Globecom 2012 - Wireless Networking Symposium
Exploring Frequency Diversity with Interference Alignment in Cognitive Radio Networks †
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Youwen Yi , Jin Zhang , Qian Zhang , Tao Jiang † Department of Computer Science and Engineering Hong Kong University of Science and Technology, Hong Kong {ywyi, zjzj, qianzh}@cse.ust.hk ‡ Department of Electronics and Information Engineering Huazhong University of Science and Technology, Wuhan, China
[email protected]
Abstract—The available spectrum in cognitive radio networks is usually discontinuous but wide, which provides abundant frequency domain diversity. In this paper, we identify the opportunity of leveraging the newly-emerged technique interference alignment to exploit such diversity to support concurrent transmission and improve the network throughput in secondary networks. To enable interference alignment, independent-fading subcarriers should be grouped together to provide sufficient dimensions for intended signals and non-intended interferences at the receiver side. We formulate the subcarrier grouping problem for interference alignment to maximize the number of concurrent transmissions, and propose a greedy-based algorithm to solve it, which is proved to be optimal. Simulation results show that using the proposed scheme, the total throughput in cognitive radio networks can be greatly improved.
I. I NTRODUCTION The traditional fixed long-term spectrum allocation is inefficient because the actual licensed spectrum is largely underutilized in vast temporal and geographic dimensions. Over the past decade a great deal of research has been conducted to solve the spectrum under-utilization problem. The concept of dynamic spectrum access is proposed to exploit the idle spectrum opportunities, and thus improve the spectrum efficiency. It is commonly believed that cognitive radio technology is a promising approach to enable dynamic spectrum access. In cognitive radio networks, unlicensed users (secondary users) are allowed to access the licensed bands owned by the legacy spectrum holders (primary users) on an opportunistic or a negotiated manner. Moreover, as point-to-point link throughput approaches Shannon capacity, it becomes increasingly important to allow concurrent transmissions to substantially improve network capacity. As a result, techniques such as cooperative multipleinput multiple-output (MIMO) and interference alignment are proposed. The key idea of interference alignment is that, by forcing interfering signals at each receiver into a reduceddimensional subspace of the received space, the receivers reserve more dimensions for interference-free desired signals. Such multi-dimensional receive space corresponds to physical space (antennas) or frequency sub-bands, or time slots [1]. Most of the current papers about interference alignment con-
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sider the theoretical analysis on the degree of freedom (dof) under different channel conditions [1], [2]. There are also some recent papers considering the practical problems when employing interference alignment in multi-antenna systems [3], [4]. Unfortunately, the number of dimensions supplied by multiple antennas is often limited by the size of the end-user devices. In addition, since the practical channels are time-varying and unpredictable, it is less realistic to use multiple symbols to provide the dimensions. Therefore, the most practical way to leverage interference alignment is using multiple subcarriers. Since the spectrum for secondary usage is usually very wide, and thus can provide sufficient number of subcarriers, it is very desirable to consider interference alignment in cognitive radio environment. In this paper, we consider the scenario that two secondary networks coexist with each other, and share the common available spectrum. However, the following challenges should be addressed if we expect to leverage interference alignment in cognitive radio networks to improve the network capacity. First, although interference alignment works well when the signal-to-noise ratio (SNR) goes infinite, it may not be able to guarantee acceptable bit error rate (BER) performance in intermediate SNR environments, which is more common for the cognitive radio systems. Because the signals may be too close to each other to decode at the intended receiver, if they are aligned to the same dimension at the non-intended receiver. To tackle this challenge, the subcarriers fading independently are grouped to provide the signal dimensions in our scheme. Second, since the spectrum for secondary usage is discontinuous due to the existence of primary users, different grouping schemes will severely vary the throughput gain provided by interference alignment. Therefore, in this paper, we seek to address this challenge by formulating the subcarrier grouping problem targeting at maximum number of concurrent transmissions in the secondary networks, and proposing a subcarrier grouping algorithm that can identify the best interference alignment opportunity. Numerical results demonstrate that by enabling concurrent transmission with the proposed algorithm in cognitive radio networks, the network throughput gain can be as high as 80%.
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The main contributions of this paper are as follows: 1) We identify the novel scenario for using interference alignment to improve the network throughput in cognitive radio networks. To the best of our knowledge, this is the first work that exploits frequency diversity in cognitive radio network with interference alignment. 2) We formulate the subcarrier grouping problem that maximizes the network throughput which is crucial for interference alignment using independent non-continuous subcarriers. 3) We propose a greedy-based algorithm to solve the problem which is proved to be optimal, and numerical results show that the network throughput is greatly increased with the proposed algorithm. The rest of the paper is organized as follows. In Section II, we introduce the system model and the interference alignment technique adopted. In Section III, we formulate the subcarrier grouping problem for interference alignment in cognitive radio networks, and propose a greedy-based algorithm. Numerical results are shown in Section IV. Finally, in Section V, we conclude the whole paper. II. S YSTEM M ODEL In this section, we introduce the system model, as well as the interference alignment technique adopted in this paper. We consider the scenario that two secondary networks coexist with each other, and there are N users in each network transmitting data to their intended AP. It is assumed that the spectrum availability for all the users in the two networks are the same, the two secondary APs will negotiate with each other to decide how to allocate the mutual idle spectrum. The total band for opportunistic access is denoted by B. To better utilize the idle spectrum, multi-subcarrier technologies are commonly employed for data transmission. In cognitive radio networks, noncontinuous orthogonal frequency-division multiplexing (NC-OFDM) technique is commonly employed because the spectrum available for secondary usage is discontinuous. We consider a general multi-subcarrier system with the distance of adjunct subcarriers denoted by d. Therefore, the total bandwidth can support K subcarriers, where K = B/d. We use si to indicate whether the i-th subcarrier is available or not, 1 < i ≤ K, with si = 1 indicating idle, and si = 0 occupied. Then, the availabilities of all K subcarriers can be described by the sequence s = {s1 , s2 , · · · , sK }. Traditionally, concurrent packets at the same subcarrier interfere due to the open characteristics of the wireless channel, and none of them can be decoded correctly, as a result, only K data streams can be supported by these subcarriers. Fortunately, it is proved in [5] that interference alignment allows K−1(K ≥ 2) data streams in each of the two networks, resulting in 2K − 2 concurrent transmissions. Therefore, the throughput gain obtained from interference alignment in such an interfering multiple access channels (IMACs) scenario is K −2 G(K) = . (1) K When there is only one subcarrier, it can only support one data stream in the network. Therefore, the number of
concurrent transmissions as a function of the number of subcarriers is described as follow, 2(K − 1), K ≥ 2 (2) C(K) = 1, K=1 It is proved that there is a duality between the uplink interfering multiple access channel and the downlink interfering broadcast channel (IBC). In this paper, we only consider uplink transmission, while the downlink case can be analyzed in a similar way. III. I NTERFERENCE A LIGNMENT IN C OGNITIVE R ADIO N ETWORKS To leverage interference alignment in cellular networks, we have to group subcarriers to provide dimensions for intended signals and non-intended interferences. Although in cognitive radio scenarios, the spectrum of interesting is very wide, which can provide sufficient number of independent subcarriers, we have to address the key challenges of subcarrier grouping if we want to leverage it in secondary cellular networks. First of all, most works on interference alignment focus on the dof (degree of freedom) of the network which make sense when the SNR is high enough. However, in cognitive radio environment, the SNR of the secondary receiver is usually intermediate due to the coexistence with primary users and other secondary users, which may cause significant BER performance degradation for systems using interference alignment. For instance, in the scenario of the previous section, when subcarriers are employed to provide the dimensions for intended signal and non-intended interference, although the signals from the users of the first cell to the second AP align to one dimension, they may also very close to each other at the first AP, if the subcarriers are highly correlated. As a result, the first AP may not be able to successfully decode the signals from it own users if the SNR is not high enough, which is similar to the MIMO systems when the antennas are correlated [6]. To address this problem, independent-fading subcarriers are grouped together to provide dimensions when leverage interference alignment in cognitive radio networks. Moreover, if the spectrum is continuous, the subcarrier grouping scheme is quite straightforward. By dividing the spectrum according to the coherence bandwidth, the subcarriers groups can be easily obtained. However, in cognitive radio networks, due to the existence of the primary users, the available spectrum for secondary usage is usually noncontinuous. As a result, a subcarrier grouping scheme should be able to identify the best interference alignment opportunity that maximizes the network throughput. In the rest of this section, we first formulate the subcarrier grouping problem to be an optimization problem. To solve the problem, we propose a greedy-based algorithm, the performance of which is demonstrated with simulation in the next section. A. Problem Formulation To identify the best interference alignment opportunity that boosts the network throughput, the subcarrier group-
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ing problem is formulated as follows: given the sequence s = {s1 , s2 , · · · , sK } indicating the spectrum availability, the coherence bandwidth D to determine whether two subcarriers fade independently or not, and sufficient number of users per network for interference alignment, we should find out a subcarrier grouping result that maximizes the total number of concurrent transmissions, max C(|∫ |) (3) S
∫ ⊆S
with following constraints,
∫j = ∅;
∀∫i , ∫j ⊆ S
,
∫i
∀si , sj ∈ ∫
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si = 1, sj = 1, |i − j| > D;
(4) (5)
where S is the set of all groups, ∫j is the j-th group that contains the grouped subcarriers. The first constraint (4) requires that each subcarrier can only be grouped once. The second constraint (5) ensures that the subcarriers corresponding to the signal space bases must be idle and irrelevant. In the second constraint, D is a preset parameter related to coherence bandwidth. Because the subcarriers should fade independently to provide signal dimensions for interference alignment in cognitive radio network, which is possible for frequencyselective channel. The channel could be frequency-selective if the transmission occurs in a multipath environment. The metric to evaluate the correlation of channel responses in the frequency domain is coherence bandwidth [7]. The coherence of two arbitrary different subcarriers with distance Δf is defined as E{H(f )H(f + Δf )∗ } ρΔf = . (6) E{|H(f )|2 } where H(f ) and H(f + Δf ) are the channel responses of the two subcarriers. The X% coherence bandwidth is the value of Δf such that ρΔf = X/100. With larger X, the channel responses of two subcarriers will be less correlated. Typically, X is set to be 50 or 90, and we choose the former value in our simulation. Suppose that the X% coherence bandwidth is Bc,X , if the distance of the i-th subcarrier and the j-th subcarrier is not less than Bc,X in the frequency domain, these two subcarriers can be viewed as fading independently, the minimum spacing of independent subcarrier indexes is
is equivalent to solving the following problem, min (|S 1 | + 2|S r |).
B. Subcarrier Grouping Algorithm The subcarrier grouping problem is a nonlinear integer programming problem (NIP), and generally there is no existing algorithm that can guarantee optimal solution within polynomial time. Although exclusive search from all possible grouping combination can ensure maximizing the objective function, its computational complexity increases exponentially with the number of idle subcarriers, which is unacceptable in practice. Therefore, to solve this problem, we propose a greedy-based grouping algorithm with low complexity. The key idea is that we always find the largest feasible subcarrier group from the residual available subcarriers set, because according to Eq. (2), the larger group size, the higher throughput. Based on the subcarrier availability sequence s, the algorithm (Alg. 1) groups the available subcarriers following grouping constraints. To distinguish different groups, each group is assigned a unique ordinal GroupID, with smaller GroupID value indicating earlier construction. For each subcarrer si , the algorithm first checks whether it is idle. If so, the function canGroup(si , ∫GroupID ) continues checking whether si can be put into the existing group ∫GroupID until the first feasible group is found. The criteria whether a specific subcarrier can be added to a given group is that, the index distances between the subcarrier and any other subcarriers in the group should be larger than or equal to coherence bandwidth D, which follows the last grouping constraint (5). If subcarrier si can be put into none of the existing groups, a new group will be added to S with the largest GroupID and si in it. This procedure is repeated until all idle subcarriers are grouped. Note that the algorithm tends to group more subcarriers to the previous groups than the latter one, which illustrates its greedy property. Algorithm 1 Greedy-based Subcarrier Grouping Algorithm while s = ∅ do if si == 1 then GroupID = 0; while GroupID < |S| do if canGroup(si , ∫GroupID )then ∫GroupID = ∫GroupID {si }; break; else GroupID + +; end if end while if GroupID = |S| then ∫GroupID = {si }; S = S {∫GroupID }; end if end if s = s \ si ; end while
Bc,X . (7) d According to relationship between the group size and the network throughput given by Eq. (2), the subcarrier groups set S can be divided into two subsets, size-1 groups subset S 1 , and the non-size-1 groups subset S r . Thus, the optimization problem can be transferred into max (2|∫ | − 2) + |∫ | = 2K − (|S 1 | + 2|S r |). (8) D = |i − j| =
∫ ⊆S r
∫ ⊆S 1
Obviously, since K is a given constant parameter denoted the total number of idle subcarriers, the above optimal problem
(9)
Obviously, the complexity of the proposed algorithm is K 2 , which is acceptable in practice. Moreover, we have the following theorem,
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Proposed Basic
Total Gain over Basic
0.9 0.8
400 Cumulative Fraction
The total number of concurrent transmissions
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0.6 0.5 0.4 0.3 0.2
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0.7
0.1
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160 180 200 220 The number of idle subcarriers
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0 0.6
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0.65 0.7 0.75 0.8 0.85 The Number of Current Transmission Gain
Fig. 2.
Fig. 1. The total number of concurrent transmissions versus the number of idle subcarriers.
0.9
CDF of the throughput gain.
240 The total number of concurrent transmissions
Theorem 1: The grouping result provided by the proposed greedy-based algorithm is optimal, when the number of users N ≥ 2(Km − 1), where Km is the largest group size. The detailed proof can be found in Appendix.A. IV. N UMERICAL R ESULTS
200 Proposed Basic
180 160 140 120 120
140
160 180 200 220 The number of total subcarriers
240
260
Fig. 3. The total number of concurrent transmissions versus the number of total subcarriers.
240 The total number of concurrent transmissions
In this section, we show the performance gain of the proposed algorithm, and the impact of some system parameters on the performance with simulation results. The setup of the simulation is as follows: there are 20 users in each of the two networks associated with the AP, and opportunistically access a common spectrum band which can totally support K subcarriers, some of which may be occupied by primary users or other secondary networks. Unless explicitly otherwise stated, The coherence bandwidth is 16 subcarriers, i.e., D = 16. K is set to be 256, the number of idle subcarrers is 128 whose positions follow uniform distribution. To demonstrate the performance gain of employing interference alignment using the subcarrier grouping algorithm we proposed, we compare the total number of concurrent transmissions in the networks using our approach with that of using traditional technique without interference alignment. In the traditional approach, which is denoted as ’Basic’, one idle subcarrier can support at most one secondary data stream. Fig. 1 shows the total number of concurrent transmissions versus the number of idle subcarriers varying from 128 to 256 (Fig. 1). The more idle subcarriers, the more number of data streams permitted. The figure also shows that, with the proposed algorithm for interference alignment, the number of concurrent transmissions is greatly increased comparing with the traditional approach, because the frequency domain diversity is exploited. Fig. 2 shows the cumulative distribution function (CDF) of the throughput gain of the proposed scheme over basic one. Our scheme provides a 80% increase with regards to the number of concurrent transmissions. Fig. 3 demonstrates the total number of concurrent transmissions versus the number of total subcarriers varying from 128 to 256. It is observed that when the number of total subcarriers increases, the throughput of our scheme increases, because
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Proposed Basic 220 200 180 160 140 120
10
20
30 40 The coherence bandwidth
50
60
Fig. 4. The total number of concurrent transmissions versus the coherence bandwidth.
the more total subcarriers, the wider the total spectrum, and the subcarrier group size can be larger eventually. With larger subcarrier group, the throughput gain will increase according to Eq. (2). Since our scheme better exploits the diversity in frequency domain, the networks can support more concurrent transmissions.
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The coherence bandwidth is a key parameter for subcarrier grouping, because it determines whether two subcarriers can be grouped together or not, and thus affects the group size. Fig. 4 shows the total throughput versus the coherence bandwidth. It is illustrated that the number of concurrent transmissions decreases when the coherence bandwidth increases, because the maximum group size that the total spectrum can support is less, which reduces the total throughput gain.
Then, we begin to analyze the results given by the optimal grouping and greedy-based one under different cases considering the number of non-size-1 groups in the two grouping results, So and Sg . r r 1 r • Case 1: If |So | > |Sg |, since |Sg | + |Sg | is minimum (Lemma 1), there must be 2|Sg1 | + |Sgr | < 2|So1 | + |Sor |,
V. C ONCLUSIONS The wide band for secondary usage can provide sufficient diversity in the frequency domain. In this paper, we identify this scenario, and leverage interference alignment to exploit the diversity, such that the allowed number of concurrent transmissions can be greatly increased. Since the available spectrum is usually discontinuous, we formulate the subcarrier grouping problem, and propose an efficient greedy-based algorithm for grouping the nonadjacent subcarriers to support interference alignment. Numerical results show that, by grouping independent subcarriers for interference alignment, the number of concurrent transmissions in the cognitive radio networks is significantly improved.
•
2|Sg1 | + |Sgr | ≤ 2|So1 | + |Sor |, •
To prove the optimality of the proposed algorithm when there are enough number of users, we first introduce the lemma as follow. Lemma 1: The total number of groups obtained by the greedy-based algorithm is minimum when the number of users is sufficient. Proof: Given the grouping results provided by the greedybased algorithm, for arbitrary subcarrier sk in the last group, there is one and only one subcarrier in each previous group, whose subcarrier index i satisfies i ∈ (k − D, k);
(10)
otherwise, if there is one group doesn’t contain one such subcarrier, subcarrier sk should have already been chosen in that group due to the greedy property of the algorithm; obviously, if there is a group with more than one such subcarriers, the last grouping constraint is violated. Thus, these subcarriers are “conflicting”, which means any two of them cannot be in the same group. Therefore, the number of the groups cannot be reduced, e.g., the number of groups obtained by the proposed algorithm is minimum. Proof is finished. Base on the above lemma, we now begin to prove Theorem 1. Proof: Suppose there is an optimal grouping result So , we denote the grouping result provided by the proposed greedybased algorithm as Sg . The size-1 groups subset and the rest non-size-1 groups subset of the two results are denoted as So1 , Sor , Sg1 , and Sgr , respectively. According to Lemma 1, we have |Sg1 | + |Sgr | ≤ |So1 | + |Sor |
(11)
our claim holds. Case 3: If |Sor | < |Sgr |, according to Eq. (9), we have |So1 | > |Sg1 |. suppose that |So1 | = 1, and |Sg1 | = 0, then, for the scenario the same with current one except that the 1 single subcarrier in So1 is occupied, we have |S o | = 0, however, the number of groups provided by the greedybased algorithm cannot be reduced. The proof is similar to that of Lemma 1. As a result, 1
r
1
r
|S g | + |S g | > |S o | + |S o |,
A PPENDIX A. The Proof of Theorem 1
which is contradict with the precondition that So is optimal. Case 2: If |Sor | = |Sgr |, according to Eq. (11), we have
which contradicts with Eq. (11). For the general cases that |Sor | < |Sgr |, by eventually removing the subcarriers, we can reach the above special case, which is contradict with our previous conclusion. We have shown that the supposed optimal solution cannot work better than the solution obtained from the proposed algorithm, therefore, they are equivalent. This completes the proof. ACKNOWLEDGMENT This research was supported in part by the National Natural Science Foundation of China with Grant 61172052 and 61173156, and the Project-sponsored by SRF for ROCS, SEM, the National & Major Project with Grant 2012ZX03003004. R EFERENCES [1] S. Jafar and S. Shamai, “Degrees of freedom region of the mimo x channel,” Information Theory, IEEE Transactions on, vol. 54, no. 1, pp. 151–170, Jan. 2008. [2] C. Yetis, T. Gou, S. Jafar, and A. Kayran, “Feasibility conditions for interference alignment,” Global Telecommunications Conference, 2009. GLOBECOM 2009. IEEE, pp. 1–6, Nov. 2009. [3] S. Perlaza, N. Fawaz, S. Lasaulce, and M. Debbah, “From spectrum pooling to space pooling: Opportunistic interference alignment in mimo cognitive networks,” Signal Processing, IEEE Transactions on, vol. 58, no. 7, pp. 3728–3741, Jul. 2010. [4] J.-M. Wu, T.-F. Yang, and H.-J. Chou, “Mimo active interference alignment for underlay cognitive radio,” Communications Workshops (ICC), 2010 IEEE International Conference on, pp. 1–5, May 2010. [5] C. Suh and D. Tse, “Interference alignment for cellular networks,” Communication, Control, and Computing, 2008 46th Annual Allerton Conference on, pp. 1037–1044, Sept. 2008. [6] H. Sampath, S. Talwar, J. Tellado, V. Erceg, and A. Paulraj, “A fourthgeneration mimo-ofdm broadband wireless system: design, performance, and field trial results,” Communications Magazine, IEEE, vol. 40, no. 9, pp. 143 – 149, Sept. 2002. [7] T. S. Rappaport, “Wireless communications,” Second Edition, Prentice Hall, 2002.
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