≥ b as Ax ≥ b where x = [x1 x2 . A is a full matrix that . . . xκ ], xi = his(i) pi (i = 1, 2, . . . , κ), and is defined as follows:
A(i,j)
1 h[ℓ] . is(j) γi = − [ℓ] h ℓ∈ Ξ ( i ) is(i) 0
if j = i; if j ∈ φ(i); else
[ℓ] where his(j)
is the transfer function for the communication channel for transmission from s(j) (which is either a base station or a relay) [ℓ] to user i on the ℓth subchannel—hij is taken to be zero if s(j) does not cause interference to user i on the ℓth subchannel. Theorem 3 (SINR Guarantees in the General Case). Consider an OFDMA network featuring multiple base stations, multiple relays, and possibly multiple shared subchannels for downlink transmissions to users. The optimization problem given by (12) is feasible if and only if
his[ℓ](j) j∈φ(i) ℓ∈Ξ (i)
[ℓ]
his(i)
γi < 1
(13)
for every user i. Furthermore, if (13) holds, then every p satisfying Ap ≥ b satisfies p > 0. Proof. The proof follows along the lines of the proof of Theorem 2.
5. SINR feasibility conditions for the general network
min
= 10 = 10 = 18 = 18
User 4
where b = [γ 1 σ12 γ 2 σ22 . . . γ κ σκ2 ], and A is suitably defined as earlier. After elementary exchanges of rows and columns, we
6. Simulation results We now present simulation results for a single-cell network comprising a base station, two relays and four users. The base station has two subchannels available to it. We assume that these subchannels have a path-loss exponent of 2. Thus, the gain of the communication channel between a node i and a node j is given as hji = 5000/d2ji , where dji is the distance between nodes i and j. The background noise on each subchannel at each node is chosen to be σ 2 = 0.1. We will look at four case scenarios in our simulation. In Case 1, the base station communicates directly with users 1 and 4 on channels 1 and 2 respectively. Relay 1 is used to communicate with users 2 and 3, on channels 1 and 2 respectively. Relay 2 is not used in this case. For the subsequent three case scenarios, relay 2 is used to forward the traffic to user 4, with a different target SINR in each case. Table 2 shows the relevant simulation parameters of node locations and target SINRs for the four cases. For Case 1, our Theorem 3 implies that the target SINR values are feasible for the users if and only if (i) [2]
(iii)
h3B [2 ] h3R1
γ 3 < 1 and (iv)
[2 ] h4R1 [2 ] h4B
[1 ] h1R1 [1] h1B
γ 1 < 1, (ii)
[1 ] h2B [1] h2R1
γ 2 < 1,
γ 4 < 1. On the basis of the locations
of the nodes in Table 2, we can see from Table 3 that the feasibility conditions are not met. As we next show, our simulation results do indeed validate this intuition facilitated by Theorem 3. To update the downlink transmit power at the base station, we use a power control algorithm described in Chiang, Hande, Lan, and Tan (2008), namely, Chiang et al. (2008, Algorithm 2.1).
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Fig. 2. Case 1: Instantaneous SINR values of all users. Since the conditions in Theorem 3 are not satisfied, the target SINR values are infeasible. Using the transmit power algorithm given by (14), although the target SINRs of users 1 and 2 could be met, the instantaneous SINRs of users 3 and 4 oscillate around a baseline value of 4.
Fig. 3. Case 2: Instantaneous SINR values of all users. Since the conditions in Theorem 3 are satisfied, the target SINR values are feasible with the addition of relay 2. Using the transmit power algorithm given by (14), these values are reached quickly.
Table 3 Feasibility condition term values—Case 1. [1] h1R1 [1] h1B
Case 1
[1] h2B [1] h2R1
γ1
0.813
γ2
[2] h3B [2] h3R1
γ3
[2] h3R1 [2] h3B
γ4
0.770
2.60
2.94
[1] h2B [1] h2R1
[2] h3R2 [2] h3R1
[2] h3R1 [2] h3R2
Table 4 Feasibility condition term values. [1] h1R1 [1] h1B
Case 2 Case 3 Case 4
γ1
0.813 1.46 1.46
γ2
0.770 0.770 1.39
γ3
0.958 1.72 1.72
γ4
0.621 0.621 1.12
Specifically, the transmit power pi of the base station (or the relay station) for the downlink transmission to user i is updated at each time instant as follows: pi [k + 1] =
γi pi [k], γi [k]
k∈Z
(14)
where γi [k] is the SINR at user i at time instant k. The initial power level is taken to be 0.1 for the base station as well as for the relay. Fig. 2 shows the target SINR values and the instantaneous SINR values for all the users for Case 1, after 40 iterations of the power update algorithm. In order to meet the target SINRs of all the users, we will attempt to use relay station 2 to transmit the traffic that is meant for user 4. So for subsequent case scenarios, we will have the base station communicating directly with user 1 on channel 1, using relay station 1 to communicate with users 2 and 3 on channels 1 and 2 respectively, and using relay station 2 to communicate with user 4 on channel 2. Under such a scenario, the feasibility conditions now become (i) [2 ] h4R1 [2 ] h4R2
[1 ] h1R1 [1] h1B
γ 1 < 1, (ii)
[1] h2B [1 ] h2R1
γ 2 < 1, (iii)
[2 ] h3R2 [2 ] h3R1
γ 3 < 1 and (iv)
γ 4 < 1. This constitutes the remaining cases, 2, 3, and 4, with
the corresponding target SINRs as shown in Table 2. Table 4 shows the new feasibility condition values with the addition of relay station 2. We see that in Case 2, even though the target SINR values remain the same as in Case 1, the feasibility conditions are now met, since all the values are less than 1. We would expect the target SINR values to now be achievable in our simulation. However, due to the different target SINR values in Cases 3 and 4 (see Table 2), not all of the feasibility conditions are met.
Fig. 4. Case 3: Instantaneous SINR values of all users. Since the conditions in Theorem 1 are not satisfied, the target SINR values are infeasible. Using the transmit power algorithm given by (14), the instantaneous SINRs of users 1 and 3 oscillate around a baseline value of 17. The instantaneous SINRs of users 2 and 4 oscillate around a baseline value of 9.5. We have verified that the instantaneous SINRs of all the users do not converge to the target value even after 100 iterations.
Figs. 3–5 show the target SINR values and the instantaneous SINR for the users of Cases 2, 3, and 4, respectively, after 40 iterations of the power update algorithm. 7. Discussion To implement our results (namely, Theorems 1–3), the base station and the relay need to know the values of his(i) and hiφ(i) for every user i. This information is not available directly. However, every base station and every relay has an estimate of his(i) for every user i served by it. In addition, it has the full knowledge of the downlink transmit power and the SINR for that user. By exchanging this information, hiφ(i) can be calculated using the SINR equations. A comprehensive overview of the radio resource management challenges in relay-enabled OFDMA networks is given in Salem, Adinoyi, Yanikomeroglu, and Falconer (2010b). An overview of the next generation wireless broadband networks is presented in Soldani and Dixit (2008). Several results exist on resource allocation for multihop OFDMA networks. For example, a sumrate maximization problem, wherein the locations of the relays and channel allocations are known, is solved in Kwak and Cioffi (2007). Assuming full information on the relay locations, a channel allocation problem subject to fixed power constraints is solved in Kim and Lee (2007).
V.V. Kulkarni et al. / Automatica 48 (2012) 1818–1824
Fig. 5. Case 4: Instantaneous SINR values of all users. Since the conditions in Theorem 1 are not satisfied, the target SINR values are infeasible. Using the transmit power algorithm given by (14), the instantaneous SINRs of all the users oscillate around a baseline value of 13. The oscillations are more pronounced compared to those in Case 3.
Salem, Adinoyi et al. (2010a) presented a comprehensive overview of using relays to improve the performances of OFDMA networks, including how such techniques relate to relaying in non-OFDMA networks. There have been a number of power and subchannel allocation solutions proposed. Sundaresan and Rangarajan (2008, 2009) highlight the gains that can be achieved in the spatial reuse of both relay and access hops. They formulate an optimization problem and propose approximation algorithms after proving its hardness. While (Sundaresan & Rangarajan, 2008, 2009) deal with a single-cell scenario (one base station), (Hua, Zhang, & Niu, 2010) focus on multi-cell OFDMA relay networks. They argue that inter-cell interference should be taken into consideration and propose a joint scheduling, subchannel and power allocation algorithm. Our results here also include considerations for multicell scenarios. Many of these solutions assume known relay positions (also Shi, Gong, & Wu, 2009 and Xiao, Zhang, Zhu, & Cuthbert, 2009). In contrast, our results determine whether the relays can help meet the SINR requirements and, if so, determine a set of appropriate locations for these relays. Relay placement in wireless networks has been investigated extensively in the past. For example, the optimal positions for placing the relays in a two-tiered wireless sensor network are studied in Guo, Huang, Lou, and Liang (2008). The relay placement in OFDMAbased networks where the source and relay nodes make use of decode-and-forward cooperative communication is studied in Lin, Ho, Xie, Shen, and Tapolcai (2010). The relay placement problem in an 802.16j network has been solved in Niyato, Hossain, Dong, and Zhu (2009) using stochastic programming after accounting for uncertainties in the user population and traffic patterns. Our approach is novel in its use of the properties of M-matrices. Several other power allocation solutions and subchannel allocation solutions assume known relay positions (see Shi et al., 2009 and Xiao et al., 2009). In contrast, our results determine whether the relays can help meet the SINR requirements and, if so, determine a set of appropriate locations for these relays. 8. Conclusion Extending the approach developed by Alpcan et al. (2008), we have investigated the feasibility of SINR guarantees in relay-enhanced OFDMA networks featuring stationary users. Our approach is based on checking the feasibility of a convex optimization problem of minimizing the total downlink transmit
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power subject to the constraints that (i) the SINR of every user exceeds a certain threshold and (ii) the transmit power is not to exceed a certain threshold. We first check the feasibility of a solution for the specific case (namely, the NM network) in which a user i served by the relay station shares at most one subchannel with a user j served by the base station. For such a network, using certain properties of M-matrices, we obtain analytical conditions (namely, Theorem 1) for the existence of a solution. Our Theorem 2 generalizes these results to the case (namely, the NM ℓ network) in which a user i served by the base station may share more than one subchannel with a user j served by the relay station. Our conditions are a function of the target SINR values and the channel gains. Our Theorem 3 generalizes these results to an arbitrarily large OFDMA network featuring multiple base stations, multiple relays, and possibly multiple (shared) OFDMA channels for downlink transmissions to the users. If the geographical positions of the base stations, the relay stations, and the users are fixed, our results can be used to determine the feasible SINR values for the users. Conversely, if these entities are not geographically fixed but the target SINR values are fixed, our results can be used to determine geographical locations for these entities that ensure that the target SINR values are realized. Acknowledgments We thank Prof. Tansu Alpcan (Berlin Technical University) for useful comments. References Alavi, S. M., Zhou, C., & Cheng, Y. (2009). Low complexity resource allocation algorithm for IEEE 802.16 OFDMA system. In IEEE international conference on communications. http://dx.doi.org/10.1109/ICC.2009.5199154. Alpcan, T., Fan, X., Basar, T., Arcak, M., & Wen, J. (2008). Power control for multicell CDMA wireless networks: a team optimization approach. Wireless Networks, 14(5), 647–657. Berman, A., & Plemmons, R. (1994). Classics in applied mathematics, Nonnegative matrices in the mathematical sciences. Philadelphia, PA: SIAM. Chiang, M., Hande, P., Lan, T., & Tan, C. (2008). Power control in wireless cellular networks. Foundations and Trends in Networking, 2(4), 381–533. Genc, V., Murphy, S., Yu, Y., & Murphy, J. (2008). IEEE 802.16j relay-based wireless access networks: an overview. IEEE Wireless Communications, 15(5), 56–63. Guo, W., Huang, X., Lou, W., & Liang, C. (2008). On relay node placement and assignment for two-tiered wireless networks. ACM Mobile Networks and Applications, 13, 186–197. Horn, R., & Johnson, C. (1991). Topics in matrix analysis. Cambridge, UK: Cambridge University Press. Hua, Y., Zhang, Q., & Niu, Z. (2010). Resource allocation in multi-cell OFDMA-based relay networks. In Proceedings of IEEE INFOCOMC (pp. 2133–2141). Shi, Jie, Gong, Ping, & Wu, Wei-ling (2009). Joint allocation of power and subcarriers for OFDMA relay network. In Proceedings of the 5th international conference on wireless communications, networking and mobile computing (pp. 34–37). Kim, M., & Lee, H. (2007). Radio resource management for a two-hop OFDMA relay system in downlink. In Proceedings of IEEE symposium on computers and communications (pp. 25–31). Kivanc, D., Li, G., & Liu, H. (2003). Computationally efficient bandwidth allocation and power control for OFDMA. IEEE Transactions on Wireless Communications, 2(6), 1150–1158. Kulkarni, V. V., Kothare, M. V., & Safonov, M. G. (2010). Decentralized dynamic nonlinear controllers to minimize transmit power in cellular networks—part I. Systems and Control Letters, 59(5), 294–298. Kwak, R., & Cioffi, J. (2007). Resource-allocation for OFDMA multi-hop relaying downlink systems. In Proceedings of IEEE global communications conference (pp. 3225–3229). Lin, B., Ho, P., Xie, L., Shen, X., & Tapolcai, J. (2010). Optimal relay station placement in broadband wireless access networks. IEEE Transactions on Mobile Computing, 9, 259–269. Niyato, D., Hossain, E., Dong, K., & Zhu, H. (2009). Joint optimization of placement and bandwidth reservation for relays in IEEE 802.16j mobile multihop networks. In Proceedings of IEEE international conference on communication. http://dx.doi.org/10.1109/ICC.2009.5199116. Salem, M., Adinoyi, A., Rahman, M., Yanikomeroglu, H., Falconer, D., Kim, YoungDoo, et al. (2010a). An overview of radio resource management in relayenhanced OFDMA-based networks. IEEE Communications Surveys and Tutorials, 12(3), 422–438. Salem, M., Adinoyi, A., Yanikomeroglu, H., & Falconer, D. (2010b). Opportunities and challenges in OFDMA-based cellular relay networks: a radio resource management perspective. IEEE Transactions on Vehicular Technology, 59(5), 2496–2510.
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Soldani, D., & Dixit, S. (2008). Wireless relays for broadband access. IEEE Communications Magazine, 46(3), 58–66. Sundaresan, K., & Rangarajan, S. (2008). On exploiting diversity and spatial reuse in relay-enabled wireless networks. In Proceedings of ACM international symposium on mobile ad hoc networking and computing. MobiHoc 2008 (pp. 13–22). Sundaresan, K., & Rangarajan, S. (2009). Efficient algorithms for leveraging spatial reuse in OFDMA relay networks. In Proceedings of IEEE INFOCOM (pp. 1539–1547). Xiao, L., Zhang, T., Zhu, Y., & Cuthbert, L. (2009). Two-hop subchannel scheduling and power allocation for fairness in OFDMA relay networks. In Proceedings of international conference on wireless and mobile communications (pp. 267– 271). Yang, Y., Hu, H., Xu, J., & Mao, G. (2009). Relay technologies for WiMAX and LTEadvanced mobile systems. IEEE Communications Magazine, 47(10), 100–105.
Vishwesh V. Kulkarni received a Ph.D. in Electrical Engineering from the University of Southern California, Los Angeles, CA, in 2001. He was a postdoctoral researcher at the Laboratory for Information and Decision Systems of the Massachusetts Institute of Technology, Cambridge, MA, during 2001–2003 and at the University of Colorado, Boulder, CO, during 2004–2005. He was an Assistant Professor at the Department of Electrical Engineering of the Indian Institute of Technology Bombay, Mumbai, India, during 2006–2010 and is now with the Department of Electrical Engineering, University of Minnesota,
Minneapolis. His research interests are nonlinear control systems, resource allocation in distributed systems, network modeling, and synthetic biological constructs. Joo Ghee Lim is currently a lecturer with the School of Electrical and Electronic Engineering at Singapore Polytechnic, Singapore. He received the B. Engg. and M. Engg. degrees in Electrical and Computer Engineering from the National University of Singapore and the Ph.D. degree in Computer Science and Engineering from the University of New South Wales, Australia. Previously, he was an Associate Scientist at the Institute for Infocomm Research, A∗ STAR, Singapore. His current research interests include MAC, QoS, and resource optimization in wireless networks. Sanjay K. Jha is a Professor and Head of the Network Group at the School of Computer Science and Engineering at the University of New South Wales. His research activities cover a wide range of topics in networking including wireless sensor networks, ad hoc/community wireless networks, resilience and multicasting in IP networks and security protocols for wired/wireless networks. Sanjay has published over 160 articles in high quality journals and conferences. He is the principal author of the book Engineering Internet QoS and a co-editor of the book Wireless Sensor Networks: A Systems Perspective. He is an associate editor of the IEEE Transactions on Mobile Computing.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Feasibility of SINR guarantees for downlink transmissions in relay-enabled OFDMA networks✩ Vishwesh V. Kulkarni a,1 , Joo Ghee Lim b , Sanjay K. Jha c a
Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455, United States
b
School of Electrical and Electronic Engineering, Singapore Polytechnic, Singapore 139651, Singapore
c
School of Computer Science and Engineering, University of New South Wales, Sydney, NSW 2033, Australia
article
info
Article history: Received 25 October 2010 Received in revised form 16 November 2011 Accepted 4 February 2012 Available online 18 June 2012 Keywords: OFDMA Downlink transmission SINR feasibility M-matrices
abstract We examine the feasibility of the signal-to-interference-and-noise-ratio (SINR) guarantees for downlink transmissions in relay-enhanced OFDMA networks that feature stationary users. The constraints are as follows: (i) the SINR of every user exceeds a certain threshold and (ii) the transmit power for each transmission is less than a certain threshold. We first derive a set of necessary and sufficient feasibility conditions for the specific case in which a user i served by the relay station shares at most one subchannel with a user j served by the base station. These conditions are a function of the target SINR values and the channel gains, and derived using a property of an M-matrix. If the geographical positions of the base station, the relay station, and the users are fixed, then our result determines the feasible SINR values for the users. Conversely, if these entities are not geographically fixed but the target SINR values are fixed, our result determines the geographical locations for these entities that ensure that the target SINR values are realized. We then extend these results to the case of networks featuring multiple base stations and multiple relays. Our conditions for checking the feasibility can be easily implemented in practice. Published by Elsevier Ltd
1. Introduction Orthogonal frequency division multiple-access (OFDMA) technology is expected to play a significant part in the next generation wireless networks such as WiMAX, 3GPP-LTE, and IEEE 802.22 WRAN (see Alavi, Zhou, & Cheng, 2009 and Kivanc, Li, & Liu, 2003). In OFDMA, the base station assigns sets of orthogonal subcarriers, i.e., subchannels, to the users. OFDMA can utilize frequency and multi-user diversities in order to maximize the system capacity. To further improve the system performance, relay stations are often introduced (see Soldani & Dixit, 2008): the examples include WiMAX (802.16j) and the 3GPP LTE-Advanced standards (see Yang, Hu, Xu, & Mao, 2009). The relays are wireless stations that (i) receive the data from a base station and transmit it to a set of users, and (ii) receive the data from a set of users and transmit it to
✩ This research was supported by the ARC Discovery Grant DP1096353. The material in this paper was partially presented at the 50th IEEE Conference on Decision and Control (CDC 2011), December 12–15, 2012, Orlando, Florida, USA. This paper was recommended for publication in revised form by Associate Editor Huijun Gao, under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (V.V. Kulkarni), [email protected] (J.G. Lim), [email protected] (S.K. Jha). 1 Tel.: +1 612 625 3300; fax: +1 612 625 4583.
0005-1098/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.automatica.2012.05.048
their designated base station (see Fig. 1). Now, if the SINR requirements of a set of users cannot be satisfied by a base station, could those be satisfied by adding a finite number of relays and, if so, where should the relays be placed? We investigate this problem in the context of downlink transmit power allocation in a relayenabled OFDMA network in which the number of users exceeds the number of subcarriers. To solve this problem, we build on the framework developed in Alpcan, Fan, Basar, Arcak, and Wen (2008) to characterize the maximum number of uplink transmissions that a base station can support in a multi-cell code division multiple-access (CDMA) network. Let the γ i denote the target signal-to-interferenceand-noise-ratio (SINR) values of user i and let L denote the CDMA spreading gain. Then, the base station can support the M uplink transmissions from M users simultaneously if i=1 γ i /(γ i + L) < 1 (see Alpcan et al., 2008, Lemma 2). We consider a similar convex optimization problem of minimizing the sum downlink transmit power subject to the constraint that the SINR exceeds the desired threshold for all users. In OFDMA networks, the CDMA spreading effect is absent, so L = 1. Hence, using the same logic, it trivially follows that, in an OFDMA network, a base station – unaided by a relay – may satisfactorily serve only as many users as there are subchannels available to it. In this paper, we make use of certain properties of M-matrices to establish the required feasibility conditions and extend these results to the general case of networks featuring multiple base stations and multiple relays.
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transmit powers pi obey the constraint pi ∈ [0, pmax ] for some predefined pmax ; we assume that pmax is sufficiently large. The channel gain from the base station to the ith user is denoted as hiB and the channel gain from the relay to the ith user is denoted as hiR . At times, we shall refer to these channel gains as his(i) , i.e., the channel gain for the transmission of the sender s(i) of user i to user i—the sender s(i) can be either the base station B or the relay station R. Both hiB and hiR are constrained to lie between 0 and 1. The received signal at the ith M1 user is xi = hiB pi + hφ(i)R pφ(i) . Likewise, the received signal at the ith M2 user is xi = hiR pi + hφ(i)B pφ(i) . Thus, the SINR for the ith M1 user is given as hiB pi
γi = Fig. 1. Two scenarios of a relay-enabled OFDMA network. The squares denote either the base station or a relay whereas the circles denote users. The relay R1 is used to serve users beyond the coverage range of the base station B while the relay R2 is used to serve the users within the coverage range of the base station who, otherwise, cannot be served due to data rate (or SINR) requirements. The dotted circles denote the coverage ranges of B, R1, and R2.
hφ(i)R pφ(i) + σi2
,
and the SINR for the ith M2 user is given as hiR pi
γi =
hφ(i)B pφ(i) + σi2
,
The paper is organized as follows. In Section 2, we introduce the notation and describe the single-cell network featuring a base station and a relay. In Section 3, the optimization problem is stated for this network. In Section 4, the SINR feasibility conditions are established for this network using certain properties of the M-matrices. In Section 5, these results are generalized to establish the SINR feasibility conditions for networks featuring multiple base stations and multiple relays. Simulation results for a single-cell network featuring one base station, one relay, and three users are presented in Section 6. The paper is concluded in Section 8 after a brief discussion in Section 7.
where σi2 is the background noise on the subchannel assigned to the ith user. We refer to this network as NM . To better illustrate our notation, let us consider the case of an OFDMA wireless network having two subchannels, say f1 and f2 , and a single base station enhanced by a relay station. Let there be three users, indexed 1, 2, 3. Thus, in this example, N = 2 and M = 3. Let us refer to this network as N3 . Suppose the base station assigns the subchannels f1 and f2 to users 1 and 2, respectively, and does not serve the third user, who is served by the relay station instead. Suppose the relay station assigns the subchannel f2 to user 3. Thus, user 1 experiences no interference whereas the users 2 and 3 interfere with each other. The SINRs for the users are as follows:
2. OFDMA system description
γ1 =
We assume that the OFDMA network differentiates between the uplink and downlink traffics, either in time (e.g., using time-domain duplexing) or frequency (e.g., using frequencydomain duplexing). In addition, we assume that the downlink transmissions from a base station to the relays (i.e., the so-called relay zones) are separated from the transmissions from the base station or relays to the users (i.e., the so-called access zones), again in time or frequency. This is commonly used, e.g. in WiMAX relay operations (see Genc, Murphy, Yu, & Murphy, 2008), so that users do not require any modifications to support relay operations. Thus, the downlink transmission progresses as the following repeating sequence of two time-slots: (i) in the first time-slot, the base station transmits the data to the relay station, and (ii) in the next time-slot, the base station and the relay transmit the data to their users (see Table 1). The OFDMA wireless network comprises a base station ‘‘B’’, a relay station ‘‘R’’, N subchannels and M users, indexed 1, 2, . . . , M, with 2N > M > N. Without loss of generality, we assume that M1 = N users receive the data directly from the base station; let us refer to this set of users as M1 . The remaining M2 = M − N users receive the data from the relay station; let us refer to this set of users as M2 . The base station assigns one orthogonal channel each to the M1 users while the relay station assigns one orthogonal channel each to the M2 users. Thus, an M1 user may experience interference from at most one M2 user and vice versa. So, given a user i, let φ(i) denote the user interfering with it. Let M denote the union of M1 and M2 . Without loss of generality, we index the users in M such that the first N users are from M1 . Let us denote the transmission power of the base station for the ith user as piB and that of the relay station for the ith user as piR . With a slight abuse of notation, we will, at times, refer to piB and piR as pi . All
h1B p1
σ
2 1
,
γ2 =
h2B p2 h2R p3 + σ
2 2
,
γ3 =
h3R p3 h3B p2 + σ32
.
3. Problem formulation Let M denote the number of users in the OFDMA networks. Then, the optimization problem that we wish to solve is as follows: min
M
p
Ci (pi )
s.t. γi ≥ γ i , 0 ≤ pi ≤ pmax ∀i,
(1)
i=1
.
where Ci (·) is a convex continuously differentiable function, p = [p1 p2 . . . pM ]T is the vector of the downlink transmit powers and γ i is the target SINR for user i. Let the channel gain for transmission to user i from the sender (base station or relay) of its interfering user φ(i) be denoted as his(φ(i)) . Let the channel gain for transmission to user φ(i) from the sender (base station or relay) of its interfering user i be denoted as hφ(i)s(i) . Let
.
Ai,j =
his(i) −his(j) γ¯j
if i = j; if j = φ(i).
(2)
.
b = [γ 1 σ12 γ 2 σ22 . . . γ M σM2 ]T ,
. Ω = {p ∈ RM : Ap ≥ b, pi ∈ [0, pmax ] ∀i}.
(3) (4)
For example, for the case of N3 , we have
. A=
h1B 0 0
0 h2B −h3B γ 3
0
−h2R γ 2 ; h3R
. b = [γ 1 σ12 γ 2 σ22 γ 3 σ32 ]T .
For the general case of an NM ℓ network in which at most ℓ > 1 users may interfere with a user i, φ(i) is not a single number but,
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Table 1 Notation. Symbol Meaning The set of users served by the base station The set of users served by the relay φ(i) The set of users interfering with user i s(i) Sender (B or R) of the downlink transmission to user i his(i) The channel gain for transmission to user i from its sender s(i) (which is either B or R) his(φ(i)) The channel gain for transmission to user i from the sender s(φ(i)) of φ(i) (which may be either B or R) hφ(i)s(i) The channel gain for transmission to user φ(i) from the sender s(i) of i (which may be either B or R) γi The target SINR for user i pi Transmit power for downlink transmission to user i diag(xi ) Diagonal matrix with xi as its elements NM A relay enabled OFDMA network with one base station and M users s.t. a user served by B may share at most one subchannel with a user served by the relay R NM ℓ A relay enabled OFDMA network with one base station and M users s.t. a user served by B may share up to ℓ subchannels with a user served by the relay R A(ij) (i, j)th element of the matrix A Z Set of non-negative integers
M1 M2
rather, a vector of indices of users that cause interference to user i. In this case, let
.
Ai,j =
his(i) −his(j) γ¯j
if i = j; if j ∈ φ(i).
(5)
. b = [γ 1 σ12 γ 2 σ22 . . . γ M σM2 ]T .
(6)
. Ω = {p ∈ RM : Ap ≥ b, pi ∈ [0, pmax ] ∀i}.
(7)
Now, the optimization problem given by (1) can be recast as M
min p
Ci (pi )
subject to p ∈ Ω .
(8)
i=1
Theorem 1 (SINR Guarantees in NM ). Consider the network NM . The optimization problem given by (2)–(4) and (8) is feasible if and only if his(φ(i)) his(i)
Proof. The objective function i=1 Ci (pi ) is clearly strictly convex in its arguments due to the strict convexity of Ci with respect to pi for all i. We now establish the convexity and compactness of the constraint set Ω . Let rowi (X ) denote the ith row of the matrix X . First, note that Ω is bounded by definition since pi ∈ [0, pmax ]. In addition, it is closed since it consists of the intersection of halfspaces, namely, rowi (A)p ≥ bi i = 1, 2, . . . , M, which include their respective separating hyperplanes, and the closed set pi ∈ [0, pmax ]. Thus, the constraint set Ω is compact. Finally, following a similar argument it is easy to see that it is also convex, since it is the intersection of convex sets. Hence the proof.
M
4. An upper bound on simultaneous downlink transmissions in NM and NM ℓ To determine the largest target SINR values that can be supported by a relay-enhanced base station in downlink transmissions in N3 , we now extend the technique used in Alpcan et al. (2008) to prove its Lemma 3.2 (also see Kulkarni, Kothare, & Safonov, 2010). We first note that a matrix C is said to be an M-matrix if it can be decomposed as C = kI − B where k is a positive real number, I is the identity matrix of suitable size, and B is a matrix comprising non-negative elements such that the spectral radius of B is strictly less than k (see Berman & Plemmons, 1994 and Horn & Johnson, 1991 for details on M-matrices).
(9)
for every user i that shares a subchannel with some other user φ(i). Furthermore, if (9) holds, then every p satisfying Ap ≥ b satisfies p > 0. Proof. To prove the result, we use the approach used in Alpcan et al. (2008) to prove Alpcan et al. (2008, Lemma 3.2). Let us rewrite . . Ap ≥ b as Ax ≥ b where x = [x1 x2 xM ]T , xi = his(i) pi (i = 1, 2, . . . , M ), and
For the sake of completeness, we now re-state Alpcan et al. (2008, Lemma 3.1) in the setting of our optimization problem. Lemma 1. Let Ω be non-empty. Then, the system problem given by (2)–(4) and (8) is a strictly convex optimization problem with a convex, compact constraint set, and hence admits a unique global minimum.
γi < 1
A(i,j)
1 h is(φ(i)) . γi = − h is(i) 0
if j = i; if j = φ(i);
(10)
else.
Now, if the condition (9) holds, then A is a nonsingular M-matrix. Note that b is componentwise positive. Hence, it follows from the result (Berman & Plemmons, 1994, Chapter 6, N39) that Ax ≥ 0 implies x ≥ 0, i.e., p ≥ 0 since the channel gains his(i) ∈ (0, 1]. This proves that if the condition (9) holds, then every p satisfying Ap ≥ b also satisfies p ≥ 0, whence Ω is non-empty, i.e., the given optimization problem is feasible. To prove that if the condition (9) does not hold then Ω is empty, let us use a contradiction. Suppose, on the contrary, that the condition (9) does not hold but Ω is non-empty. Thus, by assumption, A is not an M-matrix and there exists x ≥ 0 such that Ax ≥ b. Since each element of b is strictly positive, it follows that x and Ax are componentwise strictly positive. Hence, by Berman and Plemmons (1994, Chapter 6, I39), A is an M-matrix, which is a contradiction. Hence the proof. Remark 1. We assume that the base station and the relay have sufficiently large transmit power budgets. This assumption is important because the sum of the downlink transmit power at either the base station or the relay is constrained to be less than that threshold. If the threshold is too low, the solution p described in Theorem 1, which disregards the upper limit on transmit powers, might not be implementable in practice. In Kulkarni et al. (2010), sufficient conditions under which such a p may still lead to a stable closed-loop system are described. Next, we extend Theorem 1 to the general case of a NM ℓ network in which a user served by the base station may share up to ℓ subchannels with a user served by the relay. Let the vector of
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Table 2 Simulation parameters. Basestn (300, 300)
Relay 1 (593, 300)
Relay 2 (7, 300)
(365, 300)
User 2
Coords
User 1
(690, 350)
User 3 (670, 100)
(10, 150)
Case 1 Case 2 Case 3 Case 4
Used Used Used Used
Used Used Used Used
Unused Used Used Used
γ1 γ1 γ1 γ1
γ2 γ2 γ2 γ2
γ3 γ3 γ3 γ3
γ4 γ4 γ4 γ4
channel gains for transmission to user i from the sender (base station or relay) of its interfering users φ(i) be denoted as his(φ(i)) . Let the vector of channel gains for transmission to user φ(i) from the sender (base station or relay) of its interfering users i be denoted as hφ(i)s(i) . Note that a user i may experience interference from at most ℓ users. Then, the feasibility of SINR guarantees in NM ℓ is characterized as follows. Theorem 2 (SINR Guarantees in NM ℓ ). Consider the network NM ℓ . The optimization problem given by (5)–(8) is feasible if and only if
his(j) j∈φ(i)
his(i)
γi < 1
(11)
for every user i. Furthermore, if (11) holds, then every p satisfying Ap ≥ b satisfies p > 0. Proof. Since no two users served by the base station share the same subchannel and since no two users served by the relay R share the same subchannel, we can view the M-user system as a collection of mutually exclusive subsystems in which at most ℓ users may interfere with each other. Then, the arguments used to prove Theorem 1 can be easily adopted to prove the necessity and sufficiency of (11) for the feasibility of the optimization problem.
We now generalize the results derived in Section 4 to determine the SINR feasibility conditions for networks featuring multiple base stations, multiple relays, and possibly multiple (shared) subchannels for downlink transmission to each user. In this general case, the interference for the downlink transmission to user i on any given subchannel is caused by other base stations and other relay stations that use the same subchannel. Suppose the OFDMA network has L base stations. Let Γi denote the set of users in the ith cell. Let κ denote the number of users in the entire network. Let the users be indexed 1 through κ . For every user i, let φ(i) denote the set of interfering users—these users receive downlink transmissions either from one of the base stations or from one of the relays on the set, say, Ξ (i) of subchannels assigned to it. Then, the optimization problem of interest given by (1) is now generalized as follows: L
p
Ci (pi )
s.t. γi ≥ γ i , 0 ≤ pi ≤ pmax ∀i,
j=1 i∈Γj
.
where Ci (·) is a convex continuously differentiable function, p = [p1 p2 . . . pκ ]T is the vector of the downlink transmit powers and γ i is the target SINR for user i. This optimization problem can be rewritten as min p
L
Ci (pi ) s.t. Ap ≥ b, p ≥ 0,
(12)
j=1 i∈Γj
.
= 10 = 10 = 10 = 18
= 10 = 10 = 18 = 18
= 10 = 10 = 10 = 18
.
can rewrite the condition Ap ≥ b as Ax ≥ b where x = [x1 x2 . A is a full matrix that . . . xκ ], xi = his(i) pi (i = 1, 2, . . . , κ), and is defined as follows:
A(i,j)
1 h[ℓ] . is(j) γi = − [ℓ] h ℓ∈ Ξ ( i ) is(i) 0
if j = i; if j ∈ φ(i); else
[ℓ] where his(j)
is the transfer function for the communication channel for transmission from s(j) (which is either a base station or a relay) [ℓ] to user i on the ℓth subchannel—hij is taken to be zero if s(j) does not cause interference to user i on the ℓth subchannel. Theorem 3 (SINR Guarantees in the General Case). Consider an OFDMA network featuring multiple base stations, multiple relays, and possibly multiple shared subchannels for downlink transmissions to users. The optimization problem given by (12) is feasible if and only if
his[ℓ](j) j∈φ(i) ℓ∈Ξ (i)
[ℓ]
his(i)
γi < 1
(13)
for every user i. Furthermore, if (13) holds, then every p satisfying Ap ≥ b satisfies p > 0. Proof. The proof follows along the lines of the proof of Theorem 2.
5. SINR feasibility conditions for the general network
min
= 10 = 10 = 18 = 18
User 4
where b = [γ 1 σ12 γ 2 σ22 . . . γ κ σκ2 ], and A is suitably defined as earlier. After elementary exchanges of rows and columns, we
6. Simulation results We now present simulation results for a single-cell network comprising a base station, two relays and four users. The base station has two subchannels available to it. We assume that these subchannels have a path-loss exponent of 2. Thus, the gain of the communication channel between a node i and a node j is given as hji = 5000/d2ji , where dji is the distance between nodes i and j. The background noise on each subchannel at each node is chosen to be σ 2 = 0.1. We will look at four case scenarios in our simulation. In Case 1, the base station communicates directly with users 1 and 4 on channels 1 and 2 respectively. Relay 1 is used to communicate with users 2 and 3, on channels 1 and 2 respectively. Relay 2 is not used in this case. For the subsequent three case scenarios, relay 2 is used to forward the traffic to user 4, with a different target SINR in each case. Table 2 shows the relevant simulation parameters of node locations and target SINRs for the four cases. For Case 1, our Theorem 3 implies that the target SINR values are feasible for the users if and only if (i) [2]
(iii)
h3B [2 ] h3R1
γ 3 < 1 and (iv)
[2 ] h4R1 [2 ] h4B
[1 ] h1R1 [1] h1B
γ 1 < 1, (ii)
[1 ] h2B [1] h2R1
γ 2 < 1,
γ 4 < 1. On the basis of the locations
of the nodes in Table 2, we can see from Table 3 that the feasibility conditions are not met. As we next show, our simulation results do indeed validate this intuition facilitated by Theorem 3. To update the downlink transmit power at the base station, we use a power control algorithm described in Chiang, Hande, Lan, and Tan (2008), namely, Chiang et al. (2008, Algorithm 2.1).
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Fig. 2. Case 1: Instantaneous SINR values of all users. Since the conditions in Theorem 3 are not satisfied, the target SINR values are infeasible. Using the transmit power algorithm given by (14), although the target SINRs of users 1 and 2 could be met, the instantaneous SINRs of users 3 and 4 oscillate around a baseline value of 4.
Fig. 3. Case 2: Instantaneous SINR values of all users. Since the conditions in Theorem 3 are satisfied, the target SINR values are feasible with the addition of relay 2. Using the transmit power algorithm given by (14), these values are reached quickly.
Table 3 Feasibility condition term values—Case 1. [1] h1R1 [1] h1B
Case 1
[1] h2B [1] h2R1
γ1
0.813
γ2
[2] h3B [2] h3R1
γ3
[2] h3R1 [2] h3B
γ4
0.770
2.60
2.94
[1] h2B [1] h2R1
[2] h3R2 [2] h3R1
[2] h3R1 [2] h3R2
Table 4 Feasibility condition term values. [1] h1R1 [1] h1B
Case 2 Case 3 Case 4
γ1
0.813 1.46 1.46
γ2
0.770 0.770 1.39
γ3
0.958 1.72 1.72
γ4
0.621 0.621 1.12
Specifically, the transmit power pi of the base station (or the relay station) for the downlink transmission to user i is updated at each time instant as follows: pi [k + 1] =
γi pi [k], γi [k]
k∈Z
(14)
where γi [k] is the SINR at user i at time instant k. The initial power level is taken to be 0.1 for the base station as well as for the relay. Fig. 2 shows the target SINR values and the instantaneous SINR values for all the users for Case 1, after 40 iterations of the power update algorithm. In order to meet the target SINRs of all the users, we will attempt to use relay station 2 to transmit the traffic that is meant for user 4. So for subsequent case scenarios, we will have the base station communicating directly with user 1 on channel 1, using relay station 1 to communicate with users 2 and 3 on channels 1 and 2 respectively, and using relay station 2 to communicate with user 4 on channel 2. Under such a scenario, the feasibility conditions now become (i) [2 ] h4R1 [2 ] h4R2
[1 ] h1R1 [1] h1B
γ 1 < 1, (ii)
[1] h2B [1 ] h2R1
γ 2 < 1, (iii)
[2 ] h3R2 [2 ] h3R1
γ 3 < 1 and (iv)
γ 4 < 1. This constitutes the remaining cases, 2, 3, and 4, with
the corresponding target SINRs as shown in Table 2. Table 4 shows the new feasibility condition values with the addition of relay station 2. We see that in Case 2, even though the target SINR values remain the same as in Case 1, the feasibility conditions are now met, since all the values are less than 1. We would expect the target SINR values to now be achievable in our simulation. However, due to the different target SINR values in Cases 3 and 4 (see Table 2), not all of the feasibility conditions are met.
Fig. 4. Case 3: Instantaneous SINR values of all users. Since the conditions in Theorem 1 are not satisfied, the target SINR values are infeasible. Using the transmit power algorithm given by (14), the instantaneous SINRs of users 1 and 3 oscillate around a baseline value of 17. The instantaneous SINRs of users 2 and 4 oscillate around a baseline value of 9.5. We have verified that the instantaneous SINRs of all the users do not converge to the target value even after 100 iterations.
Figs. 3–5 show the target SINR values and the instantaneous SINR for the users of Cases 2, 3, and 4, respectively, after 40 iterations of the power update algorithm. 7. Discussion To implement our results (namely, Theorems 1–3), the base station and the relay need to know the values of his(i) and hiφ(i) for every user i. This information is not available directly. However, every base station and every relay has an estimate of his(i) for every user i served by it. In addition, it has the full knowledge of the downlink transmit power and the SINR for that user. By exchanging this information, hiφ(i) can be calculated using the SINR equations. A comprehensive overview of the radio resource management challenges in relay-enabled OFDMA networks is given in Salem, Adinoyi, Yanikomeroglu, and Falconer (2010b). An overview of the next generation wireless broadband networks is presented in Soldani and Dixit (2008). Several results exist on resource allocation for multihop OFDMA networks. For example, a sumrate maximization problem, wherein the locations of the relays and channel allocations are known, is solved in Kwak and Cioffi (2007). Assuming full information on the relay locations, a channel allocation problem subject to fixed power constraints is solved in Kim and Lee (2007).
V.V. Kulkarni et al. / Automatica 48 (2012) 1818–1824
Fig. 5. Case 4: Instantaneous SINR values of all users. Since the conditions in Theorem 1 are not satisfied, the target SINR values are infeasible. Using the transmit power algorithm given by (14), the instantaneous SINRs of all the users oscillate around a baseline value of 13. The oscillations are more pronounced compared to those in Case 3.
Salem, Adinoyi et al. (2010a) presented a comprehensive overview of using relays to improve the performances of OFDMA networks, including how such techniques relate to relaying in non-OFDMA networks. There have been a number of power and subchannel allocation solutions proposed. Sundaresan and Rangarajan (2008, 2009) highlight the gains that can be achieved in the spatial reuse of both relay and access hops. They formulate an optimization problem and propose approximation algorithms after proving its hardness. While (Sundaresan & Rangarajan, 2008, 2009) deal with a single-cell scenario (one base station), (Hua, Zhang, & Niu, 2010) focus on multi-cell OFDMA relay networks. They argue that inter-cell interference should be taken into consideration and propose a joint scheduling, subchannel and power allocation algorithm. Our results here also include considerations for multicell scenarios. Many of these solutions assume known relay positions (also Shi, Gong, & Wu, 2009 and Xiao, Zhang, Zhu, & Cuthbert, 2009). In contrast, our results determine whether the relays can help meet the SINR requirements and, if so, determine a set of appropriate locations for these relays. Relay placement in wireless networks has been investigated extensively in the past. For example, the optimal positions for placing the relays in a two-tiered wireless sensor network are studied in Guo, Huang, Lou, and Liang (2008). The relay placement in OFDMAbased networks where the source and relay nodes make use of decode-and-forward cooperative communication is studied in Lin, Ho, Xie, Shen, and Tapolcai (2010). The relay placement problem in an 802.16j network has been solved in Niyato, Hossain, Dong, and Zhu (2009) using stochastic programming after accounting for uncertainties in the user population and traffic patterns. Our approach is novel in its use of the properties of M-matrices. Several other power allocation solutions and subchannel allocation solutions assume known relay positions (see Shi et al., 2009 and Xiao et al., 2009). In contrast, our results determine whether the relays can help meet the SINR requirements and, if so, determine a set of appropriate locations for these relays. 8. Conclusion Extending the approach developed by Alpcan et al. (2008), we have investigated the feasibility of SINR guarantees in relay-enhanced OFDMA networks featuring stationary users. Our approach is based on checking the feasibility of a convex optimization problem of minimizing the total downlink transmit
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power subject to the constraints that (i) the SINR of every user exceeds a certain threshold and (ii) the transmit power is not to exceed a certain threshold. We first check the feasibility of a solution for the specific case (namely, the NM network) in which a user i served by the relay station shares at most one subchannel with a user j served by the base station. For such a network, using certain properties of M-matrices, we obtain analytical conditions (namely, Theorem 1) for the existence of a solution. Our Theorem 2 generalizes these results to the case (namely, the NM ℓ network) in which a user i served by the base station may share more than one subchannel with a user j served by the relay station. Our conditions are a function of the target SINR values and the channel gains. Our Theorem 3 generalizes these results to an arbitrarily large OFDMA network featuring multiple base stations, multiple relays, and possibly multiple (shared) OFDMA channels for downlink transmissions to the users. If the geographical positions of the base stations, the relay stations, and the users are fixed, our results can be used to determine the feasible SINR values for the users. Conversely, if these entities are not geographically fixed but the target SINR values are fixed, our results can be used to determine geographical locations for these entities that ensure that the target SINR values are realized. Acknowledgments We thank Prof. Tansu Alpcan (Berlin Technical University) for useful comments. References Alavi, S. M., Zhou, C., & Cheng, Y. (2009). Low complexity resource allocation algorithm for IEEE 802.16 OFDMA system. In IEEE international conference on communications. http://dx.doi.org/10.1109/ICC.2009.5199154. Alpcan, T., Fan, X., Basar, T., Arcak, M., & Wen, J. (2008). Power control for multicell CDMA wireless networks: a team optimization approach. Wireless Networks, 14(5), 647–657. Berman, A., & Plemmons, R. (1994). Classics in applied mathematics, Nonnegative matrices in the mathematical sciences. Philadelphia, PA: SIAM. Chiang, M., Hande, P., Lan, T., & Tan, C. (2008). Power control in wireless cellular networks. Foundations and Trends in Networking, 2(4), 381–533. Genc, V., Murphy, S., Yu, Y., & Murphy, J. (2008). IEEE 802.16j relay-based wireless access networks: an overview. IEEE Wireless Communications, 15(5), 56–63. Guo, W., Huang, X., Lou, W., & Liang, C. (2008). On relay node placement and assignment for two-tiered wireless networks. ACM Mobile Networks and Applications, 13, 186–197. Horn, R., & Johnson, C. (1991). Topics in matrix analysis. Cambridge, UK: Cambridge University Press. Hua, Y., Zhang, Q., & Niu, Z. (2010). Resource allocation in multi-cell OFDMA-based relay networks. In Proceedings of IEEE INFOCOMC (pp. 2133–2141). Shi, Jie, Gong, Ping, & Wu, Wei-ling (2009). Joint allocation of power and subcarriers for OFDMA relay network. In Proceedings of the 5th international conference on wireless communications, networking and mobile computing (pp. 34–37). Kim, M., & Lee, H. (2007). Radio resource management for a two-hop OFDMA relay system in downlink. In Proceedings of IEEE symposium on computers and communications (pp. 25–31). Kivanc, D., Li, G., & Liu, H. (2003). Computationally efficient bandwidth allocation and power control for OFDMA. IEEE Transactions on Wireless Communications, 2(6), 1150–1158. Kulkarni, V. V., Kothare, M. V., & Safonov, M. G. (2010). Decentralized dynamic nonlinear controllers to minimize transmit power in cellular networks—part I. Systems and Control Letters, 59(5), 294–298. Kwak, R., & Cioffi, J. (2007). Resource-allocation for OFDMA multi-hop relaying downlink systems. In Proceedings of IEEE global communications conference (pp. 3225–3229). Lin, B., Ho, P., Xie, L., Shen, X., & Tapolcai, J. (2010). Optimal relay station placement in broadband wireless access networks. IEEE Transactions on Mobile Computing, 9, 259–269. Niyato, D., Hossain, E., Dong, K., & Zhu, H. (2009). Joint optimization of placement and bandwidth reservation for relays in IEEE 802.16j mobile multihop networks. In Proceedings of IEEE international conference on communication. http://dx.doi.org/10.1109/ICC.2009.5199116. Salem, M., Adinoyi, A., Rahman, M., Yanikomeroglu, H., Falconer, D., Kim, YoungDoo, et al. (2010a). An overview of radio resource management in relayenhanced OFDMA-based networks. IEEE Communications Surveys and Tutorials, 12(3), 422–438. Salem, M., Adinoyi, A., Yanikomeroglu, H., & Falconer, D. (2010b). Opportunities and challenges in OFDMA-based cellular relay networks: a radio resource management perspective. IEEE Transactions on Vehicular Technology, 59(5), 2496–2510.
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Soldani, D., & Dixit, S. (2008). Wireless relays for broadband access. IEEE Communications Magazine, 46(3), 58–66. Sundaresan, K., & Rangarajan, S. (2008). On exploiting diversity and spatial reuse in relay-enabled wireless networks. In Proceedings of ACM international symposium on mobile ad hoc networking and computing. MobiHoc 2008 (pp. 13–22). Sundaresan, K., & Rangarajan, S. (2009). Efficient algorithms for leveraging spatial reuse in OFDMA relay networks. In Proceedings of IEEE INFOCOM (pp. 1539–1547). Xiao, L., Zhang, T., Zhu, Y., & Cuthbert, L. (2009). Two-hop subchannel scheduling and power allocation for fairness in OFDMA relay networks. In Proceedings of international conference on wireless and mobile communications (pp. 267– 271). Yang, Y., Hu, H., Xu, J., & Mao, G. (2009). Relay technologies for WiMAX and LTEadvanced mobile systems. IEEE Communications Magazine, 47(10), 100–105.
Vishwesh V. Kulkarni received a Ph.D. in Electrical Engineering from the University of Southern California, Los Angeles, CA, in 2001. He was a postdoctoral researcher at the Laboratory for Information and Decision Systems of the Massachusetts Institute of Technology, Cambridge, MA, during 2001–2003 and at the University of Colorado, Boulder, CO, during 2004–2005. He was an Assistant Professor at the Department of Electrical Engineering of the Indian Institute of Technology Bombay, Mumbai, India, during 2006–2010 and is now with the Department of Electrical Engineering, University of Minnesota,
Minneapolis. His research interests are nonlinear control systems, resource allocation in distributed systems, network modeling, and synthetic biological constructs. Joo Ghee Lim is currently a lecturer with the School of Electrical and Electronic Engineering at Singapore Polytechnic, Singapore. He received the B. Engg. and M. Engg. degrees in Electrical and Computer Engineering from the National University of Singapore and the Ph.D. degree in Computer Science and Engineering from the University of New South Wales, Australia. Previously, he was an Associate Scientist at the Institute for Infocomm Research, A∗ STAR, Singapore. His current research interests include MAC, QoS, and resource optimization in wireless networks. Sanjay K. Jha is a Professor and Head of the Network Group at the School of Computer Science and Engineering at the University of New South Wales. His research activities cover a wide range of topics in networking including wireless sensor networks, ad hoc/community wireless networks, resilience and multicasting in IP networks and security protocols for wired/wireless networks. Sanjay has published over 160 articles in high quality journals and conferences. He is the principal author of the book Engineering Internet QoS and a co-editor of the book Wireless Sensor Networks: A Systems Perspective. He is an associate editor of the IEEE Transactions on Mobile Computing.