Resource Allocation in Topology Management of Asymmetric Wireless Interference Networks †
Xinyu Zhang† , Nan Zhao† , F. Richard Yu∗ , and Victor C.M. Leung§
School of Inform. and Commun. Eng., Dalian University of Technology, Dalian, Liaoning, P. R. China Department of Systems and Computer Engineering, Carleton University, Ottawa, ON, K1S 5B6, Canada § Department of Electrical and Computer Engineering, the University of British Columbia, Vancouver, BC V6T 1Z4 Canada Email:
[email protected],
[email protected];
[email protected];
[email protected] ∗
Abstract—Most research works of interference alignment (IA) focus on the symmetric networks. When the difference of path loss is considered in asymmetric networks, topology management (TM) needs to be carefully designed for IA-based interference networks, through separating the network into an IA subnetwork and some spatial multiplexing (SM) subnetworks. Nevertheless, the resource allocation problem has been largely ignored in previous works on TM for IA-based networks. In this paper, antenna selection (AS) and power allocation (PA) are exploited to further improve the performance of IA-based networks. We first apply AS technique to the IA subnetwork, through fully utilizing the redundant antennas. Then the transmitted power is allocated among the transmitters of both the IA and SM subnetworks, to optimize the spectrum efficiency. Based on these two techniques, the joint optimization of AS and PA is developed through a stepped resource allocation optimization strategy to further improve the performance with low computational complexity. Simulation results are presented to show the effectiveness of the proposed schemes. Index Terms—Interference alignment, topology management, asymmetric network, antenna selection, power allocation.
I. I NTRODUCTION Interference is a major factor that restricts the performance of modern wireless systems, and interference alignment (IA) was recently proposed by Cadambe and Jafar in [1], [2] as an effective interference management method, which has attracted significant interests. In IA-based networks, interferences are aligned into certain subspaces at unintended receivers, so that the desired signal can be retrieved in the remaining subspaces free of interference. In conventional IA-based networks, the network topology is assumed to be symmetric. However, in practical systems, the symmetry is difficult to be satisfied due to the different location and mobility of users. Therefore, several works have been concentrated on asymmetric IA-based networks [3], [4]. In [3], through effective clustering algorithms, the asymmetric interference network was divided into several disjoint IA-based clusters, and the proposed scheme can achieve a balance between mitigating the signaling overhead and maximizing the achievable rate. In [4], a spectrum-efficient topology management (TM) scheme This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 61201224, and the Fundamental Research Funds for the Central Universities under DUT14QY44. Nan Zhao is the corresponding author.
combining IA and spatial multiplexing (SM) was proposed for asymmetric interference networks. Through eliminating the strong interference by IA and treating the weak interference as noise, the whole network would be divided into an IA subnetwork and several SM subnetworks. Although the performance of asymmetric IA-based network can be improved by the TM scheme, its resource allocation has not been considered, e.g., antenna selection (AS), power allocation (PA), etc. AS is a powerful technique to improve the performance of multi-input and multi-output (MIMO) systems, which can obtain the selection diversity with the quality of service (QoS) improved [5] and has been introduced into several kinds of IA-based networks [6], [7]. Koltz and Sezgin jointly considered the IA and AS techniques in wireless interference networks [6], and proposed several antenna selection criteria based on different properties of the channel to improve the performance. In [7], Li et al. investigated the AS technique in the IA-based cognitive radio network, and proposed a novel algorithm based on discrete stochastic optimization to deal with the problems of imperfect channel state information (CSI) and high computational complexity. On the other hand, the PA strategy is also an important method of resource allocation in wireless interference networks, through which the transmitted power is allocated among users to optimize the performance. In [8], the power control strategy on interference channels under sum power constraint was considered by Hassan et al., and two low-complexity suboptimal algorithms were proposed to optimize the sum rate of the networks. The PA problem was studied for IA-based networks in [9], and through the proposed algorithm, the energy efficiency of the network can be improved significantly. In this paper, we study the resource allocation problem in topology management [4] for IA-based interference networks. Specifically, AS and PA techniques are jointly utilized to further improve the spectrum efficiency of IA-based interference networks. To reduce computational complexity, a stepped resource allocation optimization strategy is proposed to solve the joint optimization problem. Simulation results are presented to show the effectiveness of the proposed schemes. Notation: A† and ∣A∣ are the Hermitian transpose and determinant of matrix A, respectively. ℂ𝑀 ×𝑁 is the space of complex 𝑀 × 𝑁 matrices. 𝒞𝒩 (a, A) is the complex Gaussian
978-1-5090-1698-3/16/$31.00 ©2016 IEEE
distribution with mean a and covariance matrix A. Tr(A) is the trace of the matrix A. II. S YSTEM D ESCRIPTION Consider an asymmetric interference network with 𝐾-user as shown in Fig. 1, and for the 𝑘-th pair, 𝑀 [𝑘] and 𝑁 [𝑘] antennas are equipped at its transmitter and receiver, respectively. Due to the asymmetric distribution of the users, the largescale fading gain from from the 𝑖-th Tx to the 𝑗-th Rx 𝜌[𝑗𝑖] is introduced into the model as 𝜌[𝑗𝑖] = 𝑟[𝑗𝑖]−𝛼 ,
(1)
where 𝑟[𝑗𝑖] is the distance between the 𝑖-th Tx and the 𝑗-th Rx, and 𝛼 represents the path-loss exponent determined by the wireless environment. Hence, the received signal with 𝑑[𝑗] data streams at the 𝑗-th Rx can be given as y
[𝑗]
=
𝐾 √ ∑
𝜌[𝑗𝑖] U[𝑗]† H[𝑗𝑖] V[𝑖] x[𝑖]
+U
[𝑗]† [𝑗]
z ,
(2)
𝑖=1 [𝑗]
[𝑖]
where H[𝑗𝑖] ∈ ℂ𝑁 ×𝑀 denotes the small-scale fading gain from the 𝑖-th Tx to the 𝑗-th Rx, and each entity of H[𝑗𝑖] is independent and identically distributed (i.i.d.) with the complex Gaussian distribution 𝒞𝒩 (0, 1). The unitary ma[𝑖] [𝑖] [𝑖] [𝑖] trices V[𝑖] ∈ ℂ𝑀 ×𝑑 and U[𝑖] ∈ ℂ𝑁 ×𝑑 represent the precoding and decoding matrices of the 𝑖-th pair, respectively. x[𝑖] is the data streams from the 𝑖-th[ Tx to the ] 𝑖-th[𝑘]Rx with a equal power constraint 𝑃𝑡 , i.e., E ∣∣x[𝑖] ∣∣2 = 𝑃𝑡 = 𝑃𝑡 . The additive Gaussian noise with distribution 𝒞𝒩 (0, 𝜎 2 I𝑁 [𝑗] ) [𝑗] at the 𝑗-th Rx is denoted as z[𝑗] ∈ ℂ𝑁 ×1 . In [4], a TM scheme was proposed to improve the spectrum efficiency of asymmetric IA-based interference networks. In the proposed scheme, according to the distribution of the location, the pairs close to each other will jointly comprise one IA subnetwork 𝒜, in which linear IA is adopted to eliminate the strong interference among IA pairs, and the weak intersubnetwork interference is treated as noise. On the other hand, those pairs (the 3-th and 5-th pairs as shown in Fig. 1), which are far away from the other pairs, will act as some SM subnetworks independently (belonging to the set 𝒮), in which SM is applied and the weak inter-subnetwork interference is also treated as noise. For simplicity and convenience, we assume that the antenna configuration of each pair is the same in the rest of this paper, i.e., 𝑀 [𝑗] = 𝑀 , 𝑁 [𝑗] = 𝑁 . Besides, according to the TM scheme, when the 𝑗-th pair acts as an IA pair, the number of its data streams 𝑑[𝑗] is set to 𝑑, whereas when it acts as an ˆ In a feasible IA-based network with SM pair, 𝑑[𝑗] is set to 𝑑. 𝑀 + 𝑁 = 𝑑(𝐾 + 1), after implementing the TM scheme, the whole network is divided into several subnetworks, i.e., one IA subnetwork 𝒜 and several SM subnetworks 𝒮. Therefore, the antennas at each user in the IA subnetwork will be redundant because the number of IA pairs 𝐾IA is smaller than the number of all the users in the network 𝐾. To further improve the performance of the network, the resource allocation in the TM scheme is considered, and the
Fig. 1. A 𝐾-pair asymmetric IA-based interference network with topology management.
two techniques, i.e., AS and PA, are adopted in this paper, which are summarized as follows. ∙ Antenna Selection: When the antennas are redundant in the IA-based network, the performance of the network can be optimized by selecting the optimal combination of antennas at all the users. ∙ Power Allocation: The transmitted power of the users can be allocated to optimize the spectrum efficiency of the IA-based network. III. A NTENNA S ELECTION IN IA S UBNETWORK In this section, the IA strategy in the asymmetric IA subnetwork is first presented. Then we develop the AS algorithm in the IA subnetwork of the TM scheme. A. IA Strategy in the TM scheme In the IA subnetwork, to eliminate the interferences among IA pairs perfectly, the following conditions should be satisfied [10], [11]. U[𝑗]† H[𝑗𝑖] V[𝑖] = 0, ∀𝑖 ∕= 𝑗, 𝑖, 𝑗 ∈ 𝒜, ) ( rank U[𝑖]† H[𝑖𝑖] V[𝑖] = 𝑑.
(3) (4)
When each data stream is allocated with equal transmit power, the average transmission rate of the 𝑗-th pair in the IA subnetwork can be expressed as } { } { 𝜌[𝑗𝑗] 𝑃𝑡 [𝑗𝑗] [𝑗𝑗]† [𝑗] H H E 𝑅IA = E log2 I𝑑 + , (5) [𝑗] 𝑑(𝜎 2 + 𝐼IA ) [𝑗𝑗]
where H = U[𝑗]† H[𝑗𝑗] V[𝑗] ∈ ℂ𝑑×𝑑 is the effective channel matrix. In the TM scheme, the inter-subnetwork interference from other subnetworks is treated as noise, and the power of [𝑗] the equivalent noise 𝐼IA can be expressed as ∑ [𝑘] ∑ [𝑗] 𝐼IA = 𝑃𝑡 𝜌[𝑗𝑘] = 𝑃𝑡 𝜌[𝑗𝑘] . (6) 𝑘∈𝒮
𝑘∈𝒮
Lemma 1: When 𝐾IA < 𝐾, there will be more optional antenna combinations for IA pairs in IA subnetwork. The average performance of the IA cases based on these combinations
TABLE I N UMBER OF O PTIONAL A NTENNA C OMBINATIONS FOR THE 𝑗- TH IA PAIR Configuration [𝑗] 𝑀IA [𝑗] 𝑀IA
= =
Number of Antenna Combinations
[𝑗] 3, 𝑁IA [𝑗] 3, 𝑁IA
= 2, 𝑑 = 1
3
= 3, 𝑑 = 1
1
Ω∈Φ
is all the same with the following condition satisfied [𝑗]
[𝑗]
𝑀IA + 𝑁IA ≥ 𝑑 ⋅ (𝐾IA + 1), [𝑗]
[𝑗]
𝑑 ≤ min {𝑀IA , 𝑁IA }, [𝑗]
in each time slot, and the optimization problem can be expressed as } {𝐾 IA ∑ [𝑘] ∗ 𝑅IA (Ω) . (9) Ω = arg max
(7)
𝑘=1
In this paper, the exhaustive search method is applied to solve the optimization problem (9), which can be summarized as Algorithm 1. Besides, as the size of Φ becomes larger, the computational complexity of exhaustive search method will become extremely high, and some effective combinatorial optimization algorithms can be adopted to solve the problem with low complexity.
[𝑗]
where 𝑀IA and 𝑁IA are the number of antennas participating in the IA subnetwork at the Tx and Rx of the 𝑗-th pair, respectively. Proof: In the IA subnetwork, when designing the precoding matrix V[𝑗] and decoding matrix U[𝑗] for the 𝑗-th pair based on the MinIL algorithm [11], V[𝑗] and U[𝑗] are both independent of H[𝑗𝑗] . Therefore, the joint distribution of each [𝑗𝑗] = U[𝑗]† H[𝑗𝑗] V[𝑗] is equal entity of the effective channel H [𝑗𝑗] according to the properties of Wishart matrix to that of H [12]. Therefore, the changes of 𝑀IA and 𝑁IA will not affect the expectation of the received signals power at the 𝑗-th pair, which can be expressed as { } 𝑃𝑡 ( [𝑗𝑗] [𝑗𝑗]† ) Tr H H (8) E = 𝑑𝑃𝑡 . 𝑑 Thus the average transmission rate of the IA subnetwork will not be affected by the changes of 𝑀IA and 𝑁IA as long as (7) can be met. Remark 1: The average performance of the IA cases with all the antenna combinations is the same when it is feasible. However, in each time slot, the performance of each antenna combination may be quite different. Therefore, based on Lemma 1, the performance of the IA subnetwork can be further improved through selecting the optimal antenna combination to perform IA in each time slot. B. Antenna Selection in the IA Subnetwork of TM scheme Take the interference network with (𝑀, 𝑁, 𝑑, 𝐾) = (3, 3, 1, 5) as an example to analyze the implementation of AS in the IA subnetwork. Assume that after the TM scheme, there exist one IA subnetwork with 𝐾IA = 4 pairs and 𝐾SM = 1 SM subnetwork in the network. Considering the computational complexity and CSI feedback overhead, AS is just performed at the Rx side of each IA pair. According to (7), when AS is applied in the IA subnetwork, the number of optional antenna combinations for the 𝑗-the IA pair can be summarized as Table I. Hence, there will be 𝑧 = (3 + 1)4 = 256 available antenna combinations in the IA subnetwork, and the set of all 𝑧 possible antenna combinations is denoted as Φ = {Ω1 , Ω2 , ⋅ ⋅ ⋅ , Ω𝑧 }. Through AS, we can select the best antenna combinations from the set Φ to optimize the performance of IA subnetwork
Algorithm 1 Exhaustive search method to find Ω∗ 1: 2:
3: 4: 5: 6:
for each antenna combination Ω𝑖 ∈ Φ do Perform the IA scheme in the IA subnetwork, and calculate the precoding matrices V and decoding matrices U. ∑𝐾IA [𝑘] Obtain 𝑅(Ω𝑖 ) = 𝑘=1 𝑅IA (Ω𝑖 ). end for Search for the maximum value of 𝑅(Ωmax ) in the set {𝑅(Ω𝑖 )∣𝑖 = 1, 2, ⋅ ⋅ ⋅ , 𝑧}. Output the optimal antenna combination Ω∗ = Ωmax .
IV. P OWER A LLOCATION FOR TM SCHEME In this section, we first present the SM strategy for SM subnetworks in the TM scheme. Then the PA algorithm is proposed for the TM scheme, and finally the joint resource allocation of AS and PA is developed. A. SM Strategy in the TM scheme In the SM subnetwork, when each data stream is allocated with equal transmit power, the average transmission rate of the 𝑗-th pair can be represented as } { } { 𝜌[𝑗𝑗] 𝑃𝑡 [𝑗] [𝑗] W E 𝑅SM = E log2 I𝑑ˆ + , (10) ˆ 2 + 𝐼 [𝑗] ) 𝑑(𝜎 SM where 𝑑ˆ = min (𝑀, 𝑁 ) represents the maximum rank of the channel matrix H, and the central Wishart matrix W[𝑗] can described as { [𝑗𝑗] [𝑗𝑗]† H H 𝑁 < 𝑀, (11) W[𝑗] = 𝑁 ≥ 𝑀. H[𝑗𝑗]† H[𝑗𝑗] [𝑗]
In the SM subnetwork, the power of the equivalent noise 𝐼SM is similar to (6), and can be written as [𝑗]
𝐼SM =
𝐾 ∑ 𝑘=1,𝑘∕=𝑗
[𝑘]
𝑃𝑡 𝜌[𝑗𝑘] =
𝐾 ∑ 𝑘=1,𝑘∕=𝑗
𝑃𝑡 𝜌[𝑗𝑘] .
(12)
B. Power Allocation for the TM Scheme
where
Due to the asymmetric distribution of users, the path-loss in different wireless links may be various, which means that the level of the received signal-to-interference-plus-noise ratio (SINR) of pairs may be quite different. Therefore, taking the advantage of asymmetric distribution, the PA strategy among pairs can be applied to optimize the performance of the network in each time slot. In Section II, we assume that each pair follows a equal power constraint 𝑃𝑡 . Nevertheless, to perform PA strategy in more practical scenarios, the individual power constraint can be replaced by the sum power constraint as 𝐾 ∑
[𝑘]
𝑃𝑡
[𝑘]
≤ 𝐾𝑃𝑡 , 𝑃𝑡
≥ 0, ∀𝑘 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝐾}.
(13)
𝑘=1
Therefore, in the PA strategy, the equations (5) and (10) should be modified as 𝜌[𝑗𝑗] [𝑗𝑗] [𝑗] [𝑗𝑗]† [𝑗] ˆ (14) 𝑅IA = log2 I𝑑 + H P H , 𝑡 [𝑗] (𝜎 2 + 𝐼 ) IA
ˆ [𝑗] 𝑅 SM
𝜌[𝑗𝑗] [𝑗] ˆ W = log2 I𝑑ˆ + , [𝑗] (𝜎 2 + 𝐼SM )
where
{ ˆ W
[𝑗]
=
[𝑗]
H[𝑗𝑗] P𝑡 H[𝑗𝑗]† [𝑗] H[𝑗𝑗]† P𝑡 H[𝑗𝑗]
(15)
𝑁 < 𝑀, 𝑁 ≥ 𝑀.
(16)
[𝑗]
The diagonal matrix P𝑡 is the PA matrix of the 𝑗-th pair, whose diagonal elements are the transmit power allocated to each data stream, and the nondiagonal elements are zero. Based on the description above, the PA problem among pairs can be defined as ⎛ ⎞ ∑ [𝑗] ∑ [𝑗] ˆ + ˆ ⎠ ⎝ 𝑅 𝑅 max IA SM [1] [2] [𝐾] (17) 𝑃𝑡 ,𝑃𝑡 ,...,𝑃𝑡 𝑗∈𝒜 𝑗∈𝒮 𝑠.𝑡.
Constraint (13).
To elaborate the problem (17) explicitly, we can redescribe the expressions of the rate of IA pairs and SM pairs in a more consolidated form. According to the following identical equation in the matrix theory 𝑑 ( ) ∑ log2 I𝑑 + 𝛽APA† = log2 1 + 𝛽𝑝(𝑗)𝜆[𝑗] .
(18)
𝑗=1 [𝑗]
where 𝜆 is the 𝑗-th eigenvalue of the matrix AA and 𝑝(𝑗) is the 𝑗-th diagonal element of the diagonal matrix P, the problem (17) can be rewritten as ⎞ ⎛ 𝐾 ∑ 𝑑[𝑘] ( ) ∑ ⎝ max log2 1 + 𝛽 [𝑘] 𝑝(𝑘, 𝑗)𝜆[𝑘,𝑗] ⎠ 𝑝(1,1),...,𝑝(𝐾,𝑑[𝐾] ) (19) 𝑘=1 𝑗=1 Constraint (13),
and
𝑘 ∈ 𝒜, 𝑘 ∈ 𝒮,
(20)
[𝑘]
𝑑 ∑
[𝑘]
𝑝(𝑘, 𝑗) = 𝑃𝑡 .
(21)
𝑗=1
ˆ and 𝑝(𝑘, 𝑗) is the transmit power allocated to the 𝑑[𝑘] ∈ {𝑑, 𝑑}, 𝑗-th data stream of the 𝑘-th pair. Therefore, the PA problem among pairs is transformed into an equivalent PA problem among data streams based on (19). Due to the non-convexity nature of the problem (19), it is difficult to find the optimal PA solutions through convex optimization techniques. As an alternative approach, the interpoint method is adopted to solve the PA problem in this paper. C. Joint Resource Allocation The resource allocation of AS and PA can be jointly performed to further improve the performance of the TM scheme. Nevertheless, according to Section III-B and IV-B, the AS is a combinatorial optimization problem, while the PA is a continuous optimization problem, which makes the joint optimization of AS and PA much more difficult to solve. As a replacement of optimizing the joint optimization problem directly, a stepped resource allocation optimization strategy is adopted for the TM scheme in each time slot, which can be described as follows. ∙ Step 1: In the asymmetric network, we first utilize the TM scheme to obtain the network topology. ∙ Step 2: Based on the network topology, the AS technique is applied to select the optimal antenna combination in the IA subnetwork. ∙ Step 3: According to the network topology and the optimal antenna combination of the IA subnetwork, the PA among data streams is performed in the whole network. Remark 2: ∙ When no SM subnetworks exist, Step 2 will be omitted. This is because there are no redundant antennas under the condition 𝑀 + 𝑁 = 𝑑(𝐾 + 1). ∙ After PA in certain time slots, the allocated transmit power to some pairs may be zero. These pairs will keep silent or turn into sleep mode in these slots accordingly. ∙ The implementation of AS and PA will not affect the topology of the network in each time slot. V. S IMULATION R ESULTS AND D ISCUSSIONS
†
𝑠.𝑡.
𝛽 [𝑘]
⎧ ) ( ⎨ 𝜌[𝑘𝑘] / 𝜎 2 + 𝐼 [𝑘] IA ) ( = ⎩ 𝜌[𝑘𝑘] / 𝜎 2 + 𝐼 [𝑘] SM
In this paper, We consider an asymmetric interference networks with the configuration (𝑀, 𝑁, 𝑑, 𝐾) = (3, 3, 1, 5). Assume that perfect CSI can be obtained at each pair, and the path-loss exponent 𝛼 is set to 3 according to [13]. All the pairs are randomly and uniformly distributed in a 10 × 10 km square area, and the distance between the 𝑖-th Tx and its corresponding Rx is set to 1 km, ∀𝑖 ∈ {1, 2, ⋅ ⋅ ⋅ , 𝐾}. In Fig. 2, we present a certain topology of an asymmetric
consider five MinIL algorithm based schemes, including the MinIL scheme, the MinIL-based TM scheme, the MinILbased TM scheme with AS, the MinIL-based TM scheme with PA, the MinIL-based TM scheme with AS and PA. From the results, we can observe that separately applying one resource allocation technique (e.g., AS or PA) in the MinILbased TM scheme can effectively improve the performance of the network compared with the conventional TM scheme. Meanwhile, the MinIL-based TM scheme with AS and PA can achieve the optimal performance among the five schemes due to the joint optimization of AS and PA.
10 Rx1 9 Tx1
IA Subnetwork
8 Rx4 Location in Y (km)
7 Tx4 6 Rx2
5
Tx2
Rx3 Tx3
4
SM Subnetwork
3
VI. C ONCLUSIONS
Tx5
2
Rx5
Tx Rx
1 0
0
2
4 6 Location in X (km)
8
10
Fig. 2. A certain topology of the interference network with 5 pairs when the transmit SNR is 15 dB.
50 45
Spectrum Efficiency (bits/s/hz)
40
MinIL MinIL−based TM scheme MinIL−based TM scheme with AS MinIL−based TM scheme with PA MinIL−based TM scheme with AS and PA
R EFERENCES
35 30 25 20 15 10 5 0 −10
−5
0
5
10 SNR (dB)
15
In this paper, we combined the two resource allocation techniques, i.e., AS and PA, to further improve the TM scheme for asymmetric IA-based interference networks. In the IA subnetwork, antennas are redundant after performing IA, and more antenna combinations were exploited to improve the performance of IA subnetwork through the diversity gain by AS. On the other hand, the transmit power can be allocated among data streams to enhance the spectrum efficiency of the network. Thus, we jointly optimized the AS and PA problem, which can further improve the performance of TM scheme for the IA-based network. Finally, simulation results were presented to show the effectiveness of the proposed schemes.
20
25
30
Fig. 3. The spectrum efficiency versus the transmit SNR in an asymmetric interference network with 5 pairs using different TM schemes.
interference network through TM scheme with 5 pairs when the transmit SNR is 15 dB. From the results, we can find that after performing the TM scheme at 15 dB, the four pairs, i.e., the 1-st, 2-nd, 3-rd and 4-th pair, gather together as one IA subnetwork, and the 5-th pair acts as an SM subnetwork independently. Note that Fig. 2 is just an example for demonstrating the TM scheme, and the topology of the network may be varying according to the different SNRs and the locations of the users. The spectrum efficiency of different schemes in the asymmetric IA-based interference network with 5 pairs (corresponding to that in Fig. 2) is compared in Fig. 3. We
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