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Closed-Loop Cross-Layer SDMA Designs with Outdated CSIT Rui Wang and Vincent K. N. Lau

Abstract—In this paper, we propose a novel closed-loop approach for robust downlink multi-antenna cross-layer design with outdated channel state information at the transmitter (CSIT). Based on the ACK/NAK feedbacks from the mobiles, the proposed cross-layer design does not require the knowledge of CSIT error statistics. We formulate the cross-layer design as a mixed combinatorial search and Markov decision process (MDP). While it is well-known that general solutions for MDP are very complex, one important contribution of this paper is that we obtain simple closed-form power, rate and user allocation policies which are asymptotically optimal for small target frame error rate (FER). Simulation results illustrate that the performance of the proposed closed-loop cross-layer design is very robust with respect to the outdated CSIT, CSIT error model mismatch as well as channel variations due to Doppler. Index Terms—Closed-loop, multiple antennas, outdated CSIT, cross-layer.

I. I NTRODUCTION

R

ECENTLY, cross-layer scheduling in multiuser multiantenna systems has received tremendous attentions. High spectral efficiency can be achieved by exploiting the multiuser selection diversity and spatial multiplexing[1]–[3]. To exploit the multiuser selection diversity, knowledge of Channel State Information is required at the base station (CSIT). However, obtaining perfect CSIT is very challenging at the base station especially for large number of transmit antennas nT or large number of users K. When the CSIT is imperfect 1 at the base station, the actual instantaneous mutual information is unknown to the base station, and as a result, there may be packet errors (despite of powerful channel coding) whenever the scheduled data rate exceeds the instantaneous mutual information [6]. Moreover, the efficiency of the multiuser scheduling is reduced because the wrong set of users may be selected for transmission. Most of the existing crosslayer designs address the imperfect (outdated) CSIT issue Manuscript received October 28, 2007; revised April 8, 2008 and July 10, 2008; accepted September 4, 2008. The associate editor coordinating the review of this paper and approving it for publication was S. Affes. The authors are with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technologies, Clear Water Bay, Hong Kong, China (e-mail: {wray, eeknlau}@ust.hk). This work was supported by the Research Grants Council of the Hong Kong Government through the grant RGC 615606. Digital Object Identifier 10.1109/TWC.2009.071200 1 There are two interpretations of imperfect CSIT in the literature. The first meaning refers to the incomplete knowledge of CSIT such as the partial CSIT or limited feedback of CSIT[4], [5]. Yet, the incomplete CSIT is received error-free and without delay and hence, there is no issue of packet outage. The second meaning refers to the "erroneous CSIT" in which the CSIT obtained has errors or suffer from delay (no longer updated). In this case, there will be uncertainty on the instantaneous mutual information and hence, issues of packet outage. In this paper, we shall focus on the second meaning behind imperfect CSIT.

based on heuristic approach. For example, in [7], the crosslayer scheduler is designed for perfect CSIT (naive scheduler) and the effect of imperfect CSIT is evaluated by simulations. 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 crosslayer scheduler (designed as if the CSIT were perfect) is very sensitive to the imperfect or outdated CSIT even at very small CSIT errors [8]. In [6], [9], the authors discuss the optimal resource allocation for multi-antenna systems with imperfect CSIT. In [8], [10], [11], the authors discuss the optimal cross-layer design for multiuser systems with imperfect CSIT. However, in all these works, the base station requires the knowledge of CSIT error statistics (such as error variance and the CSIT error distribution), which may not be easily obtained in practice. In all the works mentioned above, the cross-layer design is classified as open loop. In open-loop designs, the power, rate and user adaptation are determined based on particular CSIT error model. If there is any mismatch on the assumed parameters (e.g. CSIT error variance) versus the actual parameters, the open-loop design will not be able to automatically correct for those, and therefore, the performance will be sensitive to the mismatch of the CSIT error model. In all the above works, the ACK/NAK feedback is not fully utilized in the cross-layer design. In fact, the ACK/NAK feedback can provide a distinctive "closed-loop" feedback information which allows the base station to automatically correct for potential model or parameter mismatch or different receiver sensitivity2 . As a result, closed-loop power control is widely used in IS95 and UMTS systems[12], [13]. There are some existing works using the ACK/NAK to facilitate the closed-loop adaptation [14]–[17]. In [14], the authors present a power and rate control policy for a point-to-point SISO system with delay constrained traffic based on ACK/NAK feedback. However, the cross-layer scheduling (user selection) issue is not addressed and the power and rate adaptation policy cannot be extended to MIMO scenario. In [15], the authors present a heuristic rate control and randomized scheduling algorithm for flat-fading SISO channels based on automata learning. However, in all these works, the solutions are heuristic and it is not clear what should be the optimal policy and how far the schemes are from optimal performance. Furthermore, these solutions cannot be applied to MIMO systems and these 2 For example, not all the mobile users in a cellular system have the same receiver design (RF and baseband) and hence, they may have different sensitivity. Using the ACK/NAK feedback, one can naturally accommodate different receivers in the cross-layer scheduling as well.

c 2009 IEEE 1536-1276/09$25.00 

WANG and LAU: CLOSED-LOOP CROSS-LAYER SDMA DESIGNS WITH OUTDATED CSIT

policies cannot guarantee a target FER for wireless links, which is a very important requirement for applications. In this paper, we shall propose a robust closed-loop multiantenna cross-layer design for downlink systems with outdated CSIT in slow fading channels. In addition to the outdated CSIT, we shall utilize the ACK/NAK feedback from the mobiles to adjust the power allocation, rate allocation so as to maintain a target FER. Unlike other conventional design, no knowledge of the CSIT error statistics is required at the base station and the performance of the proposed closedloop design is very robust with respect to CSIT errors. Due to the absence of knowledge on the accurate distribution of SINR 3 (under SDMA) at the BS, we shall impose a canonical model on SINR distribution, formulate the problem as Markov decision process (MDP) and derive the optimal control policy (optimal w.r.t.4 the imposed canonical model). Finally, simulation results illustrate that the average goodput performance of the proposed closed-loop cross-layer design is very robust with respect to outdated CSIT, SINR model mismatch as well as channel variation due to Doppler. This paper is organized as follows. In section II, we outline the multiuser MISO system model as well as the outdated CSIT model. In section III, we shall define the system goodput and formulate the closed-loop cross-layer design as a Markov decision process in the presence of the outdated CSI. In section IV, we shall present the optimal solution for the closeloop cross-layer scheduling problem. In section V, numerical results are presented and discussed. Finally, we give a brief summary in section VI. II. MULTIUSER MISO 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 flat fading channel. We assume the base station is equipped with nT transmit antennas and each mobile user is equipped with only one receive antenna. We consider a scheduling slot structure which consists of multiple packet bursts, and assume the channel is quasi-statistic within a scheduling slot. Let Yk,n be the received signal at the k-th mobile in the n-th packet burst in a scheduling slot. The K × 1 dimension vector of the aggregate received signals Yn in the n-th packet burst is given by ⎤ ⎡ ⎤ ⎤ ⎡ Y1,n Z1,n H1 ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ Yn = ⎣ ... ⎦ = ⎣ ... ⎦ Xn + ⎣ ... ⎦ . YK,n HK ZK,n ⎡

where Xn is the nT × 1 transmit symbol from the base station to the K mobiles, Hk is the 1×nT channel coefficients where each element is i.i.d. complex Gaussian with zero mean and unit variance, Zk,n is the i.i.d. complex Gaussian noise with variance σz2 . 3 Even if the BS knows about the CSIT error statistics, the accurate SINR distribution (induced by the CSIT errors) is very complicated and does not lead to any tractable solution. We shall illustrate in this paper that brute force optimization with respect to accurate SINR distribution is not necessary. 4 w.r.t. is short for "with respect to"

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B. CSIT Error Model In TDD systems, the estimated CSIT is usually outdated due to duplexing delay. The outdated CSIT can be modeled as: Hbk = Hk + ΔHbk where Hk is the actual CSI and ΔHbk is the CSIT error matrix 2 with variance of each element given by σΔH . In this paper, we shall propose a robust closed-loop cross-layer design where no 2 ) is needed. knowledge of the CSIT error statistics (σΔH C. Multiuser Physical Layer Model We consider the system design with ZF linear precoder at the base station. At the base station, the K streams of information data for the K individual users (some users may be assigned zero rate if not selected by the scheduler) are channel encoded independently at the base station. During the n-th packet burst, the K × 1 vector of encoded symbols, Un = [U1,n , ..., UK,n ]T , are further processed by the K × K diagonal power control matrix Pn = diag(p1,n , ..., pk,n , ..., pK,n ) and the nT × K spatial multiplexing matrix W = [w1 , ..., wk , ..., wK ], where pk,n ≥ 0 is the transmit power and wk is the nT × 1 complex spatial multiplexing weight for the k-th user. In the n-th packet burst, the received signal of the k-th user Yk,n is given by √ √ Yk,n = pk,n Hk wk Uk,n + pj,n Hbk wj Uj,n 

 j=k 

 Information Multiuser Interference I √ − pj,n ΔHbk wj Uj,n +Zk,n , (1) j=k



 Multiuser Interference II where Hbk denotes the estimated CSIT, the first term contains the desired signal, the second term represents the multiuser interference I due to simultaneous transmission of independent information streams for different users, and the third term represents the multiuser interference II due to CSIT error. Since the base station only has the knowledge of the estimated CSIT Hb = {Hb1 , ..., HbK }, the spatial multiplexing weights cannot be chosen to eliminate completely both multiuser interference terms in (1). Using the zero-forcing (ZF) approach, the spatial multiplexing weight wk is selected to optimize the following maximization problem: max Hbk wk wk

s.t.

wk∗ wk = 1 Hbj wk = 0 ∀j ∈ A, j = k.

In the above equations, A is the set of admitted user indices (users with non-zero allocated power and rate) and the operator ∗ denotes complex conjugate transpose. With the ZF approach, the multiuser interference I is zero-out. Due to the limitation of the zero-forcing processing, we have |A| ≤ nT , thus, the transmitter can select at most nT users simultaneously for each packet slot. In physical layer, the spatial multiplexing weights {wk } are calculated at the

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beginning of a scheduling slot based on the outdated CSIT and keep unchanged during the entire scheduling slot. D. Maximum Achievable Data Rate The maximum achievable data rate of the k-th user in the n-th packet burst is given by the maximum mutual information between Yk,n and Uk,n conditional on CSIR Hk :

 pk,n |Hk wk |2 Ck,n = log2 1 + 2  . (2) σz + j=k pj,n |ΔHk wj |2 Note that Ck,n is a function of Hk and ΔHk which are unknown to the base station. E. Packet Outage and Average Goodput Let rk,n be the scheduled data rate for the user k in the n-th packet. The instantaneous goodput of the k-th user in the n-th subcarrier is given by

where g¯n denotes the conditional average goodput (conditioned on the CSIT Hb and the ACK/NAK feedback sequence Sn−1 ). Therefore, the closed-loop cross-layer scheduling prob1 lem with outdated CSIT can be summarized as the following optimization problem: Prob 1 (Cross-Layer Problem Formulation): Given any realization of the estimated CSIT for all mobile users {Hb1 , ..., HbK }, determine the optimal admitted user set A, the optimal power allocation policy {pk,n } and the optimal rate allocation policy {rk,n } to maximize the conditional total goodput, G(P0 , Hb , A, {rk,n }, {pk,n }). That is, 

∗

G (P0 , H , A, {rk,n }, {pk,n }) =

 = EHb

n=1 k=1 N K



rk,n Pr[Ck,n ≥ rk,n |H ] b

n=1 k=1

  = EHb G(P0 , Hb , A, {rk,n }, {pk,n }) ,

(3)

where EHb [X] denotes the expectation of the random variable X w.r.t. Hb ; P0 , A, {rk,n }, {pk,n } denote the total transmit power constraint, the user selection policy, the rate allocation policy and the power allocation policy respectively; and G(·) denotes the conditional system goodput (conditioned on the CSIT Hb ). Due to the outdated CSIT and the associated packet outage, we shall design the cross-layer scheduler to optimize the total average system goodput U (as well as the conditional system goodput G). III. C ROSS -L AYER D ESIGN F ORMULATION WITH O UTDATED CSIT For notation convenience, we denote qk,i = {0, 1} as the ACK/NAK feedback from mobile user k after the i-th packet transmission: qk,i = 1 if an ACK is received and 0 otherwise. Let Si = {qk,i : ∀k ∈ A} denote the collection of all the ACK/NAK feedbacks from the selected users after the i-th packet transmissions and SN i = (Si , ..., SN ). The conditional goodput G(·) in (3) can be expressed as G(P0 , Hb , A, {rk,n }, {pk,n })  N K  b n−1   = ES N rk,n Pr Ck,n ≥ rk,n H , S1 1  = ES N 1

n=1 k=1  N

g¯n

n=1

,

max

{pk,n },{rk,n },A

ESN 1

N n=1



g¯n (4)

where the power allocation, rate allocation policies {rk,n }, {pk,n } are subject to the following constraints: •

ρk,n = rk,n I[Ck,n ≥ rk,n ], where I(E) is the indicator function which is equal to 1 if the event E 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 K  ρk,n U (P0 , A, {rk,n }, {pk,n }) = E

b

• • •

Causality Constraint: The power and rate allocation should be a causal function of the ACK/NAK feedbacks Sn . i.e. pk,n = pk,n (Sn−1 ) and rk,n = rk,n (Sn−1 ) 1 1 N   Total Transmit Power Constraint: pk,n ≤ P0 ; n=1 k∈A

Cardinality Constraint:|A| ≤ nT ; Quality of Service (QoS) Requirement: The conditional packet outage probability of the users is less than a target outage probability .

The optimization variables in problem 1 include combinatorial variables (A) as well as real variables ({pk,n }, {rk,n }) and hence, it is a mixed convex and combinatorial optimization problem. To solve the mixed combinatorial and convex optimization problem, we shall separate the solution into two steps. In the first step, we shall determine the optimal power and rate allocations based on a given admitted user set A. In the second step, a combinatorial search is performed over all combinations of A to determine the optimal set. These two steps are elaborated in the following section.

IV. C LOSED -L OOP C ROSS - LAYER S CHEDULING S OLUTION Given an admitted user set A, we shall determine the optimal power allocation scheme {pk,n } and the optimal rate allocation scheme {rk,n } based on the outdated CSIT and the ACK/NAK feedbacks from the mobiles. To obtain more insight into the structure of the optimization problem, we shall show that the optimal objective G∗ (P0 , Hb , A) can be divide and conquer into a set of recursive equations. In this section, we shall derive the recursive relationship first, and then, solve the convex optimization problem based on this recursive structure. The optimal conditional average goodput G∗ (P0 , Hb , A) in (4) can be expressed into a recursive form as summarized by the following lemma. Lemma 1 (Recursive Formulation): Let Fn∗ (P, Hb , Sn−1 ) 1 be the total optimal average goodput from the n-th packet burst to the N -th packet burst conditional on the CSIT and

WANG and LAU: CLOSED-LOOP CROSS-LAYER SDMA DESIGNS WITH OUTDATED CSIT

the first n − 1 ACK/NAK feedbacks Sn−1 . i.e., 1 ) Fn∗ (P, Hb , Sn−1 1 =

max

{pk,n ,..,pk,N }, {rk,n ,..,rk,N }

...+



 Pr(Sn |Sn−1 )¯ gn+1 (·) + g¯n (·) + 1 Sn



n−1 Pr(SN )¯ gN (·) n |S1

,

(5)

SN n

where

N   i=n k∈A

pk,n = P . Fn∗ (P, Hb , Sn−1 ) can be expressed 1

recursively as

=

) Fn∗ (P, Hb , Sn−1 1  max )+ g¯n ({pk,n }, {rk,n }, Hb , Sn−1 1 {pk,n },{rk,n }  b ∗ b n Pr(Sn |Sn−1 , H )F (P − p , H , S ) , (6) n n+1 1 1 Sn



where pn = k∈A pk,n for all n ∈ [1, N ], and FN∗ +1 = 0. Moreover, the optimal conditional goodput in (4) is given by G∗ (P0 , Hb , A) = F1∗ (P0 , Hb ).

(7)

Proof 1: Please refer to [18]. As a result of Lemma 1, the optimization problem with respect to {rk,n }, {pk,n } (given any CSIT realization Hb ) can be divided and conquer into N steps. The recursive equation in (6) is also called the Bellmen’s equation[19] 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. The offline recursion is to determine the power allocation and rate allocation policies for all possible ACK/NAK feedbacks. This is not a realtime process. On the other hand, the online strategy is a realtime algorithm which selects the optimal power and rate allocation for the n−th packet burst upon receiving the previous ACK/NAK feedbacks from the mobiles. The instantaneous mutual information Ck,n in (2) suffers  from spatial 2interference term in the denominator j=k pj,n |ΔHk wj | . This is due to the absence of perfect CSIT at the base station, and hence, the SDMA cannot be perfect and there is residual spatial interference between the SDMA channels. When the BS knows about CSIT error 2 , it can potentially calculate the power and data statistics σΔH rate given a PER constraint by taking into account of the spatial interference. However, in this paper, we do not even have the knowledge of CSIT error variance because it’s hard to obtain in practice. As a result, this poses a great challenge in modelling the packet outage probability in open-loop crosslayer designs[8]. Yet, using the closed-loop cross-layer design approach, we can get around the issue because the base station can track the distribution of the mutual information Ck,n using the ACK/NAK feedbacks. To obtain tractable solution, we shall impose a "structure" on our solution framework and then optimizes the "parameters of the imposed structure" as well as the system parameters ( power/rate/user selection) using Markov decision process (MDP). Specifically, the instantaneous mutual information Ck,n can be modelled by the following.

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Model 1: A Statistical Model for Instantaneous Mutual Information (8) Ck,n = log2 (1 + pk,n Bk ), where pk,n Bk denotes the instantaneous SINR of the k-th user during the n-th packet burst. In addition, the model has the following properties. • Bk is quasi-static within a scheduling slot. • The a priori conditional pdf of Bk (before any ACK/NAK feedback) is denoted by fk,1 (Bk |Hb ). Given the ACK/NAK feedback sequence up to the n-th packet Sn−1 , the conditional pdf on Bk is denoted by 1 fk,n (Bk |Hb , Sn−1 ). 1 Based on the model elaborated in the above, we shall derive the scheduling solution in the following sections. A. Backward Recursion for the Rate and Power Adaptation Policies In the offline strategy, we shall partition the optimization for the average goodput G∗ (P, Hb ) with respect to the power allocation policy {pk,1 }, {pk,2 },...,{pk,N } and the rate allocation policy {rk,1 }, {rk,2 }, ..., {rk,N } (for the N packet bursts) into N recursive optimizations using the recursive relationship ∗ ∗ in (6). Let {rk,n }, {p∗k,n } be the optimized of Fn∗ and Fn+1 rate and power allocation policies with respect to all possible ACK/NAK feedback events. These optimal policies 5 will be used for the online algorithm when the actual ACK/NAK feedbacks are received. • Step 1. For n = N , the optimization on rate and power ∗ ∗ {p are describe by the following subproblem Subproblem k,N , rk,N }1: ∗ {rk,N (SN−1 )}, {p∗k,N (SN−1 )} 1 1

=arg =arg

max

{rk,N },{pk,N }

max

{rk,N },{pk,N }



FN∗ (PN , Hb , SN−1 ) 1 rk,N · Pr(Ck,N ≥ rk,N |Hb , SN−1 ) 1 k∈A

where pN = k∈A pk,N . In order to solve the subproblem 1, we notice that rk,N (1 − ) g¯N (pN , {fk,N }) = k∈A

where  is the target packet outage probability. To satisfy ∗ } the target outage, the scheduled data rate policy {rk,N is given by ∗ rk,N = log2 (1 + pk,N θk,N ),

(9)

where θk,N is the scaling factor is given by the root of the following equation (when n = N )      (10) Pr Bk ≥ θk,n fk,n = 1 − . Let Φk,N be the corresponding CDF of the distribution fk,N , the general solution of (10) is given by θk,N = Φ−1 k,N (). 5 The "optimality" is for the MDP problem using Model 1. Since in practice, the actual distribution of Ck,n may be different from Model 1, we shall study the performance degradation due to model mismatch by extensive simulation.

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To determine the optimal power allocation policies, {p∗k,N }, we form the Lagrangian function as L=

k∈A

and N 1 1 1 P + − . N − n + 1 N − n i=n θk,i θk,n k∈A k∈A (16) Similarly, we have

1− (1 − ) log2 (1 + pk,N θk,N ) − pk,N . λN ln 2

pn =

k∈A

Using standard approach, the optimal power allocation policy is given by

+ 1 1 ∗ pk,N = − , (11) λN θk,N

Fn∗ (P, {fk,n }) N  1  (N−n+1)|A|

i=n k∈A θk,i P + =(1 − ) log2 N −n+1 N −n+1 N   θk,i  i=n k∈A  . (17) +(1 − ) log2 |A|(N−n+1)|A|

where (X)+ = max{0, X} and λN is the Lagrangian multiplier given by 1  1  1 = pN + (12) λN |A| θk,n

Notice that in (10), in order to solve θk,n , we should derive the density function fk,n according to the density function of the previous transmission fk,n−1 in the following lemma: Lemma 2 (Density Evolution): The conditional pdf of the SINR Bk (in the instantaneous mutual information Ck,n ), fk,n+1 (Bk |Hb , Sn1 ), can be expressed recursively as: ⎧ n−1 fk,n (Bk |Hb ,S 1 ) ⎪ ⎪   rk,n  ⎪ ⎪ 2 −1  b n−1 ⎪ ⎪ ⎪ Pr pk,n ≤Bk H ,S1 ⎨ fk,n+1 (Bk |Hb , Sn1 ) = 0 ⎪ n−1 ⎪ fk,n (Bk |Hb ,S ⎪ 1 ) ⎪ ⎪   . ⎪ ⎪ ⎩ Pr Bk ≤ 2rpk,n −1 Hb ,Sn−1 1  k,n (18) where the first equation is for q = 1 and B ≥ k,n k 2rk,n −1 ; the third equation is for q = 0 and B ≤ k,n k pk,n

k∈A

FN∗

for sufficiently large P0 . Hence, the closed-form for is given by

1 |A| ∗ FN (pN , fk,N ) = (1 − ) log2 pN + θk,N  k∈A θk,N 

k∈A +(1 − ) log2 . |A||A| •

Step 2. The optimization for rate and power for n = {N − 1, N − 2, .....1} is described by the following subproblem Subproblem 2: ∗ (Sn−1 )}, {p∗k,n (Sn−1 )} {rk,n 1 1  = arg max g¯n ({pk,n }, {rk,n }, fn ) {pk,n },{rk,n }  ∗ Pr(Sn |fn )Fn+1 (P − pn , fn+1 ) +

2rk,n −1 pk,n ;

and the second equation is for otherwise. Proof 2: Please refer to [18].

Sn



where pn = k∈A pk,n . Given a target outage probability , the probability Pr(Sn |{fk,n }) consists of a summation of the terms (1−)a b , where a is the total number of ACK feedbacks and b is the total number of NAK feedbacks in Sn . Since  is usually chosen to be very small, most of the terms in Pr(Sn |{fk,n }) are very small except the one when a = |A| and b = 0 (In this case, there is no transmission outage). Hence, we have

B. Online Solution for the Rate, Power and User Selection Adaptations The online processing is given below. •

A∗ = max argA G∗ (P0 , Hb , A)

Fn∗ (P, {fk,n })   ∗ g¯n (pn , {fk,n }) + Fn+1 (P − pn , {fk,n+1 }) . = max {pk,n },{rk,n }

Using the results in Step 1, the optimal power and rate ∗ } are given by allocation policies, {p∗k,n } and {rk,n p∗k,n

where

=

1 1 − λn θk,n

(13)

∗ rk,n = log2 (1 + p∗k,n θk,n ),

(14)

1 1 1 (pn + = ), λn |A| θk,n

(15)

k∈A

•

+ ,

Step 1. Before packet transmission, select the admitted user set according to

•

(19)

Alternatively, realtime genetic search [10] can be used to simplify the search in (19). Step 2. At the first packet burst, the optimal power ∗ }, {p∗k,1 } based on the estimated and rate allocation {rk,1 b CSIT H is obtained according to (13) and (14). Step 3. Before transmitting the n-th packet burst (n = {2, 3, ....N }), the base station has already obtained the specific ACK/NAK feedbacks of the pre. The optimal power and vious n − 1 packets Sn−1 1 rate allocation for the n-th packet is obtained from ∗ (Sn−1 )}, {p∗k,n (Sn−1 )} according to (13), (14) or {rk,n 1 1 (9), (11) in the offline recursion.

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V. N UMERICAL R ESULT AND D ISCUSSION

2 2 (n + 1) = σB (n) μBk (n + 1) = μBk (n), σB k k  rk,n max{LBk (n), 2 pk,n−1 } if qk,n = 1, LBk (n + 1) = LBk (n) otherwise. (20) and  rk,n min{U Bk (n), 2 pk,n−1 } if qk,n = 0, U Bk (n + 1) = U Bk (n) otherwise. (21) Figure 1 illustrates the average system goodput (bit/Sec/Hz) versus the number of transmit antennas nT at CSIT error 2 σΔH = 0.1, SNR = 23 dB and K = 10. The duration of the scheduling slot is 4ms. Observe that the system goodput increases significantly as nT increases due to the spatial multiplexing gains. We also observe that there is a significant goodput gain of the proposed closed-loop scheduler over the regular round robin scheduler. This illustrates the multiuser diversity gain and the proposed closed-loop scheduler offers robust and significant goodput gains at high CSIT errors and moderate Doppler frequency. Furthermore, there is also a significant goodput gain of the proposed closed-loop scheduler over the naive scheduler (scheduler designed for perfect CSIT and treats the estimated CSIT as the perfect CSIT). Figure 2 shows the per-packet average goodput and the average mutual information (averaged over 1000 channel realizations at the packet slot) for the proposed closed-loop scheduler as well as the round robin and the naive scheduler. The number of transmit antenna nT is 2 and the total transmit power of a scheduling time slot is 23dB. We observe that although there is about 1.5 bit/Sec/Hz performance degradation, which is due to the packet transmission outage and fact that the simple mutual information model 1 is not the exact model, the goodput performance can still track the time variations of the instantaneous mutual information even at high Doppler frequency fd = 10Hz. Figure 3 shows the transient response of the loop. The instantaneous mutual information and the instantaneous scheduled rate (rk,n ) are plotted against the packet time slot at fd = 1 and fd = 10 Hz. We can observe from Figure 3 that in both cases, the scheduled data rate of the proposed closed-loop cross-layer design tracks the instantaneous mutual

fd=10Hz,closed−loop fd=10Hz,naive scheduler

Average System Goodput (bit/Sec/Hz)

12

fd=10Hz,round robin

10

8

6

4

2

0

2

3 4 Number of Transmit Antennas

5

Fig. 1. Average goodput performance of the proposed closed-loop scheduler versus nT at K = 10, fd = 10Hz, σΔH = 0.01, P0 = 23dB.

9 Average mutual information, closed−form

8 Bandwidth Efficiency (bit/sec/Hz)

In this section, we shall illustrate the performance of the closed-loop scheduler designs. In our simulation, the total number of receivers in the system K is 10. The duration of the packet slot is 0.2ms. The Doppler frequency fd is 1Hz or 10Hz. Furthermore, the target packet outage probability is fixed to be  = 0.01. Each point in the figures is obtained by averaging over 1000 independent fading realizations. The conditional density of Bk in the instantaneous mutual infor) is assumed to be truncated mation model fk,n (Bk |Hb , Sn−1 1 2 Gaussian with parameters μBk (n) and σB (n) (mean and k variance of the Gaussian part) as well as the lower and upper bounds on Bk (LBk (n), U Bk (n)) respectively. As a result, the evolution of the conditional pdf of Bk given ACK/NAK feedbacks (from Lemma 2) is equivalent to the following evolution of parameters:

14

Average goodput, closed−form 7 6 5 4

Average goodput, naive

3

Average goodput, round robin

2 1

2

4

6

8 10 12 14 Index of Packet Burst

16

18

20

Fig. 2. The average goodput and the average mutual information per packet slot versus the packet time slot number with nT = 2,fd = 10Hz,K = 10, P0 = 23dB, σΔH = 0.01.

information quite well. This justifies the robustness of our closed-loop scheduler with respect to the CSIT error, model mismatch and the channel variation due to Doppler. VI. S UMMARY In this paper, we propose a robust closed-loop cross-layer design for downlink multi-antenna systems with outdated CSIT. Using the ACK/NAK feedbacks from the mobiles, the closed-loop cross-layer scheduler does not require any knowledge of the CSIT error statistics. To take into account the potential packet outage (due to outdated CSIT), we define system goodput, which measures the average b/s/Hz successfully delivered to the mobiles, as the optimization objectives. We formulate the cross-layer design as a mixed combinatorial search and Markov decision problem. Based on the backward recursion and the forward recursion algorithms, we derive closed-form solutions (which is asymptotically optimal) for the power and rate allocations. Simulations illustrate that the proposed closed-loop cross-layer scheduler has very robust

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9 fd=1Hz, instantaneous mutual information 8

Bandwidth Efficiency (bit/Sec/Hz)

7 fd=1Hz, instantaneous data rate

6 5 4

fd=10Hz, instantaneous mutual information

3

fd=10Hz, instantaneous data rate

2 1 0

10

20

30 40 50 Index of Packet Burst

60

70

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Fig. 3. The transient of the instantaneous scheduled data rate versus the actual instantaneous mutual information of each packet burst with nT = 2,K = 10, P0 = 26dB at fd = 1 and fd = 10 Hz.

goodput performance at moderate to high CSIT errors and pedestrian mobility. R EFERENCES [1] H. Viswanathan and K. Kumaran, “Rate scheduling in multiple antenna downlink„" in Proc. Allerton Conf. Commun. Control, Monticello, IL, Oct. 2001. [2] K. N. Lau, Y. Liu, and T. A. Chen, “Optimal space-time scheduling for wireless communications with partial power feedback," Bell Labs Technical J., Nov. 2002. [3] R. B. R. Gozali and B. Woerner, “On the performance of scheduling over space-time architectures," in Proc. IEEE Veh. Technol. Conf., pp. 420-424, Sept. 2002. [4] D. J. Love and J. R. W. Heath, “Limited feedback unitary precoding for orthogonal space-time block codes," IEEE Trans. Signal Processing, vol. 53, pp. 64-73, Jan. 2005. [5] J. Choi, B. Mondal, and R. W. Heath, “Interpolation based unitary precoding for spatial multiplexing MIMO-OFDM with limited feedback," IEEE Trans. Signal Processing, vol. 54, pp. 4730-4740, Dec. 2006. [6] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming," IEEE J. Select. Areas Commun., vol. 24, pp. 528-541, Mar. 2006. [7] 3GPP, “Physical layer aspects of utra high speed downlink packet access," TR 25.848 V4.0.0. [8] 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, vol. 3, pp. 1628-1632, Sept. 2004. [9] E. Baccarelli and M. Biagi, “Power-allocation policy and optimized design of multiple-antenna systems with imperfect channel estimation," IEEE Trans. Veh. Technol., pp. 136-145, Jan. 2004.

[10] 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. [11] W. Huang and K. B. Letaief, “A cross-layer resource allocation and scheduling for multiuser space-time block coded MIMO/OFDM systems," in Proc. 2005 IEEE International Conf. Commun., pp. 2655-2659, May 2005. [12] L. Harte, CDMA Is-95 for Cellular and PCs: Technology, Applications, and Resource Guide. McGraw-Hill Professional, 1999. [13] H. Holma and A. Toskala, WCDMA for UMTS: Radio Access for Third Generation Mobile Communications. Wiley, 2000. [14] 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, vol. 5, pp. 3366-3371, Apr. 2002. [15] M. A. Haleem and R. Chandramouli, “Joint adaptive rate control and randomized scheduling for multimedia wireless systems," in Proc. IEEE ICC 2004, vol. 3, pp. 1500-1504, June 2004. [16] A. K. Karmokar, D. V. Djonin, and V. K. Bhargava, “Delay constrained rate and power adaptation over correlated fading channels," in Proc. IEEE Global Telecommun. Conf., 2004., vol. 6, pp. 3448-3453, Nov. 2004. [17] 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. [18] R. Wang and V. K. N. Lau, “Closed-loop cross-layer SDMA designs with outdated CSIT," [Online]. Available: http://www.ee.ust.hk/∼wray. [19] M. L. Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley and Sons, 2005. Rui Wang obtained B.Eng in computer science from the University of Science & Technology of China (USTC, 2000-2004) and Ph.D. in electronic and computer engineering from the Hong Kong University of Science & Technology (HKUST, 20042008). He is currently a post-doctoral researcher in HKUST. His current research interests include cross-layer optimization, wireless ad-hoc network, and cognitive radio. He is also interested in the standardization of wireless systems, i.e. IEEE 802.22, IEEE 802.16m and IMT-Advanced. Vincent K. N. Lau obtained B.Eng (Distinction 1st Hons) from the University of Hong Kong (19891992) and Ph.D. from the Cambridge University (1995-1997). He joined the Bell Labs - Lucent Technologies as member of technical staff from 1997-2003 and the Department of ECE, Hong Kong University of Science and Technology (HKUST) as Associate Professor afterwards. His current research focus includes robust cross layer scheduling for MIMO/OFDM wireless systems with imperfect channel state information, communication theory with limited feedback as well as delay-sensitive cross layer optimizations. He is currently an associate editor of IEEE T RANSACTIONS ON W IRELESS C OMMUNICATIONS, IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNI CATIONS , EUARSIP W IRELESS C OMMUNICATIONS AND N ETWORKING .