IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 7, SEPTEMBER 2013

Robust Power Allocation for Multicarrier Amplify-and-Forward Relaying Systems Anlei Rao, Student Member, IEEE, M. D. Nisar, Member, IEEE, and M.-S. Alouini, Fellow, IEEE Abstract—It has been shown that adaptive power allocation can provide a substantial performance gain in wireless communication systems when perfect channel state information (CSI) is available at the transmitter. In the case when only imperfect CSI is available, the performance may significantly degrade, and robust power-allocation schemes are developed to save this kind of degradation. In this paper, we investigate powerallocation strategies for multicarrier systems, in which each subcarrier employs single amplify-and-forward (AF) relaying scheme. Optimal power-allocation schemes are proposed by maximizing the approximated channel capacity under an aggregate power constraint (APC) and a separate power constraint (SPC). By comparison with the uniform power-allocation (UPA) scheme and the best channel power-allocation (BCPA) scheme, we confirm that both the APC and SPC schemes achieve a performance gain over benchmark schemes. In addition, the impact of channel uncertainty is also considered in this paper by modeling the uncertainty regions as bounded sets, and results show that the uncertainty can degrade the worst case performance significantly. Index Terms—Multicarrier system, robust power allocation.

I. I NTRODUCTION To improve the throughput and power efficiency of communication systems, the transmission strategy is often adapted according to the channel conditions. When perfect channel state information (CSI) is available at the transmitter, the transmission power can be adapted to optimize a certain objective function, such as the transmission rate, the SNR, or the channel capacity. For multiple parallel channels, the optimal power-allocation strategy is given by the water filling algorithm [1]. The perfect CSI, however, often requires errorless channel estimation and reliable feedback, which is usually unrealistic for practical systems. In practice, the available CSI is often partial or subject to some kind of uncertainty. As such, the problem is how to design robust power-allocation algorithms with partial or imperfect CSI. The CSI uncertainty is typically modeled in two ways. The first is the stochastic framework, in which the uncertainty of the channel fading itself is assumed to be following a certain distribution, and the design of algorithms is based on the mean and covariance of the distribution [2], [3]. The deterministic framework, on the other hand, assumes that the channel uncertainty is deterministic and taking

Manuscript received April 25, 2012; revised August 21, 2012, November 22, 2012 and February 14, 2013; accepted February 19, 2013. Date of publication March 11, 2013; date of current version September 11, 2013. The review of this paper was coordinated by Prof. M. Uysal. A. Rao and M.-S. Alouini are with the Computer, Electrical, and Mathematical Engineering and Science Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia (e-mail: anlei. [email protected]; [email protected]). M. D. Nisar is with Intel Corporation, Mobile & Communications Group, 85579 Munich, Germany. The work reported in this paper was done in his private capacity. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2013.2252212

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values from a known set, and the uncertainty regions lie around these nominal channel coefficients [4]–[8]. This framework does not require the statistical knowledge of the CSI as for the stochastic framework but need the estimations of instantaneous CSI. The stochastic framework gives the optimal algorithm with respect to the average performance based on the channel fading distributions, whereas the deterministic framework designs robust algorithms with respect to the uncertainty region based on instantaneous channel conditions. Nisar and Utschick [4] considered a spherical and unbiased channel uncertainty model, and derived the optimal power-allocation strategy by maximizing the received SNR. In [5], the band model, or the confidence-intervalbased model, was used for spectral uncertainty, for robust equalization design, and for robust matched filter design. In [6], a variety of uncertainty models, including the high-dimensional-band model (the cubic model), was investigated to design precoders by maximizing the worst case received SNR in the multiple-input–multiple-output (MIMO) systems. In addition, the band model was also adopted in [7] to design the Tomlinson–Harashima precoders for broadcast channels. In [8], the band model was used to derive robust power split between the transmitter and the relay in a single-carrier relaying system. For relay systems, the amplify-and-forward (AF) scheme [9] is often adopted due to its simplicity. In [10], robust power-allocation schemes for coherent and noncoherent AF relay networks were designed under imperfect global CSI by maximizing the output SNR, and the authors used ellipsoidal uncertainty sets for the imperfect CSI model. In [11], optimal transmit power allocations at the transmitter and the relay are obtained based on the average SNR and the average bit error rate (BER) in an AF relaying systems where the direct link is in deep fading. In [12], a number of computational strategies were presented to evaluate both the problems of minimizing power consumption with a rate constraint and maximizing the throughput with a power constraint in the MIMO relay systems. Zhang et al. [13] investigated a novel multicarrier relay scheme in which the subcarrier power allocation is jointly optimized with the relay scaling coefficients of different subcarriers. A quadratic complexity algorithm and a suboptimal algorithm with linear complexity are proposed to solve the optimization problem. In this paper, we consider the power-allocation problem for multicarrier AF-relaying systems, in which each subcarrier performs data transmission via a single relay. The power-allocation strategies are approached by maximizing the approximated channel capacity under an aggregate power constraint (APC) and a separate power constraint (SPC), respectively. The CSI channel uncertainty in this paper still follows a deterministic framework, but unlike the spherical model in [4] or the ellipsoidal model in [10], we adopt the band model to allow the uncertainty of each link to fluctuate within a region around its actual values, as it did in [8]. In addition, we also assume that the direct links are in deep fading, and discard the signal from the direct links and only consider the relay link transmission. Some extension work on this paper can be focused on power allocation and optimal relay precoding design regarding quality-of-service constraints, such as the sum rate and the mean square error, in the MIMO relay networks [14], or furthermore, in the MIMO multirelay networks. II. S YSTEM M ODEL We consider a multicarrier system with N subcarriers in total, and each subcarrier transmits data via a single relay employing AF scheme. For each subcarrier, it is single-input single-output (SISO) with the direct transmitter–receiver link negligible due to large path attenuation. This choice is motivated by the fact that the relay link plays a much more important role when the channel conditions of the direct link

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are of bad quality. The channel gains for the transmitter–relay link and relay–receiver link of the ith subcarrier are denoted by hi and gi , respectively.1 With xi sent from the ith transmitter, the ith relay receives zi = hi xi + n1i , and with a relay gain Ai , the ith receiver finally has yi = Ai gi (hi xi + n1i ) + n2i . n1i ∼ N (0, σ12 ) and n2i ∼ N (0, σ22 ) are the additive white Gaussian noise (AWGN) at the ith relay and the ith receiver, respectively. For the channel-assisted relays, the relay gain is determined accordpi h2i + ing to the CSI of the transmitter–relay link as [15] A2i = q¯i /(¯ σ12 ), where p¯i = |xi |2 is the power allocated for the ith transmitter, and q¯i is the power allocated for the ith relay. The SNR for the ith subcarrier is then given by p¯i h2i q¯i gi2 . γi = 2 2 σ1 σ2 + σ22 p¯i h2i + σ12 q¯i gi2

(1)

Taking pi and qi as the normalized power, i.e., pi = p¯i /σ12 and qi = q¯i /σ22 , so that the SNR for the transmitter–relay link and relay–receiver link of the ith subcarrier are γ1i = pi h2i and γ2i = qi gi2 , respectively. Thus, the SNR for the ith subcarrier can be approximated by γi =

1+

pi h2i qi gi2 pi h2i + qi gi2

 (3)

whereas the channel capacity for the whole multicarrier system is given by CΣ ({pi }, {qi }) =

N  i=1

 log 1 +

pi h2i qi gi2 pi h2i + qi gi2

max

{pi ,qi }∈S



S=

CΣ ({pi }, {qi }) ,

{pi , qi }

N  

σ12 pi

+

where

σ22 qi

 (4)

We choose the natural-logarithm-based channel capacity in (4) as our objective function, and maximize (4) with given power constraints to find the optimal power-allocation schemes. III. P OWER A LLOCATION W ITH P ERFECT C HANNEL E STIMATION Here, we assume errorless estimation and reliable feedback of the channel conditions, so that perfect CSI of both the transmitter–relay link and relay–receiver link is available to the transmitters. Then, the performance gain can be achieved by adapting the power allocation across subcarriers for the transmitters and relays according to the CSI.

≤ S, pi > 0, qi > 0

min max L1 ({pi }, {qi }, λ) , λ>0 {pi ,qi }

(5)

L1 ({pi }, {qi }, λ) =

N  i=1

where



pi h2i qi gi2 log 1 + pi h2i + qi gi2



S−

N  

σ12 pi

+



σ22 qi



.

(6)

i=1

The optimal power allocation {p∗i , qi∗ } for the inner maximization of (6) can be achieved by directly taking the first partial derivatives of L1 ({pi }, {qi }, λ) over pi and qi , respectively, i.e., ∂L1 /∂pi = 0 and ∂L1 /∂qi = 0, which are equivalent to 2

h2i (qi gi2 ) = λσ12 (pi h2i + qi gi2 + pi h2i qi gi2 ) (pi h2i + qi gi2 ) 2

(pi h2i

+

qi gi2

gi2 (pi h2i ) = λσ22 . + pi h2i qi gi2 ) (pi h2i + qi gi2 )

(7)

From (7), it is obvious to see that pi hi σ1 = qi gi σ2 , which shows that the status of the transmitter and relay in one subcarrier are always the same. Taking this simple relation into (7), we could derive the optimal assignment of power to each transmitter and relay as3



1 + αi 1 − σ1 hi λ αi α

1 + 1 i − qi∗ = σ2 gi λ αi

(8)

where αi = hi gi /(σ2 hi + σ1 gi ), and (·)+ = max(·, 0). The results are similar to the water filling strategy. For each subcarrier, it is active only if the channel conditions of this subcarrier are good enough to satisfy αi2 > λ, and αi is determined by the channel gain and the noise level of both links. From (8), we see that whether one subcarrier is active or not depends on the Lagrangian multiplier λ, which is determined by the outer minimization in (6). First, the channel capacity in (4) can be written as CΣ ({p∗i }, {qi∗ }, λ) =

 T1

1 In practical systems, the channel coefficients h ˜ i and g˜i for each link are ˜ i xi + n ˜ i = hi ejφi complex, such that z˜i = h ˜ 1i and y˜i = g˜i zi + n ˜ 2i . Let h ˜ i | and gi = |˜ and g˜i = gi ejψi , where hi = |h gi |, respectively. We assume that ˜ i and g˜i . the relay and the receiver have perfect information of the phase of h This assumption serves at least as an approximation for the scenario when the relay/receiver CSI is of sufficient reliable quality. As phase information φi or ψi are obtained, they are applied to cancel the phase in z˜i and y˜i , such that zi = e−jφi z˜i = hi xi + n1i and yi = e−jψi y˜i = gi zi + n2i .

.

As the power constraint in S forms a convex set, and the channel capacity in (4) is concave on {pi , qi },2 the maximization is convex and can be rewritten as a Lagrangian dual problem, i.e.,

p∗i = .



i=1

(2)

where γi is a tight upper bound of γi , particularly under high SNR environments when γ1i and γ2i are large. This tight bound is very useful and can translate to a tight lower bound to the average BER and outage probability [16]. In such cases, the end-to-end SNR for each subcarrier is given by (2). If we adopt the natural-logarithmbased channel capacity (in nats/Hz), a tight upper bound of the channel capacity for the ith subcarrier with given power allocations pi and qi yields Ci (pi , qi ) = log (1 + γi ) = log

The total power available for the whole system is often limited; therefore, we constrain the aggregate Npower available for all the trans(¯ p + q¯i ) < S. Our resulting mitters and relays within S, i.e., i=1 i optimization problem can be written as



γ1i γ2i γ1i γ2i Δ  ≤ = γi 1 + γ1i + γ2i γ1i + γ2i



A. Power Allocation With APCs



log

αi2 λ



(9)

2 It is very easy to prove that the Hessian matrix of (4) is negative semidefinite when pi > 0 and qi > 0; therefore, the channel capacity (4) is concave on {pi , qi : pi > 0, qi > 0, i = 1, 2, . . . , N }. 3 Please note that the power allocations given in (8) for APC and in (15) for SPC are only optimal to the channel capacity with approximated SNR but may not be optimal to the capacity with accurate SNR. In Section V-B, we discuss the impact of the SNR approximation on the optimal power allocation.

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where T1 = {t1 , t2 , . . . , tN1 } is the index set in which the subcarriers are in active mode, and N1 is the cardinality of the set T1 . Considering that the power assigned to each subcarrier is σ12 p∗i + σ22 qi∗ = ((1/λ) − (1/αi2 ))+ , we then have L1 ({p∗i }, {qi∗ }, λ) = λS +





log

T1

αi2 λ



−1+

λ αi2

N

S+

1

1 T1 α2 i

.

.

(11)

B. Power Allocation With SPCs Here, we consider a more rigid situation that the power available for transmitters and relays has separate constraints, the optimization problem in this case then yields max{pi }∈P,{qi }∈Q CΣ ({pi }, {qi }), where the power constraints are given by P=

 Q=

N 



N 

qi∗

u,v>0 {pi ,qi }

.

=

N 



L2 ({p∗i }, {qi∗ }, u, v)

 P Q = u 2 + v2 2 + σ1 σ2

log

i=1



+ v2

Q  − qi σ22 N





 +u

2

 P − pi 2 σ1 N



i=1

.

(13)

In the same way, the inner maximization gives the equations that {pi } and {qi } must satisfy 2

+

qi gi2

(15)

 

log βi2

(16)

 2

log βi

T2

1 −1+ 2 βi

 .

(17)

T2

 T2

h2i (qi gi2 ) = u2 + pi h2i qi gi2 ) (pi h2i + qi gi2 )

T2

 1 hi −u =v ugi + vhi hi gi T2





 1 P + σ12 h2i T2

 1 Q + 2 σ2 gi2



.

(18)

T2

Multiplying the first equation in (18) by u and the second equation by v, and then adding them, we have



P  1 + σ12 h2i





2

u +

Q  1 + σ22 gi2



T2

v2 +

 2uv T2

hi gi

= N2

(19)

IV. C HANNEL U NCERTAINTY For practical communication systems, the perfect CSI at the transmitter side is often unavailable. Due to erroneous, outdated, or quantized feedback, the CSI obtained by the transmitters is typically subject to uncertainty. Here, we investigate the error analysis of the CSI at the transmitter side. The channel conditions are deterministically modeled as hi = hti + hui

2

gi2 (pi h2i ) = v2 . (pi h2i + qi gi2 + pi h2i qi gi2 ) (pi h2i + qi gi2 )

 1 gi −v =u ugi + vhi hi gi

which means the solution (u, v) for (18) lies on an ellipse in the u−v plane in a form of au2 + bv 2 + 2cuv = d, with which we can solve u from (18) and have the corresponding v from (19) iteratively.

i=1

(pi h2i

+





2

T2

pi h2i qi gi2 1+ pi h2i + qi gi2

1 βi − βi

where T2 = {t1 , t2 , . . . , tN2 } is the index set in which the subcarriers are active. Note that u2 p∗i + v 2 qi∗ = (1 − (1/βi2 ))+ ; thus, we have L2 ({p∗i }, {qi∗ }, u, v) as

(12)

where

L2 ({pi }, {qi }, u, v)



+

T2



The power for all transmitters and all relays are limited within P and Q, respectively, and the power constraints P and Q are convex sets; therefore, optimization problem max{pi }∈P,{qi }∈Q CΣ ({pi }, {qi }) is convex and can be written with two Lagrangian multipliers u2 and v 2 as max L2 ({pi }, {qi }, u, v) ,

1 βi

βi −

CΣ ({p∗i }, {qi∗ }, u, v) =

i=1

min

1 = vgi



where βi = hi gi /(vhi + ugi ). The subcarrier with a channel condition satisfying βi > 1 will be able to transmit, whereas others are suspended to save power for the active subcarriers. In addition, note that βi is not only determined by channel gains hi and gi but also by the Lagrangian multipliers u and v. With the optimal power allocation in (15), the channel capacity in (4) is then simplified to

σ22 qi ≤ Q, qi > 0

1 uhi

By minimizing (17) with respect to u and v, the solution for u∗ and v ∗ can be determined by numerically solving the set of two equations as follows:

σ12 pi ≤ P, pi > 0

i=1

p∗i =

(10)

As the optimal λ∗ in (11) depends on the active user set T1 , whereas T1 is decided jointly by the channel conditions {hi , gi } and threshold λ∗ , the value of λ∗ is iteratively solved with (11) by supposing that all the users are initially active.



with pi hi u = qi gi v leads to the optimal power allocations for all the transmitters and relays as



With the optimal power allocations p∗i and qi∗ , the optimization problem in (6) then yields minimization problem minλ>0 L1 ({p∗i }, {qi∗ }, λ), and the optimal Lagrangian multiplier λ can be obtained via the first-order optimality condition of (10) as λ∗ =

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(14)

Note that it is very similar to (7), and we also have pi hi u = qi gi v, which indicates that the transmitter and the relay in the same subcarrier are always in the same mode, either active or inactive. Solving (14)

gi = gti + gui

(20)

where hti and gti are the actual channel gains for the transmitter–relay link and the relay–receiver link, whereas hui and gui are the associated channel uncertainty for each link meeting the requirements

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that |hui | < hti and |gui | < gti . Uncertainty hui and uncertainty gui are the errors that cause the mismatch of the CSI obtained at the transmitters and the instantaneous actual CSI. The uncertainty regions for hui and gui are given as bounded sets, i.e., Uh = {hui : hui ∈ [−1i , 1i ]} Ug = {gui : gui ∈ [−2i , 2i ]} .

(21)

Please note that this uncertainty model is not aimed to reflect the performance of any practical estimators but is based on the notion of confidence interval. A quantity estimated from some measurements cannot be perfectly accurate; however, we can define a “confidence interval,” just like sets Uh and Ug in (21), within which the actual quantity lies with a very high probability based on the measurement/estimation quality. The notion of confidence interval works regardless of the type of estimator employed. Under such channel uncertainty, the optimal power allocation {p∗i , qi∗ } can be determined by solving the following minimax problem: {p∗i , qi∗ } = arg max

min

{pi ,qi } {hui }∈Uh ,{gui }∈Ug

CΣ ({pi }, {qi }) (22)

where the inner minimization over Uh and Ug gives the worst case channel capacity for the multicarrier system. For such an inner minimization, it is easy to show that ∂CΣ /∂hui > 0 and ∂CΣ /∂gui > 0 within the constraints Uh and Ug . This way, the inner minimization in (22) gives the uncertainty (hui , gui ) = (−1i , −2i ), so that the worst case happens only when the hui and gui reach the lower bounds of their uncertainty regions. In this case, the worst case channel capacity is given by CΣ−wc =

N  i=1

 log

1+

ˆ 2 qi gˆ2 pi h i i 2 ˆ + qi gˆ2 pi h i

 (23)

i

ˆ i = hti − 1 and gˆi = gti − 2 , and (23) takes the same form where h as the channel capacity in (4). To this end, we can obtain the robust power-allocation schemes for APC and SPC by optimizing the worst case channel capacity in (23) for the uncertainty case. Using the same procedure in Section III, the robust power allocation are still given by ˆ i , gˆi ). (8) and (15) only with (hi , gi ) replaced by (h To show that the power allocation given by (22) is robust, we must confirm that the minimax theorem holds for the optimization problem in (22). For the problem minx maxy f (x, y), the minimax theorem holds if minx maxy f (x, y) and maxy minx f (x, y) lead to identical solution (x∗ , y ∗ ), or equivalently f (x∗ , y) ≤ f (x∗ , y ∗ ) ≤ f (x, y ∗ ) [17]. Point (x∗ , y ∗ ) is called a saddle point of f (x, y); therefore, the minimax theorem holds if and only if the saddle point exists. We write the channel capacity given in (4) with hi = hti + hui and gi = gti + gui as CΣ ({pi , qi }, {hui , gui }), or for short, CΣ . We now show that its saddle point exists, and this saddle point is (x∗ , y ∗ ) = ({−1i , −2i }, {p∗i , qi∗ }). For the max–min problem max{pi ,qi } minUh ,Ug CΣ , it is easy to show that ∂CΣ /∂hui > 0 and ∂CΣ /∂gui > 0 with hui and gui constrain by Uh and Ug ; therefore, the channel capacity CΣ is monotonically increasing with hui and gui , i.e., the inner minimization minUh ,Ug CΣ gives x∗ = {−1i , −2i }. Then, consider the outer maximization max{pi ,qi } CΣ ({pi , qi }, {−1i , −2i }). Using the same process in Section III, we have the optimal power allocation y ∗ = {p∗i , qi∗ } given by (8) for APC and (15) for SPC, with hi = hti − 1i and gi = gti − 2i . On the other hand, we now show that the min–max problem minUh ,Ug max{pi ,qi } CΣ also gives the result (x∗ , y ∗ ) = ({−1i , −2i }, {p∗i , qi∗ }). First, the inner maximization gives the optimal power allocation y ∗ ({hui , gui }) = {p∗i ({hui , gui }), qi∗ ({hui , gui })},

where the values of optimal power allocation {p∗i , qi∗ } depend on the uncertainty levels {hui , gui }. Now, we consider the following three cases: • Case 1: The channel uncertainty is x1 = {hui,1 , gui,1 }, and the corresponding optimal power allocation is given by y1∗ = {p∗i (x1 ), qi∗ (x1 )} with the channel capacity CΣ (x1 , y1∗ ). • Case 2: The channel uncertainty is x2 = {hui,2 , gui,2 }, and the optimal power allocation regarding to x2 is y2∗ = {p∗i (x2 ), qi∗ (x2 )}, whereas the channel capacity is CΣ (x2 , y2∗ ). We suppose that the channel conditions of x2 is a little worse than that of x1 , denoted byx2 < x1 , which means that hui,2 ≤ hui,1 and gui,2 ≤ gui,1 for all the index i = 1, 2, . . . N , but the equality does not hold for all i. • Case 3: The channel uncertainty is x1 = {hui,1 , gui,1 }, but the given power allocation is y2∗ = {p∗i (x2 ), qi∗ (x2 )}; then, the corresponding channel capacity is CΣ (x1 , y2∗ ). Compare case 1 and case 3; they have the same channel uncertainty but different power allocation. As y1∗ is the optimal power allocation with associated channel uncertainty x1 , therefore, we have CΣ (x1 , y1∗ ) > CΣ (x1 , y2∗ ). Then, compare case 2 and case 3; they have the same power allocation. As shown earlier, the channel capacity CΣ is monotonically increasing with hui and gui when the power allocation are fixed. As x1 > x2 , then CΣ (x1 , y2∗ ) > CΣ (x2 , y2∗ ). This way, we have CΣ (x1 , y1∗ ) > CΣ (x1 , y2∗ ) > CΣ (x2 , y2∗ ) or CΣ (x1 , y1∗ ) > CΣ (x2 , y2∗ ). Compare case 1 and case 2; we see that the case 2 has worse channel conditions (x2 < x1 ), and the maximized channel capacity is also less. Then, it is easy to say that the channel capacity for the system CΣ with power allocation {p∗i (hui , gui ), qi∗ (hui , gui )} reaches it minimum at {hui , gui } = {−1i , −2i }. This way, the outer minimization of the min–max problem gives the results (x∗ , y ∗ ) = ({−1i , −2i }, {p∗i , qi∗ }). Now, we have shown that the max–min problem and the min–max problem max{pi ,qi } minUh ,Ug CΣ minUh ,Ug max{pi ,qi } CΣ lead to identical solution (x∗ , y ∗ ) = ({−1i , −2i }, {p∗i , qi∗ }), and this indicates that the minimax theorem holds for the optimization in (22) and that (x∗ , y ∗ ) is a saddle point of CΣ ({pi , qi }, {hui , gui }), which means that the power allocation in (22) is robust. V. S ELECTED N UMERICAL R ESULTS Here, we show some numerical results for our proposed powerallocation schemes. The channel gains of each subcarrier are supposed to be deterministic with some uncertainty. The interference between any two subcarriers and between the transmitter–relay link and the relay–receiver link are ignored, to focus on the impact of the channel uncertainty. A. Comparison of Different Allocation Schemes In Fig. 1, we ignore the channel uncertainty and compare the performance of different power-allocation schemes. These power-allocation schemes include the following: • APC scheme: The aggregate power is a constraint within S, and the power allocated for each transmitter and relay follows (8). • SPC scheme: The power constraints for the transmitters and relays are P and Q with P = Q = S/2. The optimal power allocation follows (15). • Uniform power-allocation (UPA) scheme: The power is uniformly distributed to each transmitter and relay, i.e., pi = qi = S/(2N ). • Best channel power-allocation (BCPA) scheme: The total power is allocated to the subcarrier with the best channel conditions, i.e.,

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Fig. 1.

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Channel capacity for different power-allocation schemes over Rayleigh fading (N = 10).

with the largest normalized SNR when the transmitter and the relay in the same subcarrier receive equal power. The power allocation to the selected subcarrier is pk∗ = qk∗ = S/2, where k∗ is defined as k∗ = arg max{h2i gi2 /(h2i + gi2 ), i = 1, 2, . . . , N }. The channels for the transmitter–relay links and the relay–receiver links are supposed to be subject to independent identically distributed (i.i.d.) Rayleigh fading with the average channel gains E[hi ] and E[gi ]. The Gaussian noise variance are handled as a unit, i.e., σ12 = σ22 = 1. The number of subcarriers in Fig. 1 is given as N = 10. From the figure, we see that the APC and SPC schemes perform almost the same when the transmitter–relay links and the relay–receiver links have equal average channel gains. When the discrepancy between the average channel gains E[hi ] and E[gi ] increases, the performance gap between the APC and SPC schemes becomes larger, whereas the performance of BCPA gradually exceeds that of UPA. In addition, it is not surprising to see that the APC and SPC schemes outperform UPA and BCPA, and it is necessary to point out that, for SPC, it is also of importance to split the power reasonably between transmitters and relays when the total power budget (P + Q) is fixed. More power should be distributed to the links with better channel conditions, so that the SPC scheme performs closer to the APC scheme. B. Approximated SNR Versus Accurate SNR In Section II, we mentioned that the approximated SNR γi (not γi ) in (2) is used for the channel capacity as our objective function. However, the optimal power allocation derived from the objective function with an approximated SNR and accurate SNR may be different. In Fig. 2, we investigate the average channel capacity for different

Fig. 2. Channel capacity for different power-allocation schemes over Rayleigh fading with E[hi ] = 1 and E[gi ] = 1.

power constraint levels with an approximated SNR and an accurate SNR. In this figure, we investigate three cases: the approximated case (the objective function uses the approximated SNR), the accurate case (the objective function uses the accurate SNR), and the accurate/ approximated case (the channel capacity is measured with accurate SNR, but the power allocation is from the approximated case). The channel for each link is assumed i.i.d. Rayleigh fading with unit average channel gain. In the figure, we plot the performance for the APC scheme of a two-carrier system, the SPC scheme of the two-carrier system, and the APC scheme of a single-carrier system. As we can see,

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links will be smaller, and the difference of the channel gains between the transmitter–relay links and the relay–receiver links decreases. For the APC, the total power allocated for all the transmitters will be close to that for all the relays, i.e., the power allocated for all the transmitters and all the relays will approach to be equal (the SPC scheme). In this case, the power-allocation scheme for APC will be closer to that for SPC, so that the performance gap between APC and SPC diminishes.

VI. C ONCLUSION

Fig. 3. Impact of uncertainty on channel capacity for APC and SPC over Rayleigh fading with E[hi ] = 2 and E[gi ] = 1/2 (N = 10).

the approximated channel capacity is a tight bound of the accurate capacity, and as the power budget increases (the SNR will be higher), the gap between the approximated and accurate capacity is smaller. Compare the dash curves (the accurate case) and the circles/squares (the accurate/approximated case), we find that, at a low-SNR region (small power budget), there exists small discrepancy between them, whereas at a high-SNR region (large power budget), this performance discrepancy almost disappears. This shows that the power allocation derived for the approximated case may be suboptimal to the accurate case, but the performance difference between these two powerallocation schemes is so small that it can be almost ignored, and the optimal power allocation for the approximated case can also be regarded “optimal” to the accurate case. C. Impact of Uncertainty on Channel Capacity In Fig. 3, we consider the impact of uncertainty on the worst case channel capacity. The uncertainty for each link is measured by ρji , which is defined as ρ1i = 1i /hti ρ2i = 2i /gti .

(24)

For simplicity, we take an equal normalized uncertainty level for all links, i.e., ρ1i = ρ2i = ρ for all i. When ρ = 0, there is no channel uncertainty. On the other hand, ρ = 1 corresponds to the maximum uncertainty that leads to the worst case channel capacity dropping to zero. In this figure, the channels are subject to Rayleigh fading with E[hi ] = 2 and E[gi ] = 1/2. The noise level is the same with that in Fig. 1. For the SPC, we equally distributed the total power to the transmitters and relays, i.e., P = Q = S/2. The uncertainty level of ρ = 0.2, ρ = 0.5, and ρ = 0.8 correspond to low uncertainty, medium uncertainty, and high uncertainty, respectively. As we can see in this figure, the channel capacity rises as we provide more power. The uncertainty can greatly decrease the worst case channel capacity. For the medium-level uncertainty, the worst case capacity is less than 50% of the capacity without uncertainty, whereas for the high-level uncertainty, the worst case capacity is less than 15%. For the low or zero uncertainty, increasing the power constraints achieves a remarkable improvement in the performance. Under the high uncertainty, however, providing more power also brings difference, but this difference seems fruitless when compared with the zero uncertainty and the low uncertainty. In addition, the gap between APC and SPC also diminishes when the uncertainty level increases. As the uncertainty level increases, the channel gains of all

In this paper, optimal power-allocation algorithms are proposed for multicarrier AF relaying systems with both APC and SPC by maximizing the approximated channel capacity. Some selected numerical results show that the performances of the APC and SPC schemes outperform considerably that of the UPA and BCPA schemes when the perfect CSI is available. When only imperfect CSI is obtained, the worst case throughput is employed to develop robust powerallocation schemes. The channel uncertainty levels are shown to have a substantial impact on the performance. When the uncertainty rises from a low level to a high level, a large amount of the worst case throughput is lost.

R EFERENCES [1] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. New York, NY, USA: Cambridge Univ. Press, 2005. [2] A. Narula, M. Lopez, M. Trott, and G. Wornell, “Efficient use of side information in multiple-antenna data transmission over fading channels,” IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1423–1436, Oct. 1998. [3] S. Jafar, S. Vishwanath, and A. Goldsmith, “Channel capacity and beamforming for multiple transmit and receive antennas with covariance feedback,” in Proc. IEEE ICC, Helsinki, Finland, Jun. 2001, pp. 2266–2270. [4] M.-D. Nisar and W. Utschick, “Minimax robust power allocation over parallel communication channels,” in Proc. 19th EUSIPCO, Barcelona, Spain, Aug. 2011. [5] S. Kassam and H. Poor, “Robust techniques for signal processing: A survey,” Proc. IEEE, vol. 73, no. 3, pp. 433–481, Mar. 1985. [6] A. Pascual-Iserte, D. Palomar, A. Pérez-Neira, and M. Lagunas, “A robust maximin approach for MIMO communications with imperfect channel state information based on convex optimization,” IEEE Trans. Signal Process., vol. 54, no. 1, pp. 346–360, Jan. 2006. [7] M. Shenouda and T. Davidson, “Tomlinson-Harashima precoding for broadcast channels with uncertainty,” IEEE J. Sel. Areas Commun., vol. 25, no. 7, pp. 1380–1389, Sep. 2007. [8] M.-D. Nisar and M.-S. Alouini, “Minimax robust power split in AF relays based on uncertain long-term CSI,” in Proc. IEEE VTC, San Francisco, CA, USA, Sep. 2011, pp. 1–5. [9] Y. Zhao, R. Adve, and T. Lim, “Improving amplify-and-forward relay networks: optimal power allocation versus selection,” in Proc. IEEE ISIT, Seattle, WA, USA, Jul. 2006, pp. 1234–1238. [10] T. Quek, M. Win, H. Shin, and M. Chiani, “Robust power allocation for amplify-and-forward relay networks,” in Proc. IEEE ICC, Glasgow, U.K., Jun. 2007, pp. 957–962. [11] H. Rasouli, “Performance of power allocation schemes in a two-hop AF relay system with faded direct link,” in Proc. 7th IWCMC, Istanbul, Turkey, Jul. 2011, pp. 749–753. [12] Y. Yu and Y. Hua, “Power allocation for a MIMO relay system with multiple-antenna users,” IEEE Trans. Signal Process., vol. 58, no. 5, pp. 2823–2835, May 2010. [13] W. Zhang, U. Mitra, and M. Chiang, “Optimization of amplify-andforward multicarrier two-hop transmission,” IEEE Trans. Commun., vol. 59, no. 5, pp. 1434–1445, May 2011. [14] L. Sanguinetti, A. D’Amico, and Y. Rong, “A tutorial on the optimization of amplify-and-forward MIMO relay systems,” IEEE J. Sel. Areas Commun., vol. 30, no. 8, pp. 1331–1346, Sep. 2012. [15] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062–3080, Dec. 2004.

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 62, NO. 7, SEPTEMBER 2013

[16] M. Hasna and M. Alouini, “End-to-end performance of transmission systems with relays over Rayleigh-fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 6, pp. 1126–1131, Nov. 2003. [17] D. Bertsekas, A. Nedi, and A. Ozdaglar, Convex Analysis and Optimization. Belmont, MA, USA: Athena Scientific, 2003.

Mobile Data Offloading With Policy and Charging Control in 3GPP Core Network Sok-Ian Sou, Member, IEEE Abstract—The Third-Generation Partnership Project (3GPP) has recently standardized the policy and charging control (PCC) system to provide dynamic policy enforcement for network resource allocation combined with real-time charging. The upward trend in smartphone use and the subsequent explosion of data traffic on third-generation (3G) networks have pressed carriers to offload the data traffic from the cellular domain. One key challenge is to elaborate on PCC functionality and mobility management for offloading sessions. This paper proposes an enhanced Wi-Fi offloading model to bring mobile Internet Protocol (IP) integration to a core network with PCC. Specifically, we develop a comprehensive analytical model to quantify the performance of data offloading concerning the amount of 3G resources saved by offloading and the deadline assurance for measuring the quality of user experience with PCC support. Numerical results demonstrate that deadline assurance can be satisfied while saving a significant amount of 3G resources in many situations. Index Terms—Cellular core, deadline miss, mobility, policy and charging control (PCC), signaling overhead.

I. I NTRODUCTION Over the past several years, the increasing use of Wi-Fi-enabled smartphones and third generation (3G)-capable tablets has lifted mobile broadband data traffic to a great extent around the world. The Wi-Fi offloading solution has been intensively discussed as an essential technology to alleviate traffic load for mobile data. Incorporating nonThird-Generation Partnership Project (3GPP) access technologies in 3GPP standards, such as loose coupling and tight coupling with Wi-Fi access, has been studied in the past decade [1], [2]. In tight coupling, the 3GPP system specifies rerouting of cellular network signaling through Wi-Fi that is considered as a de facto 3GPP radio access network. In loose coupling, the 3GPP system uses Wi-Fi access to transfer and tunnel Internet Protocol (IP) data between a mobile device and the operator’s core network; the only integration and interworking point between networks is the common authentication architecture. Unlike early coupling-based integration, in mobile data offloading, the goal is to dynamically redirect (offload) selected Internet traffic toward a low-cost access radio network to alleviate data congestion while delivering positive user experience. Because there is an urgent need for offloading, 3GPP has been very active recently in developing a new Manuscript received September 2, 2012; revised January 14, 2013; accepted March 18, 2013. Date of publication March 29, 2013; date of current version September 11, 2013. This work was supported in part by the National Science Council, Taiwan, under Contract NSC 100-2221-E-006-200 and Contract NSC 101-2221-E-006-216. The review of this paper was coordinated by Dr. T. Taleb. The author is with the Department of Electrical Engineering, Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan 701, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2013.2255899

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standard to support the simultaneous use of cellular access [Long-Term Evolution (LTE)], femtocells, and non-3GPP access (such as Wi-Fi) [3]–[5]. Discussions on 3GPP offloading include the definition of new architecture in addition to issues regarding security, charging, mobility, and traffic control/handling [6], [7]. The IP-based policy and charging control (PCC) system has been standardized for dynamic policy control combined with real-time charging since 3GPP/LTE release 8. Delivering appropriate PCC functionalities to mobile nodes with offloading is essential [8]. One key challenge is to elaborate on the PCC functionality and mobility management of an offloading session in the service data flow level, which is significantly different from that in the IP packet level widely discussed before. Although offloading traffic is not targeted for high-quality services such as traditional cellular services, customer satisfaction, including performing admission control and providing a minimum degree of completion deadline, plays a critical role in the quality evaluation criterion [9], [10]. This paper intends to answer three essential questions. 1) How do we offload data with PCC support for service data flow mobility? 2) How much data can actually be offloaded? 3) How does the PCC system serve the offloading session in terms of quality of experience (QoE), such as deadline assurance that provides a measurement of QoE in terms of session completion within a given deadline period? In the current literature, there are only a few studies on offloading. For instance, Balasubramanian et al. in [11] leveraged delay tolerance of applications and used fast switching to the context of augmenting 3G with Wi-Fi in mobile environments. Lee et al. in [12] conducted an experimental testbed and simulation study to exploit offloading efficiency with delayed transmission. Zhou et al. in [13] reported that many operators perform deprioritization of heavy users in wireless networks for congestion management. In [14], Han et al. proposed to delay the delivery of information over cellular networks and offload it through the free opportunistic communications. In [15], Oliva et al. presented the advantages and drawbacks of two approaches to enable service data flow mobility based on Internet Engineering Task Force and 3GPP standards. The previous studies revealed that a significant amount of data can be offloaded to Wi-Fi networks when it is properly deployed; however, a number of protocol issues, such as service-level mobility management, charging record generation, policy control management, and the performance on QoE guaranteed, involved in core networks are not taken into consideration. Before sophisticated deployment on micro Wi-Fi offloading areas, it is crucial to implement appropriate policy control and to raise the quality of user experience. This paper proposes an enhanced offloading framework with PCC support for a Wi-Fi-based offloading solution. In addition, we investigate how to raise QoE by evaluating the deadline assurance in offloading [16]. In particular, we make the following contributions: First, we present a generic framework to support Wi-Fibased offloading with PCC functionality and dynamic service data flow mobility. Second, we investigate the probability of missing a deadline and the amount of 3G resources saved by offloading. Third, we conduct extensive simulations considering the traffic flow characteristics of mobile environments. The remainder of this paper is organized as follows: Section II proposes a Wi-Fi offloading reference model with service data flow mobility. Section III formulates an analytical model for computing output metrics. Section IV gives numerical results considering various network parameters. Section V provides our conclusions.

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