1

Optimal Resource Allocation in OFDM Multihop System Wenyi Wang, and Renbiao Wu Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin E-mail: wenyi [email protected], [email protected]

Abstract—OFDM multihop system is efﬁcient scheme to save energy and extend coverage. In this paper, optimal resource allocation in OFDM multihop system, including arbitrary number of hops and subcarriers, is studied. By joint subcarrier matching and power allocation, our goal is to maximize system capacity with the system-wide power constraint. First, we formulate the optimal subcarrier matching and power allocation problem as an optimization problem, which is a mixed binary integer programming problem and prohibitive to ﬁnd the global optimum. Then, by utilizing the special structure of the system, we propose a low complexity resource allocation scheme, which is proved to be globally optimal. The simulations prove that the proposed scheme improves the system capacity. Index Terms—multihop system, resource allocation, system capacity.

I. I NTRODUCTION

T

HE inter-symbol interference (ISI) caused by multi-path fading is one of the main challenges in single carrier high-speed data transmission system. The ISI is even increased because it reduces the symbol period to below channel coherence time. In orthogonal-frequency-duplex-multiplexing (OFDM) system, by dividing the total bandwidth into multiple sub-bands and increasing lengthen of symbol period, ISI can be mitigated. With many advantages, use of OFDM architectures is becoming a common approach for high speed wireless communication systems. At the same time, multihop system attracts more and more attention, as it can save energy and extend coverage. Therefore, the multihop system based on OFDM is an efﬁcient way for wideband system to save energy and extend coverage. In OFDM multihop system, to reduce the complexity of system architecture, we assume that the bits on a subcarrier can be forwarded on another subcarrier, which avoids the reallocation of the bits among the subcarriers at every hop. However, the subcarriers through different hop experience independent fading, the subcarrier with deep fading through a hop may be not deep fading through another hop. System capacity will be reduced without correct subcarrier matching. It means that resource allocation for OFDM multihop system should consider subcarrier matching together with power allocation. OFDM multihop system has been widely investigated [1][7]. In [1]-[3], resource allocation problem for three-node OFDM relay links were investigated. For cooperative OFDMA

This work was supported in part by the National Natural Science Foundation of China (No. 60879019, 60825104) and the startup foundation of Civil Aviation University of China (No. 08QD11X).

systems, resource allocation and scheduler problem were considered in [4]-[6]. Adaptive relaying scheme for OFDM that taking channel state information at the relay node into account has been proposed in [7], where subcarrier matching was considered for OFDM amplify-and-forward scheme and power allocation was not considered. Performances of the two-hop system with and without subcarrier matching were analyzed in [8]-[9], which reveal the performance improvement through subcarrier matching. However, in order to further improve system performance, the subcarrier matching and power allocation should be considered jointly. In [10]-[11], the joint subcarrier matching and power allocation was considered for only two-hop system with separate and system-wide power constraint. But, the most of papers did not consider subcarrier matching except [9]-[11] and all algorithms can not be directly extended to the system including arbitrary number of hops and subcarriers. In this paper, we consider joint subcarrier matching and power allocation for OFDM multihop system including arbitrary number of hops and subcarriers. Firstly, we formulate an optimization problem, which is a mixed binary integer programming problem and prohibitive in terms of complexity to ﬁnd a global optimum. Then, by utilizing the special structure of the system, we propose a low complexity joint subcarrier matching and power allocation scheme, and prove that it is optimal. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model In this paper, a wideband multihop system is considered including more than one relays. In order to mitigate the ISI, OFDM is utilized. Therefore, without loss of generality, the different subcarrier is assumed experience independent fading. The subcarriers with severe fading through one hop may have small fading through another hop. We assume that the number of hops is T , and the number of subcarriers is N . It means that the number of relay is T − 1. The number of subcarrier is large enough to make the fading of every subcarrier ﬂat. The relay strategies of all internal nodes are decode-and-forward. It is assumed that all internal nodes can only receive signal from the fore one-hop node but not from the other nodes because of distance or obstacle. The cooperative diversity is not considered in this paper. The different hop utilizes different time slot to forward the signal. This means that the time is divided to T equal

2

time slots. The communication between the adjacent nodes covers one time slot. The spatial reuse is not considered in this paper. We assume that every node holds the complete channel state information (CSI). The CSI can be obtained by channel estimation and feedback. The total system power is constrained. In order to improve system capacity, every relay has to rematch subcarrier and reallocate power. The total system power should be allocated among different hops and subcarriers. As before, we assume that different channels experience independent fadings. The different subcarriers through the same hop experience independent fadings. The power spectral densities of additive white Gaussian noise (AWGN) are equal at all nodes. The codes achieving the Shannon capacity are adopted. Therefore, the channel capacity of the subcarrier i through the hop t is given t Pit hti 1 t Ri Pi = log2 1 + 2 (1) T σ0 where Pit is the power allocated to the subcarrier i through the t hop, hti is the corresponding channel power gain, σ02 is the power spectral density of AWGN. The parameter T1 is because of every hop lasts for T1 of time of the whole communication process. B. Problem Formulation When the subcarrier i through the ﬁrst hop is matched to the given subcarrier through all other hops, we call these subcarrier as i matched subcarrier group. Therefore, the channel capacity of i matched subcarrier group is the minimum of capacities of all subcarrier, which can be expressed as ⎛ ⎞ N N Ri = min ⎝Ri1 Pi1 , ρ2ij Rj2 Pj2 , ..., ρTij RjT PjT ⎠ j=1

j=1

(2) where, ρtij ∈ {0, 1} denotes the result of subcarrier matching, is one when the subcarrier j through the t hop is matched to the subcarrier i through the ﬁrst hop, otherwise is zero. The upper equation means that the channel capacity of the matched subcarrier is limited by the worst subcarrier in the matched subcarrier group, which implies that the subcarrier matching is necessary to maximize the system capacity. The system capacity is the capacity sum of all matched subcarrier groups, which can be expressed as Rsum =

N

Ri

(3)

i=1

The power allocated to all subcarriers through all hops is constained, i.e., system-wide power constraint N T

Pit ≤ Ptot

(4)

t=1 i=1

Another constraint is the one-to-one subcarrier matching. The bits through one subcarrier can only transmitted by one

subcarrier. It simply the system architecture. N

ρtij = 1

(5)

j=1

In order to match subcarrier and allocate power optimal, we can formulate it as an optimization problem as following max t t

N

Ri Pi ,ρij i=1 N T s.t.

Pit ≤ t=1 i=1 N ρtij = 1 j=1 Pit ≥ 0, ρtij =

Ptot

{0, 1}

(6)

∀i, j, t

The variables are the power allocated to every subcarrier Pit and the subcarrier matching variable ρtij . Because the subcarrier matching variables are constrained to be one or zero, it is a mixed binary integer programming problem and NP-hard [12]. This means that it is very difﬁcult to ﬁnd global optimum in terms of complexity. We can ﬁnd the global optimum by searching the all possibilities about the subcarrier matching variables. When the subcarrier matching is constant, the upper optimization problem is convex without the last two constraints. The optimal power allocation can be obtained by solving the convex optimization problem. For every possibility, we obtain the optimal power allocation. Then, by comparing the all possibilities, the greatest system capacity is optimal. The corresponding power allocation and the subcarrier matching are also optimal. But, when the number of subcarriers and hops becomes great, the complexity is not achievable. In the following section, by making use of the special structure of the system, the low complexity scheme is proposed and proved to be globally optimal. III. L OW- COMPLEXITY O PTIMAL R ESOURCE A LLOCATION S CHEME Before giving the low complexity optimal resource allocation scheme, we can get two lemmas from the reference [10]. Without loss of generality, we assume that hti ≥ hti+1 . Lemma 1: For any two-hop system including arbitrary number of subcarriers, the matched subcarrier pair can be equivalent to a single subcarrier. The relation of channel power gains among them can be expressed as following 1 1 1 = 1+ 2 (7) hi hi hj where hi is the channel power gain of the equivalent subcarrier. Lemma 2: For the two-hop system with arbitrary number of subcarriers, the optimal subcarrier matching is to match subcarrier according to the channel power gains, i.e., h1i ∼ h2i . Together with the optimal power allocation for this subcarrier matching, they are optimal joint subcarrier matching and power allocation. The details of proofs about the Lemma 1 and the Lemma 2 can be obtained in the reference [10].

3

The other equivalent subcarrier channel power gains can be obtained iteratively by making use of the fore equivalent subcarrier channel power gains. ht+1,i = f ht,i , ht+1 (9) i

First Hop

Second Hop

Third Hop

Final Subcarrier Matching and Equivalent Channel

Last Hop

Fig. 1.

Block diagram of iteratively subcarrier matching and equivalence

Using aforementioned lemmas, we can get the following proposition, which states the optimal subcarrier matching result between any adjacent two hops. Proposition 1: For the global optimum, the optimal subcarrier matching through the any adjacent two hops is according to the order of the subcarrier power gains, i.e., hti ∼ ht+1 . i Proof: This proposition will be proved in the contrapositive form. Assuming that there is a subcarrier matching scheme including two matched subcarrier pairs hti ∼ ht+1 j and htj ∼ ht+1 (j = i), and the channel capacity of this two i subcarrier pairs is greater than that of the matching scheme in proposition 1. When the power allocated to other subcarrier pairs and the other subcarrier matching are constant, the channel capacity of this two subcarrier pairs is the optimum according to the subcarrier matching in proposition 1. It means that there is no subcarrier matching scheme better than the subcarrier matching scheme in propostion 1, e.g, hti ∼ ht+1 and htj ∼ ht+1 i j . Corresponding, the system capacity is greater than that of any subcarrier matching. It is contrary to the assumption. Therefore, there is no subcarrier matching scheme better than the scheme in proposition 1. It means that the optimal subcarrier matching between the any adjacent two hops is according to the order of the subcarrier power gain, i.e., hti ∼ ht+1 . i From Proposition 1, we can get the following result directly. For the global optimum, the optimal subcarrier matching is h1i ∼ h2i ∼ ... ∼ hTi . Here, by utilizing the special structure of the system, the main idea is that the matched subcarrier group through arbitrary number of hops can be matched and equivalent to a subcarrier iteratively, which can be demonstrated in Fig. 1. From the ﬁrst hop, getting the optimal subcarrier matching by lemma 2; then the equivalent subcarrier as the ﬁrst hop, matching the subcarrier with the third hop, and so on. Incorporate the Lemma 1 and the Proposition 1, we get the equivalent subcarrier for the matched subcarrier group i. We xy . Therefore, the channel deﬁne the function f (x, y) = x+y power gain of the equivalent subcarrier can be obtained iteratively. For the ﬁrst hop, the equivalent subcarrier channel power gains are themselves. h1,i = h1i

(8)

By now, we obtain the optimal subcarrier matching and the channel equivalent subcarrier for all hops, so the power allocation among the hops can be equivalent to allocate power among the equivalent subcarriers. Then, the optimal power allocation problem can be formulated as the following optimization problem

N PT ,i hT ,i 1 max 1 + log 2 T σ2 PT ,i i=1 N

s.t.

i=1

0

(10)

PT,i ≤ Ptot

PT,i ≥ 0

∀i

It is clear the upper optimization problem is convex. The optimal power allocation can be obtained by solving the convex optimization problem. The optimal power allocation is water-ﬁlling. + 1 σ02 PT,i = (11) − λ hT,i +

where (a) = max (a, 0). The parameter λ can be found by the following equation N

PT,i = Ptot

(12)

i=1

The power allocation among the subcarrier matching groups is obtained. The power allocation among the subcarriers in a subcarrier matching group is to make the subcarrier capacity of all subcarriers equivalent as follow Ri1 Pi1 = Ri2 Pi2 = ... = RiT PiT (13) T

Pit = PT,i

(14)

t=1

Because the upper subcarrier matching is globally optimal and the optimal power allocation is based on this subcarrier matching, the power allocation together with the subcarrier matching is optimal joint subcarrier matching and power allocation. The global optimum of the optimization problem (6) is also the optimal joint subcarrier matching and power allocation. This means that it is the globally optimum for the optimization problem (6). IV. N UMERICAL E XAMPLES In simulations, we compare the capacity of the optimal joint subcarrier and power allocation with that of several other schemes. These schemes include: (I) No subcarrier matching and no power allocation; (II) Power allocation and no subcarrier matching; (III) Subcarrier matching and no power allocation. Here, the methods of subcarrier matching and power allocation are the same as that of the optimal joint subcarrier matching and power allocation. No power allocation

4

1.8 Optimally Joint Scheme III Scheme II Scheme I

1.6

Capacity(bits/s/Hz)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10

15 SNR(dB)

20

25

30

Fig. 2. Comparison of different schemes for joint subcarrier matching and power allocation with different SN R

6 Optimally Joint Scheme III Scheme II Scheme I

Capacity(bits/s/Hz)

5

4

3

scheme improves the system capacity considerably, compared with all other schemes. Fig. 3 shows that the total channel capacity versus the number of hops. The same conclusions can be obtained about the comparison of all schemes. The capacity of the scheme I, where there is no subcarrier matching and no power allocation, is the least one compared with that of the other schemes. If other conditions remain unchanged, subcarrier matching or power allocation can improve the total channel capacity by comparing the capacity of scheme I with the scheme II and III. The proposed scheme of the optimal joint subcarrier and power allocation scheme improves the system capacity considerably, compared with all other schemes. The system capacities of all schemes decrease as increasing the number of hops. The reason is that the worst subcarrier will limit the capacity for every matched subcarrier group. More is the number of hops, more possible is the deep fading. V. C ONCLUSION In OFDM multihop system including arbitrary number of hops and subcarriers, this paper considered optimal resource allocation, e.g., joint subcarrier matching and power allocation. Though the formulated optimization problem is a mixed binary integer problem and prohibitive to ﬁnd global optimum in terms of complexity, we proposed a low complexity scheme by utilizing the special structure of the system, which achieves to the global optimum. R EFERENCES

2

1

0

1

2

3

4

5

6

7

8

9

10

Number of hops

Fig. 3. Comparison of different schemes for joint subcarrier matching and power allocation with different number of hops

means that the system power is allocated equally among the all subcarriers through all hops. Fig. 2 shows that the total channel capacity versus SN R. We employ a four-hop system including 16 subcarriers, which means that T = 4 and N = 16. In the computer simulations, we assume that each subcarrier undergoes Rayleigh fading independently. The SN R is deﬁned as SN R = σP2tot , where 0B B is the total bandwidth and σ02 is the power spectral density of AWGN. The capacity of the scheme I, where there is no subcarrier matching and no power allocation, is the least one compared with that of the other schemes. If other conditions remain unchanged, subcarrier matching or power allocation can improve the total channel capacity. Specially, subcarrier matching can improve the capacity when comparing the capacity of the scheme III to that of the scheme I. The system capacity can be improved by power allocation when comparing the capacity of the scheme II to that of scheme I. The proposed scheme of the optimal joint subcarrier and power allocation

[1] I. Hammerstrom and A. Wittneben, “On the Optimal Power Allocation for Nonregenerative OFDM Relay Links,” IEEE International Conference on Communications, ICC’06, pp. 4463-4468, June 2006. [2] B. Gui and L. J. Cimini, Jr., “Bit Loading Algorithms for Cooperative OFDM Systems,” EURASIP Journal on Wireless Communications and Networking, vol. 2008, Article ID 476797, 9 pages, 2008. [3] Y. Ma, N. Yi, and R. Tafazolli, “Bit and Power Loading for OFDMBased Three-Node Relaying Communications,” IEEE Transactions on Signal Processing, vol. 56, no. 7, pp. 3236-3247, July 2008. [4] S. -J. Kim, X. Wang, and M. Madihian, “Optimal Resource Allocation in Multi-hop OFDMA Wireless Networks with Cooperative Relay,” IEEE Transactions on Wireless Communications, vol. 7, no. 5, pp. 1833-1838, May 2008. [5] M. Pischella, and J. -C. Belﬁore,, “Power Control in Distributed Cooperative OFDMA Cellular Networks,” IEEE Transactions on Wireless Communications, vol. 7, no. 5, pp. 1900-1906, May 2008. [6] M. Kaneko and P. Popovski., “Radio resource allocation algorithm for relay-aided cellular OFDMA system,” IEEE International Conference on Communications, ICC’07, pp. 4831-4836, June, 2007. [7] M. Herdin, “A chunk based OFDM amplify-and-forward relaying scheme for 4G mobile radio systems,” IEEE International Conference on Communications, ICC’06, pp. 4507 -4512, June, 2006. [8] H. A. Suraweera, and J. Armstrong, “Performance of OFDM-Based DualHop Amplify-and-Forward Relaying,” IEEE Communications Letter,vol. 11, no. 9, pp. 726-728, Sept. 2007. [9] C. Athaudage, M. Saito, J. Evans, “Performance Analysis of Dual-Hop OFDM Relay Systems with Subcarrier Mapping, ” IEEE International Conference on Communications, ICC’08, May, 2008. [10] Wenyi Wang, Shefeng Yan, and Shuyuan Yang. “Optimally Joint Subcarrier Matching and Power Allocation in OFDM Multihop System,” EURASIP Journal on Advances in Signal Processing, vol. 2008, Article ID 241378, 8 pages, 2008. [11] Wenyi Wang, Shuyuan Yang, and Li Gao. “Comparison of Schemes for Joint Subcarrier Matching and Power Allocation in OFDM Decode-andForward Relay System,” IEEE International Conference on Communications, ICC’08, May, 2008. [12] B. Korte, J. Vygen, and J. Vygen, Combinatorial Optimization: Theory and Algorithms, 3rd ed. New York: Springer-Verlag, 2002.

Optimal Resource Allocation in OFDM Multihop System Wenyi Wang, and Renbiao Wu Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin E-mail: wenyi [email protected], [email protected]

Abstract—OFDM multihop system is efﬁcient scheme to save energy and extend coverage. In this paper, optimal resource allocation in OFDM multihop system, including arbitrary number of hops and subcarriers, is studied. By joint subcarrier matching and power allocation, our goal is to maximize system capacity with the system-wide power constraint. First, we formulate the optimal subcarrier matching and power allocation problem as an optimization problem, which is a mixed binary integer programming problem and prohibitive to ﬁnd the global optimum. Then, by utilizing the special structure of the system, we propose a low complexity resource allocation scheme, which is proved to be globally optimal. The simulations prove that the proposed scheme improves the system capacity. Index Terms—multihop system, resource allocation, system capacity.

I. I NTRODUCTION

T

HE inter-symbol interference (ISI) caused by multi-path fading is one of the main challenges in single carrier high-speed data transmission system. The ISI is even increased because it reduces the symbol period to below channel coherence time. In orthogonal-frequency-duplex-multiplexing (OFDM) system, by dividing the total bandwidth into multiple sub-bands and increasing lengthen of symbol period, ISI can be mitigated. With many advantages, use of OFDM architectures is becoming a common approach for high speed wireless communication systems. At the same time, multihop system attracts more and more attention, as it can save energy and extend coverage. Therefore, the multihop system based on OFDM is an efﬁcient way for wideband system to save energy and extend coverage. In OFDM multihop system, to reduce the complexity of system architecture, we assume that the bits on a subcarrier can be forwarded on another subcarrier, which avoids the reallocation of the bits among the subcarriers at every hop. However, the subcarriers through different hop experience independent fading, the subcarrier with deep fading through a hop may be not deep fading through another hop. System capacity will be reduced without correct subcarrier matching. It means that resource allocation for OFDM multihop system should consider subcarrier matching together with power allocation. OFDM multihop system has been widely investigated [1][7]. In [1]-[3], resource allocation problem for three-node OFDM relay links were investigated. For cooperative OFDMA

This work was supported in part by the National Natural Science Foundation of China (No. 60879019, 60825104) and the startup foundation of Civil Aviation University of China (No. 08QD11X).

systems, resource allocation and scheduler problem were considered in [4]-[6]. Adaptive relaying scheme for OFDM that taking channel state information at the relay node into account has been proposed in [7], where subcarrier matching was considered for OFDM amplify-and-forward scheme and power allocation was not considered. Performances of the two-hop system with and without subcarrier matching were analyzed in [8]-[9], which reveal the performance improvement through subcarrier matching. However, in order to further improve system performance, the subcarrier matching and power allocation should be considered jointly. In [10]-[11], the joint subcarrier matching and power allocation was considered for only two-hop system with separate and system-wide power constraint. But, the most of papers did not consider subcarrier matching except [9]-[11] and all algorithms can not be directly extended to the system including arbitrary number of hops and subcarriers. In this paper, we consider joint subcarrier matching and power allocation for OFDM multihop system including arbitrary number of hops and subcarriers. Firstly, we formulate an optimization problem, which is a mixed binary integer programming problem and prohibitive in terms of complexity to ﬁnd a global optimum. Then, by utilizing the special structure of the system, we propose a low complexity joint subcarrier matching and power allocation scheme, and prove that it is optimal. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model In this paper, a wideband multihop system is considered including more than one relays. In order to mitigate the ISI, OFDM is utilized. Therefore, without loss of generality, the different subcarrier is assumed experience independent fading. The subcarriers with severe fading through one hop may have small fading through another hop. We assume that the number of hops is T , and the number of subcarriers is N . It means that the number of relay is T − 1. The number of subcarrier is large enough to make the fading of every subcarrier ﬂat. The relay strategies of all internal nodes are decode-and-forward. It is assumed that all internal nodes can only receive signal from the fore one-hop node but not from the other nodes because of distance or obstacle. The cooperative diversity is not considered in this paper. The different hop utilizes different time slot to forward the signal. This means that the time is divided to T equal

2

time slots. The communication between the adjacent nodes covers one time slot. The spatial reuse is not considered in this paper. We assume that every node holds the complete channel state information (CSI). The CSI can be obtained by channel estimation and feedback. The total system power is constrained. In order to improve system capacity, every relay has to rematch subcarrier and reallocate power. The total system power should be allocated among different hops and subcarriers. As before, we assume that different channels experience independent fadings. The different subcarriers through the same hop experience independent fadings. The power spectral densities of additive white Gaussian noise (AWGN) are equal at all nodes. The codes achieving the Shannon capacity are adopted. Therefore, the channel capacity of the subcarrier i through the hop t is given t Pit hti 1 t Ri Pi = log2 1 + 2 (1) T σ0 where Pit is the power allocated to the subcarrier i through the t hop, hti is the corresponding channel power gain, σ02 is the power spectral density of AWGN. The parameter T1 is because of every hop lasts for T1 of time of the whole communication process. B. Problem Formulation When the subcarrier i through the ﬁrst hop is matched to the given subcarrier through all other hops, we call these subcarrier as i matched subcarrier group. Therefore, the channel capacity of i matched subcarrier group is the minimum of capacities of all subcarrier, which can be expressed as ⎛ ⎞ N N Ri = min ⎝Ri1 Pi1 , ρ2ij Rj2 Pj2 , ..., ρTij RjT PjT ⎠ j=1

j=1

(2) where, ρtij ∈ {0, 1} denotes the result of subcarrier matching, is one when the subcarrier j through the t hop is matched to the subcarrier i through the ﬁrst hop, otherwise is zero. The upper equation means that the channel capacity of the matched subcarrier is limited by the worst subcarrier in the matched subcarrier group, which implies that the subcarrier matching is necessary to maximize the system capacity. The system capacity is the capacity sum of all matched subcarrier groups, which can be expressed as Rsum =

N

Ri

(3)

i=1

The power allocated to all subcarriers through all hops is constained, i.e., system-wide power constraint N T

Pit ≤ Ptot

(4)

t=1 i=1

Another constraint is the one-to-one subcarrier matching. The bits through one subcarrier can only transmitted by one

subcarrier. It simply the system architecture. N

ρtij = 1

(5)

j=1

In order to match subcarrier and allocate power optimal, we can formulate it as an optimization problem as following max t t

N

Ri Pi ,ρij i=1 N T s.t.

Pit ≤ t=1 i=1 N ρtij = 1 j=1 Pit ≥ 0, ρtij =

Ptot

{0, 1}

(6)

∀i, j, t

The variables are the power allocated to every subcarrier Pit and the subcarrier matching variable ρtij . Because the subcarrier matching variables are constrained to be one or zero, it is a mixed binary integer programming problem and NP-hard [12]. This means that it is very difﬁcult to ﬁnd global optimum in terms of complexity. We can ﬁnd the global optimum by searching the all possibilities about the subcarrier matching variables. When the subcarrier matching is constant, the upper optimization problem is convex without the last two constraints. The optimal power allocation can be obtained by solving the convex optimization problem. For every possibility, we obtain the optimal power allocation. Then, by comparing the all possibilities, the greatest system capacity is optimal. The corresponding power allocation and the subcarrier matching are also optimal. But, when the number of subcarriers and hops becomes great, the complexity is not achievable. In the following section, by making use of the special structure of the system, the low complexity scheme is proposed and proved to be globally optimal. III. L OW- COMPLEXITY O PTIMAL R ESOURCE A LLOCATION S CHEME Before giving the low complexity optimal resource allocation scheme, we can get two lemmas from the reference [10]. Without loss of generality, we assume that hti ≥ hti+1 . Lemma 1: For any two-hop system including arbitrary number of subcarriers, the matched subcarrier pair can be equivalent to a single subcarrier. The relation of channel power gains among them can be expressed as following 1 1 1 = 1+ 2 (7) hi hi hj where hi is the channel power gain of the equivalent subcarrier. Lemma 2: For the two-hop system with arbitrary number of subcarriers, the optimal subcarrier matching is to match subcarrier according to the channel power gains, i.e., h1i ∼ h2i . Together with the optimal power allocation for this subcarrier matching, they are optimal joint subcarrier matching and power allocation. The details of proofs about the Lemma 1 and the Lemma 2 can be obtained in the reference [10].

3

The other equivalent subcarrier channel power gains can be obtained iteratively by making use of the fore equivalent subcarrier channel power gains. ht+1,i = f ht,i , ht+1 (9) i

First Hop

Second Hop

Third Hop

Final Subcarrier Matching and Equivalent Channel

Last Hop

Fig. 1.

Block diagram of iteratively subcarrier matching and equivalence

Using aforementioned lemmas, we can get the following proposition, which states the optimal subcarrier matching result between any adjacent two hops. Proposition 1: For the global optimum, the optimal subcarrier matching through the any adjacent two hops is according to the order of the subcarrier power gains, i.e., hti ∼ ht+1 . i Proof: This proposition will be proved in the contrapositive form. Assuming that there is a subcarrier matching scheme including two matched subcarrier pairs hti ∼ ht+1 j and htj ∼ ht+1 (j = i), and the channel capacity of this two i subcarrier pairs is greater than that of the matching scheme in proposition 1. When the power allocated to other subcarrier pairs and the other subcarrier matching are constant, the channel capacity of this two subcarrier pairs is the optimum according to the subcarrier matching in proposition 1. It means that there is no subcarrier matching scheme better than the subcarrier matching scheme in propostion 1, e.g, hti ∼ ht+1 and htj ∼ ht+1 i j . Corresponding, the system capacity is greater than that of any subcarrier matching. It is contrary to the assumption. Therefore, there is no subcarrier matching scheme better than the scheme in proposition 1. It means that the optimal subcarrier matching between the any adjacent two hops is according to the order of the subcarrier power gain, i.e., hti ∼ ht+1 . i From Proposition 1, we can get the following result directly. For the global optimum, the optimal subcarrier matching is h1i ∼ h2i ∼ ... ∼ hTi . Here, by utilizing the special structure of the system, the main idea is that the matched subcarrier group through arbitrary number of hops can be matched and equivalent to a subcarrier iteratively, which can be demonstrated in Fig. 1. From the ﬁrst hop, getting the optimal subcarrier matching by lemma 2; then the equivalent subcarrier as the ﬁrst hop, matching the subcarrier with the third hop, and so on. Incorporate the Lemma 1 and the Proposition 1, we get the equivalent subcarrier for the matched subcarrier group i. We xy . Therefore, the channel deﬁne the function f (x, y) = x+y power gain of the equivalent subcarrier can be obtained iteratively. For the ﬁrst hop, the equivalent subcarrier channel power gains are themselves. h1,i = h1i

(8)

By now, we obtain the optimal subcarrier matching and the channel equivalent subcarrier for all hops, so the power allocation among the hops can be equivalent to allocate power among the equivalent subcarriers. Then, the optimal power allocation problem can be formulated as the following optimization problem

N PT ,i hT ,i 1 max 1 + log 2 T σ2 PT ,i i=1 N

s.t.

i=1

0

(10)

PT,i ≤ Ptot

PT,i ≥ 0

∀i

It is clear the upper optimization problem is convex. The optimal power allocation can be obtained by solving the convex optimization problem. The optimal power allocation is water-ﬁlling. + 1 σ02 PT,i = (11) − λ hT,i +

where (a) = max (a, 0). The parameter λ can be found by the following equation N

PT,i = Ptot

(12)

i=1

The power allocation among the subcarrier matching groups is obtained. The power allocation among the subcarriers in a subcarrier matching group is to make the subcarrier capacity of all subcarriers equivalent as follow Ri1 Pi1 = Ri2 Pi2 = ... = RiT PiT (13) T

Pit = PT,i

(14)

t=1

Because the upper subcarrier matching is globally optimal and the optimal power allocation is based on this subcarrier matching, the power allocation together with the subcarrier matching is optimal joint subcarrier matching and power allocation. The global optimum of the optimization problem (6) is also the optimal joint subcarrier matching and power allocation. This means that it is the globally optimum for the optimization problem (6). IV. N UMERICAL E XAMPLES In simulations, we compare the capacity of the optimal joint subcarrier and power allocation with that of several other schemes. These schemes include: (I) No subcarrier matching and no power allocation; (II) Power allocation and no subcarrier matching; (III) Subcarrier matching and no power allocation. Here, the methods of subcarrier matching and power allocation are the same as that of the optimal joint subcarrier matching and power allocation. No power allocation

4

1.8 Optimally Joint Scheme III Scheme II Scheme I

1.6

Capacity(bits/s/Hz)

1.4 1.2 1 0.8 0.6 0.4 0.2 0 0

5

10

15 SNR(dB)

20

25

30

Fig. 2. Comparison of different schemes for joint subcarrier matching and power allocation with different SN R

6 Optimally Joint Scheme III Scheme II Scheme I

Capacity(bits/s/Hz)

5

4

3

scheme improves the system capacity considerably, compared with all other schemes. Fig. 3 shows that the total channel capacity versus the number of hops. The same conclusions can be obtained about the comparison of all schemes. The capacity of the scheme I, where there is no subcarrier matching and no power allocation, is the least one compared with that of the other schemes. If other conditions remain unchanged, subcarrier matching or power allocation can improve the total channel capacity by comparing the capacity of scheme I with the scheme II and III. The proposed scheme of the optimal joint subcarrier and power allocation scheme improves the system capacity considerably, compared with all other schemes. The system capacities of all schemes decrease as increasing the number of hops. The reason is that the worst subcarrier will limit the capacity for every matched subcarrier group. More is the number of hops, more possible is the deep fading. V. C ONCLUSION In OFDM multihop system including arbitrary number of hops and subcarriers, this paper considered optimal resource allocation, e.g., joint subcarrier matching and power allocation. Though the formulated optimization problem is a mixed binary integer problem and prohibitive to ﬁnd global optimum in terms of complexity, we proposed a low complexity scheme by utilizing the special structure of the system, which achieves to the global optimum. R EFERENCES

2

1

0

1

2

3

4

5

6

7

8

9

10

Number of hops

Fig. 3. Comparison of different schemes for joint subcarrier matching and power allocation with different number of hops

means that the system power is allocated equally among the all subcarriers through all hops. Fig. 2 shows that the total channel capacity versus SN R. We employ a four-hop system including 16 subcarriers, which means that T = 4 and N = 16. In the computer simulations, we assume that each subcarrier undergoes Rayleigh fading independently. The SN R is deﬁned as SN R = σP2tot , where 0B B is the total bandwidth and σ02 is the power spectral density of AWGN. The capacity of the scheme I, where there is no subcarrier matching and no power allocation, is the least one compared with that of the other schemes. If other conditions remain unchanged, subcarrier matching or power allocation can improve the total channel capacity. Specially, subcarrier matching can improve the capacity when comparing the capacity of the scheme III to that of the scheme I. The system capacity can be improved by power allocation when comparing the capacity of the scheme II to that of scheme I. The proposed scheme of the optimal joint subcarrier and power allocation

[1] I. Hammerstrom and A. Wittneben, “On the Optimal Power Allocation for Nonregenerative OFDM Relay Links,” IEEE International Conference on Communications, ICC’06, pp. 4463-4468, June 2006. [2] B. Gui and L. J. Cimini, Jr., “Bit Loading Algorithms for Cooperative OFDM Systems,” EURASIP Journal on Wireless Communications and Networking, vol. 2008, Article ID 476797, 9 pages, 2008. [3] Y. Ma, N. Yi, and R. Tafazolli, “Bit and Power Loading for OFDMBased Three-Node Relaying Communications,” IEEE Transactions on Signal Processing, vol. 56, no. 7, pp. 3236-3247, July 2008. [4] S. -J. Kim, X. Wang, and M. Madihian, “Optimal Resource Allocation in Multi-hop OFDMA Wireless Networks with Cooperative Relay,” IEEE Transactions on Wireless Communications, vol. 7, no. 5, pp. 1833-1838, May 2008. [5] M. Pischella, and J. -C. Belﬁore,, “Power Control in Distributed Cooperative OFDMA Cellular Networks,” IEEE Transactions on Wireless Communications, vol. 7, no. 5, pp. 1900-1906, May 2008. [6] M. Kaneko and P. Popovski., “Radio resource allocation algorithm for relay-aided cellular OFDMA system,” IEEE International Conference on Communications, ICC’07, pp. 4831-4836, June, 2007. [7] M. Herdin, “A chunk based OFDM amplify-and-forward relaying scheme for 4G mobile radio systems,” IEEE International Conference on Communications, ICC’06, pp. 4507 -4512, June, 2006. [8] H. A. Suraweera, and J. Armstrong, “Performance of OFDM-Based DualHop Amplify-and-Forward Relaying,” IEEE Communications Letter,vol. 11, no. 9, pp. 726-728, Sept. 2007. [9] C. Athaudage, M. Saito, J. Evans, “Performance Analysis of Dual-Hop OFDM Relay Systems with Subcarrier Mapping, ” IEEE International Conference on Communications, ICC’08, May, 2008. [10] Wenyi Wang, Shefeng Yan, and Shuyuan Yang. “Optimally Joint Subcarrier Matching and Power Allocation in OFDM Multihop System,” EURASIP Journal on Advances in Signal Processing, vol. 2008, Article ID 241378, 8 pages, 2008. [11] Wenyi Wang, Shuyuan Yang, and Li Gao. “Comparison of Schemes for Joint Subcarrier Matching and Power Allocation in OFDM Decode-andForward Relay System,” IEEE International Conference on Communications, ICC’08, May, 2008. [12] B. Korte, J. Vygen, and J. Vygen, Combinatorial Optimization: Theory and Algorithms, 3rd ed. New York: Springer-Verlag, 2002.