Performance Analysis of Heterogeneous Feedback Design in an OFDMA Downlink with Partial and Imperfect Feedback
aa r X i v : . [ c s . I T ] J a n TO APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 1
Performance Analysis of HeterogeneousFeedback Design in an OFDMA Downlinkwith Partial and Imperfect Feedback
Yichao Huang,
Member, IEEE , and Bhaskar D. Rao,
Fellow, IEEE
Abstract
Current OFDMA systems group resource blocks into subband to form the basic feedback unit.Homogeneous feedback design with a common subband size is not aware of the heterogeneous channelstatistics among users. Under a general correlated channel model, we demonstrate the gain of matching thesubband size to the underlying channel statistics motivating heterogeneous feedback design with differentsubband sizes and feedback resources across clusters of users. Employing the best-M partial feedbackstrategy, users with smaller subband size would convey more partial feedback to match the frequencyselectivity. In order to develop an analytical framework to investigate the impact of partial feedbackand potential imperfections, we leverage the multi-cluster subband fading model. The perfect feedbackscenario is thoroughly analyzed, and the closed form expression for the average sum rate is derived forthe heterogeneous partial feedback system. We proceed to examine the effect of imperfections due tochannel estimation error and feedback delay, which leads to additional consideration of system outage.Two transmission strategies: the fix rate and the variable rate, are considered for the outage analysis.We also investigate how to adapt to the imperfections in order to maximize the average goodput underheterogeneous partial feedback.
Index Terms
Copyright (c) 2012 IEEE. Personal use of this material is permitted. However, permission to use this material for any otherpurposes must be obtained from the IEEE by sending a request to [email protected] research was supported by Ericsson endowed chair funds, the Center for Wireless Communications, UC Discovery grantcom09R-156561 and NSF grant CCF-1115645. The material in this paper was presented in part at the 45th Asilomar Conferenceon Signals, Systems, and Computers, Pacific Grove, CA, November 2011.The authors are with Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA92093-0407, USA (e-mail: [email protected]; [email protected]).
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 2
Heterogeneous feedback, OFDMA, partial feedback, imperfect feedback, average goodput, multiuserdiversity
I. I
NTRODUCTION
Leveraging feedback to obtain the channel state information at the transmitter (CSIT) enables a wirelesssystem to adapt its transmission strategy to the varying wireless environment. The growing number ofwireless users, as well as their increasing demands for higher data rate services impose a significant burdenon the feedback link. In particular for OFDMA systems, which have emerged as the core technologyin 4G and future wireless systems, full CSIT feedback may become prohibitive because of the largenumber of resource blocks. This motivates more efficient feedback design approaches in order to achieveperformance comparable to a full CSIT system with reduced feedback. In the recent years, considerablework and effort has been focused on limited or partial feedback design, e.g., see [1] and the referencestherein. To the best of our knowledge, most of the existing partial feedback designs are homogeneous,i.e., users’ feedback consumptions do not adapt to the underlying channel statistics. In this paper, wepropose and analyze a heterogeneous feedback design, which aligns users’ feedback needs to the statisticalproperties of their wireless environments.Current homogeneous feedback design in OFDMA systems groups the resource blocks into subband[2] which forms the basic scheduling and feedback unit. Since the subband granularity is determined bythe frequency selectivity, or the coherence bandwidth of the underlying channel, it would be beneficial toadjust the subband size of different users according to their channel statistics. Empirical measurements andanalysis from the channel modeling field have shown that the root mean square (RMS) delay spread whichis closely related to the coherence bandwidth, is both location and environment dependent [3], [4]. Thetypical RMS delay spread for an indoor environment in WLAN does not exceed hundreds of nanoseconds;whereas in the outdoor environment of a cellular system, it can be up to several microseconds. Intuitively,users with lower RMS delay spread could model their channel with a larger subband size and require lessfeedback resource than the users with higher RMS delay spread. Herein, we investigate this heterogeneousfeedback design in a multiuser opportunistic scheduling framework where the system favors the userwith the best channel condition to exploit multiuser diversity [5], [6]. There are two major existingpartial feedback strategies for opportunistic scheduling, one is based on thresholding where each userprovides one bit of feedback per subband to indicate whether or not the particular channel gain exceedsa predetermined or optimized threshold [7]–[10]. The other promising strategy currently considered inpractical systems such as LTE [11] is the best-M strategy, where the receivers order and convey the M
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 3 best channels [12]–[19]. The best-M partial feedback strategy is embedded in the proposed heterogeneousfeedback framework. Apart from the requirement of partial feedback to save feedback resource, the studyof imperfections is also important to understand the effect of channel estimation error and feedback delayon the heterogeneous feedback framework. These imperfections are also considered in our work.
A. Focus and Contributions of the Paper
An important step towards heterogeneous feedback design is leveraging the “match” among coherencebandwidth, subband size and partial feedback. Under a given amount of partial feedback, if the subbandsize is much larger than the coherence bandwidth, then multiple independent channels could exist withina subband and the subband-based feedback could only be a coarse representative of the channels. On theother hand, if the subband size is much smaller than the coherence bandwidth, then channels in adjacentsubbands are likely to be highly correlated and requiring feedback on adjacent subbands could be awaste of resource; or a small amount of subband-based partial feedback may not be enough to reflectthe channel quality. In order to support this heterogeneous framework, we first consider the scenario ofa general correlated channel model with one cluster of users with the same coherence bandwidth. Thesubband size is adjustable and each user employs the best-M partial feedback strategy to convey the Mbest channel quality information (CQI) which is defined to be the subband average rate. The simulationresult shows that a suitable chosen subband size yields higher average sum rate under partial feedbackconforming the aforementioned intuition. This motivates the design of heterogeneous feedback to “match”the subband size to the coherence bandwidth. The above-mentioned study, though closely reflects therelevant mechanism, is not analytically tractable due to two main reasons. Firstly, the general correlatedchannel model complicates the statistical analysis of the CQI. Secondly, the use of subband average rateas CQI makes it difficult to analyze the multi-cluster scenario. Therefore, a simplified generic channelmodel is needed that balances the competing needs of analytical tractability and practical relevance.In order to facilitate analysis, a subband fading channel model is developed that generalizes thewidely used frequency domain block fading channel model. The subband fading model is suited forthe multi-cluster analysis. According to the subband fading model, the channel frequency selectivity isflat within each subband, and independent across subbands. Since the subband sizes are different acrossdifferent clusters, the number of independent channels are heterogeneous across clusters and this yieldsheterogeneous partial feedback design. Another benefit of the subband fading model is that the CQIbecomes the channel gain and thus facilitate further statistical analysis. Under the multi-cluster subband
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 4 fading model and the assumption of perfect feedback, we derive a closed form expression for the averagesum rate. Additionally, we approximate the sum rate ratio for heterogeneous design, i.e., the ratio of theaverage sum rate obtained by a partial feedback scheme to that achieved by a full feedback scheme, inorder to choose different best-M for users with different coherence bandwidth. We also compare anddemonstrate the potential of the proposed heterogeneous feedback design against the homogeneous caseunder the same feedback constraint in our simulation study.The average sum rate helps in understanding the system performance with perfect feedback. In practicalfeedback systems, imperfections occur such as channel estimation error and feedback delay. Theseinevitable factors degrade the system performance by causing outage [21], [22]. Therefore, rather thanusing average sum rate as the performance metric, we employ the notion of average goodput [23]–[25]to incorporate outage probability. Under the multi-cluster subband fading model, we perform analysis onthe average goodput and the average outage probability with heterogeneous partial feedback. In additionto examining the impact of imperfect feedback on multiuser diversity [26], [27], we also investigate howto adapt and optimize the average goodput in the presence of these imperfections. We consider both thefixed rate and the variable rate scenarios, and utilize bounding technique and an efficient approximationto derive near-optimal strategies.To summarize, the contributions of this paper are threefold: a conceptual heterogeneous feedbackdesign framework to adapt feedback amount to the underlying channel statistics, a thorough analysisof both perfect and imperfect feedback systems under the multi-cluster subband fading model, andthe development of approximations and near-optimal approaches to adapt and optimize the systemperformance. The rest of the paper is organized as follows. The motivation under the general correlatedchannel model and the development of system model is presented in Section II. Section III deals withperfect feedback, and Section IV examines imperfect feedback due to channel estimation error andfeedback delay. Numerical results are presented in Section V. Finally, Section VI concludes the paper.II. S YSTEM M ODEL
A. Motivation for Heterogeneous Partial Feedback
This part provides justification for the adaptation of subband size with one cluster of users under thegeneral correlated channel model, and motivates the design of heterogeneous partial feedback for themulti-cluster scenario in Section II-B. Consider a downlink multiuser OFDMA system with one base An initial treatment of a two-cluster scenario was first presented in [20].
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 5 station and K users. One cluster of user is assumed in this part and users in this cluster are assumedto experience the same frequency selectivity. The system consists of N c subcarriers. H k,n , the frequencydomain channel transfer function between transmitter and user k at subcarrier n , can be written as: H k,n = L − X l =0 σ l F k,l exp (cid:18) − j πlnN c (cid:19) , (1)where L is the number of channel taps, σ l for l = 0 , . . . , L − represents the channel power delay profileand is normalized, i.e., P L − l =0 σ l = 1 , F k,l denotes the discrete time channel impulse response, which ismodeled as complex Gaussian distributed random processes with zero mean and unit variance CN (0 , and is i.i.d. across k and l . Only fast fading effect is considered in this paper, i.e., the effects of path lossand shadowing are assumed to be ideally compensated by power control . The received signal of user k at subcarrier n can be written as: u k,n = p P c H k,n s k,n + v k,n , (2)where P c is the average received power per subcarrier, s k,n is the transmitted symbol and v k,n is theadditive white noise distributed as CN (0 , σ n c ) . From (1), it can be shown that H k,n is distributed as CN (0 , . The channels at different subcarriers are correlated, and the correlation coefficient betweensubcarriers n and n can be described as follows: cov( H k,n , H k,n ) = L − X l =0 σ l exp (cid:18) − j πl ( n − n ) N c (cid:19) . (3)In general, adjacent subcarriers are highly correlated. In order to reduce feedback needs, R c subcarriersare formed as one resource block, and η resource blocks are grouped into one subband . Thus, there are N = N c R c resource blocks and Nη subbands . In this manner, each user performs subband-based feedbackto enable opportunistic scheduling at the transmitter. Since the channels are correlated and there is oneCQI to represent a given subband, the CQI is a function of the all the individual channels within that This assumption has been employed in [10], [17], [27] to simplify the scheduling policy. With the same average SNR, theopportunistic scheduling policy is also long-term fair. When different average SNR is assumed, the proportional-fair schedulingpolicy [6] can be utilized. E.g., in LTE, one resource block consists of subcarriers, and one subband can contain to resource blocks [28]. Throughout the paper, N c , N and η are assumed to be a radix number. A more general treatment is possible but this willresult in edge effects making for more complex notation without much insight. O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 6
Number of Users A ve r a g e S u m R a t e ( bp s / H z ) M = 2 ( η = 1)M = 2 ( η = 2)M = 2 ( η = 4)M = 4 ( η = 1)M = 4 ( η = 2)M = 4 ( η = 4) Fig. 1. Comparison of average sum rate for different subband sizes ( η = 1 , , ) and partial feedback ( M = 2 , ) with respectto the number of users. A general correlated channel model is assumed with an exponential power delay profile. ( N c = 256 , N = 32 , L = 16 , δ = 4 , P c σ n c = 10 dB) subband. Herein, we employ the following subband (aggregate) average rate S k,r as the functional form [34], [35] of the CQI for user k at subband r : S k,r , ηR c rηR c X n =( r − ηR c +1 log (cid:18) P c | H k,n | σ n c (cid:19) . (4)Each user employs the best-M partial feedback strategy and conveys back the M best CQI valuesselected from S k,r , ≤ r ≤ Nη . A detailed description of the best-M strategy can be found in [15], [17],[19]. After the base station receives feedback, it performs opportunistic scheduling and selects the user k for transmission at subband r if user k has the largest CQI at subband r . Also, it is assumed that if nouser reports CQI for a certain subband, scheduling outage happens and the transmitter does not utilize itfor transmission.Now we demonstrate the need to adapt the subband size to achieve the potential “match” amongcoherence bandwidth, subband size and partial feedback through a simulation example. The channel is This functional form employs the capacity formula and the resulting effective SNR has a geometric mean interpretation.Other functional forms of the CQI exist in practical systems such as exponential effective SNR mapping (EESM) [29]–[31] andmutual information per bit (MMIB) [32], [33] to map the effective SNR to the block-error-rate (BLER) curve. The intuitionsare similar: to obtain a representative CQI as a single performance measure corresponding to the rate performance.
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 7
Fig. 2. Illustration of the multi-cluster subband fading channel model for two different clusters with resource blocks. Thesubband sizes equal and for the two different clusters respectively. According to the subband fading model, the channelfrequency selectivity is flat within each subband, and independent across subbands. The subband sizes can be heterogeneous acrossclusters, and this leads to heterogeneous channel frequency selectivity across clusters. The subband fading model approximatesthe general correlated channel model, and is useful for statistical analysis. modeled according to the exponential power delay profile [36]–[38]: σ l = − exp( − /δ )1 − exp( − L/δ ) exp (cid:0) − lδ (cid:1) for ≤ l < L , where the parameter δ is related to the RMS delay spread. The simulation parameters are: N c = 256 , N = 32 , L = 16 , δ = 4 , P c σ n c = 10 dB. The subband size η can be adjusted and ranges from to resource blocks. We consider partial feedback with M = 2 and M = 4 . The average sum rate ofthe system for different subband sizes and partial feedback with respect to the number of users is shownin Fig. 1. Under the given coherence bandwidth, several observations can be made. Firstly, the curveswith η = 4 has the smallest increasing rate because a larger subband size gives a poor representationof the channel. Secondly, the curve with η = 1 , M = 2 has the smallest average sum rate because asmall amount of partial feedback is not enough to reflect the channel quality. Thirdly, the two curves η = 1 , M = 4 and η = 2 , M = 2 possess similar increasing rate. This is because the underlying channelis highly correlated within resource blocks and thus having M -best feedback with η = 2 yields similareffect as having M -best feedback with η = 1 . From the above observations, η = 2 matches the frequencyselectivity and there would be performance loss or waste of feedback resource when a subband size isblindly chosen. In a multi-cluster scenario where users in different clusters experience diverse coherencebandwidth, this advocates heterogeneous subband size and heterogeneous feedback.The general correlated channel model as well as the non-linearity of the CQI, though useful todemonstrate the need for heterogeneous feedback, does not lend itself to tractable statistical analysis.To develop a tractable analytical framework, an approximated channel model is needed. A widely used O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 8 model is the block fading model in the frequency domain [39], [40] due to its simplicity and capabilityto provide a good approximation to actual physical channels. According to the block fading model, thechannel frequency selectivity is flat within each block, and independent across blocks [9], [17], [19].Herein, we generalize the block fading model to the subband fading model for the multi-cluster scenario.We assume that users possessing similar frequency selectivity are grouped into a cluster and the subbandsize is perfectly matched to the coherence bandwidth for a given cluster . According to the subband fadingmodel, for a given cluster with a perfectly matched subband size, the channel frequency selectivity is flatwithin each subband, and independent across subbands. Fig. 2 demonstrates the subband fading modelfor two different clusters with different subband sizes under a given number of resource blocks. B. Multi-Cluster Subband Fading Model
We now present the multi-cluster subband fading model. Consider a downlink multiuser OFDMAsystem with one base station and G clusters of users. The system consists of N resource blocks and thetotal number of users equals K . Users in cluster K g are indexed by the set K g = { , . . . , k, . . . , K g } for ≤ g ≤ G , with |K g | = K g and P Gg =1 K g = K . In our framework, users in the same cluster group theirresource blocks into subbands in the same manner while each cluster can potentially employ a differentgrouping which enables the subband size to be heterogeneous between clusters. Denote η g as the subbandsize for cluster K g , and η g ∈ { , , . . . , N } . The η g ’s are ordered such that η < · · · < η G . Based onthe assumption for η g , the number of subbands in cluster K g equals Nη g .Let H ( g ) k,r be the frequency domain channel transfer function between transmitter and user k in cluster K g at subband r , where ≤ k ≤ K g , ≤ r ≤ Nη g . H ( g ) k,r is distributed as CN (0 , . According tothe subband fading model, H ( g ) k,r is assumed to be independent across users and subbands in cluster K g . The feedback for different clusters is at different granularity, and so to model the channel for thedifferent clusters of users at the same basic resource block level, some additional notation is needed. Let ˜ H ( g ) k,n = H ( g ) k, ⌈ nηg ⌉ with ≤ n ≤ N denote the resource block based channel transfer function. Then thereceived signals of user k in cluster K g at resource block n can be represented by: u ( g ) k,n = √ P ˜ H ( g ) k,n s ( g ) k,n + v ( g ) k,n , (5) In practical systems, since the coherence bandwidth is determined by the channel statistics which vary on the order of tensof seconds or more, the cluster information can be learned and updated through infrequent user feedback. Therefore, the clusteris formed dynamically but in a slow way compared to the time variation of the fast fading effect which is on the order ofmilliseconds.
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 9 where P is the average received power per resource block, s ( g ) k,n is the transmitted symbol and v ( g ) k,n isadditive white noise distributed with CN (0 , σ n ) .Let Z ( g ) k,r , | H ( g ) k,r | denote the CQI for user k in cluster K g at subband r . In order to reduce thefeedback load, users employ the best-M strategy to feed back their CQI. In the basic best-M feedbackpolicy, users measure CQI for each resource block at their receiver and feed back the CQI values of the M best resource blocks chosen from the total N values. For each resource block, the scheduling policyselects the user with the largest CQI among the users who fed back CQI to the transmitter for that resourceblock. However, in our heterogeneous partial feedback framework, since the number of independent CQIfor cluster K g is Nη g , a fair and reasonable way to allocate the feedback resource is to linearly scale thefeedback amount for users in cluster K g . To be specific, user k in K G (i.e., the cluster with the largestsubband size) is assumed to feed back the M best CQI selected from { Z ( G ) k,r } , ≤ r ≤ Nη G , whereas user k in K g conveys the η G η g M best CQI selected from { Z ( g ) k,r } , ≤ r ≤ Nη g . After receiving feedback from allthe clusters, for each resource block the system schedules the user for transmission with the largest CQI.It is useful to note that the user feedback is based on the subband level, while the base station schedulestransmission at the resource block level.III. P ERFECT F EEDBACK
In this section, the CQI are assumed to be fed back without any errors and the average sum rate isemployed as the performance metric for system evaluation. We derive a closed form expression for theaverage sum rate in Section III-A for the multi-cluster heterogeneous feedback system. In Section III-Bwe analyze the relationship between the sum rate ratio and the choice of the best-M.
A. Derivation of Average Sum Rate
According to the assumption, the CQI Z ( g ) k,r is i.i.d. across subbands and users, and thus let F Z denotethe CDF. Because only a subset of the ordered CQI are fed back, from the transmitter’s perspective, if itreceives feedback on a certain resource block from a user, it is likely to be any one of the CQI from theordered subset. We now aim to find the CDF of the CQI seen at the transmitter side as a consequenceof partial feedback. Let ˜ Y ( g ) k,n denote the reported CQI viewed at the transmitter for user k in K g atresource block n . Also, let Y ( g ) k,r represent the subband-based CQI seen at the transmitter for user k in K g at subband r , then ˜ Y ( g ) k,n = Y ( g ) k, ⌈ nηg ⌉ . The following lemma describes the CDF of ˜ Y ( g ) k (the index n isdropped for notational simplicity), which is denoted by F ˜ Y ( g ) k . O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 10
Lemma . The CDF of ˜ Y ( g ) k is given by: F ˜ Y ( g ) k ( x ) = ηGηg M − X m =0 ξ g ( N, M, η , m )( F Z ( x )) Nηg − m , (6)where the vector η , ( η , · · · , η g , · · · , η G ) and ξ g ( N, M, η , m ) = ηGηg M − X i = m η G η g M − i η G η g M (cid:18) Nη g i (cid:19)(cid:18) im (cid:19) ( − i − m . (7) Proof:
The proof is provided in Appendix A.Let k ∗ n demote the selected user at resource block n , then according to the scheduling policy: k ∗ n = arg max k ∈U n { ˜ Y (1) k,n , · · · , ˜ Y ( g ) k,n , · · · , ˜ Y ( G ) k,n } , (8)where U n , {U (1) n , · · · , U ( g ) n , · · · , U ( G ) n } is the set of users who convey feedback for resource block n ,with |U ( g ) n | = τ g representing the number of users belonging to U n in cluster K g . It can be easily seen thatin the full feedback case, i.e., M = M F , Nη G , |U ( g ) n | = K g . For the general case when ≤ M < M F ,the probability mass function (PMF) of U n is given by: P ( U n ) = G Y g =1 (cid:18) K g τ g (cid:19) (cid:18) η G MN (cid:19) P Gg =1 τ g (cid:18) − η G MN (cid:19) K − P Gg =1 τ g , ≤ τ g ≤ K g . (9) Remark:
Only the largest subband size η G appears in the expression of P ( U n ) instead of the vector η .This is due to our heterogeneous partial feedback design to let users in cluster K g convey back the η G η g M best CQI out of Nη g values.Now we turn to determine the conditional CDF of the CQI for the selected user at resource block n ,conditioned on the set of users providing CQI for that resource block. Since users are equiprobable tobe scheduled according to the fair scheduling policy, the condition on k ∗ n is not described explicitly, andso we denote the conditional CDF as F X n |U n , where X n |U n is the conditional CQI of the selected userat resource block n . Notice from Lemma 1 that ˜ Y ( g ) k possess a different distribution for different g dueto the heterogeneous feedback from different clusters. Using order statistics [41] yields F X n |U n as: F X n |U n ( x ) = G Y g =1 ( F ˜ Y ( g ) k ( x )) τ g . (10)Then the polynomial form of F X n |U n can be obtained, which is stated in the following theorem. Theorem . The CDF of F X n |U n is given by: F X n |U n ( x ) = Φ( M, η , τ ) X m =0 Θ G − ( N, M, η , τ , m )( F Z ( x )) P Gg =1 Nηg τ g − m , (11) O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 11 where the vector τ , ( τ , · · · , τ g , · · · , τ G ) , Φ( M, η , τ ) , P Gg =1 τ g (cid:16) η G η g M − (cid:17) , Θ g ( N, M, η , τ , m ) = m P i =0 Λ ( N, M, η , τ , i )Λ ( N, M, η , τ , m − i ) , g = 1 m P i =0 Θ g − ( N, M, η , τ , i )Λ g +1 ( N, M, η , τ , m − i ) , ≤ g < G (12) Λ g ( N, M, η , τ , m ) = ( ξ g ( N, M, η , τ g , m = 0 mξ g ( N,M, η , P min (cid:16) m, ηGηg M − (cid:17) ℓ =1 (( τ g + 1) ℓ − m ) × ξ g ( N, M, η , ℓ )Λ g ( N, M, η , τ , m − ℓ ) , ≤ m < τ g ( η G η g M − ξ g ( N, M, η , η G η g M − τ g , m = τ g ( η G η g M − . (13) Proof:
The proof is provided in Appendix A.After obtaining the conditional CDF F X n |U n , let C P ( M ) denote the average sum rate and it can becomputed using the following procedure. C P ( M ) = 1 N N X n =1 E [log (1 + X n )] ( a ) = E U (cid:20)Z ∞ log (1 + ρx ) d ( F X |U ( x )) (cid:21) ( b ) = E U Φ( M, η , τ ) X m =0 Θ G − ( N, M, η , τ , m ) Z ∞ log (1 + ρx ) d ( F Z ( x )) P Gg =1 Nηg τ g − m ( c ) = X τ = P ( U ) Φ( M, η , τ ) X m =0 Θ G − ( N, M, η , τ , m ) I ρ, G X g =1 Nη g τ g − m , (14)where ρ , Pσ n and P ( U ) is given by (9). (a) follows from the conditional expectation of X n |U n and theidentically distributed property (let X and U represent X n and U n respectively), (b) follows from (11) inTheorem 1, (c) follows from (9), and define I ( a, b ) , R ∞ log (1 + ax ) d ( F Z ( x )) b . I ( a, b ) is computedin Appendix A to be: I ( a, b ) = b ln 2 b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ ℓ + 1 exp (cid:18) ℓ + 1 a (cid:19) E (cid:18) ℓ + 1 a (cid:19) , (15)where E ( x ) = R ∞ x exp( − t ) t − dt is the exponential integral function [42].The average sum rate for the full feedback is a special case and is given by: C P ( M F ) = Z ∞ log (1 + ρx ) d ( F Z ( x )) K = I ( ρ, K ) . (16) Remark:
It is noteworthy to mention that the functional form of C P ( M ) in (14) consists of two mainparts. The first part, which involves P ( U ) and Θ G − ( · , · , · , · , · ) , accounts for the randomness of the set of O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 12 users who convey feedback as well as the scheduling policy. This part is inherent to the heterogeneouspartial feedback strategy, and is independent of the system metric for evaluation, such as the average sumrate employed in this paper. The second part I ( · , · ) depends on statistical assumption of the underlyingchannel and the system metric, and it is impacted by partial feedback as well. B. Sum Rate Ratio and Best-M
We now examine how to determine the smallest M that results in almost the same performance, interms of average sum rate, as the full feedback case. Applying the same technique as in [15], [19], define γ P as the spectral efficiency ratio and the problem can be formulated as: Find the minimum M ∗ , s.t. γ P = C P ( M ∗ ) C P ( M F ) ≥ γ. (17)The above problem can be numerically solved by substituting the expressions for C P ( M ) and C P ( M F ) .In order to obtain a simpler and tractable relationship between M and K given η , i.e., the tradeoffbetween the amount of partial feedback and the number of users given existing heterogeneity of channelstatistics in frequency domain, an approximation is utilized similar to that in [19], by observing that I ( a, b ) in (15) is slowly increasing in b with fixed a (This phenomenon is due to the saturation ofmultiuser diversity [43]). Observing P Φ( M, η , τ ) m =0 Θ G − ( N, M, η , τ , m ) = 1 and employing the binomialtheorem yields the approximation for the spectral efficiency ratio as: γ P ≃ − (cid:18) − η G M ∗ N (cid:19) K . (18)From (17) and (18), the minimum required M ∗ can be obtained as follows: M ∗ ≥ Nη G (cid:16) − (1 − γ ) K (cid:17) . (19) Remark:
It can be seen that M ∗ depends on the system parameters ( N, K, γ ) as well on the largestsubband size η G . It is also a consequence of our heterogeneous partial feedback assumption to let usersin cluster K g convey back the η G η g M best CQI out of Nη g values. This results in the fact that obtainingfeedback information from users belonging to different clusters have almost the same statistical influenceon scheduling performance. IV. I MPERFECT F EEDBACK
After analyzing the heterogeneous partial feedback design with perfect feedback, we turn to examinethe impact of feedback imperfections in this section. We develop the imperfect feedback model due tochannel estimation error and feedback delay in Section IV-A, and investigate the influence of imperfections
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 13 on two different transmission strategies in Section IV-B and IV-C. Then we propose how to optimize thesystem performance to adapt to the imperfections in Section IV-D.
A. Imperfect Feedback Model
The imperfect feedback model is built upon the subband fading model for the perfect feedback case. Todifferentiate from the notation for the perfect feedback case and focus on the imperfect feedback model,the resource block index is dropped. Let h k denote the frequency domain channel transfer function ofuser k (users in different clusters are not temporally distinguished to avoid notational overload). Due tochannel estimation error, the user only has its estimated version ˆ h k , and the relationship between h k and ˆ h k can be modeled as: h k = ˆ h k + w k , (20)where w k ∼ CN (0 , σ w k ) is the channel estimation error. The channel of each resource block is assumed tobe estimated independently, which yields the channel estimation errors w k i.i.d. across users and resourceblocks, i.e., w k ∼ CN (0 , σ w ) . It is clear that the base station makes decision on scheduling and adaptivetransmission depending on CQI, a function of ˆ h k . Thus this information can be outdated due to delaybetween the instant CQI is measured and the actual instant of use for data transmission to the selecteduser. Let ˜ h k be the actual channel transfer function and we employ a first-order Gaussian-Markov model[22], [25], [27] to describe the time evolution and to capture the relationship with the delayed version asfollows: ˜ h k = α k (ˆ h k + w k ) + q − α k ε k , (21)where ε k accounts for the innovation noise and is distributed as CN (0 , . The delay time between ˜ h k and ˆ h k is not explicitly written for notational simplicity, and α k ∈ [0 , is used to model thecorrelation coefficient. Since the feedback delay is mainly caused by the periodic feedback interval andprocessing complexity [25], the innovation noise ε k are i.i.d. across users and a common α is assumed.Moreover, w k and ε k are assumed independent. Therefore, for notational simplicity, the user index k inthe aforementioned parameters is dropped and ˆ Z , | ˆ h | is denoted as CQI.Let ˜ χ, χ and ˆ χ represent: the actual CQI of the selected user for transmission, its outdated version, andits outdated estimate respectively ( ˆ χ corresponds to X for the perfect feedback case in Section III-A).Notice that the PDF of the outdated estimate ˆ χ depends on the heterogeneous feedback design and thescheduling strategy, whereas the conditional PDF of ˜ χ | ˆ χ only depends on α and σ w . Employing the same O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 14 method in [26], [27], the conditional PDF is obtained as follows: f ˜ χ | ˆ χ ( x | ˆ χ ) = α w (cid:18) − α w x + α w α ˆ χ (cid:19) I ( α w α p ˆ χx ) , (22)where α w = q α σ w +1 − α , and I ( · ) is the zeroth-order modified Bessel function of the first kind [42].Since the feedback is imperfect, there are two types of issues that arise. The first is the choice of theincorrect user to serve. However, because of the i.i.d nature of the errors this does not compromise thefairness and also does not complicate the determination of the CDF. The second problem is that of outagebecause the rate adaptation is made by the base station based on the erroneous CQI. Because of the errorin the CQI, the rate chosen may exceed the rate that the channel can support and so the base station hasto take steps to mitigate this effect of outage. A conservative strategy will result in less outage but underutilization of the channel while an aggressive strategy will result in good utilization of the channel butonly for a small fraction of the time. We now present two transmission strategies to address the outageissue. B. Fix Rate Strategy
In the fix rate conservative scenario, a system parameter β is chosen for rate adaptation, and outageresults under the following condition: Declare outage if : { ˜ χ ≤ β | ˆ χ } . (23)The system average goodput is defined as the total average bps/Hz successfully transmitted [23]. Wederive the average goodput and average outage probability for a given choice of system parameter β inthe following procedure.Firstly the conditional outage probability is expressed as: P ( ˜ χ ≤ β | ˆ χ ) = 1 − Q ( α w α p ˆ χ, α w p β ) , (24)where Q ( a, b ) = R ∞ b t exp( − t + a ) I ( at ) dt is the first-order Marcum-Q function [44]. Denote R ( β , M ) as the average goodput for the heterogeneous partial feedback system, which is written according todefinition: R ( β , M ) = E U (cid:2) E ˆ χ |U [ P ( ˜ χ ≥ β | ˆ χ ) log (1 + ρβ )] (cid:3) . (25) O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 15
Then, from (9) and (14), R ( β , M ) can be computed as: R ( β , M )= E U Φ( M, η , τ ) X m =0 Θ G − ( N, M, η , τ , m ) Z ∞ Q ( α w α √ x, α w p β ) log (1 + ρβ ) d ( F ˆ Z ( x )) P Gg =1 Nηg τ g − m = X τ = P ( U ) Φ( M, η , τ ) X m =0 Θ G − ( N, M, η , τ , m ) log (1 + ρβ ) I β , G X g =1 Nη g τ g − m , (26)where I ( a, b ) , R ∞ Q ( α w α √ x, α w √ a ) d ( F ˆ Z ( x )) b . I ( a, b ) is computed in Appendix B to be: I ( a, b ) = 2 b (1 − σ w ) ln 2 b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ ζ ℓ (cid:18) exp( − ϑ − ζ ℓ ϑ ̟ + ζ ℓ ) )(1 − exp( − ̟ ϑ ̟ + ζ ℓ ) )) (cid:19) , (27)where ̟ = α w α , ϑ = α w √ a , ζ ℓ = ℓ +1)1 − σ w .The average outage probability P ( β , M ) for the heterogeneous partial feedback design can be directlycalculated from definition and (26) as follows: P ( β , M ) = E U (cid:2) E ˆ χ |U [ P ( ˜ χ ≤ β | ˆ χ )] (cid:3) = X τ = P ( U ) Φ( M, η , τ ) X m =0 Θ G − ( N, M, η , τ , m ) − I β , G X g =1 Nη g τ g − m . (28)The average goodput and average outage probability for the full feedback scenario is a special caseand is given by: R ( β , M F ) = log (1 + ρβ ) I ( β , K ) ,P ( β , M F ) = 1 − I ( β , K ) . (29) C. Variable Rate Strategy
Instead of choosing a conservative system parameter to account for the fix rate scenario as in theprevious subsection, we consider an approach we refer to as the variable rate strategy. In the variablerate scenario, a system parameter β is chosen and outage results under the following condition: Declare outage if : { ˜ χ ≤ β ˆ χ | ˆ χ } , (30)where β can be regarded as the backoff factor. The system average goodput and average outageprobability can be derived utilizing the following procedure. O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 16
Now under the variable rate scenario, the conditional outage probability is expressed as: P ( ˜ χ ≤ β ˆ χ | ˆ χ ) = 1 − Q ( α w α p ˆ χ, α w p β ˆ χ ) . (31)Using the same method as (25) and (26), let R ( β , M ) denote the average goodput for the variable ratescenario whose expression can be written as follows: R ( β , M ) = E U (cid:2) E ˆ χ |U [ P ( ˜ χ ≥ β ˆ χ | ˆ χ ) log (1 + ρβ ˆ χ )] (cid:3) = X τ = P ( U ) Φ( M, η , τ ) X m =0 Θ G − ( N, M, η , τ , m ) I β , G X g =1 Nη g τ g − m , (32)where I ( a, b ) , R ∞ Q ( α w α √ x, α w √ ax ) log (1 + ρax ) d ( F ˆ Z ( x )) b .For the full feedback case, the average goodput is given by: R ( β , M F ) = I ( β , K ) . (33)Note that unlike I ( a, b ) , I ( a, b ) can not be written in closed form. Therefore, bounding techniqueand suitable approximation are attractive to find closed form alternatives for I ( a, b ) . The followingproposition presents a valid closed form upper bound for I ( a, b ) in the low SNR regime.
Proposition . In the low
SNR regime, I ( a, b ) can be efficiently upper bounded by: I UB3 ( a, b ) = 4 ρab (1 − σ w ) ln 2 b − X ℓ =0 ( − ℓ ζ ℓ ϑ ϕ ℓ ̟ ϕ ℓ F (cid:18) ,
32 ; 2; 4 ̟ ϑ ϕ ℓ (cid:19) − F (cid:18) ,
1; 1; 4 ̟ ϑ ϕ ℓ (cid:19) + 2 ζ ℓ ϕ ℓ ̟ ϕ ℓ F (cid:18) ,
2; 2; 4 ̟ ϑ ϕ ℓ (cid:19) − F (cid:18) ,
32 ; 1; 4 ̟ ϑ ϕ ℓ (cid:19) !!! , (34)where ̟ = α w α , ϑ = α w √ a , ζ ℓ = ℓ +1)1 − σ w , ϕ ℓ = ̟ + ϑ + ζ ℓ , and F ( · , · ; · ; · ) is the Gaussianhypergeometric function [42]. Proof:
The proof is provided in Appendix B. I UB3 ( a, b ) is valid and tight especially for the low SNR regime. In order to track I ( a, b ) over the whole SNR regimes, we propose the following approximation method by leveraging Jensen’s inequality [45].Recall the definition of I ( a, b ) = E [ Q ( α w α √ ˇ χ, α w √ a ˇ χ ) log (1 + ρa ˇ χ )] , where the random variable ˇ χ is defined to have CDF ( F ˆ Z ( x )) b . Firstly, E [ ˇ χ ] can be computed and is given by: E [ ˇ χ ] = Z ∞ x b − σ w b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ exp (cid:18) − ( ℓ + 1) x − σ w (cid:19) dx = b − σ w b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ (cid:18) − σ w ℓ + 1 (cid:19) . (35) O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 17
Then plugging (35) into Q ( α w α √ x, α w √ ax ) log (1 + ρax ) yields: I A ( a, b ) = Q (cid:16) α w α p E [ ˇ χ ] , α w p a E [ ˇ χ ] (cid:17) log (1 + ρa E [ ˇ χ ]) . (36)Note that I A ( a, b ) would serve as an upper bound from Jensen’s inequality if the function of interest Q ( α w α √ x, α w √ ax ) log (1 + ρax ) was concave in x . Properties of this function such as monotonicityand concavity are of interest and lead to rigorous arguments in support of this bound. If outage doesnot occur, extensive analysis can be carried out due to the well known properties of the log( · ) function.However, the concavity (or log-concavity) of Q ( α w α √ x, α w √ β x ) in x (notice that x appears in bothentries of Q ( · , · ) ) still remains an important open problem [46]. Our numerical evidence suggests that Q ( α w α √ x, α w √ β x ) log (1 + ρβ x ) is concave and monotonically increasing in x for practical choicesof β . For any given β preserving the aforementioned property, Jensen’s inequality yields an upper bound,whose tightness is of interest and discussed in the following proposition. The word practical is used toexclude the situation when β approaches its maximum which in turn enables Q ( · , · ) to dominatethe goodput to incur extreme outage. This makes intuitive sense according to the definition of averagegoodput. Proposition . Let { ˇ χ b } be the family of positive i.i.d. random variables. If Q ( α w α √ x, α w √ β x ) log (1+ ρβ x ) is concave and monotonically increasing in x for any given β , then the Jensen bound is asymp-totically tight, i.e., as b → ∞ , E [ Q ( α w α √ ˇ χ b , α w √ β ˇ χ b ) log (1 + ρβ ˇ χ b )] Q ( α w α p E [ ˇ χ b ] , α w p β E [ ˇ χ b ]) log (1 + ρβ E [ ˇ χ b ]) → . (37) Proof:
The proof is provided in Appendix B.Nonetheless, when the aforementioned property is not preserved (e.g., β approaches ), Jensen’sinequality does not hold but the expression has been experimentally found to be a good approximationand so can still be used. Therefore, (36) is denoted as Jensen approximation. We conduct a numerical studyand demonstrate the tightness of Jensen approximation in Fig. 3. It is observed that the approximationmethod is very tight for moderate (even small) number of users and for all values of β ∈ [0 , , whichshows its potential in accurately tracking the performance of average goodput.Now we calculate the average outage probability. Since it does not involve the log( · ) function, it can O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 18
Variable Rate Parameter β A ve r a g e G oodpu t ( bp s / H z ) Numerical Evaluation (M = 2)Numerical Evaluation (M = 4)Numerical Evaluation (M = 16)Jensen Approximation (M = 16) ρ = 20 dB ρ = 10 dB Fig. 3. Calculating the average goodput from numerical evaluation ( M = 2 , , ) and Jensen approximation ( M = 16 ) forthe variable rate scenario under different ρ . ( N = 64 , η = (1 , , K = K = K/ , α = 0 . , σ w = 0 . , ρ = 10 dB,and dB) be computed into closed form as follows: P ( β , M ) = E U (cid:2) E ˆ χ |U [ P ( ˜ χ ≤ β ˆ χ | ˆ χ )] (cid:3) = X τ = P ( U ) Φ( M, η , τ ) X m =0 Θ G − ( N, M, η , τ , m ) − I β , G X g =1 Nη g τ g − m , (38)where I ( a, b ) , Z ∞ Q ( α w α √ x, α w √ ax ) d ( F ˆ Z ( x )) b ( a ) = 2 b (1 − σ w ) b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ Z ∞ Q ( α w αx, α w √ ax ) exp (cid:18) − ( ℓ + 1) x − σ w (cid:19) xdx ( b ) = b (1 − σ w ) b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ ζ ℓ (cid:18) ψ ℓ ς ℓ (cid:19) , (39) ̟ = α w α , ϑ = α w √ a , ζ ℓ = ℓ +1)1 − σ w , ϕ ℓ = ̟ + ϑ + ζ ℓ , ψ ℓ = ̟ − ϑ + ζ ℓ , ς ℓ = q ϕ ℓ − ̟ ϑ . (a)follows from change of variables; (b) follows from applying [47, B.48]. O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 19
In the case of full feedback, the average outage probability P ( β , M F ) becomes: P ( β , M F ) = 1 − I ( β , K ) . (40) D. Optimization and Adaptation to Imperfections
We have obtained the relationship between the system parameter ( β or β ) and the system averagegoodput, and we now aim to maximize the average goodput by adapting the system parameters.Consider the optimization of R ( β , M ) to obtain the optimal backoff factor β ∗ . It is observed from(32) that directly optimizing R ( β , M ) is tedious, and a near-optimal method is now proposed to obtain β ∗ . This method is inspired by the results in Section III-B, which show that the minimum required M ∗ can be chosen to achieve almost the same performance as a system with full feedback. Thus an optimal β ∗ for the full feedback scenario can be optimized first, and then M ∗ is obtained to “match” the systemperformance. Looking again at Fig. 3 with emphasis on different number M of partial feedback, as M gets larger, the optimal β converges to the full feedback case. In this example, M ∗ = 4 is adequateto match the system performance. It is noteworthy to mention that this adaptation philosophy can beapplied to partial feedback systems wherein system parameters are optimized according to full feedbackassumption first and minimum required partial feedback is chosen subsequently.Note that a closed form approximation has been obtained to track R ( β , M F ) in Section IV-C, whichis denoted as R A ( β , M F ) , I A3 ( β , K ) . The following proposition demonstrates the optimal propertyof β when optimizing R A ( β , M F ) . Proposition . There exists a unique global optimal β that maximizes R A ( β , M F ) . Proof:
The proof is provided in Appendix B.From the above analysis, the optimization strategy can be described as: β ∗ = arg max ≤ β ≤ R A ( β , M F ) ≃ arg max ≤ β ≤ R ( β , M F ) . (41)Since it is proved in Proposition 3 that R A ( β , M F ) is quasiconcave [45] in β , numerical approachsuch as Newton-Raphson method can be applied to obtain β ∗ . As discussed before, once β ∗ is found,the minimum required M ∗ can be obtained by solving (17) or relying on (19).The same strategy can be carried over to the optimization of β , which is presented as follows: β ∗ = arg max β R ( β , M F ) . (42)The impact of imperfections on system parameter adaptation, and the comparison between the fixed rateand variable rate strategies will be examined through simulations in Section V-B. O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 20
Number of Users R e qu i r e d M i n i m u m M Exact minimum M ( γ = 0.99)Appro minimum M ( γ = 0.99) Appro minimum M ( γ = 0.99)Exact minimum M ( γ = 0.9)Appro minimum M ( γ = 0.9) Appro minimum M ( γ = 0.9) Ratio of Users in the Basic Cluster (K / K) R e qu i r e d M i n i m u m M K + K = 8K + K = 16K + K = 24 (a) (b) Fig. 4. The required minimum M for heterogeneous perfect feedback design: (a) Comparison of the required minimum M between numerically solving (17) and using approximation (19) under different γ with respect to the number of users; ( N = 64 , η = (1 , , K = K = K/ , ρ = 10 dB) (b) Computing the required minimum M with respect to different number of userswhen varying the ratio of the number of users in cluster K . ( N = 64 , η = (1 , , ρ = 10 dB, γ = 0 . ) V. N
UMERICAL R ESULTS
In this section, we conduct a numerical study to verify the results developed and to draw some insight.
A. Perfect Feedback Scenario
The number of resource blocks N is assumed to be for simulations throughout this section. We firstconsider a -cluster system. Fig. 4 (a) plots the minimum required M obtained by directly solving (17)and alternatively by the approximation (19) for two thresholds: γ = 0 . and . . Note that the resultfrom (19) is rounded with the ceiling function since the required M is an integer. The other simulationparameters are η = (1 , (i.e., M F = 16 ), and ρ = 10 dB. It is observed that the results from theapproximate expression matches quite well with the exact computation. The question of whether therequired M is sensitive to the partition of users in the system is examined in Fig. 4 (b) wherein the ratioof the number of users in cluster K is changed and the minimum required M is depicted for differenttotal number of users with threshold γ = 0 . . Interestingly, the result turns out to be “uniform”. Asdiscussed in Section III, it is due to the heterogenous feedback design assumption to let users in cluster K consume η G η M ( M in this simulation) feedback which results in the fact that obtaining feedbackinformation from users belonging to different clusters have almost the same influence on scheduling O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 21
Number of Users A ve r a g e S u m R a t e ( bp s / H z ) Heterogeneous (Joint: M = 2)Heterogeneous (Separate: M = 2)Homogeneous (subband size = 1: M = 2)Homogeneous (subband size = 2: M = 2)Heterogeneous (Joint: M = 4)Heterogeneous (Separate: M = 4)Homogeneous (subband size = 1: M = 4)Homogeneous (subband size = 2: M = 4)
Fig. 5. Comparison of the average sum rate for a -cluster system under different feedback strategies with respect to thenumber of users ( N = 64 , η = (1 , , , , K = K = K = K = K/ , M = 2 , , ρ = 10 dB) performance. Therefore, the representative simulation setup K g = K/G can be employed when thesystem performance metric is investigated with respect to the total number of users.We now consider a -cluster system with subband size vector η = (1 , , , (i.e., M F = 8 ). Fig.5 demonstrates the benefit of using heterogeneous feedback design. One of the competing strategies isalso heterogeneous, but treats users from each cluster separately. In particular, the system firstly clustersthe users based on their channel statistics, and then serves the clusters one by one requiring feedbackonly from the served cluster of users. In this way, the feedback amount is varying over time dependingon the partition of users. This strategy is denoted as separate heterogeneous feedback compared to our joint heterogeneous feedback design. The other competing strategies are homogeneous without takingadvantage of the channel statistics of different users. To maintain at least the same feedback amount for faircomparison, each user in the homogeneous case is assumed to feed back ⌈ P Gg =1 η G η g MG ⌉ CQI values. Twosubband sizes are assumed for the homogeneous feedback. It is clear that for the homogeneous case, usersin cluster K have more independent feedback while users in cluster K suffer from redundant feedback.The average sum rate for two different values of M are shown in Fig. 5. The separate heterogenous O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 22
Normalized Parameter β A ve r a g e G oodpu t ( bp s / H z ) K = 10, 20, 30, 40Variable Rate with β Fix Rate with β Normalized Parameter β A ve r a g e O u t a g e P r ob a b ili t y K = 10, 20, 30, 40 Fix Rate with β Variable Rate with β (a) (b) Fig. 6. Comparison of fixed rate and variable rate strategies under normalized parameter β ( β = β = β / ) for differentnumber of users K ( N = 64 , α = 0 . , σ w = 0 . , ρ = 10 dB): (a) Comparison of average goodput; (b) Comparison ofaverage outage probability. feedback is observed to have the worst performance from a sum rate perspective because it does not fullyexploit multiuser diversity, but it consumes the least feedback. Our joint heterogenous feedback design isshown to perform much better than the two homogeneous strategies for the -cluster system. It is due tothe fact that by considering the existing heterogeneity among users, the proposed heterogeneous designcan make the best use of the degrees of freedom in the frequency domain in order to enhance the systemperformance as well as reduce feedback needs. B. Imperfect Feedback Scenario
Fig. 6 exhibits the comparison between the fix rate and variable rate outage scenarios as well as theeffect of the number of users on the optimization of β and β . In order to show the system performanceof the two scenarios in one figure, a normalized parameter β is defined. While examining the variablerate plots β = β , and when considering the fixed rate plots β = β / . The system parameters are: α = 0 . , σ w = 0 . , and ρ = 10 dB. It can be seen that for both scenarios, larger number of users K yields better system performance, i.e., higher average goodput and lower average outage probability.This is a consequence of increased multiuser diversity gain to combat the imperfections in the feedbacksystem.Fig. 7 illustrates the effect of channel estimation error σ w and feedback delay α on the optimal value O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 23 σ w2 α O p t i m a l β σ w2 α O p t i m a l β (a) (b) Fig. 7. The effect of channel estimation error ( σ w ) and feedback delay ( α ) on the optimal value of β and β ( ρ = 10 dB, K = 10 , σ w = 0 : 0 .
005 : 0 . , α = 0 . .
005 : 0 . ): (a) The optimal fix rate parameter β with respect to σ w and α ; (b)The optimal variable rate parameter β with respect to σ w and α . of β and β . Here σ w is varied from to . , and α is varied from . to . in steps of . . Itcan be observed from the changing profiles that both the optimal values of β and β get smaller as theimperfections become worse. Therefore, the system should adjust the system parameters to adapt to theencountered imperfections.Now we consider the adaptation of system parameters ( β or β ) and partial feedback in a -clusterheterogeneous feedback system. The system parameters are: η = (1 , , , , α = 0 . , σ w = 0 . , and ρ = 10 dB. For both transmission strategies and for a given number of users K , the optimal value of β ∗ or β ∗ is first optimized according to the full feedback case discussed in Section IV-D. Then, a minimumrequired M ∗ is obtained by matching the system performance to that in the full feedback case. Fig. 8demonstrates the average goodput for both transmission strategies with M ∗ and β ∗ (or β ∗ ). We observethat there is almost a constant performance gain for the variable rate strategy compared with the fix rateone. This is due to the fact that for the variable rate scenario, the system is adapting the transmissionparameters conditioned on the past memory even if it is the outdated one. If the channel estimation errorand feedback delay are not severe, the imperfections can be compensated by multiplying with the backofffactor and relying on the past feedback. O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 24
Number of Users A ve r a g e G oodpu t ( bp s / H z ) Fix Rate StrategyVariable Rate Strategy
Fig. 8. Comparison of the average goodput for a -cluster system with fix rate and variable rate strategies using optimized β ∗ and β ∗ . The average goodput is calculated using the best-M partial feedback scheme when the minimum required M ∗ iscomputed after obtaining β ∗ or β ∗ . ( N = 64 , η = (1 , , , , K = K = K = K = K/ , α = 0 . , σ w = 0 . , ρ = 10 dB) VI. C
ONCLUSION
In this paper, we propose and analyze a heterogeneous feedback design adapting the feedback resourceaccording to users’ frequency domain channel statistics. Under the general correlated channel model, wedemonstrate the gain by achieving the potential match among coherence bandwidth, subband size andpartial feedback. To facilitate statistical analysis, we employ the subband fading model for the multi-cluster heterogeneous feedback system. We derive a closed form expression of the average sum rateunder perfect partial feedback assumption, and provide a method to obtain the minimum heterogeneouspartial feedback required to obtain performance comparable to a scheme using full feedback. We alsoanalyze the effect of imperfections on the heterogeneous partial feedback system. We obtain a closedform expression for the average goodput of the fix rate scenario, and utilize a bounding technique andtight approximation to track the performance of the variable rate scenario. Methods adapting the systemparameters to maximize the average system goodput are proposed. The heterogeneous feedback designis shown to outperform the homogeneous one with the same feedback resource. With imperfections, thesystem adjusting the transmission strategy and the amount of partial feedback is shown to yield better
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 25 performance. The developed analysis provides a theoretical reference to understand the approximatebehavior of the proposed heterogeneous feedback system and its interplay with practical imperfections.Dealing with the general channel correlation and the corresponding nonlinear nature of the CQI areinteresting directions for the heterogeneous feedback system.A
PPENDIX A Proof (sketch) of Lemma 1:
The methodology is an extension of the work in [19] which deals withthe homogeneous feedback case with one cluster of users and one specific subband size.Let F Y ( g ) k denote the CDF of Y ( g ) k , Y ( g ) k,r . Substituting the subband size Nη g and the number of reportedCQI η G η g M for user k in cluster K g makes F Y ( g ) k satisfy (6). It can be shown that F ˜ Y ( g ) k ( x ) = P ( ˜ Y ( g ) k,n ≤ x ) = P ( Y ( g ) k, ⌈ nηg ⌉ ≤ x ) = F Y ( g ) k ( x ) , which concludes the proof. Proof of Theorem 1:
Substituting the expressions of F ˜ Y ( g ) k from Lemma 1 and combining (10) yield: F X n |U n ( x ) = ( F Z ( x )) P Gg =1 Nηg τ g G Y g =1 ηGηg M − X m =0 ξ g ( N, M, η , F Z ( x )) m τ g . (43)Applying [48, 0.314] to a finite-order power series in (43), (cid:18)P ηGηg M − m =0 ξ g ( N,M, η , F Z ( x )) m (cid:19) τ g can be expressedas P τ g ( ηGηg M − m =0 Λ g ( N,M, η , τ ,m )( F Z ( x )) m , where the expression for Λ g ( N, M, η , τ , m ) is described in Theorem 1.Note that the coefficients of F Z ( x )) m can be computed in a recursive manner.Then we employ [48, 0.316] for the multiplication of power series. For g = 1 , Θ ( N, M, η , τ , m ) canbe calculated from Λ ( N, M, η , τ , m ) and Λ ( N, M, η , τ , m ) as: Θ ( N, M, η , τ , m ) = m X i =0 Λ ( N, M, η , τ , i )Λ ( N, M, η , τ , m − i ) . For ≤ g < G , Θ g ( N, M, η , τ , m ) can be computed from Θ g − ( N, M, η , τ , m ) and Λ g +1 ( N, M, η , τ , m ) in the following manner: Θ g ( N, M, η , τ , m ) = m X i =0 Θ g − ( N, M, η , τ , i )Λ g +1 ( N, M, η , τ , m − i ) . This concludes the proof.
Derivation of I ( a, b ) : From the definition of Z , F Z ( x ) = 1 − exp( − x ) and f Z ( x ) = exp( − x ) . Thus d ( F Z ( x )) b = b ( F Z ( x )) b − f Z ( x ) dx = b P b − ℓ =0 (cid:0) b − ℓ (cid:1) ( − ℓ exp( − ( ℓ + 1) x ) dx , where the last equalityfollows from the binomial theorem.Therefore, R ∞ log (1 + ax ) d ( F Z ( x )) b = b ln 2 P b − ℓ =0 (cid:0) b − ℓ (cid:1) ( − ℓ R ∞ ln(1 + ax ) exp( − ( ℓ + 1) x ) dx .Applying [48, 4.337.2] yields (15). O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 26 A PPENDIX B Derivation of I ( a, b ) : From the definition of ˆ Z , it can be shown that F ˆ Z ( x ) = 1 − exp( − − σ w x ) and f ˆ Z ( x ) = − σ w exp( − − σ w x ) . Then I ( a, b ) can be calculated as: I ( a, b ) ( a ) = 2 b (1 − σ w ) ln 2 b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ Z ∞ Q ( α w αx, α w √ a ) exp (cid:18) − ( ℓ + 1) x − σ w (cid:19) xdx ( b ) = 2 b (1 − σ w ) ln 2 b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ ζ ℓ Q (0 , ϑ ) + exp( − ζ ℓ ϑ ̟ + ζ ℓ ) )(1 − Q (0 , ̟ϑ p ̟ + ζ ℓ )) ! ( c ) = 2 b (1 − σ w ) ln 2 b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ ζ ℓ (cid:18) exp( − ϑ − ζ ℓ ϑ ̟ + ζ ℓ ) )(1 − exp( − ̟ ϑ ̟ + ζ ℓ ) )) (cid:19) , (44)where ̟ = α w α , ϑ = α w √ a , ζ ℓ = ℓ +1)1 − σ w . (a) is obtained by substituting the expression of d ( F ˆ Z ( x )) b and using change of variables; (b) follows from applying [47, B.18]; (c) follows from using the fact that Q (0 , ϑ ) = exp( − ϑ ) . Proof of Proposition 1: I ( a, b ) ( a ) = b (1 − σ w ) ln 2 b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ Z ∞ Q ( α w α √ x, α w √ ax ) ln(1 + ρax ) exp (cid:18) − ( ℓ + 1) x − σ w (cid:19) dx ( b ) < ρab (1 − σ w ) ln 2 b − X ℓ =0 (cid:18) b − ℓ (cid:19) ( − ℓ Z ∞ Q ( α w αx, α w √ ax ) exp (cid:18) − ( ℓ + 1) x − σ w (cid:19) x dx ( c ) = 4 ρab (1 − σ w ) ln 2 b − X ℓ =0 ( − ℓ ζ ℓ ϑ ϕ ℓ ̟ ϕ ℓ F (cid:18) ,
32 ; 2; 4 ̟ ϑ ϕ ℓ (cid:19) − F (cid:18) ,
1; 1; 4 ̟ ϑ ϕ ℓ (cid:19) + 2 ζ ℓ ϕ ℓ ̟ ϕ ℓ F (cid:18) ,
2; 2; 4 ̟ ϑ ϕ ℓ (cid:19) − F (cid:18) ,
32 ; 1; 4 ̟ ϑ ϕ ℓ (cid:19) !!! , (45)where ̟ = α w α , ϑ = α w √ a , ζ ℓ = ℓ +1)1 − σ w , ϕ ℓ = ̟ + ϑ + ζ ℓ , and F ( · , · ; · ; · ) is the Gaussianhypergeometric function [42]. (a) is obtained by substituting the expression of d ( F ˆ Z ( x )) b ; (b) followsfrom the fact that when ρ ≪ , ρax is a tight upper bound for ln(1 + ρax ) ; note that change of variablesare used; (c) follows from applying [47, B.60]. Proof (sketch) of Proposition 2:
Define s ( ˇ χ b ) , Q ( α w α √ ˇ χ b , α w √ β ˇ χ b ) log (1 + ρβ ˇ χ b )] . Firstly itmust be shown that s (ˇ χ b ) s ( E [ ˇ χ b ]) converges to in probability. For ∀ ǫ > , it is now shown that: P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) s ( ˇ χ b ) s ( E [ ˇ χ b ]) − (cid:12)(cid:12)(cid:12)(cid:12) ≥ ǫ (cid:19) = P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) s ( ˇ χ b ) − s ( E [ ˇ χ b ]) s ( E [ ˇ χ b ]) (cid:12)(cid:12)(cid:12)(cid:12) ≥ ǫ (cid:19) ( a ) ≤ P (cid:18) s ( | ˇ χ b − E [ ˇ χ b ] | ) s ( E [ ˇ χ b ]) ≥ ǫ (cid:19) ( b ) → , (46) O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 27 where (a) follows from the concave and monotonically increasing property of s ( · ) : | s ( x ) − s ( y ) |
It must be shown that I A3 ( β , K ) = Q (cid:16) α w α p E [ ˇ χ ] , α w p β E [ ˇ χ ] (cid:17) log (1 + ρβ E [ ˇ χ ]) is strictly quasiconcave in β .This property can be proved by log-concavity [45]. It is shown in [46], [50] that Q ( √ a, √ b ) is log-concave in b ∈ [0 , ∞ ) for a ≥ . Also, log(1 + b ) is concave thus log-concave in b ∈ [0 , ∞ ) . Sincelog-concavity is maintained in multiplication, Q ( √ a, √ b ) log(1 + b ) is log-concave in b ∈ [0 , ∞ ) . Fromthe definition of I A3 ( β , K ) , it is now proved to be log-concave in β ∈ [0 , ∞ ) since E [ ˇ χ ] is irrelevantto β . Therefore, it is quasiconcave in β ∈ [0 , ∞ ) because log-concave functions are also quasiconcave.In addition, it is clear that lim β → I A3 ( β , K ) = 0 . Also, it is now shown that: ≤ lim β →∞ I A3 ( β , K ) ( a ) ≤ lim β →∞ exp − ( α w p β E [ ˇ χ ] − α w α p E [ ˇ χ ]) ! log (1 + ρβ E [ ˇ χ ]) ( b ) = lim β →∞ ρ α w ln 2 1(1 + ρβ E [ ˇ χ ]) (cid:16) − α √ β (cid:17) exp − ( α w p β E [ ˇ χ ] − α w α p E [ ˇ χ ]) ! = 0 , (47)where (a) follows from the upper bound Q ( a, b ) ≤ exp (cid:16) − ( b − a ) (cid:17) for b > a ≥ [47]; (b) followsfrom applying L’Hospital’s rule. Therefore, there exists a unique global optimal β which maximizes I A3 ( β , K ) . A CKNOWLEDGMENT
The authors want to express their deep appreciation to the anonymous reviewers and the AssociatedEditor for their many valuable comments and suggestions, which have greatly helped to improve thispaper.
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 28 R EFERENCES [1] D. J. Love, R. W. Heath, V. K. N. Lau, D. Gesbert, B. D. Rao, and M. Andrews, “An overview of limited feedback inwireless communication systems,”
IEEE J. Sel. Areas Commun. , vol. 26, no. 8, pp. 1341–1365, Oct. 2008.[2] H. Zhu and J. Wang, “Chunk-based resource allocation in OFDMA systems-part I: chunk allocation,”
IEEE Trans. Commun. ,vol. 57, no. 9, pp. 2734–2744, Sept. 2009.[3] H. Asplund, A. A. Glazunov, A. F. Molisch, K. I. Pedersen, and M. Steinbauer, “The COST 259 directional channelmodel-part II: macrocells,”
IEEE Trans. Wireless Commun. , vol. 5, no. 12, pp. 3434–3450, Dec. 2006.[4] Y. Huang and B. D. Rao, “Awareness of channel statistics for slow cyclic prefix adaptation in an OFDMA system,”
IEEEWireless Commun. Lett. , vol. 1, no. 4, pp. 332–335, Aug. 2012.[5] R. Knopp and P. A. Humblet, “Information capacity and power control in single-cell multiuser communications,” in
Proc.IEEE International Conference on Communications (ICC) , Jun. 1995, pp. 331–335.[6] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,”
IEEE Trans. Inf. Theory ,vol. 48, no. 6, pp. 1277–1294, Jun. 2002.[7] S. Sanayei and A. Nosratinia, “Opportunistic downlink transmission with limited feedback,”
IEEE Trans. Inf. Theory ,vol. 53, no. 11, pp. 4363–4372, Nov. 2007.[8] V. Hassel, D. Gesbert, M. S. Alouini, and G. E. Oien, “A threshold-based channel state feedback algorithm for moderncellular systems,”
IEEE Trans. Wireless Commun. , vol. 6, no. 7, pp. 2422–2426, Jul. 2007.[9] J. Chen, R. Berry, and M. Honig, “Limited feedback schemes for downlink OFDMA based on sub-channel groups,”
IEEEJ. Sel. Areas Commun. , vol. 26, no. 8, pp. 1451–1461, Oct. 2008.[10] M. Pugh and B. D. Rao, “Reduced feedback schemes using random beamforming in MIMO broadcast channels,”
IEEETrans. Signal Process. , vol. 58, no. 3, pp. 1821–1832, Mar. 2010.[11] S. Sesia, I. Toufik, and M. Baker,
LTE–The UMTS Long Term Evolution , 2nd ed. Wiley, 2011.[12] B. C. Jung, T. W. Ban, W. Choi, and D. K. Sung, “Capacity analysis of simple and opportunistic feedback schemes inOFDMA systems,” in
Proc. International Symposium on Communications and Information Technologies (ISCIT) , Oct.2007, pp. 203–208.[13] J. Y. Ko and Y. H. Lee, “Opportunistic transmission with partial channel information in multi-user OFDM wireless systems,”in
Proc. IEEE Wireless Communications and Networking Conference (WCNC) , Mar. 2007, pp. 1318–1322.[14] J. G. Choi and S. Bahk, “Cell-throughput analysis of the proportional fair scheduler in the single-cell environment,”
IEEETrans. Veh. Technol. , vol. 56, no. 2, pp. 766–778, Mar. 2007.[15] Y. J. Choi and S. Bahk, “Partial channel feedback schemes maximizing overall efficiency in wireless networks,”
IEEETrans. Wireless Commun. , vol. 7, no. 4, pp. 1306–1314, Apr. 2008.[16] K. Pedersen, T. Kolding, I. Kovacs, G. Monghal, F. Frederiksen, and P. Mogensen, “Performance analysis of simple channelfeedback schemes for a practical OFDMA system,”
IEEE Trans. Veh. Technol. , vol. 58, no. 9, pp. 5309–5314, Nov. 2009.[17] J. Leinonen, J. Hamalainen, and M. Juntti, “Performance analysis of downlink OFDMA resource allocation with limitedfeedback,”
IEEE Trans. Wireless Commun. , vol. 8, no. 6, pp. 2927–2937, Jun. 2009.[18] S. Donthi and N. Mehta, “Joint performance analysis of channel quality indicator feedback schemes and frequency-domainscheduling for LTE,”
IEEE Trans. Veh. Technol. , vol. 60, no. 7, pp. 3096–3109, Sept. 2011.[19] S. Hur and B. D. Rao, “Sum rate analysis of a reduced feedback OFDMA downlink system employing joint schedulingand diversity,”
IEEE Trans. Signal Process. , vol. 60, no. 2, pp. 862–876, Feb. 2012.
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 29 [20] Y. Huang and B. D. Rao, “Environmental-aware heterogeneous partial feedback design in a multiuser OFDMA system,”in
Proc. Asilomar Conference on Signals, Systems, and Computers , Nov. 2011, pp. 970–974.[21] P. Piantanida, G. Matz, and P. Duhamel, “Outage behavior of discrete memoryless channels under channel estimationerrors,”
IEEE Trans. Inf. Theory , vol. 55, no. 9, pp. 4221–4239, Sept. 2009.[22] Y. Isukapalli and B. D. Rao, “Packet error probability of a transmit beamforming system with imperfect feedback,”
IEEETrans. Signal Process. , vol. 58, no. 4, pp. 2298–2314, Apr. 2010.[23] V. Lau, W. K. Ng, and D. S. W. Hui, “Asymptotic tradeoff between cross-layer goodput gain and outage diversity inOFDMA systems with slow fading and delayed CSIT,”
IEEE Trans. Wireless Commun. , vol. 7, no. 7, pp. 2732–2739, Jul.2008.[24] T. Wu and V. Lau, “Design and analysis of multi-user SDMA systems with noisy limited CSIT feedback,”
IEEE Trans.Wireless Commun. , vol. 9, no. 4, pp. 1446 –1450, Apr. 2010.[25] S. Akoum and R. W. Heath, “Limited feedback for temporally correlated MIMO channels with other cell interference,”
IEEE Trans. Signal Process. , vol. 58, no. 10, pp. 5219–5232, Oct. 2010.[26] Q. Ma and C. Tepedelenlioglu, “Practical multiuser diversity with outdated channel feedback,”
IEEE Trans. Veh. Technol. ,vol. 54, no. 4, pp. 1334–1345, Jul. 2005.[27] A. Kuhne and A. Klein, “Throughput analysis of multi-user OFDMA-systems using imperfect CQI feedback and diversitytechniques,”
IEEE J. Sel. Areas Commun. , vol. 26, no. 8, pp. 1440–1450, Oct. 2008.[28] E. Dahlman, S. Parkvall, and J. Skold,
4G LTE/LTE-Advanced for Mobile Broadband . Academic Press, 2011.[29] Ericsson, “System-level evaluation of OFDM - further consideration,” 3GPP, TSG-RAN WG1R1-031303, Tech. Rep., Nov.2003.[30] H. Song, R. Kwan, and J. Zhang, “Approximations of EESM effective SNR distribution,”
IEEE Trans. Commun. , vol. 59,no. 2, pp. 603–612, Feb. 2011.[31] S. N. Donthi and N. B. Mehta, “An accurate model for EESM and its application to analysis of CQI feedback schemesand scheduling in LTE,”
IEEE Trans. Wireless Commun. , vol. 10, no. 10, pp. 3436–3448, Oct. 2011.[32] L. Wan, S. Tsai, and M. Almgren, “A fading-insensitive performance metric for a unified link quality model,” in
Proc.IEEE Wireless Communications and Networking Conference (WCNC) , Apr. 2006, pp. 2110–2114.[33] J. Fan, Q. Yin, G. Y. Li, B. Peng, and X. Zhu, “Adaptive block-level resource allocation in OFDMA networks,”
IEEETrans. Wireless Commun. , vol. 10, no. 11, pp. 3966–3972, Nov. 2011.[34] G. D. Forney Jr and G. Ungerboeck, “Modulation and coding for linear Gaussian channels,”
IEEE Trans. Inf. Theory ,vol. 44, no. 6, pp. 2384–2415, Oct. 1998.[35] N. Al-Dhahir and J. M. Cioffi, “Optimum finite-length equalization for multicarrier transceivers,”
IEEE Trans. Commun. ,vol. 44, no. 1, pp. 56–64, Jan. 1996.[36] S. H. Muller-Weinfurtner, “Coding approaches for multiple antenna transmission in fast fading and OFDM,”
IEEE Trans.Signal Process. , vol. 50, no. 10, pp. 2442–2450, Oct. 2002.[37] M. R. McKay, P. J. Smith, H. A. Suraweera, and I. B. Collings, “On the mutual information distribution of OFDM-basedspatial multiplexing: exact variance and outage approximation,”
IEEE Trans. Inf. Theory , vol. 54, no. 7, pp. 3260–3278,Jul. 2008.[38] M. Eslami and W. A. Krzymien, “Net throughput maximization of per-chunk user scheduling for MIMO-OFDM downlink,”
IEEE Trans. Veh. Technol. , vol. 60, no. 9, pp. 4338–4348, Nov. 2011.
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 30 [39] R. McEliece and W. E. Stark, “Channels with block interference,”
IEEE Trans. Inf. Theory , vol. 30, no. 1, pp. 44–53, Jan.1984.[40] M. M´edard and R. G. Gallager, “Bandwidth scaling for fading multipath channels,”
IEEE Trans. Inf. Theory , vol. 48, no. 4,pp. 840–852, Apr. 2002.[41] H. A. David and H. N. Nagaraja,
Order Statistics , 3rd ed. Wiley-Interscience, 2003.[42] M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables .Dover, 1972.[43] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,”
IEEE Trans. Inf.Theory , vol. 51, no. 2, pp. 506–522, Feb. 2005.[44] A. H. Nuttall, “Some integrals involving the Q-function,” Naval Underwater Systems Center, Tech. Rep., Apr. 1972.[45] S. P. Boyd and L. Vandenberghe,
Convex Optimization . Cambridge Univ Pr, 2004.[46] Y. Yu, “On log-concavity of the generalized Marcum Q function,”
Arxiv Preprint , 2011. [Online]. Available:arXiv:1105.5762[47] M. K. Simon,
Probability Distributions Involving Gaussian Random Variables: A Handbook For Engineers and Scientists .Springer Netherlands, 2002.[48] I. S. Gradshteyn and I. M. Ryzhik,
Tables of Integrals, Series and Products , 7th ed., D. Zwillinger and A. Jeffrey, Eds.Academic Press, 2007.[49] P. Billingsley,
Probability and Measure , 3rd ed. John Wiley & Sons, 1995.[50] H. Finner and M. Roters, “Log-concavity and inequalities for Chi-square, F and Beta distributions with applications inmultiple comparisons,”
Statistica Sinica , vol. 7, pp. 771–788, 1997.
Yichao Huang (S’10–M’12) received the B.Eng. degree in information engineering with highest honorsfrom the Southeast University, Nanjing, China, in 2008, and the M.S. and Ph.D. degrees in electricalengineering from the University of California, San Diego, La Jolla, in 2010 and 2012, respectively. Hethen join Qualcomm, Corporate R&D, San Diego, CA.He interned with Qualcomm, Corporate R&D, San Diego, CA, during summer 2011 and summer2012. He was with California Institute for Telecommunications and Information Technology (Calit2), SanDiego, CA, during summer 2010. He was a visiting student at the Princeton University, Princeton, NJ, during spring 2012. Mr.Huang received the Microsoft Young Fellow Award in 2007 from Microsoft Research Asia. He received the ECE DepartmentFellowship from the University of California, San Diego in 2008, and was a finalist of Qualcomm Innovation Fellowship in2010. His research interests include communication theory, optimization theory, wireless networks, and signal processing forcommunication systems.
O APPEAR IN IEEE TRANSACTIONS ON SIGNAL PROCESSING 31