Exploiting Multiuser Diversity With Imperfect One-Bit Channel State Feedback

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 1, JANUARY 2007 183

Exploiting Multiuser Diversity With ImperfectOne-Bit Channel State FeedbackYisheng Xue, Member, IEEE, and Thomas Kaiser, Senior Member, IEEE

Abstract—It has been well recognized that significant through-put gains can be leveraged in multiuser wireless communicationsystems by exploiting multiuser diversity with a smart scheduler.This scheduler collects channel state information (CSI) from allusers and allocates the resources to the user(s) experiencing favor-able channel conditions. However, for a frequency-division-duplexsystem with a large number of users, how to efficiently collectthe required CSI will be a challenging task, especially when thefeedback links are of limited capacity. In this paper, we propose ascheduling algorithm to exploit multiuser diversity with possiblyimperfect one-bit channel state feedback. The basic idea is todefine a threshold λ and let each user report one-bit information tothe scheduler about the comparison between its measured channelfading level and λ. Correspondingly, the scheduler uses these feed-back bits to classify all users into two sets and assigns the channelto one user belonging to the set experiencing favorable channelconditions. Several implementation schemes are developed by at-tacking the optimization of λ under different system configura-tions, covering both the case when the one-bit feedback is perfectand those when the one-bit feedback is imperfect. Computer sim-ulations show that when the user number is large, say, more thanten users, the proposed scheduling supports significantly largerdata rate over the round-robin scheduling, while in comparisonwith the optimum scheduling with complete CSI, the performanceloss is limited if the one-bit feedback is of high reliability. Inaddition, our studies show that we can effectively enhance therobustness against feedback imperfectness by incorporating thefeedback reliability into optimization of λ.

Index Terms—Fading channel, feedback, information rate,land mobile radio cellular systems, radio resource management.

I. INTRODUCTION

MULTIUSER diversity refers to one type of diversitypresent across different users in a fading environment.

This diversity can be exploited by transmission schedulingso that the data are transmitted on the favorable channels.In this way, the system performance is dictated by the peakchannel state rather than the average [1]. Multiuser diver-sity underlies much of the recent work on “opportunistic”downlink scheduling, as in Qualcomm’s high-data-rate (HDR)

Manuscript received October 31, 2003; revised April 13, 2004, January 15,2005, February 5, 2006, and February 20, 2006. This work was supported bythe Wolfgang Paul Award Program of the Alexander von Humboldt Foundation.The review of this paper was coordinated by Dr. E. Larsson.

Y. Xue was with the Department of Communication Systems (NTS),University Duisburg–Essen, 45141 Duisburg, Germany. He is now withCorporate Technology, Siemens, Ltd., China, Beijing 100102, China (e-mail:[email protected]).

T. Kaiser was with the Department of Communication Systems (NTS),University Duisburg–Essen, 45141 Duisburg, Germany. He is now with LeibnizUniversity of Hannover, 30419 Hannover, Germany (e-mail: [email protected]).

Digital Object Identifier 10.1109/TVT.2006.883784

system [2]. It has also been generalized to multiantenna sys-tems, e.g., [3]–[5].

Usually, a central controller is employed in a multiuser di-versity system that collects the channel state information (CSI)from all users to schedule the data transmission. However, for asystem with a large number of users, how to efficiently collectthe required CSI will be a challenging task, especially whenthe system works in frequency division duplex (FDD) mode,and the feedback links are of limited capacity. Such a taskwill be more difficult when multi-input multi-output (MIMO)architectures are employed. The studies in [3] have pointed outthat scheduling the transmit antennas independently will offersignificant throughput improvement, but such improvement isobtained at the penalty of increased channel state feedback.In addition, in practical communication systems, the feedbackinformation is rarely perfect.

To deal with this problem, in this paper, we propose ascheduling algorithm that exploits multiuser diversity withpossibly imperfect one-bit channel state feedback. The basicidea is to define a threshold λ, and let each user report one-bit information to the scheduler about the comparison betweenits measured channel fading level and λ. Correspondingly, thescheduler uses these feedback bits to partition all users into twosets and assigns the channel to one user belonging to the setexperiencing favorable channel conditions. Intuitively, with acarefully selected λ, only the users experiencing (at least) betterthan average channel conditions are allowed to compete for thechannel, and hence, some degree of multiuser diversity benefitcan be achieved.

The determination of λ depends on the pursued objective.Our objective in this paper is to maximize the data rate withsome degree of fairness. We deal with the issue of fairness byadopting the weighted signal-to-noise-ratio (SNR) scheduling[6], which can be regarded as a variant of the well-knownproportionally fair (PF) scheduling [4], [7], to ensure fairresource allocation in the sense of providing all users with thesame asymptotic channel access. By attacking the optimizationof λ with such an objective function, we develop severalimplementation schemes under different system configurations.Our investigations cover not only the case with perfect one-bit feedback but also the case when the one-bit feedback iserroneous due to the noisy feedback link and the case whenthe one-bit feedback is produced on the basis of an imprecisechannel estimate. We support the developed schemes with com-puter simulations, which show that the proposed schedulingoffers significant data rate improvement over the round-robinscheduling when the user number is large, while, in com-parison with the optimum scheduling with complete CSI, the

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184 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 1, JANUARY 2007

performance loss is limited when the one-bit feedback is of highreliability. Furthermore, our studies show that we can effec-tively enhance the robustness against feedback imperfectnessby incorporating the feedback reliability as a priori knowledgeinto optimization of λ.

Pursuing multiuser diversity with low-bit-rate feedback is nota new idea. For example, it was briefly reported in [2] that theQualcomm’s HDR system will use four-bit feedback to mapthe measured absolute SNR to one of the candidate signalingmodes. Recently, [8] reported that the combination of four-bit full channel feedback and one-bit differential channel feed-back was adopted by the Revision C of the IS-2000 standard(1xEV-DV) to quantize the estimate of the strength of the basestation’s (BS’s) pilot. Three more relevant contributions withdetailed technical investigations are in [9]–[11]. On the basis ofprobability studies and using a Rayleigh MIMO system as anexample, [9] argued that severe rate quantization can be used onthe feedback links to exploit multiuser diversity. How to exploitmultiuser diversity with adaptive signaling driven by low-bit-rate feedback in a single-input single-output orthogonal-frequency-division multiplexing (OFDM) system was exploredin [10]. Assuming a Rayleigh flat-fading channel, [11] inves-tigated the effect of feedback quantization (including one-bitquantization) on the throughput of a multiuser diversity schemewhen the maximum-SNR scheduling is employed. In compari-son with [9]–[11], we make the following distinct contributionsin this paper. First, we consider a more practical channel model,where not only the small-scale fading but also the shadowingand the path loss [13] are taken into consideration, and weaddress the issue on fairness. Second, and more importantly,we investigate the problems caused by the noisy feedback linkor the employment of an imprecise channel estimator, anddevelop scheduling schemes with enhanced robustness againstsuch feedback imperfectness. Moreover, we have noted theindependent work of [12], which demonstrated that multiuserdiversity can be exploited with reduced feedback rate by onlycollecting CSI from the users experiencing favorable channelconditions.

The rest of this paper is organized as follows. The systemmodel is introduced in Section II. Next, in Section III, wepropose the basic idea of our scheduling algorithm. Section IVis devoted to the optimization of λ under different systemconfigurations, followed by simulation and numerical results inSection V. Finally, we conclude this paper in Section VI.

II. SYSTEM MODEL

Assume a single cell system with K mobile users (MUs)that are distributed at random in a circular cell whose radiusis normalized to 1. These MUs communicate with the BSlocated at the center of the cell. Each MU always1 demandssome high-rate data services, such as e-mail retrieval and/orweb browsing, from the BS. The nature of such data traffic isdecidedly asymmetric and then how to increase the downlinkdata rate is the main challenge for the considered system.

1By always, we mean that at any time block each MU has service requestswith probability 1. This happens in an overloaded communication scenario.

Suppose that the channel is frequency flat and block sta-tic. We denote the downlink channel gain between the BSand the kth MU, k ∈ {1, . . . ,K}, at the lth time block, l ∈{−∞, . . . , 0, 1, 2, . . .}, as hk(l). Suppose further that the BStransmits with a constant power PB and that the additivewhite Gaussian noise (AWGN) at each MU is zero-mean withvariance σ2. Then, when the BS transmits to the kth MU at thelth block, we have the downlink instantaneous SNR (ISNR)

ρk(l) =|hk(l)|2 PB

σ2(1)

where | · | represents the amplitude.We characterize |hk(l)|2 further by taking into account three

effects [13]: the small-scale fading αk(l), the shadowing βk(l),and the attenuation due to the distance rk(l)

|hk(l)|2 = αk(l)βk(l) [rk(l)]−η (2)

where η, which is the path loss exponent, assumes valuesbetween 2 and 4 and is typically taken equal to 4 in land mobileradio environments. For the small-scale fading, we assume thatfor any fixed l, αk(l) are independent identically distributed(i.i.d.) random variables (RVs) from exponential distribution,i.e., we have the following probability density function (PDF):

fα(x) ={e−x, for x ≥ 00, otherwise.

(3)

As for the shadowing βk(l), we assume a log-normal distribu-tion [13], which can be described by the relation

βk(l) = 100.1ξk(l) (4)

where for fixed l, ξk(l) are i.i.d. zero-mean Gaussian withvariance σ2

SH. Finally, we assume a uniform traffic density, i.e.,for any fixed l and k, rk(l) has a PDF [14]

fr(x) ={

2x, for 0 ≤ x ≤ 10, otherwise.

(5)

In addition, we assume that for any given k and l, αk(l), βk(l),and rk(l) are statistically independent. By substituting (2) into(1), we rewrite the ISNR

ρk(l) = αk(l)βk(l)[rk(l)]−η PB

σ2. (6)

In our system, the BS broadcasts a sequence at the beginningof each time block. From the pilot embedded in the broadcast-ing sequence, each MU can estimate the channel state at thelth time block. We also assume that the BS perfectly knowsthe number K of MUs and embeds this information into thebroadcasting sequence.

XUE AND KAISER: EXPLOITING MULTIUSER DIVERSITY WITH IMPERFECT ONE-BIT CHANNEL STATE FEEDBACK 185

III. PROPOSED SCHEDULING

When the BS assigns the channel to the kth MU at thelth time block, from (6), we can map the channel to the datarate as in

Rk(l) = log2 (1 + ρk(l)) . (7)

Here, we emphasize that Rk(l) is an upper bound of the datarate, since we have calculated it as the instantaneous AWGNchannel capacity, but in our system the BS is not informed ofexact ISNR ρk(l). The use of such an information theoreticmeasure frees us from the details of adaptive coding/modulationimplementation and allows us to focus on the problem ofscheduling design.

A line of work following [1] has made it clear that topursue the maximum throughput the system resources shouldbe allocated to the user with the peak ISNR. This will causeunfairness among the MUs with different distances from theBS. To alleviate this problem, the PF scheduling was devised in[4] and [7], which helps to eliminate such fairness violation byletting each MU compete for the system resource not directlybased on its observed channel condition but on how strong theobserved condition is relative to the average. In this paper, weadopt the weighted-SNR algorithm [6], which can be explainedas a variant of the prototype PF algorithm, to devise a fair sched-uler. Mathematically, denote ρ̄k as the time-averaged value ofρk(l), and define a test variable

µk(l) =ρk(l)ρ̄k

. (8)

We propose to assign the channel at the lth time block to thek̂th MU if and only if

k̂ = arg maxk∈{1,...,K}

{µk(l)}. (9)

It is relevant to point out that by adopting such a channelassignment policy, we have neglected the issues on absolutedelay requirements and queue stability (some discussions onthese issues of the PF scheduling can be found in [15]). As aconsequence, the data rates with additional delay and stabilityconstraints will likely be less than the results reported in thispaper.

When the implementation of (8) and (9) is of concern, thefirst issue is how to obtain ρ̄k. This can be done by filtering{ρk(s), s = −∞, . . . , l} with a low-pass filter, e.g., the expo-nentially weighted low-pass filter used in [4]. Here, instead ofdiving into the details of the low-pass filter design, we assumethat we are given an ideal estimator Ψ[·], so that

ρ̄k =Ψ [ρk(s), s = −∞, . . . , l]

≈ 1S

l∑s=l−S+1

ρk(s) (10)

where S is the concerned time scale for scheduling.The second issue is how to implement the comparisons

required by (9) under the constraint that each MU can onlyreport one-bit information to the BS. We attack this problem

by defining a threshold λ and letting each MU report to the BSthe following one-bit information

Ik(l) ={

1, if µk(l) > λ0, otherwise.

(11)

Correspondingly, upon the received feedback bits, the BS clas-sifies all MUs into two sets: the qualified set, which consists ofthe MUs with feedback bit “1,” and the unqualified set, whichconsists of the MUs with feedback bit “0.” Suppose that thefeedback bits correctly reflect the comparison between µk(l)sand λ (this will be relaxed in the next section) and that the set ofqualified MUs is nonempty. Then, by assigning the channel toone qualified MU, we can obtain the multiuser diversity benefit,because the index of any qualified MU can be regarded as anapproximate solution of (9).

We summarize the proposed scheduling as follows.

1) Suppose that each MU perfectly knows the number K ofMUs and the employed threshold λ. Moreover, a round-robin MU is specified for the lth time block.

2) Each MU measures the downlink channel and reportsIk(l) to the BS.

3) According to the received feedback bits, the BS findsthe qualified set. If the qualified set has just one MU,this qualified MU wins the channel; if the qualified setconsists of two or more MUs, the BS assigns the channelto one qualified MU at random; otherwise, the BS servesthe round-robin MU.

The value of λ is important for the proposed scheduling.Intuitively, with a too small λ, the qualified set will be conta-minated by MUs experiencing unfavorable channel conditions,while with a too large λ, we have to risk an empty qualified set.Furthermore, a straightforward implementation λ = 1 can bedevised with the following motivation. Suppose that the feed-back bit Ik(l) is perfect. Then, by setting λ = 1, we only allowthe MUs experiencing better than average channel conditionsto use the channel, and hence, some data rate improvementover the round-robin scheduling can be expected. We callthis implementation Scheme I and use it as a benchmark toinvestigate how much further performance improvement can beobtained by performing optimization on λ.

IV. PARAMETER OPTIMIZATION

A. Preliminaries

Let us begin with a reexamination of (7). We assume thatall MUs have a relative large SNR in the downlink, which isensured by setting a large enough PB so that

10 log10 (E [ρk(l)]) � 0 dB, ∀k, ∀l. (12)

Here, E[·] stands for statistical expectation. When the ISNRsatisfies ρk(l) � 1, we have

Rk(l) ≈ log2 (ρk(l)) (13)

= log2 (µk(l)) + log2(ρ̄k). (14)


We introduce the approximation in (13) because of thefollowing.

1) First, we have assumed that all MUs have a relativelylarge SNR. Therefore, the condition in (12) will be sat-isfied with a large probability.

2) Second, the key of the proposed scheduling is to definean appropriate λ so that the BS can find an appropriatequalified set of MUs. In this sense, the key of the schedul-ing is to control the feedback action of the MUs with arelatively large µk(l) and, as a consequence, a relativelylarge ISNR.

Further, when the weighted-SNR scheduling [see (8) and (9)]is employed, (14) tells us that the benefit of multiuser diversityis approximately determined by log2(µk(l)). For convenienceof presentation, in the rest of this paper, we call this term thedynamic data rate contribution. Hence, we propose to use

J(λ, . . .) = E[log2

(µk̂(l)

)](15)

as an objective function to perform an approximate optimiza-tion on λ, where k̂ is the index of the MU that wins the channelat the lth time block. By saying approximate, we have explicitlynoted that the so obtained λ may not be the optimum one thatmaximizes the data rate in (7). However, the example studiespresented in the next section show that this indirect parameteroptimization approach has satisfactory efficiency.

Let us demonstrate our approach in a case with ideal assump-tions. We assume that the feedback bit Ik(l) is produced on thebasis of a precise estimate of µk(l) and that the feedback linksare noiseless. In addition, throughout this section, we assumethat for any given time block l, µk(l) are i.i.d. with PDF fµ(x).With these assumptions, we can derive the aforementionedobjective function as

JII(λ,K) = PQ(0, λ,K)RU(λ) + [1 − PQ(0, λ,K)]RQ(λ).(16)

Here

PQ(k, λ,K) =(K

k

)[pQ(λ)]k [1 − pQ(λ)]K−k

is the probability that, among K MUs, there are k qualified onesunder the assumed λ, where

pQ(λ) =

∞∫λ

fµ(x)dx

RU(λ) =E[log2 (µk(l))|µk(l)<λ

]

=

λ∫0

log2(x)fµ(x)∫ λ

0 fµ(y)dydx

is the expected dynamic data rate contribution from an un-qualified MU, whereas

RQ(λ) =E[log2 (µk(l))|µk(l)>λ

]

=

∞∫λ

log2(x)fµ(x)∫ ∞

λ fµ(y)dydx

is the expected dynamic data rate contribution from a qualifiedMU. The idea behind (16) can be explained as follows: Whenall MUs are unqualified, the BS assigns the channel to theround-robin MU (in this case the expected dynamic data ratecontribution is RU(λ)); otherwise, the channel will be assignedto one qualified MU, of which the expected dynamic data ratecontribution is RQ(λ).

By solving the constrained optimization problem

λII(K) = arg maxλ

{JII(λ,K)}subject to λ > 0 (17)

we can obtain the optimal λ for Scheme II.However, the ideal assumptions made for Scheme II can

rarely be satisfied in practical communication systems. In par-ticular, the feedback link is inherently noisy, and all MU cannotobtain a precise estimate of µk(l). Therefore, it is highly de-sirable to know how much the proposed scheduling will sufferfrom such feedback imperfectness and, more importantly, howto develop implementation schemes with enhanced robustnessagainst such imperfectness. In the next section, we treat theproblem due to a noisy feedback link, whereas in Section IV-C,we discuss how to optimize λ when Ik(l) is produced on thebasis of an imprecise estimate of µk(l).

B. Optimization Against Noisy Feedback Link

In this section, we assume that Ik(l) is produced on thebasis of a precise estimate of µk(l), but the BS only receivesan erroneous copy due to the noise of the feedback link. Wemodel the noisy feedback link by assuming that for any receivedfeedback bit, it is correct with the probability (1 − pE) andwrong with the probability pE.

Taking such feedback imperfectness into account, we modify(16) to a new objective function as

JIII(λ,K, pE)

= PQ(0, λ,K)RU(λ) +K∑

k=1

PQ(k, λ,K)

×{P1(0, k, pE)P2(K − k,K − k, pE)R1(λ, k,K)

+ P1(0, k, pE) [1 − P2(K − k,K − k, pE)]RU(λ)

+k∑

s=1

P1(s, k, pE)K−k∑q=0

P2(q,K − k, pE)

×R2(λ, s, q, k,K)}. (18)

Here, P1(s, k, pE) is the probability that s out of k qualifiedMUs send their feedback bits correctly, P2(q,K − k, pE) is theprobability that q out of (K − k) unqualified MUs send theirfeedback bits correctly, and

P1(s, t, pE) =P2(s, t, pE)

={ (

ts

)(1 − pE)spE

t−s, if s ≤ t0, otherwise.


The ideas behind (18), together with other two new notationsR1(λ, k,K) and R2(λ, s, q, k,K), are explained as follows.The first term in the right-hand side (RHS) of (18) is theexpected dynamic data rate contribution when all MUs areunqualified, while the second term considers the cases whenthere exist(s) 1 ≤ k ≤ K qualified MU(s). In particular, thesecond line of (18) addresses the case when the BS findsan empty qualified set and assigns the channel to the round-robin MU. In this case, since, in fact, there exist(s) k qualifiedMUs, the expected dynamic data rate contribution of the round-robin MU is

R1(λ, k,K) =k

KRQ(λ) +

K − k

KRU(λ).

The third line of (18) is about the case when the BS findsa nonempty qualified set but that all set members are, infact, unqualified. Finally, the fourth line manipulates the moregeneral case when the BS has a contaminated qualified setconsisting of s qualified MUs and (K − k − q) unqualifiedMUs. Correspondingly, we have the expected dynamic data ratecontribution

R2(λ, s, q, k,K)

=s

s + (K − k − q)RQ(λ) +

K − k − q

s + (K − k − q)RU(λ).

Furthermore, it is easy to verify that when pE = 0, (18) willreduce to (16).

Given pE, we can obtain the optimal λ for Scheme III bysolving

λIII(K, pE) = arg maxλ

{JIII(λ,K, pE)}subject to λ > 0. (19)

C. Optimization Against Imperfect Estimator

In this section, we discuss the optimization of λ when thefeedback link is noiseless but Ik(l) is produced on the basis ofan imprecise estimate of µk(l). In particular, we assume thatIk(l) is produced by

Ik(l) =

1, if µk(l) > λ(1 + γ)0, if 0 ≤ µk(l) < λ(1 − γ)ξ, otherwise

where 0 ≤ γ < 1, and ξ is a Bernoulli RV that takes the valueof 1 or 0 with equal probability. The rationality of such a modelcan be explained as follows. When the estimator for µk(l)is of acceptable accuracy, it is reasonable to assume that theobtained estimate µ̂k(l) is highly correlated with µk(l). In otherwords, µ̂k(l) will locate in the vicinity of µk(l). Therefore, onthe basis of µ̂k(l), an erroneous feedback bit will more likelybe produced when µk(l) falls into [λ(1 − γ), λ(1 + γ)], whichwe call the ambiguity range. Obviously, γ is a parameter thatreflects the reliability of the feedback bits.

Taking such feedback imperfectness into consideration, wemodify (16) to a new objective function that depends on λ, K,and γ2:

JIV(λ,K, γ)

= P3(0, 0,K, λ, γ)RU (λ(1 − γ))

+K∑

t=1

P3(0, t,K, λ, γ)RQ (λ(1 + γ))

+K∑

s=1

P3(s, 0,K, λ, γ) [1 − P4(0, s)]RA(λ, γ)

+K∑

s=1

P3(s, 0,K, λ, γ)P4(0, s)R3(λ, s,K, γ)

+K−1∑s=1

K−s∑t=1

P3(s, t,K, λ, γ)P4(0, s)RQ (λ(1 + γ))

+K−1∑s=1

K−s∑t=1

P3(s, t,K, λ, γ)s∑

q=1

P4(q, s)R4(λ, q, t, γ)

(20)

where

P3(s, t,K, λ, γ) =(K

t

)

∞∫λ(1+γ)

fµ(x)dx

t (K − t

s

)

×

λ(1+γ)∫λ(1−γ)

fµ(x)dx

s

λ(1−γ)∫0

fµ(x)dx

K−s−t

is the probability that, among the K MUs, there are s onesfalling into the ambiguity range, while t ones satisfy thatµk(l) > λ(1 + γ)

P4(q, s) =(s

q

)12s

is the probability that q out of the s MUs that fall into theambiguity range report bit “1” to the BS, while

RA(λ, γ) =E[log2 (µk(l))|λ(1−γ)≤µk(l)≤λ(1+γ)

]

=

λ(1+γ)∫λ(1−γ)

log2(x)fµ(x)

λ(1+γ)∫λ(1−γ)

fµ(y)dy

dx

is the expected dynamic data rate contribution from a MUfalling into the ambiguity range.

2Here, we assume that although each MU cannot obtain a precise estimate ofµk(l), the MU that wins the channel can perfectly estimate ρk̂(l) by employinga more sophisticated estimator and working on the whole transmission burst.


TABLE IOPTIMIZED PARAMETERS WHEN fµ(x) = fα(x)

We explain the ideas behind (20), together with the othertwo new notations R3(λ, s,K, γ) and R4(λ, q, t, γ), as follows.In (20), we first address two particular cases (in the first twoterms) when there is no MU falling into the ambiguity range.Then, in the third term, the case when the qualified set onlyconsists of MUs falling into the ambiguity range is considered.The fourth term is about the case when there exist(s) MU(s) inthe ambiguity range and the BS finds an empty qualified set. Inthis case, the BS will assign the channel to the round-robin MU,of which the expected dynamic data rate contribution is

R3(λ, s,K, γ) =s

KRA(λ, γ) +

K − s

KRU (λ(1 − γ)) .

The fifth term on the RHS of (20) manipulates the case whenall members in the nonempty qualified set satisfy µk(l) >λ(1 + γ). Finally, we deal with the case when the qualifiedset consists of t MUs satisfying µk(l) > λ(1 + γ) and q MUsin the ambiguity range. When the channel is assigned to oneMU belonging to this contaminated qualified set, we have theexpected dynamic data rate contribution

R4(λ, q, t, γ) =t

t + qRQ (λ(1 + γ)) +

q

t + qRA(λ, γ).

In addition, it can be shown that when γ = 0, (20) is equivalentto (16).

Then, given γ, we can obtain the optimal λ for theScheme IV by solving

λIV(K, γ) = arg maxλ

{JIV(λ,K, γ)}subject to λ > 0. (21)

V. EXAMPLE STUDIES

In previous studies, we have proposed a scheduling algorithmto exploit multiuser diversity with one-bit channel state feed-back and developed four implementation schemes. In particular,three of them are developed by performing optimization on λunder the assumption that we are given fµ(x), which is thePDF of the test variable µk(l). For any given practical sys-tem, such knowledge can be obtained by performing intensivefield measurements. In this section, we study two particularscenarios for which an approximate fµ(x) can be reasonablyobtained. By performing computer simulations under these twoexample scenarios, we demonstrate the benefit of the proposedscheduling and verify the efficiency of the developed parameteroptimization approach.

A. Benefit From Small-Scale Fading

First, we consider an example scenario where for every k,βk(l)[rk(l)]−η does not change in the concerned time scale

[defined in (10)], while αk(l) changes randomly following thePDF in (3). This happens when artificial small-scale fadingis introduced, e.g., the opportunistic beamforming [4], in acellular system that serves some stationary users. Under theseassumptions, we can derive from (10)

ρ̄k ≈E [αk(l)]βk(l) [rk(l)]−η PB

σ2

=βk(l) [rk(l)]−η PB

σ2

where we have assumed that αk(l) is ergodic. In other words,we have µk(l) ≈ αk(l). Then, by substituting fµ(x) withfα(x), we numerically solve the problems in (17), (19), and(21). Part of the obtained results are presented in Table I.Throughout this paper, all optimization problems (i.e., theoptimization of λII, λIII, and λIV) are numerically solved withthe MATLAB function fminbnd, and hence, the optimum ofthese solutions cannot be guaranteed.

We examine the obtained schemes in Table I with computersimulations. First, we consider the case when Ik(l) is producedon the basis of a precise estimate of µk(l) but transmitted to theBS through a possibly noisy feedback link. The obtained resultsare shown in Fig. 1. Throughout this paper, the simulationresults are obtained through L = 1000 000 Monte Carlo trials.Denoting the simulated data rate as RAV, the simulation can bemathematically stated as

RAV =1L

L∑l=1

log2

(1 + αk̂(l)βk̂(l)

[rk̂(l)

]−η PB

σ2

)

where k̂ is the index of the MU that wins the channel at the lthtrial. In each trial, αk(l) are generated randomly following thePDF in (3), βk(l) following (4), and rk(l) following (5). For theother parameters, we set

σSH = 8η = 4 (22)

which are typical in macrocellular systems [16], and10 log10(PB/σ

2) = 5 dB (some further discussions on thissetting can be found in the Appendix).

Fig. 1(a) shows the simulated RAV, both in the case pE = 0and the case pE = 0.02. To offer a comparison, in the figure, wealso plot the curves of the “optimum” scheduling, which solves(9) with perfect knowledge of {µk(l), k = 1, . . . ,K} and thoseof the round-robin scheduling. From the figure, we can observethe following.

O1) All developed scheduling schemes (including thebenchmark Scheme I) support significantly larger datarate over the round-robin scheduling, especially whenK is relatively large.


Fig. 1. Studies of the developed scheduling schemes under a scenario where fµ(x) ≈ fα(x). Here, we assume that each MU produces Ik(l) on the basis ofa precise estimate of µk(l) but the feedback bit is transmitted over a possibly noisy feedback link. (a) Data rate against MU number K. (b) Dynamic data ratecontribution against MU number K.

O2) When the feedback link is noiseless, Scheme II out-performs Scheme I, and the performance improvementis substantial when the value of |λII(K) − 1| is rela-tively large. This confirms the parameter optimizationapproach we developed in (17). Furthermore, in com-parison with the optimum scheduling, Scheme II onlysuffers a limited performance loss.

O3) The noisy feedback link will cause performancedegradation to the proposed scheduling. In particular,Scheme II seems to be more sensitive to such erroneousfeedback than Scheme I does. However, the curveof Scheme III tells us that such impairments can bealleviated by incorporating a priori knowledge aboutpE into optimization of λ, as we have done in (19).

To further examine the developed parameter optimizationapproach, we investigate the dynamic data rate contributiondirectly and report our studies in Fig. 1(b). In the figure, ∆Ris defined as the difference between the dynamic data ratecontribution of the considered scheduling scheme and that ofthe round-robin scheduling. For the developed Schemes I–IV,the analytical expression of ∆R can be obtained from thecorresponding objective function defined in (16), (18), and (20),respectively. For instance, when Scheme I under pE = 0 is ofconcern, we have

∆RI(K) = JII(1.000,K) −∞∫

0

log2(x)fµ(x)dx.

For the optimum scheduling, by resorting to order statistics[17], we get

∆ROPT(K)

=

∞∫0

log2(x)K

x∫0

fµ(y)dy

K−1

fµ(x)dx−∞∫

0

log2(x)fµ(x)dx.

As for the Monte Carlo simulation of ∆R, it can be mathemat-ically stated as

∆RAV =1L

L∑l=1

log2

(µk̂(l)µkRR(l)

)

where kRR stands for the index of the round-robin MU. InFig. 1(b), we show both the simulated ∆R and the numericalevaluation of the corresponding analytical expression. The fineagreement between the simulation and the analysis tells us that(16) and (18) are convincing.

In Fig. 1, we can observe that in the simulated scenarios,Scheme II seems to be more sensitive to the noisy feedbacklink than Scheme I is, especially when K is large. This can beintuitively explained by some straightforward calculations un-der the case K = 128 and pE = 0.02. In particular, in averagethere will approximately be 128 × pQ(1.000) ≈ 47, and

128 × pQ(3.598) ≈ 3.504 ≈ 4 (23)

qualified MUs for Scheme I and Scheme II, respectively. How-ever, the noisy feedback link with pE = 0.02 will make thequalified set of Scheme I and that of Scheme II contaminated by(128 − 47) × pE = 1.62 ≈ 2 and (128 − 4) × pE = 2.48 ≈ 2unqualified MUs, respectively. Then, it is obvious that thecontaminating unqualified MUs will play a more destructiverole on the data rate of Scheme II3 than it does on Scheme I.In other words, reducing threshold is a good idea to improvethe robustness against feedback imperfectness due to a noisyfeedback link.

Then, we study the case when the feedback link is noiseless,but Ik(l) is produced on the basis of an imprecise estimate of

3Please note that RQ(3.598) = 2.173 and RU(3.598) = −0.917. There-fore, the event of assigning the channel to unqualified MU (with the probability≈ 1/3) will catastrophically reduce the data rate.


Fig. 2. Studies of the developed scheduling schemes under a scenario where fµ(x) ≈ fα(x). Here, we assume that the feedback link is noiseless but that thefeedback bit is produced on the basis of a possibly imprecise estimate of µk(l). (a) Data rate against MU number K. (b) Dynamic data rate contribution againstMU number K.

µk(l). The obtained results are shown in Fig. 2. In particu-lar, Fig. 2(a) shows the simulated RAV of various schedulingschemes, while in Fig. 2(b), we study ∆R to validate theanalysis developed in (20). Besides O1) and O2), we can makeanother observation from Fig. 2(a).

O4) The imprecise estimate of µk(l) will cause perfor-mance degradation. However, by incorporating thefeedback reliability into optimization of λ, as we havedone in (21), we can alleviate such impairments.

Furthermore, the fine agreement between the simulated ∆R andthe corresponding analysis shown in Fig. 2(b) tells us that theanalytical expression developed in (20) is convincing.

By comparing Fig. 2 to Fig. 1, we can observe that the noisyfeedback link seems to be a more annoying impairment, incomparison with the imprecise estimate of µk(l). This is intu-itive because the latter only contaminates the qualified set withsome MUs whose µk(l)s fall into the ambiguity range, whilethe former may cause the channel to be assigned to an MUexperiencing very unfavorable channel condition. This under-standing, with the aid of some straightforward calculations, canalso intuitively explain why Scheme IV increases the thresholds(in comparison with Scheme II) to improve the robustnessagainst the imprecise channel estimate. As aforementioned, theobjective of optimizing λ is to obtain an appropriate qualifiedset of MUs. Let us take the case where K = 128 as an exampleand assume that λII = 3.598 can give the appropriate numberof MUs [which is calculated in (23)]. However, due to the im-perfect estimate of µk(l) (represented by γ = 0.3), the averagenumber of MUs in the qualified set will approximately be

128∑s=0

128−s∑t=0

(s

2+ t

)P3(s, t, 128, 3.598, 0.3) ≈ 5.752 ≈ 6.

In other words, on average, there will be more than enoughMUs in the qualified set, and more annoyingly, some of theseMUs are not really qualified. Intuitively, to combat against theimprecise estimate, the thresholds should be increased slightly,as with the solutions of (21) shown in Table I and those in thefollowing Table II.

B. Benefit From Joint Small-Scale Fading and Shadowing

In the second scenario, we assume that for every k, [rk(l)]−η

does not change in the concerned time scale, while αk(l) andβk(l) changes randomly following (3) and (4). These assump-tions can be approximately satisfied when each MU movesnearby a fixed point so that it experiences different small-scale fading and shadowing, but the path loss remains constant.Under these assumptions, we derive from (10)

ρ̄k ≈E [αk(l)]E [βk(l)] [rk(l)]−η PB

σ2

=C0 [rk(l)]−η PB

σ2

where C0def= E[βk(l)], and we have assumed that αk(l) and

βk(l) are ergodic. Hence, in the considered scenario, fµ(x) canbe obtained by studying the distribution of αk(l)βk(l)/C0.

After some manipulations with (3) and (4), we obtain thePDF of µ in (24), shown on the next page. Then, we resolvethe optimization problems in (17), (19), and (21) and tabulatepart of the obtained results in Table II.

We examine the obtained schemes with computer simula-tions and show the results in Fig. 3. In particular, we first showin Fig. 3(a) the simulated RAV for various scheduling schemes.The presented results tell us that the observations O1)–O4) stillhold in this scenario. Then, we report our studies on ∆R inFig. 3(b). From this figure, we can find that there is a trivialdifference between the simulation results and the corresponding


TABLE IIOPTIMIZED PARAMETERS WHEN fµ(x) FOLLOWS (24)

Fig. 3. Studies of the developed scheduling schemes under a scenario where fµ(x) follows (24). (a) Data rate against MU number K. (b) Dynamic data ratecontribution against MU number K.

analyses. In other words, we have again established the sound-ness of (16), (18), and (20).

C. Possible Application in MIMO–OFDM

In developing this one-bit feedback-based scheduling,the primary application we keep in mind is a multiuserMIMO–OFDM system. As aforementioned, when multiuserdiversity is pursued in a multiuser MIMO system with linearreceivers, it was shown in [3] that scheduling the transmitantennas independently can offer significant throughput im-provement, but such an improvement is obtained at the penaltyof increased channel state feedback. On the other hand, thestudies reported in [18]–[21] show that subcarrier assignmentis a very rewarding technique for improving data rate in themultiuser OFDM systems. These motivate us to explore thejoint transmit antenna and subcarrier assignment in the multi-user MIMO–OFDM environments. However, how to efficientlycollect the required CSI will be very challenging for an FDDsystem.

The scheduling algorithm proposed in this paper providesone possible solution to this problem. For instance, supposethat both the BS and all MUs are equipped with M antennas,and each MU employs linear receivers, e.g., the zero-forcing-based receiver or the linear-minimum-mean-square-error-basedreceiver, on each subcarrier to receive the M data streamsindependently transmitted by the BS (please refer to [3]). Then,the downlink data transmission can be decomposed into MN

subsystems (where N is the number of the OFDM subcarriers)that are in principle similar to the one presented in the section.Hence, the proposed scheduling algorithm can be employed,but the details go beyond the scope of this paper. It is relevantto point out that such a straightforward application ignoresthe coherence in frequency. Therefore, it will be interesting tostudy how to incorporate such coherence into consideration todevise an efficient low-rate channel state feedback scheme forthe multiuser MIMO–OFDM systems.

VI. CONCLUSION AND FUTURE WORK

In this paper, we proposed a scheduling algorithm to exploitmultiuser diversity with possibly imperfect one-bit channelfeedback. The basic idea is to define a threshold λ and leteach user report one-bit information to the scheduler about thecomparison between its measured channel fading level and thethreshold. Correspondingly, the scheduler uses these feedbackbits to partition all users into two sets and assigns the channelto one user belonging to the set experiencing favorable chan-nel conditions. Under different assumptions about the systemconfiguration, we developed four implementation schemes:

fµ(x)=

10ln(10)

∫ ∞0

C0√2πσSHt2

× exp(−C0x

t − (10 log10(t))2

2σ2SH

)dt, x ≥ 0

0, otherwise.(24)


Fig. 4. Data rate of Scheme II against λ when K = 16. The studies are per-formed under the scenario where fµ(x) ≈ fα(x). Moreover, in the simulation,we assume pE = 0 and γ = 0.

We supported the proposed scheduling with computer sim-ulations. The results show that all developed schemes supportsignificantly larger data rate over the round-robin schedulingwhen the user number is large. Further, in comparison withthe optimum scheduling with complete CSI, our schedulingsuffers limited loss if the one-bit feedback is of high reliabil-ity. In addition, impairments caused by noisy feedback linkor imprecise channel estimate can be effectively alleviatedby incorporating the feedback reliability into the optimiza-tion of λ.

This paper raises several questions that deserve further in-vestigations, namely, how to design an appropriate adaptivesignaling method to exploit the multiuser diversity benefitreported here with an information theoretic measure. Howmuch does the proposed scheduling sacrifice additional servicedelay in exchange for using one-bit feedback? The loss in datarate, in comparison with the optimum scheduling, reported inFigs. 1–3 tells us that some additional delay is unavoidable, yetan exact but compact analysis is missing. We are working onthese problems to obtain a more in-depth understanding of theproposed scheduling algorithm.

APPENDIX

In this paper, we assumed that all MUs have a relatively highSNR [see (12)] so that the approximation in (13) is reasonablewith a large probability. By examining the SNR of the MUslocated at the boundary of the cell, we have

E[ρk(l)|rk(l)=1

]= C0

PB

σ2

where 10 log10 C0 ≈ 11.5 dB under (3), (4), and (22). Hence,by setting 10 log10(PB/σ

2) = 5 dB, we can examine the de-veloped scheduling schemes when the SNR assumption issatisfied.

We also examine the developed scheduling under differentsettings of PB/σ

2. In Fig. 4, we show the simulation results of

Scheme II when 10 log10(PB/σ2) = 5 dB, 10 log10(PB/σ

2) =−5 dB, and 10 log10(PB/σ

2) = −15 dB, respectively. Thefigure tells us that the proposed scheme can be used in theconsidered single cell system for a wide range of PB/σ

2.

ACKNOWLEDGMENT

The authors would like to thank the Associate Editor andthe four anonymous reviewers, whose constructive commentsallowed them to improve the technique quality and the legibilityof this paper. In particular, the authors are obliged to the firstreviewer for pointing out [9], [12], and [15].

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Yisheng Xue (M’03) received the Ph.D. degree inelectronic engineering from Tsinghua University,Beijing, China, in 2002.

His research interests include signal processingand protocol design for wireless communications.

Thomas Kaiser (SM’04) received the Ph.D. (withdistinction) and German habilitation degrees inelectrical engineering, both from Gerhard–MercatorUniversity, Duisburg, Germany, in 1995 and 2000,respectively.

From April 2000 to March 2001, he was theHead of the Department of Communication Systems,Gerhard–Mercator University, and from April 2001to March 2002, he was the Head of the Departmentof Wireless Chips and Systems, Fraunhofer Instituteof Microelectronic Circuits and Systems, Duisburg.

In summer 2005, he joined the Smart Antenna Research Group, StanfordUniversity, Stanford, CA, as a Visiting Professor. Currently, he is Coleaderwith Smart Antenna Research Team at the University of Duisburg–Essen. Hehas published more than 80 papers and coedited four forthcoming books onultrawideband and smart antenna systems. He is the founding Editor-in-Chiefof the IEEE Signal Processing Society e-letter. His current research interestfocuses on applied signal processing with emphasis on multiantenna systems,especially its applicability to ultrawideband systems.

Exploiting Multiuser Diversity With Imperfect One-Bit Channel State Feedback

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