3436 IEEE TRANSACTIONS ON WIRELESS โฆnextgenwrl/papers/donthi_2011_twc.pdfย ยท Index...
Transcript of 3436 IEEE TRANSACTIONS ON WIRELESS โฆnextgenwrl/papers/donthi_2011_twc.pdfย ยท Index...
3436 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 10, OCTOBER 2011
An Accurate Model for EESM and itsApplication to Analysis of
CQI Feedback Schemes and Scheduling in LTESushruth N. Donthi and Neelesh B. Mehta, Senior Member, IEEE
AbstractโThe Effective Exponential SNR Mapping (EESM)is an indispensable tool for analyzing and simulating nextgeneration orthogonal frequency division multiplexing (OFDM)based wireless systems. It converts the different gains of multiplesubchannels, over which a codeword is transmitted, into a singleeffective flat-fading gain with the same codeword error rate.It facilitates link adaptation by helping each user to computean accurate channel quality indicator (CQI), which is fed backto the base station to enable downlink rate adaptation andscheduling. However, the highly non-linear nature of EESMmakes a performance analysis of adaptation and schedulingdifficult; even the probability distribution of EESM is not knownin closed-form. This paper shows that EESM can be accuratelymodeled as a lognormal random variable when the subchannelgains are Rayleigh distributed. The model is also valid whenthe subchannel gains are correlated in frequency or space. Withsome simplifying assumptions, the paper then develops a novelanalysis of the performance of LTEโs two CQI feedback schemesthat use EESM to generate CQI. The comprehensive model andanalysis quantify the joint effect of several critical componentssuch as scheduler, multiple antenna mode, CQI feedback scheme,and EESM-based feedback averaging on the overall systemthroughput.
Index TermsโEffective exponential SNR mapping (EESM),long term evolution (LTE), orthogonal frequency division mul-tiplexing (OFDM), channel quality feedback, multiple antennadiversity, frequency-domain scheduling, adaptive modulation andcoding, lognormal random variable.
I. INTRODUCTION
ORTHOGONAL frequency division multiplexing(OFDM) is the downlink access technique of choice
in wideband wireless cellular standards such as Long TermEvolution (LTE) [1] and IEEE 802.16e WiMAX. In anOFDM downlink, the base station (BS) simultaneously servesmultiple users on orthogonal subcarriers. The scheduler at theBS exploits the frequency-selective nature of the widebandchannel and multi-user diversity. By adapting the data rate,
Manuscript received December 19, 2010; revised May 8, 2011 and July 1,2011; accepted July 1, 2011. The associate editor coordinating the review ofthis paper and approving it for publication was I.-M. Kim.
S. N. Donthi is with Broadcom Corp., Bangalore, India. He was withthe Dept. of Electrical Communication Eng., Indian Institute of Sci-ence (IISc), Bangalore, India during the course of this work (e-mail:[email protected]).
N. B. Mehta is with the Dept. of Electrical Communication Eng., IISc,Bangalore, India (e-mail: [email protected]).
This work was partially supported by a project funded by the Ministry ofInformation Technology, India.
Digital Object Identifier 10.1109/TWC.2011.081011.102247
it also takes advantage of the time-varying nature of thechannel.
A key step in rate adaptation involves determining themodulation and coding scheme (MCS). It is chosen in orderto maximize the data rate to the scheduled user subject toa constraint on the probability of codeword error [2]. Sincea codeword is encoded across multiple, say ๐ , subcarriers,this determination would first require a characterization of thecodeword error rate as a function of the ๐ different subcarriergains. Thus, rate adaptation requires an ๐ -dimensional lookup table, which is cumbersome to generate, store, and use.
This difficulty has motivated the widespread use of theEffective Exponential Signal-to-noise-ratio (SNR) Mapping(EESM). EESM maps the SNRs of the ๐ subcarriers,๐1, ๐2, . . . , ๐๐ , to a single effective SNR ๐eff as follows [3],[4]:1
๐eff = โ๐ ln(
1
๐
๐โ๐=1
๐โ๐๐๐
), (1)
where ๐ is a parameter that is empirically calibrated as afunction of the MCS and packet length. The effective SNRis interpreted as the SNR seen by the codeword as if it weretransmitted over a flat-fading channel. While the reduction of๐ variables to a single variable is empirical and not infor-mation lossless, several studies have shown that EESM is anaccurate indicator of the codeword error rate [4]. Thus, only asingle variable, ๐eff, needs to be dealt with for rate adaptationand scheduling. EESM is also a critical component in OFDMsystem-level simulators, which simulate the performance ofa cellular system that consists of multiple cells and BSs thatserve multiple users, and makes large system-level simulationscomputationally feasible. Consequently, it was extensivelyused during the LTE standardization deliberations [6].
Another important and different use of EESM, which mo-tivates this paper, arises in the generation of channel qualityinformation by the user equipments (UEs). It is fed back by theUEs via the uplink and is then used by the BS for frequency-domain scheduling and for rate adaptation on the downlink.Such feedback is needed because the uplink and downlinkchannels are not reciprocal.2 As a result, the BS does not
1Note that other functional forms for EESM have also been proposed,e.g., [5]. We use (1) given its widespread use.
2The uplink and downlink channels are clearly not reciprocal in thefrequency division duplex (FDD) mode. Even in the time division duplex(TDD) mode, they are not reciprocal due to asymmetry in interference andtransmit and receive radio circuitries.
1536-1276/11$25.00 cโ 2011 IEEE
DONTHI and MEHTA: AN ACCURATE MODEL FOR EESM AND ITS APPLICATION TO ANALYSIS OF CQI FEEDBACK SCHEMES AND SCHEDULING . . . 3437
know a priori the channel gains of the subcarriers for any UE.Since subcarrier-level feedback consumes considerable up-
link bandwidth, several feedback reduction techniques areused in practice [7]โ[9]. For example, in the subband-levelchannel quality indicator (CQI) feedback scheme specifiedin LTE, each UE feeds back only one 4-bit CQI for everysubband. A subband is defined to be a collection of ๐ โฅ 2physical resource blocks (PRBs), and each PRB consists of12 contiguous subcarriers. While the frequency response ofthe channel is typically flat across a 180 kHz wide PRB, itis not flat across a subband, which can have a bandwidth aslarge as 1 MHz. Therefore, the UE uses EESM to computethe effective SNR for each subband and then determines theappropriate MCS. When the channel is frequency-flat acrossa PRB, the effective subband SNR is a function of ๐ SNRvalues. The computed MCS is fed back to the frequency-domain scheduler at the BS, which then assigns appropriateUEs to different PRBs. In the UE selected subband feedbackscheme of LTE, which is described in detail below, the EESMgets computed over an even larger number of PRB SNRs.
The performance of the CQI feedback schemes, therefore,depends on the statistics of the effective SNR generated byEESM. However, the non-linear nature of (1) makes an exactanalysis intractable. In fact, even the probability distributionfunction (PDF) of EESM is not known in closed-form. Whileexpressions for its moment generating function (MGF) andmoments are available [10], they are quite involved. TheEESM was approximated by a Gaussian or a logarithm ofa Gaussian (log-Gaussian) random variable (RV) in [11];however, its accuracy, as we shall see, is poor.3
One can, therefore, see that it is quite a challenge toanalyze the performance of next generation OFDM systems,such as LTE, that use a spectrally efficient combination ofrate adaptation, multiple antenna techniques, and frequency-domain scheduling, and employ EESM to generate CQI feed-back. Consequently, most papers that deal with scheduling,adaptation, or CQI feedback for the LTE downlink resortto simulations [9], [12]โ[14]. For example, in [13], contigu-ous and distributed subcarrier allocations are compared bysimulations, and their effects on the throughput of greedyand proportional fair (PF) schedulers is evaluated. In [9], theperformance of a PF scheduler was studied assuming that ithas access to PRB-level CQI feedback.
In [15], a scheme in which each user feeds back the indicesof a pre-specified number of subcarriers with the highestgains and the BS uses BPSK modulation and a round-robinscheduler to transmit, was analyzed. However, rate adaptation,multiple antenna diversity, and channel-aware scheduling werenot modeled. The analysis in [16] quantified the performancegains from MIMO and rate adaptation in a cellular system,but did not consider OFDM. For a MIMO-OFDM system thatuses orthogonal space-frequency block codes, [17] analyzedthe performance of a greedy scheduler and used simulations tostudy a PF scheduler. However, one-bit feedback without rateadaptation was assumed. Neither the coarse frequency granu-
3In [11], the sum 1๐
โ๐๐=1 ๐
โ๐๐๐ was approximated by lognormal and
Gaussian RVs. Since the effective SNR is the logarithm of this sum, weshall instead refer to these as Gaussian and log-Gaussian approximations,respectively.
larity of feedback nor EESM-based CQI feedback generationare considered by the above analysis papers.
A modified PF scheduler was proposed for a scheme similarto the UE selected feedback scheme in [18]. However, EESMwas not considered and the analysis was limited to determin-ing the fading-averaged probability that channel informationabout a given number of clusters of subcarriers is fed back.While [19]โ[21] did model the coarseness, the arithmetic mean(AM) of the subcarrier SNRs was used. The AM is lessaccurate than the EESM [3], but entails a simpler and easieranalysis. While [11] did consider EESM, aspects such asdynamic rate adaptation, scheduling, feedback, and multipleantenna diversity were not analyzed. Altogether, the simplecell planning model of [11] is quite different from ours.
A. Contributions of the Paper
Motivated by the goal of analyzing the CQI feedbacktechniques of LTE, we propose the use of an analyticallytractable lognormal distribution to characterize the PDF ofthe effective SNR. This is shown to be accurate when thefrequency-flat channel response across a PRB is a RayleighRV and the channel responses of different PRBs are mutuallyindependent [8], [15], [22], [23]. It is also accurate when mul-tiple antenna diversity techniques, which affect the statistics ofthe SNR of each PRB, are employed. Furthermore, the modelis valid even in the presence of frequency-domain correlationor spatial correlation. Note that the lognormal model has beenused earlier in the literature, for example, to approximate thedistribution of sums of lognormal or Suzuki RVs [24], [25].However, to the best of our knowledge, this is the first timethat it has been successfully applied to model EESM.
The second major contribution of the paper is a novel andcomprehensive analysis of the throughput of LTE feedbackschemes that use EESM to generate CQI to enable schedulingand rate adaptation at the BS. While we focus on the PF sched-uler due to space constraints, the analysis can be generalizedto handle the greedy and round-robin schedulers, which trade-off fairness and throughput differently. The paper analyzesboth the subband-level and the UE selected CQI feedbackschemes of LTE and accounts for its different multi-antennadiversity modes. Another important aspect of this analysisis its handling of the coarse frequency granularity of theEESM-based CQI feedback, which can occasionally lead to anincorrect choice of the MCS. This aspect is novel comparedto conventional rate adaptation and scheduling problems [2],[17].
Altogether, the analysis quantifies, for the first time, thejoint effect of several critical components such as scheduler,multiple antenna mode, CQI feedback scheme, and EESM-based feedback averaging on the throughput of an OFDMsystem that is as technologically rich and as practically rele-vant as LTE. These components have hitherto been analyzedin isolation in the literature. The analysis also enables an opti-mization of the parameters associated with the CQI feedbackscheme. The proposed lognormal model for EESM plays acrucial role in this analysis. Therefore, the lognormal modelis accurate, analytically tractable, and useful.
The paper is organized as follows. We first discuss the CQImodel of LTE in Sec. II. In Sec. III, the lognormal model
3438 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 10, OCTOBER 2011
for EESM is proposed and used to analyze the LTE CQIfeedback schemes. Numerical results and our conclusions fol-low in Sec. IV and Sec. V, respectively. Several mathematicalderivations are relegated to the Appendix.
II. MODEL FOR EESM-BASED CQI FEEDBACK IN LTE
We now briefly describe relevant details of the LTE down-link, such as its frame structure and its CQI feedback mecha-nisms, and set up the system model and notation. We use thefollowing notation henceforth. The probability of an event ๐ดis denoted by Pr(๐ด). For an RV ๐ , ๐๐(.) denotes its PDF and๐ผ [๐ ] its mean. The conditional expectation of ๐ given event๐ด is denoted by ๐ผ [๐ โฃ๐ด], and Pr(๐ตโฃ๐ด) denotes the conditionalprobability of event ๐ต given ๐ด. For a set โ, โฃโโฃ shall denoteits cardinality. The set {๐ฅ1, ๐ฅ2, . . . , ๐ฅ๐} is denoted by {๐ฅ๐}๐๐=1.
In LTE, each downlink frame is 10 ms long and consistsof ten subframes. Each subframe has a duration of 1 ms, andconsists of two 0.5 ms slots. Each slot contains seven OFDMsymbols. In the frequency domain, the system bandwidth, ๐ต,is divided into several orthogonal subcarriers. Each subcarrierhas a bandwidth of 15 kHz. A set of twelve consecutivesubcarriers over the duration of one slot is called a PhysicalResource Block (PRB). Let ๐PRB denote the total number ofPRBs available over the system bandwidth. Let ๐พ denote thenumber of UEs in a cell.
The BS is equipped with ๐t transmit antennas and eachUE has ๐r receive antennas. In LTE, both single-stream andmultiple-stream transmissions are possible. In a single-streamtransmission, diversity based multiple antenna techniques areused to transmit one codeword of data. Multiple-stream trans-mission refers to the use of spatial multiplexing techniques totransmit two codewords of data simultaneously. In this paper,we focus on single-stream transmission in order to avoid di-gressing into design issues related to codebook-based transmitprecoding of LTE. We cover the following modes of operation:single input single output (SISO) (๐t = ๐r = 1), single inputmultiple output (SIMO) (๐t = 1 and ๐r โฅ 2), closed-loopand open-loop multiple input single output (MISO) (๐t โฅ 2and ๐r = 1), and single-stream multiple input multiple output(MIMO) (๐t โฅ 2 and ๐r โฅ 2).
Since the PRB bandwidth is only 180 kHz, it is justifiableto assume that the channel response is frequency-flat acrossall the 12 subcarriers of the PRB.4 Let โ๐,๐(๐, ๐) denote thechannel gain from the ๐ th transmit antenna of the BS to the ๐th
receive antenna of the ๐th UE for the ๐th PRB. It is modeledas a zero-mean circular symmetric complex Gaussian RV withvariance ๐2
๐ , which implies that its amplitude is Rayleighdistributed. The variance depends upon shadowing and thedistance between the ๐th UE and the BS. The SNR of the๐th PRB of the ๐th UE is denoted by ๐พ๐,๐, and depends onโ๐,๐(๐, ๐), 1 โค ๐ โค ๐๐, 1 โค ๐ โค ๐๐ก, and the multiple antennamode used.
A. Channel Quality Indicator (CQI) Feedback
The CQI is a 4-bit value that indicates an estimate of theMCS that the UE can receive reliably from the BS. It is
4This assumption will not hold for highly dispersive channels with a delayspread greater than 5 ๐s, which is close to the normal cyclic prefix durationof an OFDM symbol in the LTE downlink.
typically based on the measured received signal quality, whichcan be estimated, for example, using the reference signalstransmitted by the BS. The number of MCSs is denoted by ๐ฟand equals 24 = 16. These are tabulated in [26, Tbl. 7.2.3-1].The BS controls how often and when the UE feeds back CQI.The finest possible frequency resolution for CQI reportingis a subband, which consists of ๐ contiguous PRBs, where2 โค ๐ โค 8. The total number of subbands is ๐ = โ๐PRB/๐โ,where โ.โ denotes the ceil function.
Let ๐๐ denote the rate in bits/symbol achieved by the MCScorresponding to the ๐th CQI value. To simplify notation, thesubband that contains the ๐th PRB is denoted by ๐ (๐). PRBs1, . . . , ๐ shall belong to subband 1, PRBs ๐ + 1, . . . , 2๐ shallbelong to subband 2, and so on.
B. CQI Feedback Generation
LTE specifies two different feedback schemes to facilitatefrequency-domain scheduling:5
โ UE selected subband CQI feedback, in which the UEreports the positions of ๐ < ๐ subbands that have thehighest CQIs and only a single CQI value that indicatesthe channel quality that is averaged over all these ๐subbands.
โ Subband-level CQI feedback, in which the UE reportsthe CQI for each of the ๐ subbands.
Thus, the subband-level feedback scheme generates morefeedback. Both the schemes use EESM to generate the CQIvalue(s) from the SNRs of the PRBs that constitute the sub-bands. They are illustrated in Figure 1, and are describedmathematically below.
1) UE Selected Subband Feedback: The subband SNR,๐พsub๐ ,๐, of the ๐th UE for subband ๐ is the effective SNR over
its constituent PRBs. It is computed using EESM as
๐พsub๐ ,๐ = โ๐ ln
โโ1
๐
โ๐โ๐ซโโฌ(๐ )
๐โ๐พ๐,๐
๐
โโ , (2)
where ๐ซโโฌ(๐ ) denotes the set of PRBs in subband ๐ .Remark 1: The parameter ๐ needs to be empirically fine-
tuned as a function of MCS and packet length [4]. However,in order to ensure analytical tractability, the parameter ๐ istaken to be the same for all rates. This has also been assumedin [11] to handle different codeword sizes for QPSK for which๐ is different.
The ๐th UE then orders the subband SNRs of its ๐ subbandsas ๐พsub
(1),๐ โฅ โ โ โ โฅ ๐พsub(๐),๐ โฅ โ โ โ โฅ ๐พsub
(๐),๐, where (๐),following standard order statistics notation, is the index ofthe subband with the ๐th largest SNR. It reports the setโ๐ = {(1), . . . , (๐)}, which consists of the ๐ subbands withthe highest CQIs. It also reports a single CQI, ๐ถbestM
๐ , whichcan take one of 16 possible values. It is a function of the
5The standard also defines a third wideband feedback scheme, in which justone CQI value is sent for the entire system bandwidth. We do not discussit as it not meant for frequency-domain scheduling. For both UE selectedfeedback and subband-level feedback, LTE reduces the CQI overhead further,while incurring a negligible performance loss, by using a 2-bit differentialCQI value for each subband. It also communicates one wideband CQI valuethat is averaged over the whole system bandwidth.
DONTHI and MEHTA: AN ACCURATE MODEL FOR EESM AND ITS APPLICATION TO ANALYSIS OF CQI FEEDBACK SCHEMES AND SCHEDULING . . . 3439
16151413121110987654321
Subband 3 Subband 4
180 kHz
Subband 1 Subband 2
1 ms
Generate 4 CQIs for subbandโlevel feedback
Generate 1 CQI for UE selected feedback
System bandwidth
โ
๐พ1,๐ ๐พ2,๐ ๐พ3,๐ ๐พ4,๐
๐พsub1,๐ ๐พsub
2,๐ ๐พsub3,๐ ๐พsub
4,๐
๐พrep๐
โ
Fig. 1. An illustration of the UE selected subband feedback and subband-level CQI feedback schemes of LTE for the ๐th UE (๐ = 4, ๐ = 2, and๐ = 4). The checkmarks (โ) illustrate subbands with the ๐ highest effective SNRs whose CQI is reported in the UE selected subband feedback scheme.The subband-level feedback scheme generates CQI for all the four subband SNRs. Each subband SNR itself is computed using EESM from the SNRs of the๐ = 4 PRBs that constitute the subband. Also shown is ๐พ
rep๐ , which is computed from ๐พsub
1,๐ and ๐พsub3,๐ using EESM and is used by the UE selected subband
scheme to generate one CQI.
effective SNR ๐พrep๐ that is calculated using EESM over its ๐
selected subbands as follows:
๐พrep๐ = โ๐ ln
(1
๐
โ๐ฃโโ๐
๐โ๐พsub๐ฃ,๐๐
). (3)
For example, in Figure 1, ๐ = 4, ๐ = 2, and โ๐ = {1, 3}.The CQI fed back is ๐ถbestM
๐ = ๐๐, if ๐พrep๐ โ [๐๐โ1, ๐๐). Here,
๐0, ๐1, . . . , ๐๐ฟโ1, ๐๐ฟ (where ๐๐ฟ = โ) are the link adaptationthresholds that ensure that a target block error rate is metshould the BS transmit over all its ๐ best subbands [1],[26]. For ease of explanation, we henceforth do not distinguishbetween ๐๐ and its 4-bit CQI index ๐.
2) Subband-Level Feedback: Unlike the UE selected sub-band scheme, here the ๐th UE reports a separate CQI, ๐ถsub
๐ ,๐,for every subband ๐ . It is based on ๐พsub
๐ ,๐ (which is given in (2)).The CQI fed back is ๐ถsub
๐ ,๐ = ๐๐, if ๐พsub๐ ,๐ โ [๐๐โ1, ๐๐). As before,
๐ถsub๐ ,๐ also takes one of 16 possible values. It can be different
for different subbands.
C. Frequency-Domain Scheduling
Based on the CQI reports from all the UEs, the scheduler inthe BS decides which PRB to allocate to which UE.6 The BSthen signals on the downlink control channel the specific PRBsthat are allocated to different UEs [1, Sec. 9.3.3]. The sched-uler is not specified in the standard and is implementation-dependent. Due to space constraints, we focus on the PFscheduler as it exploits multi-user diversity while also ensuringfairness [27]โ[29].
1) Using UE Selected Subband CQI Feedback: The BSuses โ๐ and ๐ถbestM
๐ reported by all the ๐พ UEs to determinewhich UE to assign to each PRB. Recall that ๐ (๐) denotes thesubband that contains PRB ๐. Let ๐ต๐ (๐) denote the subset ofUEs that have reported the subband ๐ (๐) as one of their best๐ subbands. The PF scheduler [28] assigns the ๐th PRB toUE ๐โ(๐), where
๐โ(๐) = arg max๐โ๐ต๐ (๐)
๐ถbestM๐
๐ผ[๐ถbestM๐
] , (4)
6In LTE, a pair of PRBs that span a duration of 1 ms are assigned together.For brevity, we refer to the pair as a PRB.
and ๐ผ[๐ถbestM๐
]is the average rate reported by the ๐th UE.
Thus, a PRB gets assigned to the UE whose CQI exceeds itsmean rate the most. This ensures fairness across UEs withdifferent mean rates.
Different versions of PF schedulers have been considered inthe literature. In practice, the moving window average is usedinstead of ๐ผ
[๐ถbestM๐
]in the denominator of (4) [22], [29].
Our model accurately models window averaging for windowsizes as small as 50 [30]. In [31], the ratio of instantaneousSNR to its time-averaged value is used instead. However, theseversions share similar characteristics such as allotting almostthe same amount of time to each user.
Outage: Since the CQI value corresponds to the effectiveSNR for the best ๐ subbands, the SNR of the ๐th PRB maybe less than the lower threshold of the MCS being used forthe PRB. This causes an outage, and the throughput is 0 inthat subframe. Outage for the ๐th PRB also occurs if ๐ต๐ (๐) isa null set since the PRB is then not allocated to any UE.7
2) Using Subband-Level CQI Feedback: The PF scheduleruses ๐ถsub
๐ ,๐ reported by all the UEs to allocate the PRBs in asubband. PRB ๐, which lies in subband ๐ (๐), is assigned toUE ๐โ(๐), where
๐โ(๐) = arg max1โค๐โค๐พ
๐ถsub๐ (๐),๐
๐ผ
[๐ถsub๐ (๐),๐
] . (5)
The BS then transmits data on PRB ๐ to UE ๐โ(๐) at a rate๐ถsub๐ (๐),๐โ(๐). As before, if the SNR for a PRB is less than the
lower threshold of the MCS used for it, then the PRB is inoutage.
III. STATISTICAL MODEL FOR EESM AND ITS ROLE IN
LTE THROUGHPUT ANALYSIS
For both the CQI feedback schemes, we saw that EESMplays a crucial role in the generation of CQI of a subband fromthe SNRs of its constituent PRBs. An analytically tractablestatistical characterization of EESM is, therefore, essential in
7Making a UE feed back, in addition, the CQI averaged over the entiresystem bandwidth can mitigate this outage further since it gives the BS partialknowledge about the CSI of the PRBs.
3440 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 10, OCTOBER 2011
order to analyze the performance of the feedback schemes. Tothis end, we first propose a new tractable statistical model forEESM and verify its accuracy. We then apply it to analyze thesystem throughput in Sec. III-C.
A. Statistical Model for EESM
As stated earlier, a closed-form expression for the PDF of๐eff, which is computed from the PRB SNRs ๐1, . . . , ๐๐
using (1), is not known because of the highly non-linearnature of the EESM mapping. We propose modeling ๐eff
as a lognormal RV, which can be written as ๐๐บ, where ๐บis a Gaussian RV with mean ๐ and standard deviation ฮฉ.Therefore, the PDF, ๐๐eff(๐ฅ), of ๐eff is approximated as
๐๐eff(๐ฅ) โ1
๐ฅฮฉโ2๐
๐โ(ln ๐ฅโ๐)2
2ฮฉ2 , ๐ฅ โฅ 0. (6)
By matching the first two moments of ๐eff and ๐๐บ, it canbe shown that
๐ = ln (๐ผ [๐eff])โ ฮฉ2
2, and (7)
ฮฉ =
โโโโทln
(๐ผ [๐2
eff]โ (๐ผ [๐eff])2
(๐ผ [๐eff])2 + 1
). (8)
The two parameters ๐ and ฮฉ shall henceforth be referredto as the lognormal parameters of ๐eff over its constituents๐1, . . . , ๐๐ . The two moments ๐ผ [๐eff] and ๐ผ
[๐2
eff
], which
are required in (8), are computed using one of the followingtwo different methods.
1) Using Analytical Formulae: Expressions for the momentgenerating function (MGF) and the moments of ๐eff forcorrelated Nakagami-๐ fading are derived in [10]. However,these expressions are quite involved. For example, an ๐ -foldsummation, in which each summation is over a variable setof limits, and a summation over ๐ ! permutation terms arerequired in [10, (47), (48), (50), and (52)] to compute the firsttwo moments.
2) Using Monte Carlo Integration Methods: An attractivealternative is the use of efficient Monte Carlo methods [32]to compute the moments of ๐eff. In it, several samples ofthe subcarrier SNRs are generated and empirical momentsof ๐eff are computed to approximate the actual moments.The accuracy of this method depends on the sample size,๐ . Notably, the approximation error decreases as ๐(๐โ 1
2 ),and does not depend on ๐ [32]. Such methods have beenput to good use in communication-theoretic literature, see,for example, [33], [34]. We have found that a sample size of๐ = 5000 is sufficient to compute the moments accuratelyenough for our analysis. Note that these computations need tobe carried out only once in our analysis. Further, this approachcan be easily applied for any joint probability distribution ofthe PRB SNRs.
B. Empirical Verification of Lognormal Model for EESM
We verify the model using two different methods. The firstmethod follows the convention used in lognormal approxima-tion literature [24], [25]. It compares the cumulative distri-bution function (CDF) and the complementary CDF (CCDF)
10โ1
100
101
10โ4
10โ3
10โ2
10โ1
100
x
Pr(
Xef
f โค x
)
SimulationLognormalGaussianLogโGaussian
Fig. 2. Verification of the accuracy of the CDF generated by the proposedlognormal model for EESM and comparison with other proposed distributions(๐2 = 10 dB, ๐ = 2, ๐ = 1.5, and ๐ = 4).
10โ1
100
101
10โ4
10โ3
10โ2
10โ1
100
x
Pr(
Xef
f > x
)
SimulationLognormalGaussianLogโGaussian
Fig. 3. Verification of the accuracy of the CCDF generated by lognormalmodel for EESM and comparison with other proposed distributions (๐2 =10 dB, ๐ = 2, ๐ = 1.5, and ๐ = 4).
of the lognormal model for ๐eff with its empirical CDF andCCDF, which are generated from 106 samples. Small valuesof the CDF reveal the accuracy in tracking the probability ofsmaller ๐eff values. Since all CDFs saturate to 1 for large ๐eff
values, the accuracy in tracking the probability distributionof larger ๐eff values is revealed by studying the CCDFinstead. The comparison is done in Figures 2 and 3 whenthe PRB SNRs are exponential RVs, i.e., squares of RayleighRVs (which also are Chi-square RVs with ๐ = 2). Theconstituent SNRs are independent and identically distributed(i.i.d.) exponential RVs with mean 10 dB. Also plotted are theCDF and CCDF of the Gaussian and log-Gaussian modelsof [11]. We notice that the lognormal approximation, whilenot perfect, tracks both the CDF and CCDF of EESM welland is significantly better than the Gaussian and log-Gaussianapproximations for both the CDF and the CCDF.
We have also compared the accuracy of the proposed modelwith several other common probability distributions such asGamma, Chi-square, Weibull, and ๐พ [35], with momentmatching used to determine their parameters. These are notshown in the figure to avoid clutter. In all comparisons, theproposed model gives the best match for both CDF and CCDF.The effect of increasing the number of constituent SNRs, ๐ ,is investigated in Figure 4, which plots both the CDF and
DONTHI and MEHTA: AN ACCURATE MODEL FOR EESM AND ITS APPLICATION TO ANALYSIS OF CQI FEEDBACK SCHEMES AND SCHEDULING . . . 3441
10โ1
100
101
102
10โ4
10โ3
10โ2
10โ1
100
Xeff
CD
F a
nd C
CD
F o
f Xef
f
SimulationLognormal
N=20
N = 4
CDF CCDF
Fig. 4. Effect of the number of constituent SNRs, ๐ , on the accuracy of theproposed statistical model for EESM (๐2 = 10 dB, ๐ = 2, and ๐ = 1.5).
TABLE IKL DISTANCE FOR DIFFERENT DISTRIBUTIONS (๐2 = 10 DB, ๐ = 1.5,
๐ = 2, AND ๐ = 4)
Probability distribution KL divergenceLognormal 0.024Chi-square 0.052
K-distribution 0.093Gamma 0.214Weibull 0.403
Gaussian 3.50ร 106
Log-Gaussian 1.40ร 107
CCDF. We see that the lognormal modelโs accuracy, in fact,increases as ๐ increases for both the CDF and the CCDF.
The second method of comparison that quantifies the accu-racy of the proposed model in a different manner is a mea-surement of its Kullback-Leibler (KL) divergence [36] fromthe empirically measured PDF. The KL divergence is a usefulmetric because it is zero if and only if the two distributions areidentical. It is always positive, and a smaller value indicatesa better match between the two distributions. The results areshown in Table I. Also tabulated are the KL divergence valuesfor the above mentioned common distributions. We again seethat the proposed lognormal model is the best one among all.
Recall that the square of a Rayleigh RV is an exponentialRV, which is a Chi-square RV with ๐ = 2 degrees of freedom.We now consider a more general case where the constituentSNRs are Chi-square RVs with ๐ โฅ 2 degrees of freedom,whose PDF is given by
๐๐๐ (๐ฅ) =๐ฅ
๐2 โ1๐โ
๐ฅ2
2๐2 ฮ
(๐2
) , for ๐ฅ โฅ 0. (9)
As we shall see, a larger ๐ models the effect of diversity.This can arise, for example, when multiple antenna techniquesare used. Figure 5 compares the lognormal CDF with theempirical CDF of EESM for ๐ = 2 and 8. The otherparameters are kept unchanged. Again, we observe that thelognormal model characterizes the statistics of EESM well,and is considerably better than the Gaussian and log-Gaussianmodels. The CCDF match, which is not shown due to spaceconstraints, is even more accurate.
The results presented thus far assumed that the PRB SNRsare uncorrelated. We now investigate the effect of correlation
10โ1
100
101
102
10โ4
10โ3
10โ2
10โ1
100
x
CD
F: P
r(X
eff โค
x)
SimulationGaussianLogโGaussianLognormal
ฯ = 8ฯ = 2
Fig. 5. Plot of the CDF of the lognormal model of EESM for differentdegrees of freedom, ๐ , which correspond to different multiple antenna modes(๐2 = 10 dB).
10โ1
100
101
10โ4
10โ3
10โ2
10โ1
100
x
CD
F: P
r(X
eff โค
x)
SimulationGaussianLogโGaussianLognormalฯ
f = 0.7
Fig. 6. Effect of frequency-domain correlation between PRBs on the accuracyof proposed lognormal model for EESM (๐2 = 10 dB, ๐ = 4, ๐ = 1.5, and๐ = 4).
across PRBs on the accuracy of the proposed model. Figure 6plots the empirical CDF of EESM and the CDF from the pro-posed lognormal model. Correlation between the ๐th and ๐th
PRBs is modeled using the following geometrically decayingcorrelation model [12]:
๐ผ
[โ๐,๐(๐, ๐)โ
โ๐,๐(๐, ๐)
]๐2๐
= ๐โฃ๐โ๐โฃ๐ .
Results are shown for a high correlation value of ๐๐ = 0.7.We again see that the proposed model tracks the empiricalCDF of EESM well. The corresponding CCDF curves are notshown due to space constraints. We have also verified thevalidity of the model when the channel gains are spatiallycorrelated or when a line-of-sight component is present. Inboth these cases, the PRB SNRs are no longer Chi-squareRVs. The corresponding figures are again not shown due tospace constraints.
Comments: Based on the CDF and CCDF matching and theKL divergence results, we find that the proposed lognormalmodel for EESM is accurate for a wide range of SNRs,number of constituent PRBs, and degrees of freedom (๐ ). Itworth noting from Table I that even the Chi-square distributioncomes second to the proposed model in terms of accuracy.This is an interesting observation because, at low SNRs, the
3442 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 10, OCTOBER 2011
EESM can be approximated by the arithmetic mean, which hasa Chi-square PDF when the PRB SNRs are i.i.d. Even whenthe PRB SNRs are correlated, we find the lognormal modelis quite accurate even for correlations as high as ๐๐ = 0.9.
C. Throughput Analysis of CQI Feedback Schemes
We now analyze the throughput of the UE selected subbandfeedback and subband-level feedback schemes. While thestatistical model for EESM is a significant step forward inenabling the analysis, the following additional assumptions aremade about the system model to make it analytically tractableand yet capture the interactions between the different CQIfeedback techniques, EESM, scheduler, and multiple antennadiversity. We note that several of these assumptions have alsobeen made in the related literature for analyzing relativelysimpler models, and that our results are novel and significanteven with these assumptions.
1) Assumptions: (i) The channel gains across differentPRBs are assumed to be i.i.d. This is a valid assumptionwhen the coherence bandwidth of the channel is close to the180 kHz bandwidth of a PRB [1, Sec. 5.3.2]. This has alsobeen assumed, for example, in [7], [8], [15], [22], [23], [37] toensure analytical tractability. The channel gains are assumedto be i.i.d. across different transmit-receive antenna pairs forall UEs, which is the case when the antennas are spacedsufficiently far apart in a rich scattering environment [14],[16], [28]. This scenario has also been used in LTE per-formance evaluations. Thus, the gains are i.i.d. across theantenna indices ๐ and ๐ and across the PRB index ๐. However,they need not be identically distributed across the UEs since๐2๐ = ๐ผ
[โฃโ๐,๐(๐, ๐)โฃ2], which depends on the distance of theUE from the BS and shadowing, depends on ๐. The channelgain is assumed to remain constant over a 1 ms subframe andthe feedback delays are assumed to be insignificant, whicheasily holds for UE speeds up to 30 kmph.
(ii) We assume that the scheduler can assign different MCSsto different PRBs that are assigned to a UE. However, inLTE, all the PRBs assigned to a UE use the same MCS. Thesimulation results in [38] show that the throughput differencebetween the two approaches is marginal.
(iii) To focus on CQI feedback, we assume ideal beamform-ing weight feedback for closed-loop MISO and single-streamMIMO. Clearly, no such assumption is required for SISO,SIMO, and open-loop MISO. As shown in [39], quantizationof weights typically incurs a 10% loss in throughput for๐r = ๐t = 2 compared to ideal feedback.
Extension to a Multi-cell Scenario: With the assumptionsabove, the analysis directly applies to a single cell scenario [8],[15]โ[17], [23], [28], [40]. In a multi-cell scenario, themethodology can be extended by accounting for co-channelinterference through its fading-averaged power. However, thisextension does have its limitations since there are only sixdominant first-tier interfering cells in a multi-cell OFDMsystem. A more sophisticated approach is pursued in [31],[41] (and references therein), which analyze cell throughputby conditioning on the fading experienced by co-channelinterference. However, their models cannot be generalizedto ours as they do not consider OFDM, frequency-domain
scheduling, CQI feedback, and multiple antenna diversity.While [11] also conditions on the fading experienced by theco-channel interference signals and incorporates EESM, itssimple cell planning oriented analysis does not consider rateadaptation, multiple antenna modes, and scheduling. Further,it uses simulations to determine the statistics of signal-to-interference-plus-noise-ratio (SINR).
2) Common Distribution for ๐พ๐,๐: The statistics of the SNRof ๐th PRB of ๐th UE, ๐พ๐,๐, shall play a crucial role in theanalysis as the CQI depends on it. We first present a singleunified characterization of the PDF of ๐พ๐,๐ for all the multipleantenna diversity modes, and then use it to analyze all of themtogether.
For SIMO (๐t = 1, ๐r โฅ 2), the receiver employsmaximal-ratio combining (MRC) [27, Chap. 3]. Hence, ๐พ๐,๐ =โ๐r
๐=1โฃโ๐,๐(๐, 1)โฃ2 is the summation of ๐r i.i.d. exponentialRVs, which is nothing but a Chi-square RV with ๐ = 2๐r
degrees of freedom and mean ๐r๐2๐. Recall that โ๐,๐(๐, ๐) are
i.i.d. across the transmit and receive antenna indices ๐ and๐. Clearly, SISO is a special case of SIMO with ๐r = 1.For closed-loop MISO, given the ideal beamforming weightassumption, ๐พ๐,๐ =
โ๐t
๐=1โฃโ๐,๐(1, ๐)โฃ2 is a Chi-square RVwith ๐ = 2๐t degrees of freedom and mean ๐t๐
2๐ since the
transmitter employs maximal ratio transmission (MRT) forevery PRB. For open-loop MISO (๐t = 2, ๐r = 1), theAlamouti code is used. Therefore, ๐พ๐,๐ is again a Chi-square
RV with ๐ = 2๐t degrees of freedom and mean ๐t๐2๐
2 . Thus,for all the above multi-antenna diversity modes, we can write๐พ๐,๐ as [21]
๐พ๐,๐ = ๐๐๐ + ๐, (10)
where ๐๐ is a standard Chi-square RV with ๐ degrees offreedom. The values of ๐ for SISO, SIMO, closed-loop MISO,and open-loop MISO are ๐2
๐
2 , ๐2๐
2 , ๐2๐
2 , and ๐2๐
4 , respectively, and๐ = 0 for all these modes.
For single-stream MIMO (๐t = ๐r = 2), ๐พ๐,๐ is the squareof the largest singular value of the matrix {โ๐,๐(๐, ๐)}๐,๐ andits PDF is given as [42]
๐๐พ๐,๐(๐ฅ) =
1
๐2๐
((๐ฅ
๐2๐
)2
โ 2๐ฅ
๐2๐
+ 2
)๐โ ๐ฅ
๐2๐ โ 2
๐2๐
๐โ 2๐ฅ
๐2๐ , ๐ฅ โฅ 0.
(11)Its first two moments are
๐1 โ ๐ผ [๐พ๐,๐] = 3.5๐2๐ and ๐2 โ ๐ผ
[๐พ2๐,๐
]= 15.5๐4
๐. (12)
While ๐พ๐,๐ is not a Chi-square RV, we approximate it us-ing (10) as ๐๐๐ + ๐, where ๐๐ is a Chi-square RV with๐ = 8 degrees of freedom. Matching the first and secondmoments of ๐พ๐,๐ in (10) with its moments ๐1 and ๐2, we
get ๐ =
โ๐๐2โ๐2
1
2 = 0.451๐2๐ and ๐ = ๐1 โ ๐
โ๐2โ๐2
1
2๐ =
โ0.106๐2๐. Figure 7 compares the actual and the approximate
PDFs of ๐พ๐,๐ and verifies that the model is accurate over awide range of values of ๐พ๐,๐ for single-stream MIMO.
3) Throughput of UE Selected Subband Feedback Scheme:The following two claims shall lead us to the final result forthe throughput in (17). An important issue that the analysishandles is that while the CQI feedback has a coarse frequencygranularity of a subband (๐ PRBs) or even ๐ subbands (๐๐
DONTHI and MEHTA: AN ACCURATE MODEL FOR EESM AND ITS APPLICATION TO ANALYSIS OF CQI FEEDBACK SCHEMES AND SCHEDULING . . . 3443
0 20 40 60 80 100 1200
0.005
0.01
0.015
0.02
0.025
0.03
x
PD
F o
f ฮณn,
k (f ฮณ
n,k(x
))
Actual PDFApproximation
Fig. 7. Comparison of the actual PDF and approximate Chi-square PDF(with ๐ = 8) of ๐พ๐,๐ for single-stream MIMO (๐2 = 10 dB).
PRBs), the BS schedules over a narrower PRB. This canoccasionally lead to an incorrect choice of the MCS.
Claim 1: The probability that the ๐th UE reports a CQI of๐๐, for 1 โค ๐ โค ๐ฟ, is
Pr(๐ถbestM๐ = ๐๐
)= Pr
(๐ถbestM๐ โค ๐๐
)โ Pr(๐ถbestM๐ โค ๐๐โ1
),
(13)where
Pr(๐ถbestM๐ โค ๐๐
) โ 1
๐ฝฮฉ(๐๐)๐
โ2๐
โซ ๐๐
0
๐โ(ln ๐งโ๐
(๐๐)๐ )
2
2(ฮฉ(๐๐)๐ )
2
๐ง
ร(1โ 2๐
(ln ๐ง โ ๐
(๐)๐
ฮฉ(๐)๐
))(๐โ๐)
๐๐ง. (14)
Here,
๐ฝโ 1โ๐
๐โ๐=1
๐ค๐
(1โ2๐
(โ2ฮฉ
(๐๐)๐ ๐ผ๐ + ๐
(๐๐)๐ โ ๐
(๐)๐
ฮฉ(๐)๐
))(๐โ๐)
,
(15)and {๐ค๐}๐๐=1 and {๐ผ๐}๐๐=1 are the ๐ Gauss-Hermite weightsand abscissas, respectively [43, Tbl. 25.10]. And, ๐(๐๐)
๐ andฮฉ
(๐๐)๐ are the lognormal parameters, as per Sec. III-A, of the
EESM over the SNRs of ๐๐ PRBs in any ๐ subbands. Anexample of these SNRs is {๐พโ,๐}๐๐
โ=1. Similarly, ๐(๐)๐ and ฮฉ
(๐)๐
are the lognormal parameters of EESM over the SNRs of ๐PRBs in any subband. An example of these SNRs is {๐พโ,๐}๐โ=1.
Proof: The derivation is relegated to Appendix A.Claim 2: Let the ๐th UE be selected (sel.) for the ๐th PRB
and let ๐๐ be the CQI value that it reports. Then, the conditionalprobability that ๐พ๐,๐ is less than ๐๐โ1 is
Pr(๐พ๐,๐ < ๐๐โ1โฃ๐ถbestM
๐ = ๐๐, ๐ is sel. for ๐th PRB)
โ 1
๐
(ln๐๐โ1โ๐
(๐๐)๐
ฮฉ(๐๐)๐
)โ๐
(ln๐๐โ๐
(๐๐)๐
ฮฉ(๐๐)๐
)โซ๐๐โ1โ๐
๐
โ ๐๐
๐ฆ๐2 โ1๐โ
๐ฆ2
2๐2
(๐2 โ 1
)!
ร
โกโขโขโขโขโฃ๐โโโโโโln
[โ๐ ln
(๐๐๐โ
๐๐ (๐๐โ1,๐ฆโ๐๐
)
๐ โ๐โ๐ฆโ๐๐๐
๐๐โ1
)]โ๐(๐๐โ1)
๐
ฮฉ(๐๐โ1)๐
โโโโโโ
โ๐
โโโโโโln
[โ๐ln
(๐๐๐โ
๐๐ (๐๐,๐ฆโ๐๐
)
๐ โ๐โ๐ฆโ๐๐๐
๐๐โ1
)]โ๐(๐๐โ1)
๐
ฮฉ(๐๐โ1)๐
โโโโโโ
โคโฅโฅโฅโฅโฆ๐๐ฆ,
where
๐โ(๐ฅ, ๐ฆ) = min
{max
{โ๐ ln
(โ๐ โ 1 + ๐โ
๐ฆ๐
โ๐
), ๐ฅ
},
๐ฆ + ๐ ln(โ๐)} , for ๐ = 1, 2, . . . . (16)
As used in Claim 1, ๐(๐๐)๐ and ฮฉ
(๐๐)๐ are the lognormal
parameters of EESM over any ๐๐ PRBs. And, ๐(๐๐โ1)๐ and
ฮฉ(๐๐โ1)๐ are the lognormal parameters of EESM over the
SNRs ๐พ1,๐, ๐พ2,๐, . . . , ๐พ๐๐โ1,๐.Proof: The proof is relegated to Appendix B.
Using the above results, the throughput of the PF scheduler,which is defined in (4), is as follows.
Result 1: The average throughput, ๏ฟฝฬ๏ฟฝ๐, for PRB ๐ is
๏ฟฝฬ๏ฟฝ๐ =โ๐ต๐ (๐)
Pr(๐ต๐ (๐)
) โ๐โ๐ต๐ (๐)
๐ฟโ๐=1
๐๐Pr(๐ถbestM๐ = ๐๐
)ร (
1โ Pr(๐พ๐,๐ < ๐๐โ1โฃ๐ถbestM
๐ = ๐๐, ๐ is sel. for ๐th PRB))
รโกโฃ โ๐โ๐ต๐ (๐), ๐ โ=๐
Pr(๐ถbestM๐ โค ๐๐,๐
)โคโฆ . (17)
where ๐๐,๐ is the largest rate that is strictly less
than๐ผ[๐ถbestM
๐ ]๐ผ[๐ถbestM
๐ ]๐๐, Pr
(๐ถbestM๐ โค ๐๐,๐
)is given by Claim 1,
Pr(๐พ๐,๐<๐๐โ1โฃ๐ถbestM
๐ = ๐๐, ๐ is sel. for ๐th PRB)
is given inClaim 2, andPr(๐ต๐ (๐)
)=(๐๐
)โฃ๐ต๐ (๐)โฃ (1โ๐๐
)๐พโโฃ๐ต๐ (๐)โฃ.Proof: The proof is given in Appendix C.
The above throughput expression is in the form of a singleintegral, which is numerically evaluated. This is a significantsimplification compared to either brute-force simulations or an๐ -fold integral with a considerably more involved integrandthat would arise if the proposed lognormal model is not usedfor EESM.
4) Throughput of Subband-Level Feedback Scheme: Thefollowing claim shall lead us to the final expression for thethroughput in (20).
Claim 3: Let the ๐th UE be selected for the ๐th PRB andthe CQI value reported by it be ๐๐. The probability of anoutage, i.e., ๐พ๐,๐ โค ๐๐โ1, is
Pr(๐พ๐,๐ < ๐๐โ1โฃ๐ถsub
๐ (๐),๐ = ๐๐, ๐ is sel. for ๐th PRB)
=1
Pr(๐ถsub๐ (๐),๐ = ๐๐
) โซ๐๐โ1โ๐
๐
โ ๐๐
๐ฆ๐2 โ1๐โ
๐ฆ2
2๐2
(๐2 โ 1
)!
ร
โกโขโขโขโขโขโขโฃ๐
โโโโโโโโln
โกโฃโ๐ ln
โโ ๐๐โ
๐1(๐๐โ1,๐ฆโ๐๐ )
๐ โ๐โ๐ฆโ๐๐๐
๐โ1
โโ โคโฆโ ๐
(๐โ1)๐
ฮฉ(๐โ1)๐
โโโโโโโโ
3444 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 10, OCTOBER 2011
โ๐
โโโโโโโโln
โกโฃโ๐ ln
โโ ๐๐โ
๐1(๐๐,๐ฆโ๐๐ )
๐ โ๐โ๐ฆโ๐๐๐
๐โ1
โโ โคโฆโ ๐
(๐โ1)๐
ฮฉ(๐โ1)๐
โโโโโโโโ
โคโฅโฅโฅโฅโฅโฅโฆ๐๐ฆ,
(18)
where
Pr(๐ถsub๐ (๐),๐ = ๐๐
)= ๐
(ln (๐๐โ1)โ ๐
(๐)๐
ฮฉ(๐)๐
)
โ๐
(ln (๐๐)โ ๐
(๐)๐
ฮฉ(๐)๐
), (19)
and ๐1(๐ฅ, ๐ฆ) is defined in Claim 2. Here, ๐(๐)๐ and ฮฉ
(๐)๐ , as
already used in Claim 1, are the lognormal parameters ofEESM over the SNRs of all the ๐ PRBs of a subband for the๐th UE. And, ๐(๐โ1)
๐ and ฮฉ(๐โ1)๐ are the lognormal parameters
of EESM over the SNRs of any ๐ โ 1 PRBs in a subband ofthe ๐th UE, e.g., {๐พโ,๐}๐โ1
โ=1 .Proof: The proof is similar to that in Appendix B and is
omitted to conserve space.Result 2: The average throughput, ๏ฟฝฬ๏ฟฝ๐, for PRB ๐ of the
subband-level CQI feedback scheme is
๏ฟฝฬ๏ฟฝ๐=
๐พโ๐=1
๐ฟโ๐=1
๐๐Pr(๐ถsub๐ (๐),๐ = ๐๐
)โกโฃ ๐พโ๐=1,๐ โ=๐
Pr(๐ถsub๐ (๐),๐ โค ๐๐,๐
)โคโฆร(1โPr
(๐พ๐,๐< ๐๐โ1โฃ๐ถsub
๐ (๐),๐= ๐๐, ๐ is sel. for ๐th PRB)),
(20)
where ๐๐,๐ is the largest rate among {๐1, . . . , ๐๐ฟ} that
is strictly less than๐ผ[๐ถsub
๐ (๐),๐]๐ผ
[๐ถsub
๐ (๐),๐
]๐๐, and Pr(๐ถsub๐ (๐),๐=๐๐
)and Pr
(๐พ๐,๐ < ๐๐โ1โฃ๐ถsub
๐ (๐),๐ = ๐๐, ๐ is sel. for ๐th PRB)
aregiven in Claim 3.
Proof: The proof is relegated to Appendix D.As before, the final result is in the form of a single integral,
which is numerically evaluated.
IV. LTE THROUGHPUT RESULTS AND COMPARISONS
We now use Monte Carlo simulations that average over50,000 samples to verify the analysis. We consider a singlecell with ๐พ = 6 UEs in it and a BS at the center of the cell.As discussed in Section II, the channel gains of the PRBsfor different transmit-receive antenna pairs for the ๐th UE aregenerated as independent zero-mean complex Gaussian RVswith variance ๐2
๐ . To model the non-i.i.d. channels seen bydifferent UEs, we set ๐2
๐ = ๐/๐ผ๐โ1, 1 โค ๐ โค ๐พ , where๐ผ โฅ 1. Effectively, the first UE sees the strongest channel (onaverage) and UEs 2, . . . ,๐พ see successively weaker channels(on average).
The set of link adaptation thresholds are generated usingthe coding gain loss model of [44]. They are related to therate as follows: ๐๐ = log2 (1 + ๐๐๐โ1). Here, ๐ is the codinggain loss, and is set as ๐ = 0.398 as per [44]. A smaller valueof ๐ means a lower (tighter) bit error constraint. Note that ouranalysis applies to any set of adaptation thresholds ๐0, . . . , ๐๐ฟ.
TABLE IILOGNORMAL PARAMETERS OF EESM (๐2 = 10 DB AND ๐ = 1.5)
Degrees of freedom Number of subcarriers Lognormal parameters๐ ๐ ฮฉ
2
3 1.35 0.404 1.26 0.3011 1.16 0.0912 1.16 0.08
4
3 2.26 0.274 2.19 0.2211 1.96 1.1112 1.94 0.10
8
3 3.19 0.144 3.09 0.1411 2.87 0.0812 2.25 0.08
The thresholds can alternately be chosen based on link-levelsimulation results.
A subband consists of ๐ = 4 PRBs. The number ofsubbands is ๐ = 6, which corresponds to a cell bandwidth of๐ต = 5 MHz. In the UE selected subband feedback scheme,each UE selects ๐ = 3 out of ๐ = 6 subbands. Unlessmentioned otherwise, ๐ = 10 dB, ๐ผ = 1.2, and ๐ = 1.5. Sincewe are using the lognormal model of EESM, the lognormalparameters of EESM are computed once in the analysis. Forexample, Table II enumerates these parameters for the 1st UEfor different ๐ . Since the throughput for SIMO and closed-loop MISO are the same given ๐2
๐ , we show results only forthe former.
Figure 8 plots the average throughput as a function of ๐พfor the UE selected subband scheme. This is done for SISO,SIMO (๐r = 2), open-loop MISO (๐t = 2), and single-streamMIMO. The throughput decreases marginally once ๐พ exceeds4. This is because the PF scheduler ensures fairness among theUEs even though the additional UEs have lower average SNRs.A similar effect is observed for the subband-level feedbackscheme. The throughput of single-stream MIMO, SIMO, andopen-loop MISO is 140%, 70%, and 10% more than SISO,respectively. We see that the analysis and simulation resultsdiffer by no more than 8%. The difference occurs because ofthe lognormal approximation of EESM and the approximationin Claim 1.
To compare the two CQI feedback schemes, Figure 9 plotstheir average throughput (from analysis) for SISO, SIMO, andsingle-stream MIMO. As the number of UEs increases, thethroughput of the UE selected subband scheme approachesthat of the subband-level feedback scheme for all the multipleantenna modes despite its lower feedback overhead.
Figure 10 plots the throughput of UE selected subbandfeedback (from analysis) for different values of ๐ and ๐for SISO. When ๐ is small, the throughput increases as๐ increases since the CQI is reported for more subbands.However, for larger ๐ , the throughput decreases since the UEfeeds back only one CQI value for all ๐ subbands. Further,for the same ๐ , a larger ๐ always leads to a lower throughputsince the subband-level CQI is an average of more PRBs. Wesee that for ๐ = 2, the optimum ๐ is 2, while for ๐ = 4, theoptimum ๐ is 3.
DONTHI and MEHTA: AN ACCURATE MODEL FOR EESM AND ITS APPLICATION TO ANALYSIS OF CQI FEEDBACK SCHEMES AND SCHEDULING . . . 3445
1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
A
vera
ge th
roug
hput
(bi
ts/s
ymbo
l)
Number of UEs, K
SimulationAnalysis
Singleโstream MIMO
SIMO
SISO
Openโloop MISO
Fig. 8. Comparison of the average throughput of UE selected subbandfeedback scheme from analysis and simulations for different multiple antennamodes.
1 2 3 4 5 60
0.5
1
1.5
2
2.5
3
3.5
4
Number of UEs, K
Ave
rage
thro
ughp
ut (
bits
/sym
bol)
Subbandโlevel feedbackUE selected subband feedback
SIMO
SISO
Singleโstream MIMO
Fig. 9. Comparison of the average throughputs of UE selected subbandfeedback and subband-level feedback schemes (from analysis) for differentnumbers of UEs, ๐พ .
V. CONCLUSIONS
EESM is an empirical tool that is widely used in the design,analysis, and simulation of OFDM and OFDM systems. How-ever, an analysis of systems that use EESM has been hamperedby the highly non-linear nature of EESM. Even a closed-form probability distribution function for the effective SNRis unknown. We developed an analytically tractable modelfor EESM that empirically models it as a lognormal RV, andshowed that the model is accurate under a variety of scenarios.We verified its accuracy when the PRBs channel gains areuncorrelated or correlated, and in the presence of frequency-domain correlation and spatial-domain correlation.
The lognormal model of EESM was then instrumental indeveloping an accurate analysis for the throughput of theUE selected subband CQI feedback and subband-level CQIfeedback schemes of LTE. In both these schemes, EESM isused to determine the CQI, which is averaged over manyPRBs to reduce the feedback overhead. The analysis capturedthe joint effect of multi-antenna diversity mode, scheduler,
1 2 3 4 5 60
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
M
Ave
rage
thro
ught
put (
bits
/sym
bol)
q = 4q = 2
Fig. 10. Optimization of parameters of the UE selected subband feedbackscheme as a function of the number of PRBs in a subband, ๐. The barscorresponding to the optimal values of ๐ are encircled.
CQI feedback scheme, and CQI generation. The throughputexpressions were in the form of single integrals, which is asignificant advance compared to the brute-force simulationsthat have typically been used to study this problem. It alsoenabled the optimization of the CQI feedback schemeโs param-eters.While the analysis does not obviate the need for detailedsystem-level simulations, it provides a valuable, independentand common theoretical reference to an LTE system designer.It enables the designer to quickly optimize parameters, gainintuition, and saves considerable simulation effort.
APPENDIX
A. Proof of Claim 1
From Sec. II-B1, Pr(๐ถbestM๐ โค ๐๐
)= Pr
(๐พrep๐ < ๐๐
). To
evaluate this probability, we need the PDF of the EESM of๐ ordered subband effective SNRs, which is analyticallyintractable. We circumvent this problem by deriving an ap-proximate expression that involves only a single integral asfollows [21]:
Pr(๐พrep๐ < ๐๐
) (i)=โ
๐1,...,๐๐
Pr(๐พrep๐ < ๐๐, โ๐ = {๐1, . . . , ๐๐})
(ii)=
(๐
๐
)Pr(๐พrep๐ < ๐๐, โ๐ = {1, . . . ,๐}) .
(21)
Here, (i) follows from the law of total probability. Since fora given UE, the subband SNRs are i.i.d. and there are
(๐๐
)possible combinations of best ๐ subbands, we get (ii).
Let ฮ be the event โ๐={1, . . . ,๐}. Then,
Pr(ฮ)=Pr( max๐+1โค๐โค๐
(๐พsub๐,๐
)โคmin(๐พsub1,๐, . . . , ๐พ
sub๐,๐
)). (22)
From [10], we know that
min(๐พsub1,๐, . . . , ๐พ
sub๐,๐
) โค ๐พrep๐ = โ๐ ln
(1
๐
๐โ๐=1
๐โ๐พsub๐,๐๐
).
We then get
Pr(ฮ) โค Pr
(max
๐+1โค๐โค๐
(๐พsub๐,๐
) โค ๐พrep๐
). (23)
3446 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 10, NO. 10, OCTOBER 2011
Thus, from (21),
Pr(๐ถbestM๐ โค ๐๐
)(24)
โค(๐
๐
)Pr
(max
๐+1โค๐โค๐
(๐พsub๐,๐
) โค ๐พrep๐ < ๐๐
),
=
(๐
๐
)โซ ๐๐
0
๐๐พrep๐(๐ง)
(Pr(๐พsub๐+1,๐ โค ๐ง
))(๐โ๐)๐๐ง.
(25)
At the same time, we know that Pr(๐ถbestM๐ โค ๐๐ฟ
)= 1,
since ๐๐ฟ is the highest rate. This motivates the followingapproximation that, by design, is exact for ๐ = ๐ฟ. In it,the upper bound in (25) is divided by a factor
(๐๐
)๐ฝ, where
๐ฝ is the probability that ๐ถbestM๐ = ๐๐ฟ. Using the proposed
lognormal approximation (Sec. III-A), we know that ๐พrep๐ =
โ๐ ln(
1๐
โ๐๐=1 ๐
โ ๐พsub๐,๐๐
), which is the EESM computed over
๐๐ PRB SNRs, is a lognormal RV with parameters ๐(๐๐)๐ and
ฮฉ(๐๐)๐ . Similarly, ๐พsub
๐+1,๐ is a lognormal RV with parameters
๐(๐)๐ and ฮฉ
(๐)๐ . Therefore,
๐ฝ =
โซ โ
โโ
๐โ(๐งโ๐
(๐๐)๐ )
2
2(ฮฉ(๐๐)๐ )
2
ฮฉ(๐๐)๐
โ2๐
(1โ๐
(๐ง โ ๐
(๐)๐
ฮฉ(๐)๐
))(๐โ๐)
๐๐ง.
(26)Applying Gauss-Hermite quadrature [43] to (26) results in thedesired expression for ๐ฝ.
B. Proof of Claim 2
For brevity, let
๐๐ = Pr(๐พ๐,๐<๐๐โ1โฃ๐ถbestM
๐ = ๐๐, ๐ is sel. for ๐th PRB).
(27)Applying a method similar to the one used in Appendix Aresults in a three-fold integral expression for ๐๐ that cannotbe simplified further. Hence, we develop a different methodbelow, which leads to a much simpler single-fold integralexpression.
From Bayeโs rule and (4), we get
๐๐ =
Pr
(๐พ๐,๐<๐๐โ1, ๐ถ
bestM๐ = ๐๐,
๐ถbestM๐
๐ผ[๐ถbestM๐ ]
< ๐๐๐ผ[๐ถbestM
๐ ], โ ๐ โ= ๐
)
Pr
(๐ถbestM๐ = ๐๐,
๐ถbestM๐
๐ผ[๐ถbestM๐ ]
< ๐๐๐ผ[๐ถbestM
๐ ], โ ๐ โ= ๐
) .
Since the CQIs fed back by different UEs are mutuallyindependent, the above expression simplifies to
๐๐ =Pr(๐พ๐,๐<๐๐โ1, ๐ถ
bestM๐ = ๐๐,
)Pr(๐ถbestM๐ = ๐๐
) . (28)
Further, ๐ถbestM๐ = ๐๐ if and only if the EESM of the subband
SNRs in โ๐ , ๐พrep๐ = โ๐ ln
(1๐
โ๐ฃโโ๐
๐โ๐พsub๐ฃ,๐๐
), lies in between
๐๐โ1 and ๐๐. Hence,
๐๐=
Pr
(๐พ๐,๐<๐๐โ1, ๐๐โ1โคโ๐ ln
(1๐
โ๐ฃโโ๐
๐โ๐พsub๐ฃ,๐๐
)<๐๐
)
Pr
(๐๐โ1 โค โ๐ ln
(1๐
โ๐ฃโโ๐
๐โ๐พsub๐ฃ,๐๐
)< ๐๐
) .
(29)
Note that the set โ๐ contains the subband ๐ (๐).Without loss of generality, since the PRB SNRs and
the subband effective SNRs are i.i.d., let ๐ = 1,๐ (1) = 1, and โ๐ = {1, . . . ,๐}. Since
(๐โ1๐โ1
)sets
of ๐ subbands contain subband 1, the numerator of (29)
is(๐โ1๐โ1
)Pr
(๐พ1,๐<๐๐โ1, ๐๐โ1โคโ๐ ln
(1๐
โ๐ฃโโ๐
๐โ๐พsub๐ฃ,๐๐
)<
๐๐, โ๐ = {1, . . . ,๐}). Similarly, the denominator of (29) is
(๐ โ 1
๐ โ 1
)Pr
(๐๐โ1โคโ๐ ln
(1
๐
โ๐ฃโโ๐
๐โ๐พsub๐ฃ,๐๐
)<๐๐,
โ๐ = {1, . . . ,๐}).
Substituting these in (29) and using Bayeโs rule yields
๐๐ =
Pr
โโโ๐ = {1, . . . ,๐} โฃ๐พ1,๐<๐๐โ1, ๐๐โ1โค
โ๐ ln(
1๐
โ๐๐ฃ=1 ๐
โ๐พsub๐ฃ,๐๐
)<๐๐
โโ
Pr
โโโ๐ = {1, . . . ,๐} โฃ๐๐โ1 โค
โ๐ ln(
1๐
โ๐๐ฃ=1 ๐
โ ๐พsub๐ฃ,๐๐
)< ๐๐
โโ
รPr
(๐พ1,๐<๐๐โ1, ๐๐โ1โคโ๐ ln
(1๐
โ๐๐ฃ=1 ๐
โ ๐พsub๐ฃ,๐๐
)<๐๐
)
Pr
(๐๐โ1 โค โ๐ ln
(1๐
โ๐๐ฃ=1 ๐
โ ๐พsub๐ฃ,๐๐
)< ๐๐
) .
(30)
We observe that the event โ๐ = {1, . . . ,๐} is weaklydependent on the event ๐พ1,๐ < ๐๐โ1. This is because the event๐พ1,๐ < ๐๐โ1 primarily affects the probability that subband1 is selected. Further, ๐๐ โ 1 other PRBs also affect โ๐.Neglecting ๐พ1,๐ < ๐๐โ1 in the numerator and simplifying gives
๐๐โPr
(๐พ1,๐<๐๐โ1, ๐๐โ1โคโ๐ ln
(1๐
โ๐๐ฃ=1 ๐
โ๐พsub๐ฃ,๐๐
)<๐๐
)
Pr
(๐๐โ1 โค โ๐ ln
(1๐
โ๐๐ฃ=1 ๐
โ๐พsub๐ฃ,๐๐
)< ๐๐
) .
(31)
From (2) and (3), we have โ๐ ln(
1๐
โ๐๐ฃ=1 ๐
โ๐พsub๐ฃ,๐๐
)=
โ๐ ln(
1๐๐
โ๐๐๐ฃ=2 ๐
โ ๐พ๐ฃ,๐๐ + ๐โ
๐พ1,๐๐
๐๐
), which is the EESM of
๐๐ PRB SNRs, one of which is ๐พ1,๐. Further, from (10),๐พ1,๐ = ๐๐๐ + ๐, where ๐, ๐, and ๐ depend on the multipleantenna mode. Upon writing the numerator of (31) in termsof PDF of the Chi-square RV ๐๐ (given in (9)), we get
Pr
โโ ๐พ1,๐ < ๐๐โ1, ๐๐โ1 โค
โ๐ ln(
1๐๐
โ๐๐โ1๐ฃ=1 ๐โ
๐พ๐ฃ,๐๐ + ๐โ
๐๐๐+๐๐
๐๐
)< ๐๐
โโ
=
โซ ๐๐โ1โ๐
๐
โ ๐๐
๐ฆ๐2 โ1๐โ
๐ฆ2
(2)๐2
(๐2 โ 1
)!Pr
(๐๐
(๐๐โ1,
๐ฆ โ ๐
๐
)โค
โ ๐ ln
(1
๐๐
๐๐โ1โ๐ฃ=1
๐โ๐พ๐ฃ,๐๐ +
๐โ๐ฆโ๐๐๐
๐๐
)<๐๐ (๐๐, ๐ฆ)
)๐๐ฆ,
where ๐๐ (โ , โ ) is given in the claim statement. Itfollows because, given that ๐พ๐,๐ = ๐ฆ, the term
DONTHI and MEHTA: AN ACCURATE MODEL FOR EESM AND ITS APPLICATION TO ANALYSIS OF CQI FEEDBACK SCHEMES AND SCHEDULING . . . 3447
โ๐ ln(
1๐๐
โ๐๐โ1๐ฃ=1 ๐โ
๐พ๐ฃ,๐๐ + ๐โ
๐ฆ๐
๐๐
)must lie between
โ๐ ln(
๐๐โ1+๐โ๐ฆโ๐๐๐
๐๐
)and ๐ฆโ๐
๐ + ๐ ln(๐๐).
Rearranging the terms and opening up the integrand in thenumerator, simplifies the numerator to
โซ ๐๐โ1โ๐
๐
โ ๐๐
๐ฆ๐2 โ1๐โ
๐ฆ2
(2)๐2
(๐2 โ 1
)!
ร
โกโขโขโฃPr
โโโโโ๐ ln
(1
๐๐โ1
โ๐๐โ1๐ฃ=1 ๐โ
๐พ๐ฃ,๐๐
)<
โ๐ ln(
๐๐๐โ๐๐ (๐๐,
๐ฆโ๐๐
)
๐ โ๐โ๐ฆโ๐๐๐
๐๐โ1
)โโโโ
โPr
โโโโโ๐ ln
(1
๐๐โ1
โ๐๐โ1๐ฃ=1 ๐โ
๐พ๐ฃ,๐๐
)โค
โ๐ ln(
๐๐๐โ๐๐ (๐๐โ1,
๐ฆโ๐๐
)
๐ โ๐โ๐ฆโ๐๐๐
๐๐โ1
)โโโโ โคโฅโฅโฆ ๐๐ฆ.
The above two probabilities can be written in terms of theGaussian ๐(.) function since โ๐ ln
(1
๐๐โ1
โ๐๐โ1๐ฃ=1 ๐โ
๐พ๐ฃ,๐๐
),
which is the effective SNR over ๐๐ โ 1 PRB SNRs, isa lognormal RV with parameters ๐
(๐๐โ1)๐ and ฮฉ
(๐๐โ1)๐ .
Similarly, we use the lognormal model for EESM to simplifythe denominator of (31) as well. Combining these gives therequired result.
C. Proof of Result 1
Since the SNRs of different PRBs of a UE are i.i.d., theprobability that a UE selects the subband ๐ (๐) is ๐
๐ . Hence,
Pr(๐ต๐ (๐)
)=
(๐๐
)โฃ๐ต๐ (๐)โฃ (1โ ๐๐
)(๐พโโฃ๐ต๐ (๐)โฃ). A rate ๐๐ isachieved if the UE that is selected for the PRB fed back aCQI value equal to ๐๐ and there was no outage. The probabilitythat the ๐th UE gets selected, given that it fed back the rate๐๐, is
โ๐โ๐ต๐ (๐)
๐ โ=๐
Pr(๐ถbestM๐ โค ๐๐,๐
). This is because the ๐th UE
is selected only if ๐ถbestM๐ <
๐ผ[๐ถbestM๐ ]
๐ผ[๐ถbestM๐ ]
๐๐, for ๐ โ= ๐.8 The law of
total expectation then yields (17).
D. Proof of Result 2
A rate ๐๐ is achieved if the UE, say the ๐th UE, thatis selected for the PRB fed back a CQI value equal to ๐๐and there was no outage. This results in the expression forthe average throughput in (20), where the probability thatthe ๐th UE gets selected given that it fed back the rate ๐๐isโ๐พ
๐=1๐ โ=๐
Pr(๐ถsub๐ (๐),๐ โค ๐๐,๐
). This is because the ๐th UE is
selected only if ๐ถsub๐ (๐),๐ <
๐ผ[๐ถsub๐ (๐),๐]
๐ผ
[๐ถsub
๐ (๐),๐
]๐๐, for all ๐ โ= ๐.
ACKNOWLEDGMENTS
We thank the editor and the anonymous reviewers for theircomments, which helped improve the presentation of thispaper.
8The equality case ๐ถbestM๐ =
๐ผ[๐ถbestM๐ ]
๐ผ[๐ถbestM๐ ]
๐๐ need not be considered as it
occurs with zero probability in a random deployment of UEs.
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Sushruth N. Donthi received his Bachelor of Engi-neering degree in Telecommunications Engineeringfrom Vemana Institute of Technology, Bangalore in2006. He received his Master of Science degree fromthe Dept. of Electrical Communication Engineering,Indian Institute of Science, Bangalore, India in 2011.Since then, he has been with Broadcom Commu-nications Technologies Pvt. Ltd., Bangalore, Indiaworking on design and implementation of parts ofthe LTE protocol stack. From 2006-2008, he wasat LG Soft India Pvt. Ltd., working on software
development for 3G handsets. His research interests include the design andanalysis of algorithms for wireless cellular systems, communication networksand MIMO systems.
Neelesh B. Mehta (Sโ98-Mโ01-SMโ06) receivedhis Bachelor of Technology degree in Electronicsand Communications Engineering from the IndianInstitute of Technology (IIT), Madras in 1996, andhis M.S. and Ph.D. degrees in Electrical Engineer-ing from the California Institute of Technology,Pasadena, CA, USA in 1997 and 2001, respectively.He is now an Assistant Professor in the Dept. ofElectrical Communication Eng., Indian Institute ofScience (IISc), Bangalore, India. Prior to joiningIISc, he was a research scientist in the Wireless
Systems Research group in AT&T Laboratories, Middletown, NJ, USA from2001 to 2002, Broadcom Corp., Matawan, NJ, USA from 2002 to 2003, andMitsubishi Electric Research Laboratories (MERL), Cambridge, MA, USAfrom 2003 to 2007.
His research includes work on link adaptation, multiple access protocols,WCDMA downlinks, system-level performance analysis of cellular systems,MIMO and antenna selection, and cooperative communications. He wasalso actively involved in the Radio Access Network (RAN1) standardizationactivities in 3GPP from 2003 to 2007. He has served on several TPCs.He was a TPC co-chair for WISARD 2010 and 2011, National Conferenceon Communications (NCC) 2011, the Transmission Technologies track ofVTC 2009 (Fall), and the Frontiers of Networking and Communicationssymposium of Chinacom 2008. He was the tutorials co-chair for SPCOM2010. He has co-authored 25 IEEE journal papers, 55 conference papers, andtwo book chapters, and is a co-inventor in 16 issued US patents. He is anEditor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS andis an executive committee member of the IEEE Bangalore Section and theBangalore chapter of the IEEE Signal Processing Society.