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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 5, MAY 2008

Robust Optimal Cross-Layer Designs for TDD-OFDMA Systems with Imperfect CSIT and Unknown Interference: State-Space Approach Based on 1-bit ACK/NAK Feedbacks Rui Wang and Vincent K. N. Lau

Abstract—Cross-layer designs for OFDMA systems have been shown to offer significant gains of spectral efficiency by exploiting the multiuser diversity over the temporal and frequency domains. In this paper, we shall propose a robust optimal cross-layer design for downlink TDD-OFDMA systems with imperfect channel state information at the base station (CSIT) and unknown interference in slow fading channels. Exploiting the ACK/NAK (1-bit) feedbacks from the mobiles, the proposed cross-layer design does not require knowledge of the CSIT error statistics or interference statistics. To take into account of the potential packet error due to the imperfect CSIT and unknown interference, we define average system goodput (which measures the average b/s/Hz successfully delivered to the mobile) as our optimization objective. We formulate the cross-layer design as a state-space control problem. The optimal power, optimal rate and optimal user allocations are determined as the output equations from the system states based on dynamic programming approach. Simulation results illustrate that the performance of the proposed closed-loop cross-layer design is very robust with respect to imperfect CSIT, unknown interference, model mismatch as well as channel variations due to Doppler. Index Terms—Closed-loop, multiple antennas, imperfect CSIT, cross-layer.

I. I NTRODUCTION

R

ECENTLY, cross-layer scheduling in OFDMA systems has received tremendous attentions. High spectral efficency can be achieved by exploiting the multi-user selection diversity over the temporal and frequency domains [1]–[4]. To exploit the multi-user selection diversity, knowledge of Channel State Information is required at the base station (CSIT). However, for TDD systems, obtaining perfect CSIT is very challenging, especially for large number of subcarriers or large number of users. When the base station has perfect knowledge of CSIT, the transmitted packet will be virtually error free (with powerful error correction coding) in slow fading channels, and hence, the system can achieve ergodic capacity. However, when the base station has imperfect CSIT or there is unknown interference at the mobile receivers, the

Paper approved by Y. Fang, the Editor for Wireless Networks of the IEEE Communications Society. Manuscript received March 29, 2006; revised January 31, 2007 and May 14, 2007. This work was supported by the Research Grants Council of the Hong Kong Government through the grant RGC 615606. The authors are with the Dept. of ECE, Hong Kong University of Science and Technology (e-mail: {wray, eeknlau}@ust.hk). Digital Object Identifier 10.1109/TCOMM.2008.060100.

scheduled data rate may be larger than the instantaneous channel capacity which is unknown to the base station. This results in packet transmission error even if powerful error correction code is applied. Moreover, the efficiency of the multi-user scheduling is reduced because the wrong set of users may be selected for transmission. Most of the existing cross-layer designs addressed the imperfect CSIT issue are based on heuristic approaches. For example, in [5], [6], the cross-layer schedulers are designed assuming CSIT is perfect and the effect of imperfect CSIT is evaluated by simulations. However, this approach does not offer any design insight on what should be the optimal design and performance with imperfect CSIT as the optimal design can be quite different from that with perfect CSIT. It is also found that the performance of the naive cross-layer scheduler (designed for perfect CSIT) is very sensitive to imperfect CSIT even at very small CSIT errors [7]. In [7], [8], the authors discuss the optimal cross-layer design with imperfect CSIT. However, knowledges of the CSIT error statistics (such as the error distribution or error variance) and interference statistics are required, which may not be available in practice. In all the works mentioned above, the cross-layer design is open-loop. In open-loop scheduling, the set of admitted users, the power allocation and the rate allocation are determined based on the estimated CSIT (as well as estimated interference), and remain to be the same for the entire scheduling time slot. There are some existing works on the closedloop adaptation with the ACK/NAK feedbacks [9]–[12]. For example, in [9], the authors present a power and rate control policy for a point-to-point system with delay constrained traffic based on ACK/NAK feedback. However, the crosslayer scheduling (user selection) issue is not addressed. In [10], the authors present a heuristic adaptive rate control and randomized scheduling algorithm for flat-fading channels based on learning automata. In all these works, the solutions are heuristic and there is no insight on how good the heuristic solutions approach the optimal performance. Furthermore, knowledge of CSIT error statistics are needed and they did not address the potential issue of unknown interference. In this paper, we shall propose a robust and optimal closedloop cross-layer design for downlink TDD-OFDMA systems with imperfect CSIT and unknown interference for slow fading channels. We shall utilize the ACK/NAK (1-bit) feedbacks

c 2008 IEEE 0090-6778/08$25.00 

WANG and LAU: ROBUST OPTIMAL CROSS-LAYER DESIGNS FOR TDD-OFDMA SYSTEMS

Scheduling Slot n 1

2

3

{rk,m,1} {pk,m,1} {Am,1} Packet Slot 1

Fig. 1.

4

...

......

755

Scheduling Slot n+1 N

{rk,m,N} {pk,m,N} {Am,N} Packet Slot N

1

2

{rk,m,1} {pk,m,1} {Am,1} Packet Slot 1

3

4

...

......

N

{rk,m,N} {pk,m,N} {Am,N} Packet Slot N

Illustration of scheduling slot and packet slot.

from the mobiles to adjust the power allocation, the rate allocation as well as user assignment per packet slot. No knowledge of the CSIT error statistics or interference statistics is required at the base station. To take into account of the potential packet error, we define average system goodput, which measures the average b/s/Hz successfully delivered to the mobiles, as the optimization objective. We formulate the cross-layer design as a state-space control problem, where the optimal power, optimal rate and optimal user allocations are determined as the output equations from the system states based on dynamic programming approach. Finally, simulation results illustrate that the performance of the proposed closedloop cross-layer design is very robust with respect to imperfect CSIT, unknown interference, model mismatch as well as channel variation due to Doppler. This paper is organized as follows. In section II, we outline the OFDMA system model as well as the imperfect CSIT and unknown interference model. In section III, we shall define the system goodput and formulate the closed-loop cross-layer design as a state-space control problem in the presence of imperfect CSIT and unknown interference. In section IV, we shall derive the optimal system outputs as well as the optimal state evolution in transient state based on dynamic programming approach. In section V, we shall discuss the convergence of our state-space approach. In section VI, numerical results are presented and discussed. Finally, we give a brief summary in section VII. II. OFDMA S YSTEM M ODEL A. Slow Fading Channel Model We consider a communication system with K mobile users and one base station over a slow-varying frequency selective fading channel. Let M be the number of subcarriers in the system. We consider a scheduling slot structure, which consists of N packet bursts as illustrated in Figure 1. We assume the channel is quasi-statistic within a scheduling slot in this paper. Let Xm,n be the transmit symbol on the m-th subcarrier in the n-th packet burst, the received signal Yk,m,n of the k-th user on the m-th subcarrier in the n-th packet burst can be expressed as: Yk,m,n = hk,m Xm,n + Zk,m,n + Ik,m,n

(1)

where hk,m is the channel coefficient of the m-th subcarrier and the k-th user, which is i.i.d. complex Gaussian distributed with zero mean and unit variance, Zk,m,n is the i.i.d. zeromean complex Gaussian noise with variance σz2 /M and Ik,m,n denotes the zero-mean complex Gaussian interference (due to other cell interference) at the k-th mobile receiver with variance βk2 /M .

Fig. 2.

The structure of the closed-loop cross-layer scheduler.

B. Channel Estimation Model and Maximum Achievable Data Rate In this paper, we consider the imperfect channel state information at the base station (imperfect CSIT), which can be modelled as: hbk,m = hk,m + Δk,m

(2)

where hk,m is the actual CSI and Δk,m is the CSIT estimation error. We consider the case where there is interference to the mobile receivers, which may come from the surrounding cells. As it usually being in practical systems, we assume the base station has no idea about the mobile interference power βk of the K users as well as the variance of the CSIT errors Δk,m , 2 . denoted as σΔ For simplicity, we assume the CSIR as well as interference power measurement at the mobile station is perfect for the detection of downlink packets. Hence, based on the received signal model in (1), the maximum achievable data rate of the k-th user on the m-th subcarrier in the n-th packet burst is given by the maximum mutual information between Yk,m,n and Xm,n conditional on CSIR hk,m : Ck,m,n

=

max I(Yk,m,n ; Xm,n |hk,m )   |hk,m |2 = log2 1 + pm,n 2 σz /M + βk2 /M Pr(Xm,n )

(3)

where pm,n is the corresponding transmit power. C. MAC Layer Model The MAC layer is responsible for scheduling the radio resource at each scheduling slot based on the estimated CSIT as well as the ACK/NAK feedbacks. Figure 2 illustrates the structure of the cross-layer scheduler. The outputs of the MAC scheduler include the the power allocation {pk,m,n }, the rate allocation {rk,m,n } as well as user selection {Am,n }. After the packets in the first packet slot are transmitted, the selected mobiles will send the ACK/NAK feedbacks to the base station before the next packet is delivered1. For subsequent packet bursts in a scheduling slot, the cross-layer scheduler adapts the power allocation, rate allocation as well as user selection based on the CSIT hb = {hbk,m } and the ACK/NAK feedbacks from the mobiles f1n−1 = {fAm,i ,m,i |i ∈ {1, n − 1}, ∀m} 1 For simplicity, we assume the delay of the ACK/NAK is small compared with the packet duration.

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 5, MAY 2008

(fk,m,n = 1 if an ACK is received from the k-th user after transmitting the n-th packet on the m-th subcarrier, and 0 otherwise). Hence, the MAC layer scheduler can be represented by the power allocation policy, rate allocation policy and user selection policy defined below. Definition 1. Power Allocation Policy:      b n−1  P = pk,m,n (h , f1 )∀k, m, n and pk,m,n ≤ P0 k,m,n

(4)

Rate Allocation Policy:     b n−1  R = rk,m,n (h , f1 )∀k, m, n User Selection Policy:     b n−1 A = Am,n (h , f1 ) ⊂ {1, K}∀m, n and |Am,n | ≤ 1



The causal state evolution policy S is defined as: sk,m,n = S(Sn−1 , f1n−1 )



(7)

The system outputs, including the admitted users, power allocation and rate allocation, are functions of the current system state Sn . |h

lk,m,n = inf {x|qk,m,n (x|f1n−1 , hb ) > 0}

(8)

uk,m,n = sup{x|qk,m,n (x|f1n−1 , hb ) > 0}

(9)

x

D. Packet Transmission Error and Average Goodput Let rk,m,n be the scheduled data rate for the user k on the m-th subcarrier in n-th packet. The instantaneous goodput of the k-th user on the m-th subcarrier in n-th packet, which measures the bits successfully delivered to the receiver, is given by: ρk,m,n = rk,m,n 1[Ck,m,n ≥ rk,m,n ]

(5)

where 1(I) is the indicator function which is equal to 1 if the event I is true and 0 otherwise. The average total goodput, which measures the average total b/s/Hz successfully delivered to the mobiles (averaged over ergodic realization of CSI), is defined as: N M  U (P0 , A, R, P) = E ρAm,n ,m,n n=1 m=1

=Ehb

N  M 



Eh rAm,n ,m,n 1[CAm,n ,m,n ≥ rAm,n ,m,n ]hb

n=1 m=1

=Ehb G(P0 , hb , A, R, P)

(6)

where h denotes the actual channel coefficients; Ehb [X] denotes the expectation of the random variable X w.r.t. hb . G(.) measures the conditional system goodput (conditioned on the estimated CSIT hb ). To account for the potential packet error (rk,m,n > Ck,m,n ), we shall design the cross-layer scheduler to optimize the total average system goodput U (.). III. C ROSS -L AYER D ESIGN F ORMULATION WITH I MPERFECT CSIT AND U NKNOWN I NTERFERENCE A. Closed-Loop Structure of the Cross-Layer Scheduler Figure 2 illustrates the structure of the closed-loop crosslayer scheduler. The scheduler is characterized by an internal state Sn and the state evolves based on the feedbacks of the users after each packet transmission. The scheduler outputs are uniquely determined by the system state. We first define the notations as follows: • sk,m,n denotes the state of user k on subcarrier m during the n-th packet burst, and Sn = {sk,m,n |∀k, m}.

|2

k,m From (3), the actual SINR (with unit power) σ2 /M+β 2 z k /M is a random variable with certain conditional pdf qk,m,n (x|f1n−1 , hb ) (The base station doesn’t know this distribution explicitly due to the lack of knowledge of the CSIT error statistics and interference statistics, however, the base station can make assumption on this distribution. We shall show the robustness of this assumption). We define the state sk,m,n = [lk,m,n , uk,m,n ] to be the lower bound and upper bound of the SINR given the knowledge of CSIT hb and the ACK/NAK feedbacks f1n−1 :

x

B. Optimization Objective To take into consideration of the potential packet errors, given any realization of the imperfect CSIT, we shall optimize the conditional average system goodput G(.). Since the user selection, power allocation and rate allocation are functions of the system state Sn , we rewrite (6) as: G(P0 , h , A, R, P, S) = ESN 1 b

N 

g¯n (pn , hb , Sn )

(10)

n=1

where SN 1 = {S1 , . . . , SN }, pn is the total transmit power for the n-th packet burst, g¯n denotes the conditional average goodput (conditioned on the CSIT hb and current system state Sn ) contributed by the n−th packet burst and is given by: g¯n (.) =

M 

rAm,n ,m,n Pr[CAm,n ,m,n ≥ rAm,n ,m,n |hb , Sn ]

m=1

(11) Thus, the closed-loop cross-layer scheduling problem with imperfect CSIT and unknown interference can be summarized as the following optimization problem: Prob 1 (Cross-Layer Problem Formulation with Imperfect CSIT). Given any realization of the estimated CSIT for all mobile users at all subcarriers hb = {hbk,m }, determine the optimal state evolution policy S, the optimal user selection policy A, the optimal power allocation policy P as well as the optimal rate allocation policy R such that the conditional total goodput, G(.) is maximized. That is,  N  G∗ (P0 , hb ) = max ESN g¯n (pn , hb , Sn ) (12) 1 A,R,P,S

n=1

where the power allocation, rate allocation policies are subject to the following constraints: • Total Transmit Power Constraint in (4) • Quality of Service (QoS) Requirement: The conditional packet error probability of all the users is less than a target .

WANG and LAU: ROBUST OPTIMAL CROSS-LAYER DESIGNS FOR TDD-OFDMA SYSTEMS

IV. O PTIMAL S OLUTION A. Optimal State Evolution At the base station, the actual SINR of user k on the mth subcarrier in the n-th packet burst is a random variable with density qk,m,n (x) = qk,m,1 (x|f1n−1 ). The event {f1n−1 } is equivalent to the event {LBk,m,n ≤ x ≤ U Bk,m,n }, where    rk,m,i 2 − 1  = 1 1 ≤ i ≤ n−1 and f LBk,m,n = max k,m,i i pk,m,i     rk,m,i 2 − 1  U Bk,m,n = min = 0 1 ≤ i ≤ n−1 and f k,m,i i pk,m,i  Hence,we have qk,m,n (x) = qk,m,1 (x|LBk,m,n ≤ x ≤ U Bk,m,n ). According to the definition of the system state (8,9), we get lk,m,n = LBk,m,n and uk,m,n = U Bk,m,n And the optimal state evolution in (7) is k ∈ Am,n ):  rk,m,n max{lk,m,n , 2 pk,m,n−1 } lk,m,n+1 = lk,m,n  rk,m,n min{uk,m,n , 2 pk,m,n−1 } uk,m,n+1 = uk,m,n

(13)

given by (suppose if fk,m,n = 1, otherwise.

(14) if fk,m,n = 0,

otherwise.

(15)

757

Proof 1. The proof of this lemma is based on the recursive structure of Fn (.). We omit it here due to the page limit. As a result of Lemma 1, the optimization problem with respect to {Am,n }, {pk,m,n}, {rk,m,n } (given any CSIT realization hb and current system state Sn ) can be divided and conquer into N steps. The recursive equation in (17) is also called the Bellmen’s equation [13] and the optimization problem belongs to the Markov decision problem. The general solution of the Markov decision problem involves an offline recursion and an online strategy. We elaborate these two procedures as follows. 1) Backward Recursion for User Selection Policy and Power/Rate Allocation Policies: In the offline strategy, we shall partition the optimization for the average goodput G∗ (P, hb ) with respect to the user selection policy {Am,n }, the power allocation policy {pk,m,n } and the rate allocation policy {rk,m,n } (for the N packet bursts) into N recursive ∗ optimizations using the recursive relationship of Fn∗ and Fn+1 in (17). These optimal policies will be used for the online algorithm when the actual ACK/NAK feedbacks are received. The offline recursive solution is elaborated in the following steps. • Step 1. Consider the last packet burst n = N . Recall that the channel capacity is given by:   |hk,m |2 Ck,m,N = log2 1 + pk,m,N 2 (19) σz /M + βk2 /M |2

|h

B. Optimal System Output Equations In fact, the optimization objective G(.) can be divided and conquered into a set of recursive equations. This recursive relationship is summarized in the following lemma: Lemma 1 (Recursive Formulation of the Conditional Goodput). Let Fn (P, hb , Sn ) be the total average goodput from the n-th packet burst to the N -th packet burst conditional on the CSIT and the system state Sn with total residual power P . i.e., Fn (P, hb , Sn ) =¯ gn (pn , h , Sn ) + b

N   i=n+1 Si

, Sn ) And let Fn∗ (P, hb N power constraint i=n

b

(16) Fn∗ (.)

Fn∗ (P, hb , Sn ) =    ∗ g¯n (pn ) + Pr(Sn+1 |Sn , hb )Fn+1 (P − pn , hb , Sn+1 ) max Sn+1

(17)

where pn =

M  m=1

(20)

where θk,m,N is the SINR scaling factor given by: θk,m,N = Q−1 k,m,N ()

(21)

To determine the optimal power allocation policies, {pk,m,N }, we form the Lagrangian as: M 

(1 − ) log2 (1 + pAm,n ,m,N θAm,n ,m,N )

m=1

be the optimized subject to  M ∗ p ≤ P . F n (.) can be m=1 Am,n ,m,n expressed recursively as:

{pk,m,n } {rk,m,n } {Am,n }

rk,m,N = log2 (1 + pk,m,N θk,m,N )

L=

Pr(Si |Sn , h )¯ gi (pi , h , Si ) b

k,m is a random variable with density where σ2 /M+β 2 z k /M qk,m,N (x). Let Qk,m,N (x) be the corresponding cumulative distribution function. To satisfy the packet error requirement , the scheduled data rate is given by:

pAm,n ,m,n and we assume FN∗ +1 = 0.

Furthermore, the optimal conditional goodput in (12) is given by G∗ (P0 , hb ) = F1∗ (P0 , hb ) (18)

−λN

M 

pAm,n ,m,N

m=1

Using standard optimization technique, the optimal power allocation policy is given by:  + 1 1 ∗ − (22) pAm,n ,m,N = λN θAm,n ,m,N where (X)+ = max(0, X), λN is the Lagrangian multiplier given by M   1 1 1  pN + = λN M θ m=1 Am,n ,m,N

(23)

for sufficiently large pN . Finally, substituting (22) and (23) into the objective function FN∗ (.) (17), the optimal user selection is given by: Am,N = arg max{θk,m,N } k

(24)

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 5, MAY 2008

Hence, the closed form for FN∗ (pN ) is given by: ∗ (pN ) FN∗ (pN ) = g¯N  M  =(1 − ) log2 pN +

 +(1 − ) log2

M  m=1

1

Proof 2 Please refer to Appendix A. Hence, we have

M

N  M 

θ m=1 Am,n ,m,N

Fn∗ (.) = (1 − ) log2

θAm,n ,m,N 



MM (25)



Since {θk,m,N } are functions of SN , the equations (24,22,20) give the optimal user selection, the optimal power allocation and the optimal rate allocation in terms of the system state. Step 2. Consider the packet burst n, where n = {N − 1, N − 2, .....1}. Given the target error probability , the state transition probability Pr(Sn+1 |Sn , hb ) in (17) has the form of (1−)a b , where a is the total number of ACK feedbacks and b is the total number of NAK feedbacks after the transmission of the n-th packet. Since  is usually chosen to be very small, most of state transition probabilities are very small except the one when a = |A| and b = 0 (In this case, there is no transmission error). Hence, we have: Fn∗ (P, hb , Sn )   ∗ (P − pn , hb , Sn+1 ) ≈ max g¯n (pn , hb , Sn ) + Fn+1 {pAm,n ,m,n } {rAm,n ,m,n } {Am,n }

(26)

where the state Sn+1 is derived from its previous state Sn based on the all ACK feedbacks. Similar to Step 1, the optimal power and rate allocation policies are given by:  + 1 1 pAm,n ,m,n = − (27) λn θAm,n ,m,n rAm,n ,m,n = log2 (1 + pAm,n ,m,n θAm,n ,m,n ) where

pn =

  M  1 1 1 = pn + λn M θ m=1 Am,n ,m,n

(28) (29)



1 θ m=1 Am,n ,m,n

(30)

The optimal user selection is given by the following lemma: Lemma 2 (Optimal User Selection). Let sk,m,n+1 , . . . , sk,m,N be the system states evolved from sk,m,n with all ACK feedbacks and θk,m,n , . . . , θk,m,N be the corresponding SINR scaling factors. The optimal admitted user in the m-th subcarrier and the n−th packet burst is given by: Am,n = arg max k

N  i=n

θk,m,i

P + N −n+1

(31)

θAm,i ,m,i

M (N −n+1)M

N  M 

i=n m=1

1 θAm,i ,m,i

N −n+1



+ (1 − )

(N −n+1)M (32)

2) Online Solution: The online strategy is a realtime algorithm. For instance, upon receiving the specific ACK/NAK feedbacks fn , we update the system state Sn to Sn+1 by (7), and then, select the optimal users, the optimal power and rate allocation by the optimal policies {Am,n }, {pk,m,n } and {rk,m,n } (obtained in the offline backward recursion). The online processing is illustrate below: • Step 1. At the first packet burst, the optimal users, the optimal power and rate allocation {Am,1 }, {rk,m,1 }, {pk,m,1 } based on the estimated CSIT hb is obtained according to (31,27,28). • Step 2. Before transmitting the n+1-th packet burst (n = {1, 2, ....N − 1}), the base station has already obtained the specific ACK/NAK feedbacks of the previous packet fn and updated the system state accordingly. The optimal user selection, the optimal power and rate allocation for the n + 1-th packet are obtained from (31,27,28) and (24,22,20) in the offline recursion. V. S TEADY S TATE A NALYSIS The convergence of the system state can be summarized in the following lemma: Lemma 3 For sufficiently large n and quasi-static fading channel, we have: lim Am,n = arg max

n→∞

k

|hk,m |2 . σz2 /M + βk2 /M

(33)

Furthermore, if the user j has the largest SINR in the m-th subcarrier, we have lim Sj,m,n =

n→∞

N  M  1 1 P + N − n + 1 N − n + 1 i=n m=1 θAm,i ,m,i M 

∗ log2

 i=n m=1

|hj,m |2 . + βj2 /M

σz2 /M

(34)

In other words, for sufficiently large n, the system state of the user with largest SINR will converge to the actual SINR and the user selection will converge to the best user selection (as if perfect CSIT were available). Proof 3 Please refer to Appendix B. VI. N UMERICAL R ESULT AND D ISCUSSION In this section, we shall illustrate the performance of our closed-loop cross-layer scheduler design. In our simulation, the number of users K is 5, the number of multipaths Lp is 4 and the target packet error probability  is 0.01. For simplicity, we assume the unknown interference power βk2 of each user is the same. The unknown interference is quasi-static within a scheduling slot but random between scheduling slots according to U (0, I). In the simulation, the actual CSI is generated according to complex Gaussian distribution CN (0, 1). We

WANG and LAU: ROBUST OPTIMAL CROSS-LAYER DESIGNS FOR TDD-OFDMA SYSTEMS

759

1.8 perfect CSIT, I=0.1, 2

2

1.6

proposed closed−loop, I=0.1, 2 non−adaptive closed−loop, I=0.1, 2

Bandwidth Efficiency (bit/s/Hz)

1.5

Bandwidth Efficiency (bit/s/Hz)

open−loop, I=0.1, 2

naive, I=0.1, 2 round robin, I=0.1, 2

1

1.2 closed loop scheduler

0.8 naive scheduler round robin scheduler 0.4

0.5

0 0

0

2

4 6 Average Transmit Power per Packet (dB)

8

4

8 12 Index of Packet Burst

16

20

10

2 σΔ

Fig. 3. Average goodput performance versus transmit power at = 0.1, M = 4, I = {0.1, 2}. Open-loop refers to the cross-layer design based on the imperfect CSIT knowledge obtained at the beginning of scheduling slot only. Non-adaptive closed-loop refers to the closed-loop cross-layer design where the CSIT can be updated according to the feedbacks and where there is no power adaption among the packet bursts. Perfect CSIT refers to the ideal system with perfect CSIT and this serves as performance upper bound for bench marking. Round robin scheduler refers to the naive cross-layer design assuming the CSIT is perfect while selecting user randomly. Naive scheduler refers to the cross-layer scheduler assuming the CSIT is perfect.

assume the base station does not have any knowledge on the actual interference power β, actual distribution of the SINR 2 . The base as well as the actual CSIT estimation error σΔ station has default values for these parameters (βdef = 1, 2 σΔ,def = 0.5) which is not the same as the actual parameters. We shall show by simulation that although the default parameters may not equal to the actual parameters, the system state in the proposed design can still converge to the actual SINR and the closed-loop system is very robust with respect to the mismatch even in high CSIT error and high interference power. Each point in the figures is obtained by averaging over 1000 independent fading realizations. A. Performance of the closed-loop Cross-Layer Scheduler on Static Channel We first consider the case of slow fading in which the channel fading is quasi-static within a scheduling slot. Figure 3 shows the average system goodput versus the transmit power of the proposed closed-loop scheduler at high CSIT errors 2 = 0.1, M = 4 and the maximum unknown interference σΔ power I = 0.1, 2. For comparison, we also compare our proposed design with various baselines, namely the openloop cross-layer scheduler, non-adaptive closed-loop crosslayer scheduler, naive scheduler (designed assuming perfect CSIT) and round robin scheduler. The open-loop scheduler, the round robin scheduler and the naive scheduler are considered as open-loop designs because they did not exploit the ACK/NAK feedbacks from the mobiles. The proposed closed-loop scheduler achieves a significant performance gain over these open-loop schedulers. This illustrates that with the ACK/NAK feedback, significant cross-layer gains can be

2 = 0.1, Fig. 4. Average goodput performance of each packet burst at σΔ M = 4, P0 = 23dB, I = 1. Round robin scheduler refers to the naive cross-layer design assuming the CSIT is perfect while selecting user randomly. Naive scheduler refers to the cross-layer scheduler designed for perfect CSIT.

achieved even at large CSIT errors and large unknown interference. Furthermore, the proposed closed-loop scheduler also achieves a significant performance gain over the non-adaptive closed-loop scheduler, where the CSIT is updated according to the feedbacks, however there is no power adaption among the packet bursts. This illustrates the importance of our proposed design of state-space based adaption. The proposed design is also robust to the mismatch in the channel statistics and parameters. Figure 4 illustrates the average goodput of each packet burst (averaged over multiple scheduling slots) at high CSIT 2 = 0.1, I = 1, M = 4 and P0 = 23dB. The errors σΔ average goodput of the closed-loop scheduler increases with the packet burst index. There are two reasons for this. On one hand, because the scheduler can get better estimation of the actual SINR at later packet slots after receiving more ACK/NAK feedbacks, the decisions of user selection made in the later packet slots are more accurate. Since the scheduler can explore more multiuser diversity in the later packet bursts, the performance is better. On the other hand, since the CSIT is more accurate in the later packet slots, more power will be allocated to them to explore the performance gain of multiuser diversity. As a contrary. the two reference schedulers do not have such behavior because the knowledge of the actual SINR remains to be the same at all packet slots. B. The Performance Sensitivity on Doppler Spread In this part, we consider frequency selective fading channels with Doppler frequency fd from 20Hz to 100Hz, which corresponds to a speed of 9 and 45 km/hr at 2.4GHz. The duration of the packet slot is 0.2ms. Figure 5 illustrates the average system goodput versus the doppler frequency of the proposed closed-loop scheduler, round robin scheduler and 2 = 0.1, I = 1, 2, naive scheduler at large CSIT errors σΔ M = 4 and P0 = 23dB respectively. It can be observed that significant gain of the proposed closed-loop cross-layer design can be achieved at moderate to large Doppler.

760

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 5, MAY 2008

6

1.4

close loop, I=1

1.2

instantaneous capacity, I=1

0.8

Data Rate (bit per subcarrier)

Bandwidth Efficiency (bit/s/Hz)

5

1

close loop, I= 2

0.6

round robin, I=1 round robin, I= 2

naive, I=1 naive, I= 2

0.4

0.2

0 20

40

60 Doppler Frequency (Hz)

80

100

2 = Fig. 5. Average goodput performance versus Doppler frequency at σΔ 0.1, M = 4, I = {1, 2} and P0 = 23dB. Round robin scheduler refers to the naive cross-layer design assuming the CSIT is perfect while selecting user randomly. Naive scheduler refers to the cross-layer scheduler designed for perfect CSIT.

3

4 instantaneous data rate, I= 1

instantaneous capacity, I= 2

3

instantaneous data rate, I= 2 2

1

0

0

10

20

30 40 50 Index of Packet Burst

60

70

80

Fig. 7. The transient of the instantaneous scheduled data rate and the actual instantaneous channel capacity versus time (packet slot) at fd = 20Hz, 2 = 0.1, M = 4, K = 1 and P = 29dB. σΔ 0

instantaneous capacity, I=1

We formulate the cross-layer design as a state-space control problem. The optimal power, optimal rate and optimal user allocation are determined as the output equations from the system state. Based on dynamic programming approach, we work out the optimal state evolution using backward recursion and forward recursion algorithms. Simulations illustrate that the proposed closed-loop cross-layer scheduler has very robust goodput performance at moderate to high CSIT errors, interference power and moderate Doppler.

Data Rate (bit per subcarrier)

2.5 instantaneous data rate, I=1 2

1.5

1

instantaneous data rate, I= 2

A PPENDIX A: P ROOF OF L EMMA 2

0.5

0

instantaneous capacity, I= 2

0

10

20

30 40 50 Index of Packet Burst

60

70

80

Fig. 6. The transient of the instantaneous scheduled data rate and the actual 2 = instantaneous channel capacity versus time (packet slot) at fd = 0Hz, σΔ 0.1, M = 4, K = 1 and P0 = 29dB.

C. The Convergence of the Close Loop Adaptation Figure 6 and 7 illustrate the instantaneous scheduled data rate versus time in a scheduling slot at Doppler frequencies 2 of fd = 0 and fd = 20Hz, high CSIT errors σΔ = 0.1 and high interference I = 1, 2. In the simulation, M = 4, K = 1 and P0 = 29dB. In both cases, the scheduled data rate of the proposed closed-loop cross-layer design converges to the instantaneous actual capacity quite well. This justifies the robustness of the proposed closed-loop scheduler with respect to the CSIT error, unknown interference, model mismatch and the channel variation due to Doppler. VII. S UMMARY In this paper, we propose a robust cross-layer design for the downlink OFDMA systems with imperfect CSIT and unknown interference for slow frequency selective fading channels.

Fn∗ (P, hb , Sn ) =

= max (1 − ) {Am,i }

max

{Am,i },{pk,m,i }

M  m=1

M N  

N 

 log2

(1 − )

i=n

{Am,i }

m=1

rAm,i ,m,i

m=1 i=n

θAm,i ,m,i

[M (N − n + 1)]N−n+1  )N−n+1

1 θ Am,i ,m,i i=n m=1 N M   log2 θAm,i ,m,i ≈ max (1 − ) ∗(P +

N M  

i=n

P N−n+1 [M (N − n + 1)]N−n+1



where the first equality comes from the target packet error rate constraint which is similar to (20); the second equality is obtained from standard water-filling approach over m = 1 to M and i = n to N with sufficient large power constraint P ; the last approximation is made for sufficiently large P . We can observe from the above equation that the average goodput of a subcarrier is independent of the user selection of other subcarriers. In other words, the user selection of each subcarrier can be decoupled, i.e.: {Am,n , ..., Am,N } = arg max

N  i=n

θAm,i ,m,i

∀m ∈ {1, M } (35)

WANG and LAU: ROBUST OPTIMAL CROSS-LAYER DESIGNS FOR TDD-OFDMA SYSTEMS

Since we only consider ACK feedback, as the packet index grows from n to N , the SINR scaling factor of the selected user θk,m,i will increase. However, the SINR scaling factor of the un-selected user will remain the same (because they won’t be updated by feedbacks). Hence, the optimal user selection of any subcarrier must satisfy: Am,n = Am,n+1 = ... = Am,N

∀m ∈ {1, M }

(36)

combining (35) and (36), we complete the proof of lemma IV-B1. A PPENDIX B: P ROOF OF L EMMA 3 Let’s consider another lemma first. Lemma 4 If user k is selected infinite times in the mth subcarrier , the state of this user in this subcarrier will converge to the actual SINR. Proof 4 This is because every selection will lead to update on the user state, which will make the lower bound of the state approach to the upper bound of the state. As a result, both bounds will converge to the actual SINR. We omit the detail of the proof here due to the page limit. Assume the user j has the largest SINR Bj,m in the m-th subcarrier. We can argue that only this user will be selected infinite times in the m-th subcarrier when N tends to infinity. Otherwise, suppose another user i with SINR Bi,m is selected infinite times, we have the following inconsistent conclusions: • Since the user i is selected infinite times, according to the above lemma, there should exists a packet burst indexed by n such that li,m,n ≤ Bi,m ≤ ui,m,n < Bj,m . • According to the strategy of state evolution, we have ui,m,p < Bj,m ≤ uj,m,p ∀p ≥ n. N N • Let’s compare l=p θj,m,l and l=p θi,m,l at p-th (p ≥ n) packet burst. Since θj,m,l and θi,m,l (l = p + 1, ..., N ) are derived by assuming all ACK feedbacks, we have θj,m,l → uj,m,p and θi,m,l → ui,m,p for sufficiently large l. Due to uj,m,p > ui,m,p , we can conclude that θj,m,l > θi,m,l for sufficiently large l. N N • Hence, we have l=p θj,m,l > l=p θi,m,l when N tends to infinity. According to our user selection strategy, the user i will never been selected in the p-th packet bursts. Remember that p ≥ n, the user i will never been selected after the n-th packet burst. This conflicts with the statement that user i with SINR Bi,m is selected infinite times. Hence, only the user j who has the largest SINR will be selected infinite times. Combining this result with Lemma 4, we can get the conclusion of Lemma 3. R EFERENCES [1] L. C. Wang and W. J. Lin, “Throughput and fairness enhancement for OFDMA broadband wireless access systems using the maximum C/I scheduling,” in Proc. IEEE VTC 2004, pp. 4696–4700, Sept. 2004. [2] C. Wengerter, J. Ohlhorst, and A. von Elbwart, “Fairness and throughput analysis for generalized proportional fair frequency scheduling in OFDMA,” in Proc. IEEE VTC 2005, pp. 1903–1907, May 2005. [3] M. Y. Shen, G. Q. Li, and H. Liu, “Effect of traffic channel configuration on the orthogonal frequency division multiple access downlink performance,” IEEE Trans. Wireless Commun., pp. 1901–1913, July 2005.

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[4] D. Niyato and E. Hossain, “Adaptive fair subcarrier/rate allocation in multirate OFDMA networks: radio link level queuing performance analysis,” IEEE Trans. Veh. Technol., pp. 1897–1907, Nov. 2006. [5] 3GPP, “Physical layer aspects of utra high speed downlink packet access,” TR 25.848 V4.0.0. [6] V. K. N. Lau, M. L. Jiang, and Y. J. Liu, “Cross layer design of uplink multi-antenna wireless systems with outdated CSI,” IEEE Trans. Wireless Commun., pp. 1250–1253, June 2006. [7] M. L. Jiang and V. K. N. Lau, “Performance analysis of proportional fair uplink scheduling with channel estimation error in multiple antennas system,” in Proc. IEEE PIMRC 2004, pp. 1628–1632, Sept. 2004. [8] R. Wang and V. Lau, “On the design of downlink multi-user multiantenna OFDMA systems with imperfect CSIT,” in Proc. IEEE PIMRC 2005, Sept. 2005. [9] T. Holliday, A. Goldsmith, and P. Glynn, “Wireless link adaptation policies: QoS for deadline constrained traffic with imperfect channel estimates,” in Proc. IEEE ICC 2002, pp. 3366–3371, April 2002. [10] M. A. Haleem and R. Chandramouli, “Joint adaptive rate control and randomized scheduling for multimedia wireless systems,” in Proc. IEEE ICC 2004, pp. 1500–1504, June 2004. [11] A. K. Karmokar, D. V. Djonin, and V. K. Bhargava, “Delay constrained rate and power adaptation over correlated fading channels,” in Proc. IEEE Global Telecommunications Conference, 2004., vol. 6, pp. 3448– 3453, Nov. 2004. [12] H. T. Zheng and H. Viswanathan, “Optimizing the ARQ performance in downlink packet data systems with scheduling,” IEEE Trans. Wireless Commun., vol. 4, pp. 495–506, Mar. 2005. [13] M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley and Sons, 2005.

IEEE 802.16m.

Rui Wang graduated from the Dept of Computer Science and Technology, University of Science & Technology of China, with a B.Eng in 2004. After that, he was admitted by the Dept of Electronic & Computer Engineering, Hong Kong University of Science & Technology, for PhD study. He is now Ph.D. candidate on wireless communication. His current research interests include cross-layer optimization, wireless ad-hoc network, and cognitive radio. He is also involved in the standardization of IEEE 802.22 (Wireless Regional Area Network) and

Vincent Lau graduated from the Dept of EEE, University of Hong Kong with a B.Eng (Distinction 1st Hons) in 1992. He joined the HK Telecom after graduation for three years as project engineer and later promoted to system engineer. He obtained the Sir Edward Youde Memorial Fellowship and the Croucher Foundation in 95 and went to the University of Cambridge for a Ph.D. in mobile communications. He completed the Ph.D. degree in two years and joined the Lucent Technologies - Bell labs (ASIC department) as member of technical staff in 1997. In 2004, he joined the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology. At the same time, he is a technology advisor of HK-ASTRI on the R&D of wireless LAN access infrastructure with smart antenna. He has a total of seven years industrial experience and three years of academic experience. His research interests include adaptive modulation and channel coding, information theory with state feedback, multi-user MIMO scheduling, crosslayer optimization, baseband SoC design (UMTS base station ASIC, 3G1x mobile ASIC, Wireless LAN MIMO ASIC). He is the principal author of a book on MIMO Technologies (to be published by John Wiley and Sons) as well as the chapter author of two books on wideband CDMA technologies. He has published more than 40 papers in IEEE transactions and journals and 47 papers in international conference, 14 Bell Labs Technical Memos and received two best paper awards. He has eight US patents pending and is currently a senior member of IEEE.

Robust Optimal Cross-Layer Designs for TDD-OFDMA Systems with ...

Abstract—Cross-layer designs for OFDMA systems have been shown to offer ...... is obtained from standard water-filling approach over m = 1 to M and i = n to N ...

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