Subchannel Allocation in Relay-Enhanced OFDMA Downlink With Imperfect Feedback Jouko Leinonen1 , Taneli Riihonen2 , Jyri H¨am¨al¨ainen2 , and Markku Juntti1 1

Centre for Wireless Communications, University of Oulu, P.O. Box 4500, FI–90014, Oulu, Finland [email protected], [email protected] 2 Helsinki University of Technology, P.O. Box 3000, FI-02015 TKK, Finland [email protected], [email protected]

Abstract— This paper studies the resource allocation for orthogonal frequency division multiple access (OFDMA) in a fixed amplify-and-forward (AF) relay link. We express closed-form average capacity results for the system when frequency allocation is based on limited and imperfect feedback information and multiple antennae are assumed at the relay node (RN) and the user equipment (UE). Feedback strategies based on the best-M method and resource block (RB)-wise one bit quantization are analyzed and compared. Capacity expressions are presented for space-time block coding (STBC) and transmit antenna selection (AS) schemes. The capacity results with imperfect feedback path are of particular interest. The results illustrate that promising tradeoff between the system performance, feedback overhead, and robustness against feedback errors can be achieved by using AS and limited feedback in frequency allocation.

I. I NTRODUCTION In future wireless communication networks, spectral efficiency is enhanced through opportunistic channel allocation, multiple-input multiple-output (MIMO) methods and relay nodes (RNs). Especially, connections of users near the cell edge gain from these advanced communication techniques as they are expected to offer higher data rates with lower transmit powers. The purpose of opportunistic channel allocation and MIMO methods is to benefit from the inherent time, frequency and spatial diversity of wireless environment. On the other hand, relays mitigate the effect of attenuation by dividing a long single-hop link to two shorter hops, thus reducing the path losses. Thereby, a combination of these schemes is then able to tackle the most important aspects of wireless transmission. The study of frequency allocation in orthogonal frequency division multiple access (OFDMA) based relay links has received remarkable attention in research community, see, e.g. [1]. Analytical performance evaluations have also been presented for single user relay links [2], [3] and for multiuser single-carrier relay systems [4], [5]. OFDMA multiuser systems with limited feedback information have also been considered, see, e.g., [6], [7]. Often these analyses assume ideal feedback or are based on simulations. However, the dynamic channel assignment in OFDMA based relay links with limited feedback has received less attention in existing publications and up to our knowledge the performance of such systems has not been characterized analytically. The purpose of this paper is to fill this cap.

In this paper we extend the analysis in [8], [9] for OFDMA relay links with multiple antennae in each end. Analytical study in [10] shows that the optimal subcarrier pairing provides some performance gain when the subchannels from the RNs to user equipments (UEs) are fixed. In this paper, we assume that feedback information is conveyed from the UE to the RN so that RBs can be dynamically allocated according to the strictly limited channel knowledge. We express closed-form capacity results for the system with the best-M feedback method and resource block (RB)-wise one bit feedback strategy when optimal rate adaptation is applied. Space time block coding (STBC) and antenna selection (AS) schemes are considered. The reliability of the feedback path has crucial impact on the system performance. Feedback bit error probability (FEP) is a parameter in the presented analysis so that the tradeoff between system performance, feedback overhead and robustness can be viewed using the proposed expressions. II. S YSTEM M ODEL We consider two-hop downlink OFDMA communication from a base station (BS) to mobile UEs through a fixed infrastructure-based relay node. In particular, we focus on a single-cell multiuser OFDMA system, where K UEs share N available RBs and K ≤ N . Each RB consists of several fully correlated adjacent subcarriers in some consecutive time domain OFDM symbols. The RN employs the amplify-andforward (AF) protocol and operates in the half-duplex mode such that orthogonal (in time or in frequency) channels are allocated for reception and transmission in the RN. All UEs are connected to the RN and do not receive the direct transmission from the BS. We assume that the source–relay (SR) channel is static and flat over the employed band of N subchannels. This is a realistic model, because both the BS and the RN are fixed infrastructure-based nodes and it is often possible to achieve a line-of-sight connection. The instantaneous signal-to-noise ratio (SNR) of the SR channel is denoted by γSR , which is equal to the average SNR γ¯SR . In the relay–destination (RD) link, the received SNR admits independently and identically distributed (i.i.d.) exponential fading statistics with average γ¯RD for each RB, because UEs are mobile. This assumption is not fully true in any practical

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system. However, idealized channel model has been adopted in several studies, see, e.g, [6], [7] to enable analytical performance evaluation approach. We also note that the performance of our idealized system provides an upper bound capacity results for practical systems. Furthermore, we have show in simulation results that the idealized channel is approximately valid when assuming a wide bandwidth OFDMA system where it is possible to form pools of N uncorrelated timefrequency blocks provided that N is small compared to the total number of RBs. In such system, we may form groups of K users attached to each RB pool. In addition, if users do random hopping between pools, the impact of slightly different mean powers in different parts of the frequency band is diminishing. To facilitate RB allocation, each UE estimates the RD channel and sends an instantaneous feedback information to the RN via a feedback channel. The feedback word is uncoded and feedback bit errors are uniformly distributed. Feedback bit error probability is pb and it is known by the BS. The RN then allocates RBs to UEs based on the received feedback information. We assume that allocation is fast when compared to channel coherence time, i.e., temporal feedback delay is neglected. Resource fair round robin (RR) principle is assumed in the RB allocation, i.e., each UE has the same portion of the available channels. User position in the allocation queue is denoted by k = 1, 2, . . . , K. After channel assignment, optimal rate adaptation is employed for the allocated channel. Thus, the following results illustrate the theoretical capacity of the system. A. Best-M Feedback In the best-M feedback method, the M best RBs are selected out of the total of N available RBs at the receiver. The index which indicates the selected combination N  = of the RBs is signaled back to the RN. There are M N !/M !(N − M )! different possible feedback words in the best-M  feedback scheme.Thus, the feedback method requires log2 N !/(M !(N −M )!)  bits per user. In the RB allocation, a RB is randomly selected from the indicated pool if possible. If all the M best RBs have already been allocated to the previously scheduled users, a RB is selected randomly from the available ones. The cumulative density function (CDF) of the end-to-end (E2E) SNR of the nth best RB is of the form   N  N RB FRB (γ)j (1 − FRB (γ))N −j , (1) Fn (γ) = j j=N −n+1

where FRB (γ) is the CDF of the E2E SNR of a RB. The E2E SNR is given by [2], [10] γ¯SR γRD , (2) γ= γ¯SR + γRD + 1 where γRD is the received SNR of the RD link. In the case of the full rate STBC transmission through Nt antennae at the RN and maximum ratio combining (MRC) over Nr receive antennae, the CDF of the exponentially fading SNR γRD

is expressed as FRD (γRD ) = 1 − PK (Nt γRD /¯ γRD ), where m−1 Pm (x) = i=0 (xi /i!)e−x denotes the Poisson distribution.  γ ¯SR γRD Now, we have probability P (γ < x) = P γ¯SR +γRD +1 < x =   (¯ γSR +1)x P γRD < (¯γγ¯SRSR+1)x = F . Thus, for STBC, the RD −x γ ¯SR −x CDF of the E2E SNR FRB (γ) can be expressed as   Nt (γSR + 1)γ . (3) FSTBC (γ) = 1 − PK (γSR − γ)¯ γRD For the system with antenna selection at the RN and MRC at the UE, the CDF of the SNR is presented as  Nt  (γSR + 1)γ . (4) FAS (γ) = 1 − PNr (γSR − γ)¯ γRD In the SBB-M feedback strategy the RBs are divided into sub-blocks. A sub-block includes NRBSB RBs and there are N/NRBSB sub-blocks provided that NRBSB is the same for each sub-block. The best-M feedback word is formed individually for each sub-block. At the allocation, the indicated RB (and also sub-block) is randomly selected from unoccupied alternatives. The results expressed for the best-M method can be applied for the SBB-M method by changing parameters. B. RB-Wise One Bit Feedback Strategy RB-wise one bit SNR quantization means that the SNR of each resource block is compared to a predefined threshold at the receiver. The feedback bit of each RB is conveyed from UE to the RN that forward it to the BS. One bit indicates that the SNR is below or above the threshold denoted as γT . Feedback overhead is N bits per user. In the frequency allocation, a RB from the set of the available channels whose SNR values exceed the threshold is randomly allocated to a user if possible. If good channels are not available for the user, a RB is randomly assigned from the set of available channels. III. C APACITY A NALYSIS A. RB-Wise one bit Feedback Method In this section, we derive the average capacity of the system with optimal rate adaptation and RB-wise one bit feedback. The average capacity of the kth user in the allocation queue is expressed as

q1 (k, pb ) γ¯SR fRB (γ) log2 (1 + γ)dγ C(k) = 2pQ γ 1 T  +

q0 (k, pb ) 2pQ 0


C1 (γT )



fRB (γ) log2 (1 + γ)dγ ,  C0 (γT )

where q1 (k, pb ) is the probability that the channel whose SNR exceeds γT is allocated to the kth  γ¯SRuser, q0 (k, pb ) = 1 − q1 (k, pb ), pQ ¯ )dγ, and 1 = 1 − FRB (γT ) = γT fRB (γ, γ Q 1 = 1 − p . The factor in (5) comes from the half duplex pQ 0 1 2 relaying. In the case that the SNR of used RB exceeds γT , capacity is expressed as C1 (γT )/(2pQ 1 ). If the SNR of used RB is below γT the average capacity is C0 (γT )/(2pQ 0 ). The

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expression in (5) is valid for the STBC scheme and for antenna selection with perfect antenna selection. 1) Space-Time Block Coding: Applying integration by capacity of the form parts to C1 (γT ) in (5), we obtain the  γ¯SR CSTBC 1 (γT ) = log2 (e)[pQ ln(1+γ )+ (1−FSTBC (γ))/(1+ T 1 γT γ)dγ], where FSTBC (γ) is given in (3). Similarly as in the SR +1)γ Appendix, we apply substitution t = α(γγSR γRD −γ , α = Nt /¯ and partial fraction decomposition resulting  CSTBC 1 (γT ) = log2 (e) pQ 1 ln(1 + γT )+

∞ K−1  i −t



te i! 

1 1 − α + t α(¯ γSR + 1) + t


= log2 (e) pQ 1 ln(1 + γT )+ K−1 

I2 (γT , i, 1, α)

I2 (γT , i, 1, α(¯ γSR

  + 1)) ,


i=1 SR +1)γT where γT = α(γγSR γRD and I2 is given by −γT , α = Nt /¯ (30). The capacity C0 (γT ) can be derived using (6) so that CSTBC 0 (γT ) = CSTBC 1 (0) − CSTBC 1 (γT ). 2) Antenna Selection: The average capacity with optimal rate adaptation can be expressed as [9]

C(k) = q1 (k, pb )((1 − pASE )CAS 1 (γ1 ) + pASE CASE 1 (γ1 )) + q0 (k, pb )((1 − pASE )CAS 0 (γ1 ) + pASE CASE 0 (γ1 )) (7) where pASE = 1 − (1 − pb )log2 (Nt ) is the probability of antenna selection error (ASE),

∞ fAS (γ)/pQ (8) CAS 1 (γT ) = 1 log2 (1 + γ)dγ,

γT ∞ fASE 1 (γ|γAS > γT ) log2 (1 + γ)dγ, (9) CASE 1 (γT ) = 0

γT CAS 0 (γT ) = fAS (γ)/pQ (10) 0 log2 (1 + γ)dγ, 0

γT CASE 0 (γT ) = fASE 0 (γ|γAS < γT ) log2 (1 + γ)dγ, (11) 0

and γAS is the SNR at the best antenna. The capacity CAS 1 (γT ) provides the average spectral efficiency for the channel whose SNR exceeds the threshold when the antenna selection is correct. The closed form capacity can be derived by applying integration by parts to (8) and substituting (4) to the resulting integral. Thus, the capacity is now of the form  CAS 1 (γT ) = log2 (e) pQ 1 ln(1 + γT )+



= log2 (e)

α(γSR +1)γ γSR −γ

  2 SR +1)γ − P2 α(γγSR −γ 1+γ

pQ 1


 ln(1 + γT ) + 2I1 (γT , 1) − I1 (γT , 2) , (12)

where the integral I1 is derived in the Appendix.

The capacity in (10) corresponds to a best antenna whose SNR is below the threshold. It can be easily derived as Q CAS 0 (γT ) = (CAS 1 (0) − pQ 1 CAS 1 (γT ))/p0 . For the erroneous antenna selection, the capacity is given by CASE 1 (γT ) in the case that γAS > γT . Also this integral is derived by using integration by parts. Thus, we need to know the CDF of SNR for the ASE case which is determined in [9] to be of the form  2F (γ ) ∗ = 1+FMRC γ∗ < γT , γAS > γT MRC (γ1 ) (13) FASE 1 (γ∗ |γAS ) = 2FMRC (γ∗ ) = 1+FMRC (γ∗ ) γ∗ > γT , γAS > γ∗ ,  SR +1)γ is the end-towhere FMRC (γ∗ ) = 1 − PNr α(γγSR −γ end CDF of the MRC of single antenna AF relaying. The integral CASE 1(γT ) has to be integrated so that  ∞ in two 1parts γ (γ∗ |γAS ) 1 (γ∗ |γAS ) CASE 1 (γT ) = 0 T 1−FASE1+γ dγ + γT 1−FASE1+γ dγ = log2 (e)(IP 0 (γT ) + IP 1 (γT )). The closed form solution can be easily found for the integral IP 0 (γT ) and it can be expressed as  

γT −P2 α(γSR +1)γT + 2P2 α(γSR +1)γ γSR −γT γSR −γ   dγ IP 0 (γT ) = SR +1)γT 0 2 − P2 α(γγSR (1 + γ) −γT  α(γSR +1)γT −P2 ln(1 + γT ) + 2I1 (γT , 1) γSR −γT  =− , (14) SR +1)γT 2 − P2 α(γγSR −γT where I1 is derived in the Appendix. A closed-form solution for the integral IP 1 (γT )) does not exist, but we can expand the term 1−FASE 1 (γ|γAS ), γ > γT , using a geometric series as foli ∞  ∞ 1 P2 (b) i lows 2−P = 12 P2 (b) i=0 12 P2 (b) = i=1 2i P2 (b) 2 (b) because 12 P2 (b) < 1 [9]. Thus, now we have   i α(γSR +1)γ

∞  γSR −γ 1 γ¯SR P2 IP 1 (γT ) = dγ , (15) 2i γT 1+γ i=0  I1 (γT ,i)

where I1 is derived in the Appendix. The capacity in (11) refers to the case that the antenna selection is erroneous and γAS < γT . The CDF for this case is given by [9] FASE 0 (γ∗ |γAS ) =

2FMRC (γ∗ ) , FMRC (γT ) + FMRC (γ∗ )


where γ∗ < γT and γ∗ < γAS < γT . Applying integration by parts and geometric series similarly as in (15), after some derivations, the capacity (11) can be rewritten as   2 CASE 0 (γT ) = log(1 + γT ) 1 − B   i  γT P α(γSR +1)γ ∞   2 γ −γ 1 1 SR dγ (17) −2 − i i+1 B B 1 + γ 0 i=1  where B = 2 − P2 Appendix.

I1 (0,i)−I1 (γT ,i)

α(γSR +1)γ γSR −γ

and I1 is derived in the

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B. Best-M Feedback Method 1) Space-Time Block Coding: The performance of the RB allocation according to the best-M information is characterized by the probability that a user gets a channel from the set of the M best channels. This probability is denoted by q(k, pb ) for the kth user in the allocation queue when the feedback BEP is pb . The expected capacity for the kth user can be derived as M N q(k, pb ) − M  STBC N (1 − q(k, pb )) CSTBC , C + M (N − M ) n=1 n N −M (18)  ∞ where CnSTBC = 0 fnSTBC (γ) log2 (1 + γ)dγ refers to the ca∞ pacity of the nth best RB and CSTBC = 0 fSTBC (γ) log2 (1 + γ)dγ is the capacity of a RB over fading [9]. Applying integration by parts to CnSTBC , the capacity of the nth best channel can be derived as

γ¯SR log2 (e)FnSTBC (γ) STBC dγ. (19) = log2 (1 + γ¯SR ) − Cn 1+γ 0

C(k) =

Now we need to define the CDF of the nth best RB for the system with STBC transmission at the RN and MRC at the UE. Substituting (3) into (1), this CDF can be expressed j    N α(¯ γSR + 1)γ 1 − PK = j γSR − γ j=N −n+1 N −j   α(¯ γSR + 1)γ PK (¯ γSR − γ)   j   N  α(¯ γSR +1)γ N j −(l+N −j) (¯ γSR −γ) (−1)l e = j l N 

FnSTBC (γ)

j=N −n+1


(K−1)l+N −j

βr,l+N −j


α(¯ γSR + 1)γ (¯ γSR − γ)

l  r=l−K+1



βr,j−1 I[0,(j−1)(K−1)] , (l − r)!


β0,0 = β0,n = 1, βl,1 = 1/l!, β1,j = j, I[a,b] = 1 when a ≤ r, ≤ b and I[a,b] = 0 otherwise [11]. The second expression in (20) is achieved using binomial expansion. Substituting (20) γSR +1)γ into (19) and using substitution t = α(¯ (¯ γSR −γ) similarly as in the Appendix, after some derivations we end up the result CnSTBC (γ)


  j   N j (−1)l j l

N  j=N −n+1

(K−1)l+N −j


 βr,l+N −j I2 (0, r, l + N − j, α)

r=0  − I2 (0, r, l + N − j, α(¯ γSR + 1)) ,

where I2 is given in (30).

N q(k, pb ) − M  AS C M (N − M ) n=1 n   M q(k, pb )  MRC N (1 − q(k, pb )) CAS + pASE + Cl+ N −M M l=1  M

C(k) ≈ (1 − pASE )

+ (1 − q(k, pb ))CASE ,


where the two first terms represent the capacity when AS is correct and the two last terms present the capacity for the ASE case. More precisely, CnAS denotes the capacity of the nth best RB with correct antenna selection, CAS refers to the average capacity of correct AS, the third term presents the approximate capacity when AS is erroneous and indicated channel is allocated, and the last term refers to the capacity when AS is erroneous and all indicated channels have  been occupied by N Nt MRC = N N1t −l n=l+1 CnMRC = other users. Furthermore, Cl+  l N Nt 1 MRC , where the capacity of the n=1 Cn N Nt −1 CMRC − N Nt MRC nth best RB is denoted as Cn when the MRC is applied. Now we derive capacities for each case in (23). The capacity CnAS refers to the nth best RB with correct antenna selection. Applying integration by parts, it can be derived as


γRD and the coefficient βl,j can be derived as where α = Nt /¯ βl,j =

2) Antenna Selection: In the case of the perfect antenna selection, the closed form solution can be presented. When the antenna selection is erroneous, the exact PDF or CDF of the SNR for the system with the best-M feedback method is tedious to determine. The capacity for the antenna selection is well approximated by [9]



= log2 (1 + γ¯SR ) −


γ ¯SR

log2 (e)

FnAS (γ) dγ, 1+γ


where we have applied integration by parts and FnAS (γ) refers to the CDF of the nth best channel of the best antenna. For the RN with Nt = 2 and the MRC receiver with Nr = 2, the CDF FnAS (γ) can be expressed as 2j    N α(γSR + 1)γ 1 − P2 j γSR − γ j=N −n+1     2 N −j  α(γSR + 1)γ α(γSR + 1)γ − P2 2P2 γSR − γ γSR − γ      2j N −j  N   N  2j N −j l (−1) = j l m m=0 j=N −n+1 l=0 l+m+N −j   α(γSR + 1)γ 2N −j−m (−1)m P2 (25) γSR − γ FnAS (γ) =


With the aid of the results in Appendix, the capacity (24)

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Average spectral efficiency (b/s/Hz)

2.4 2.35

Required mean RD SNR to achieve 2 b/s/Hz

1−b/RB,8 bits Best−1,3 bits Best−3,6 bits RA,0 bits

2.3 2.25 2.2 2.15 2.1 2.05


9.5 STBC 9

8.5 AS 8 1−b/RB,8 bits Best−1,3 bits Best−3,6 bits SBB−1,4 bits RA,0 bits


2 1.95



3 4 5 6 User position in the allocation queue



Fig. 1. Average spectral efficiency vs. user position in the allocation queue ¯SR = 20dB, γ ¯RD = 10dB, N = K = 8 and when Nt = Nr = 2, γ pb = 0.05. Solid curves illustrate the performance of AS scheme and dashed curves present the performance of STBC scheme.

can be rewritten as CnAS

= log2 (1 + γ¯SR ) − log2 (e)


N −j  m=0

N  j=N −n+1

  2j   N 2j j l l=0

 N − j N −j−m 2 (−1)m I1 (0, l + m + N − j) m (26)

The average capacity of AS over fading, i.e., CAS in (23) is given by CAS = CAS 1 (0) = 2I1 (0, 1) − I1 (0, 2), where CAS 1 is presented in (12). γ ) is given by (22) with parameters The capacity CnMRC (¯ K = Nr , N = N Nt and α = 1/Nt . The average capacity γ ) = C1 (0), where C1 (0) is for the MRC is derived as CMRC (¯ presented in (6) with parameters K = Nr and α = 1/Nt . The term CASE in (23) refers to the case that both the bestM feedback word and antenna selection are erroneous. The CDF of the SNR for the ASE case is given as FASE (γ) = 2FMRC (γ) − FMRC(γ)2 for Nt = 2. Applying integration by γ ¯ parts into CASE = 0 SR fASE (γ) log2 (1 + γ)dγ, we have   2

∞ P2 (γSR +1)γ (γSR −γ)¯ γRD dγ γ ) = log2 (e) (27) CASE (¯ 1 + γ 0  I1 (0,2)

γRD . where I1 is derived in the appendix with α = 1/¯ IV. P ERFORMANCE R ESULTS In numerical examples, we assume that Nt = Nr = 2 and there are eight RBs and users, i.e, N = K = 8. We consider the case where the SNR at the first hop is γ¯SR = 20 dB for each RB. A single RB is allocated for each user at the BS and the probabilities q(k, pb ) and ql (k, pb ) related to this system can be found from [12]. The threshold for






0.08 0.1 0.12 0.14 Feedback bit error probability




Fig. 2. Required γ ¯RD to achieve spectral efficiency 2 b/s/Hz vs. feedback ¯SR = 20dB and N = K = 8. Allocation BEP when Nt = Nr = 2, γ queue position fair channel assignment is applied. Solid curves illustrate the performance of AS scheme and dashed curves present the performance of STBC scheme.

one bit quantization was set so that P (γ > γT ) = 0.3125. This threshold was determined numerically to provide the best average performance when pb = 0.05 and γRD = 15 dB. Fig. 1 illustrates the performance of the expected capacity versus user position in the allocation queue when feedback BEP is 0.05 and γ¯RD = 10 dB. The performance of the last user is the same as that of the random allocation (RA) whereas the first users gets good channel at high probability. However, the capacity variation between users is smaller than that of the system in [10] with optimal pairing where the worst SR channel is paired with the worst RD channel resulting performance for the last user remarkably lower than that of the RA. As already demonstrated at the previous study [9], AS outperforms STBC also with imperfect feedback link. On the other hand, AS requires feedback information. If the index of the best antenna is requested separately for the assigned channel, one bit information is enough for AS (with two transmit antennas). In the case of the fast AS, the RB-wise feedback information for AS has been sent together with the feedback information dedicated for RB allocation when the AS requires N bits. Relatively high SNR values γ¯RD are needed to provide high data rate communications through half duplex relay protocol. Fig. 2 illustrates the required γ¯RD values that are needed to provide 2 b/s/Hz E2E spectral efficiency in queue position fair allocation, in which each user gets each queue position with equal probability. AS with RA provides performance close to that of the STBC scheme. All the considered feedback schemes provide substantial performance enhancement also with imperfect feedback. The tradeoff between feedback overhead, system performance, and robustness against feedback errors is promising in the SBB-M method with NRBSB = 4 and 2 sub-blocks. In a partially loaded case, i.e., when K < N , the allocation gain would be greater.

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We investigated information theoretical capacity results for the system with dynamic OFDMA based relay link when multiple antennae were assumed for each end. Resource blocks (RBs) were allocated according to the limited feedback information which was based on the best-M information or RBwise one bit quantization. In particular, STBC and AS multiantenna schemes were studied. Optimal rate adaptation was assumed for the assigned channels. The results indicated that simple and practical round robin allocation provides significant performance gain with very limited feedback information.

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A PPENDIX In this Appendix, we derive the closed-form solutions of the needed in terms of the exponential integral E1 (a) =  ∞ e−uintegrals du and the complementary incomplete gamma function u a ∞ defined as Γ(t, x) = x st−1 e−s ds, x ≥ 0 [13]. Especially, we need to derive the integral   i

γ¯SR P2 α(¯γSR +1)γ γ ¯SR −γ ds. (28) I1 (γT , i) = 1 +γ γT γSR +1)γ Applying binomial expansion, substitution t = α(¯ (¯ γSR −γ) and partial fraction decomposition to (28), after some calculus we have  

∞ i   1 i j −it 1 t e I1 = − dt α + t α(¯ γSR + 1) + t γT j=0 j i    i I2 (γT , j, i, α) − I2 (γT , j, i, α(¯ γSR + 1)) , (29) = j j=0

 ∞ j −it γSR +1)γT  where γT = (¯ /(α + (¯ γSR −γT ) and I2 (γT , j, i, α) = γT t e t)dt. The second required integral I2 can be expressed as I2 (γT , j, i, α) = eiα (−α)j E1 (i(γT + α)) i    i −m i (−α)i−m Γ(m, i(γT + α)), + eiα m m=1


after substitution γ = α + t, binomial expansion and substitution x = αγ.

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

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