Frequency Offset Estimation in 3G LTEPierre Bertrand

Cellular Infrastructure Systems - Texas Instruments IncorporatedVilleneuve-Loubet, FranceEmail: p-bertrand@ti.com

Abstract—3G Long Term Evolution (LTE) technology aims ataddressing the increasing demand for mobile multimedia servicesin high user density areas while maintaining good performancein extreme channel conditions such as high mobility of HighSpeed Trains. This paper focuses on the latter aspect andcompares different algorithms for the uplink frequency offsetestimation in LTE Base Stations (eNodeB). A frequency-domainmaximum-likelihood based solution is proposed, taking profitof the available interference-free OFDM symbols de-mapped(or de-multiplexed) at the output of the FFT of an OFDMAmulti-user receiver. It is shown to outperform the state-of-the-art CP correlation approach on both link-level performance andcomplexity aspects.

Index Terms—LTE, E-UTRAN, frequency estimation, synchro-nization, OFDM.

I. INTRODUCTION

The Long Term Evolution (LTE) wireless network, alsoknown as Evolved Universal Terrestrial Radio Access Network(E-UTRAN), has been standardized by the 3GPP workinggroups (WG) as part of the Release 8 specifications. OFDMAand SC-FDMA (single carrier FDMA) access schemes werechosen for the down-link (DL) and up-link (UL) of E-UTRAN, respectively [1]. User Equipments (UE’s) are timeand frequency multiplexed on a physical uplink shared channel(PUSCH), and time and frequency synchronization betweenUE’s guarantees optimal intra-cell orthogonality. In UL, fre-quency offsets (FO) can be due to local oscillator (LO) driftsat both the UE and the Base Station (BS), also referred toas eNodeB, but also to the UE speed translating into Dopplershift in line of sight (LOS) propagation conditions. The lattercan be the main FO contributor as LTE was dimensionned tosupport communications with/from high speed trains (HST)[2]. If the large sub-carrier spacing of LTE, Δf = 15kHz,makes it robust with respect to orthogonality loss due toDoppler shift (since even considering HST scenarios, it doesnot exceed 1/10th of a sub-carrier), the frequency offset stillhas a negative impact on the decoding performance, due to fastchannel variation within the minimum transmit time interval(TTI), 1 millisecond. Correlation-based frequency estimationmethods are broadly used in OFDM systems, as they make useof symbols repetition [3], such as pilot symbols, or repetitionof fraction of a symbol, as the Cyclic Prefix (CP). In particular,[4] proposes an application of this principle to LTE, combiningboth the correlations of the CP and symbol and of the two de-modulation reference symbols (DMRS) of the 1 ms LTE sub-frame [1], depicted in Figure 2. However, the LTE standardsupports a wide variety of hopping schemes so that the two

reference symbols are not necessarily identical. Moreover, theCP symbol, unlike the OFDM symbol, is not orthogonal withother CPs in the LTE multiplex and is therefore sensitive tomultiuser interference.

In this paper, we present an alternate approach for frequencyestimation in LTE based on applying the maximum likelihoodprinciple to a number of frequency hypothesis, or bins, andshow that it outperforms other solutions in both performanceand complexity aspects. The paper is organized as follows:Section II first evaluates the frequency offset range expectedat the eNodeB. Section III revisits the state of art meth-ods for estimating the frequency offset and highlights somefundamental issues when appliyng them to LTE. Section IVdescribes the new proposed method. Section V compares theperformances of both approaches and Section VI comparestheir implementation complexity before conclusions are drawnin Section VII.

II. EXPECTED FREQUENCY OFFSET RANGE

The frequency offset δf experienced at the eNodeB receiveris equal to the sum of the cumulated frequency uncertaintiesat both UE transmitter and eNodeB receiver, δfLO, and theDoppler shift resulting from the UE motion in a line of sight(LOS) radio propagation condition, δfD. Assuming maximumfrequency errors of 0.05 ppm and 0.01 ppm at eNodeB andUE local oscillators (LO) respectively and a carrier frequencyfc = 2 GHz, one can bound δfLO as δfLO ≤ 300Hz. Sincethis FO source is the only one observed on fading channels (noLOS), it also sets the FO upper bound for fading channels. Onthe AWGN channel, δfD corresponds to a UE speed s(δfD),in m/s, given by:

s(δfD) = δfDc

2fc(1)

where c is the speed of light and fc the carrier frequency, Hz.The factor two at the denominator results from the Dopplershift being accounted twice at the eNodeB: first at the UEwhen locking its local oscillator on the DL received signal,then on the UL radio link.

On top of the above mentioned fading and AWGN channels,the 3GPP Working Group RAN4 defined propagation chan-nels specifically addressing the frequency offset estimationand compensation function which worst-case scenarios areexpected to be encountered along High Speed Trains (HST)lines. Two such propagation channels are defined in Annex B.3of [2]. They are AWGN channels with the specific frequency

978-1-4244-2519-8/10/$26.00 ©2010 IEEE

offset time behaviors plotted in Figure 1. Both reflect signalsreceived at an eNodeB close to Railway tracks from UEs ina train crossing the eNodeB cell. HST-1 and HST-3 scenariosmodel open space and tunnel for multi-antennas configurationsrespectively. In the former case, the train’s speed is 350 km/h,the Base Station is located 50 m away from the tracks, andthe cell size is 1000 m. In the latter case, the train’s speedis slightly reduced, 300 km/h, but the BS antennas are closeto the tracks (2 m) and the inter-cell distance correspondingto the antennas spacing along the tunnel is only 300 m. Asa result, the FO transition and steady state intervals are morestringent with HST-3 than with HST-1.

0 5 10 15 20 25 30−1500

−1000

−500

0

500

1000

1500

Time (s)

Fre

quen

cy O

ffset

(H

z)

RAN4 HST−1 channelRAN4 HST−3 channel

Figure 1. Frequency Offset time behavior of HST channels.

In LTE, both intra-subframe and inter-subframe FrequencyHopping (FH) can be enabled [1]. The former case disallowschannel estimation interpolation between reference symbols sothat slot-based channel estimation is the only possible solutionin that case. Since FH pattern repeats every 10ms frame, inter-subframe-only FH is of no use for semi-persistent allocationswith allocation periods larger than one frame so that intra-subframe FH should be the default allocation scheme for radiobearers such as VoIP. In the context of this study, we evaluatedthe various frequency estimation methods in the worst-case(but realistic) situation of a multi-user multiplexing schemewith intra-subframe FH always enabled and slot-based channelestimation.

III. FREQUENCY OFFSET ESTIMATION:STATE OF THE ART

Most algorithms published on the topic of frequency offsetestimation in OFDM systems [5] are based on the CorrelationBased method [3] that can be used both in time and frequencydomains, and where a non-biased estimator for the normalizedfrequency offset ε = δf/fs is given by:

ε = − � ρ

2πL(2)

where ρ is the correlation in time ρ =N−1∑n=0

rnr∗n+NL (N,L

integers) of two sample sequences rn and rn+NL at samplingrate fs reflecting two N -length repeated sequences zn andzn+NL transmitted over the air and received distorted at thedemodulator unit by AWGN and a frequency offset δf . Thesame estimation is also achieved by replacing ρ with the

frequency domain correlation Γ =N−1∑k=0

RkR′∗k where Rk and

R′k are the N -length FFTs of rn and rn+NL respectively.An application of the above method to LTE is described

in [4], where both the cyclic prefix (CP) correlation of eachsymbol and the reference symbols correlation are used ina combined estimator. This is illustrated in Figure 2 whichdepicts the sub-frame structure defined in LTE [1]: one sub-frame is made of two 0.5 ms slots, each made of six DFT-SOFDM data symbols and one central demodulation referencesymbol (DMRS).

�� �slot 1 slot 2

CP Data symbol��Reference symbol

Figure 2. CP and RS correlation on the LTE sub-frame

A. Reference symbols correlation

The principle consists, for each UE, in correlating infrequency domain the two de-mapped DMRSs of each sub-frame. However, the LTE specifications allow for intra sub-frame sequence hopping [1] in which case the sequence ofthe reference symbol changes from one slot to another, so thesymbol cannot be considered anymore as a repeated symbol. Inaddition when intra sub-frame FH is enabled, the flat channelassumption of the estimator (2) is violated and the methodsimply cannot be used. As a result, in most cases, the referencesymbol correlation method is impractical for LTE.

B. CP correlation

Denote r(n) a received complete symbol (CP included),n = 0, 1, · · · , NFFT + NCP − 1, where NFFT and NCP

are the FFT size and the CP length, in samples, respectively.Denote RCP (k) and RS(k) the FFTs of the CP and the OFDMsymbol respectively:

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

RCP (k) =NCP −1∑

n=0

r(n)e−j2πkn/NF F T

RS(k) =NF F T −1∑

n=0

r(n + NCP )e−j2πkn/NF F T

(3)

The CP-correlation method consists in correlating in fre-quency domain the CP and Tail of each received and de-mapped symbol of a given UE. In principle, it requirescomputing the FFT of both the CP and the correspondingTail of the symbol to perform the correlation Γ. In anotherapproach [4], the Tail symbol can be derived from the (already

available) FFT RS(k) of the OFDM symbol, thus limitingthe resulting complexity to only one additional FFT, and Γ iscomputed as:

Γ =Nsc−1∑k=0

RS(k + k0)R∗CP (k + k0)e−j2πkNCP /NF F T (4)

where Nsc is the number of allocated sub-carriers startingfrom sub-carrier k0.

The CP-correlation method was originally designed forOFDM systems, where only one UE occupies the spectrum at atime. In that case, both the CP and the OFDM symbol are freeof interference. However, in OFDMA systems, the CP is notinterference-free anymore because, unlike the OFDM symbol,its spectrum is not limited to the UE’s allocated sub-carriers.This is illustrated in Figure 3 where it can be observed thatthe CP spectrum spills over adjacent sub-carriers where otherUE’s are expected to be scheduled. For instance in LTE UEscan be allocated frequency resources with a resource block(RB) granularity of 12 sub-carriers.

50 100 150 200 250 300 350 400 450 500

−30

−25

−20

−15

−10

−5

0

Frequency points

Am

plitu

de (

dB)

4 scheduled RBs, frequency offset = 400 Hz

SC−FDMA symbolCyclic Prefix

Figure 3. Spectrum of both the OFDM symbol and its cyclic prefix in anOFDMA multiplex

Another drawback, not assessed in this paper, is the unavoid-able erroneous estimation resulting from correlating the CPand OFDM symbol when multiple UEs are code-multiplexedin the same time-frequency resource. This is the case e.g. whenDMRSs of multiple UEs are allocated orthogonal sequencesin the same RBs (for SDMA purpose), or with uplink controlchannel (PUCCH) reference symbols and sounding referencesymbols (SRS) [1].

IV. ALTERNATE SOLUTION:FREQUENCY-BINS METHOD

The proposed alternate solution is a maximum-likelihoodalgorithm which consists in testing a number of frequencyoffset hypothesis on the frequency-domain de-mapped DMRSslocated at the centre of each slot (Figure 2):

δf = arg maxδf

{M(δf)} (5)

One possible metric M(δf) is as follows:

M(δf) =

∣∣∣∣∣∣∣Nsc−1∑k=0

Rs(k)

per bin channel estimates︷ ︸︸ ︷[Rs(k)S∗(k, δf)]∗ S∗(k, δf)

∣∣∣∣∣∣∣√√√√Nsc−1∑k=0

[S(k, δf)S∗(k, δf)]2

(6)where Rs(k) are the received de-mapped FFT output sam-

ples, and S(k, δf) are the frequency shifted replicas (or testedfrequency bins) of the expected DMRS frequency-domainsequence. At the numerator, the outer terms Rs(k)S∗(k, δf)reflect the basic maximum likelihood term which amplitude isexpected the largest for the correct frequency bin δf , while theinner terms equalize the received symbol Rs with the per-binfrequency-domain channel estimates H(k) = Rs(k)S∗(k, δf)thus providing a robust estimator on fading channels. Thedenominator is a normalization factor providing a non-biaisedestimator, which can be verified by checking that the metricis always maximum at the received frequency offset:

dM(δf)dδf

∣∣∣∣δf=δfu

= 0 (7)

where δfu is the frequency offset of the received sequenceRs(k). The metric expression (6) is further simplified to:

M(δf) =

Nsc−1∑k=0

|Rs(k)|2 |S(k, δf)|2√√√√Nsc−1∑

k=0

|S(k, δf)|4(8)

One concern with the Frequency-Bins approach is the com-plexity associated with the number of bins needed to cover asufficient frequency range with a decent granularity. This canbe solved by restricting the number of bins to only three inthe searched frequency window: [−fmax 0 fmax] and usinga parabolic interpolation to locate the position of the metricmaximum. It is easily shown that, given three Cartesian points(xi, yi), i = 1, 2, 3 such that yi = ax2

i + bxi + c, the abscissaxmax of the maximum ymax of the parabola is given by:

xmax =1

2

[x1 + x2 − (x1 − x3)(x3 − x2)(y1 − y2)

(y1 − y3)(x1 − x2) − (y1 − y2)(x1 − x3)

](9)

This reduces significantly the complexity of the Frequency-Bins approach to the computation of only three bins, andmakes it a viable alternative to correlation-based approaches.

The frequency range covered by the frequency bins isconstant during an estimation interval to allow for consistentaccumulation of the metric (8). It is dynamically controlled

Parameter Value or rangeSystem Bandwidth 5 MHz (6 UEs) / 3 MHz (12 UEs)Number of antennas 2Number of UEs 6 / 12Number of scheduled RBs per UE 4 / 1Scheduled sub-frames per UE AllRS sequences Extended Zadoff-Chu (EZC) [1] with random

selection of ZC index and cyclic shift acrossfreq estimation intervals

Scheduling scheme Frequency hopping, no sequence hoppingwithin frequency estimation interval

Frequency estimation methods CP correlation on all symbols and frequencybins on DMRS symbols only

Frequency compensation see Section VIChannels High Speed Train 1&3Channel estimation Slot-based, Frequency interpolation: Least

Squares Filter

TABLE ISIMULATION PARAMETERS.

across estimation intervals in order to narrow down, whenpossible, the scope of the searched frequency offset. Wehave optimized, through empirical simulations, the followingfrequency window adaptation scheme:

{fsearched ∈ [−fmax + fmax]

with fmax = max{

200Hz;min{

2000Hz; 3δf}}

(10)where δf is the residual frequency estimate of the previous

estimation interval.

V. PERFORMANCE EVALUATION

This section compares the Block Error Rate (BLER) perfor-mances after removal of the frequency offset estimate resultingfrom the CP correlation and Frequency-Bins approaches. Thefocus is given on HST channels due to their extreme FO rangeand variation, however, it was also verified that all schemesperform similarly under fading channels (TU-6), mainly dueto the reduced and static FO in that case (see Section II). Wesimulated two configurations: 6 UEs, 4 RBs each at 16-QAM-3/4 sharing 5 MHz spectrum and 12 UEs, 1 RB each at QPSK-1/3 sharing 3 MHz spectrum. The simulator models realistictiming errors of the UEs chosen randomly within a maximumtime uncertainty window of +/- 0.5 μs. The other UEs but theUE under test can have random power and frequency offsetswithin maximum uncertainty windows of +/- 3 dB and +/-500 Hz respectively. All UEs transmit continuously in all sub-frames. Table I provides all other parameters of the simulation.

The metric of each UE is accumulated across sub-framesand antennas during an estimation interval, at the end of whicha frequency estimate is computed. During a given estimationinterval of a given UE, the frequency estimate issued by theprevious interval may or may not be removed from ongoingsymbols. The Frequency-Bins methods always involves FOremoval before estimation. For the CP correlation method,we tested various options resulting in different performanceand complexity choices: 1) no FO removal before estimation,2) FO removal on OFDM symbol and wideband CP, 3) FOremoval on OFDM symbol and de-mapped CP. The setting

Channel type / Allocation 4 RB (16QAM 3/4) 1 RB (QPSK 1/3)HST-1 50 ms 70 msHST-3 20 ms 30 ms

TABLE IIESTIMATION INTERVAL DURATIONS.

of the estimation interval duration results from a trade-offbetween the minimum accumulated SNR to obtain a decentestimation and the ability to track fast frequency variations.Therefore, we optimized this parameter through simulationsfor the High Speed Train channels. Table II summarizes theestimation interval durations used for performance evaluations.

Finally, it should be noted that the Frequency-Bins approachalways operates on only one symbol per slot (DMRS) althoughit could benefit from an improved SNR by also operating onthe SRS, when available. The associated performance resultsare plotted in figures 4 to 5. The following ranking can bederived:

1) CP correlation with FO removal on wideband CP2) Frequency-Bins with parabolic interpolation3) CP correlation without FO removal before estimation4) CP correlation with FO removal on de-mapped CPThese results show that the CP FO removal option is key

for the CP correlation method and that removing the frequencyoffset on the de-mapped sub-carriers, although attractive froma complexity standpoint, provides the worst performance be-cause it truncates the frequency-domain CP (Section III-B),thus resulting in erroneous frequency correction.

7.5 8 8.5 9 9.5 10 10.5 11

10−2

10−1

100

SNR (dB)

BLE

R

HST−1, Freq bins w/t parabolic interp.HST−1, CP correlation, FO removed on WB CPHST−1, CP correlation, FO removed on demapped CPHST−1, CP correlation, No FO removed on CPHST−3, Freq bins w/t parabolic interp.HST−3, CP correlation, FO removed on WB CPHST−3, CP correlation, FO removed on demapped CPHST−3, CP correlation, No FO removed on CP

Figure 4. BLER performance of CP correlation and Frequency-Bins methods:6 UEs with 4 RBs, 16QAM 3/4, High Speed Train channels

VI. COMPLEXITY ESTIMATION

The complexity of the various methods is estimated innumber of complex multiplies and accumulates (CMAC) perSC-FDMA symbol in 20MHz spectrum (NFFT = 2048) with2 receive antennas. UEs are assumed evenly allocated in thespectrum. The complexity of the radix-4 FFT and time domainFO compensation are taken as (3NFFT /4)(log4(NFFT )) andNFFT CMACs per UE per antenna and per symbol respec-tively. For the metrics computations, we consider Nsc CMACs

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 010

−3

10−2

10−1

100

SNR (dB)

BLE

R

HST−1, Freq bins w/t parabolic interp.HST−1, CP correlation, FO removed on WB CPHST−1, CP correlation, FO removed on demapped CPHST−1, CP correlation, No FO removed on CPHST−3, Freq bins w/t parabolic interp.HST−3, CP correlation, FO removed on WB CPHST−3, CP correlation, FO removed on demapped CPHST−3, CP correlation, No FO removed on CP

Figure 5. BLER performance of CP correlation and Frequency-Bins methods:12 UEs with 1 RBs, QPSK 1/3, High Speed Train channels

per UE per antenna per symbol for the CP correlation method.For the Frequency-Bins method, the frequency-domain com-putation of S(k, δf ) is based on a cascaded IDFT/time-domainfrequency drift generation/DFT approach and the center fre-quency bin S(k, δf = 0) does not need any computation.For each frequency bin, the denominator of (8) takes about3 CMACs per sub-carrier, while the numerator re-uses theintermediate results from the numerator computation. Thedivision is then computed for each sub-carrier (Nsc CMACs).As a recall, another difference with the CP correlation methodis that the Frequency-Bins metrics are computed on every slotinstead of every symbol.

We first look at the frequency offset removal on the OFDMand CP symbols for the CP correlation method. The com-plexity associated with the regular OFDM symbol de-rotationis the same as for the Frequency-Bins method. On top, theCP correlation method also requires removing the FO on theCP symbol. As shown in Section V, this method providesthe best performance only when the FO is removed on thewideband CP symbol before de-mapping. In that case, the leastcomplex option is to implement it in the time domain whichonly costs NCP CMACs, which is the CP length. The burdenis transferred to the computation of as many FFTs of theCP after de-rotation as UEs. This operation also includes theweighting of the CP symbol sub-carriers by ej2πkNCP /NF F T

to compensate for the phase difference between the actual Tailsymbol and the OFDM symbol (Section III-B). It can be easilyderived that with 50 UEs in 20 MHz, the number of CMACsis in the range of one million per symbol, which makes thisapproach impractical.

In comparison, the FO removal on the de-mapped CPapplies to the UE’s allocated sub-carriers only and can ben-efit from a cascaded IDFT/time-domain frequency compen-sation/DFT approach, which requires only few thousands ofCMACs per symbol, for the same configuration. The sameapproach is also used to de-rotate the DMRS symbols for theFrequency-Bins method.

We now compare the complexity associated with the three

remaining practical options: 1) Frequency-Bins with parabolicinterpolation, 2) CP correlation without FO removal beforeestimation, 3) CP correlation with FO removal on de-mappedCP.

Figure 6 plots, in Kilo Complex Multiplications and Ac-cumulations (KCMACs) per symbol, the total complexityassociated with the above three options. As can be observed,the Frequency-Bins approach is the least complex.

01020304050607080

100 50 25 20 10 5 4 2 1

Num ber of scheduled UEs per sym bol

KCMACs per symbol

1 2 4 5 10 20 25 50 100

Num ber of RBs per UE

CP correlation (FO rem . on dem apped CP)

CP correlation (no FO rem . before estim ation)

Frequency bins

Figure 6. Complexity comparison: Frequency-Bins Vs CP correlation

VII. CONCLUSION

This paper compares the design options for the LTE ULfrequency offset estimation, from performance and complexityaspects. In particular, it is shown that the state-of-the-artCP correlation does not satisfactorily addresses the high-endperformance requirements of LTE, or at a high implementationcost, so that an alternate approach is needed. A maximum-likelihood based solution is provided, taking profit of the avail-able frequency-domain interference-free symbols de-mapped(or de-multiplexed) at the output of the FFT of an OFDMAmulti-user receiver. Specifically, when compared with thelowest complexity CP correlation approach, the proposedFrequency-Bins method shows a QPSK BLER performanceimprovement of up to 1 dB at 10% BLER in 3GPP RAN4 HighSpeed Train scenarios, with a realistic multi-user multiplexwith intra- and inter-subframe frequency hopping. Finally, acomparative complexity analysis shows that the Frequency-Bins approach is also the least complex.

REFERENCES

[1] 3GPP TS 36.211 v8.5.0, 2008-12, ”Technical Specification Group RadioAccess Network; Evolved Universal Terrestrial Radio Access (E-UTRA);Physical channels and modulation (Release 8)”

[2] 3GPP TS 36.104 v8.4.0, 2008-12, ”Technical Specification Group RadioAccess Network; Evolved Universal Terrestrial Radio Access (E-UTRA);Physical channels and modulation (Release 8)”

[3] P. H. Moose, ”A Technique for Orthogonal Division Multiplexing Fre-quency Offset Correction”, IEEE Transactions On Communications, Vol.42, N 10, pp 2908-2914, October 1994

[4] H. Cheon, ”Frequency Offset Estimation for High Speed Users in E-UTRA Uplink”, PIMRC’ 07, IEEE 2007

[5] M. Morelli, C.C. Jay Kuo, and Man-On Pun, ”Synchronization Tech-niques for Orthogonal Division Multiple Access (OFDMA): A TutorialReview”, Proceedings of the IEEE, Vol. 95, No. 7, pp 1394-1425, July2007