Low complexity estimation of multiple frequencyoffsets using optimized training signals

Tommi KoivistoNOKIA Devices R&D

Wireless Modem Systems ResearchP.O.Box 407, FIN-00045 Nokia Group, Finland

Visa KoivunenAalto University, SMARAD CoE

Department of Signal Processing and AcousticsP.O. Box 13000, FIN-00076 Aalto, Finland

Abstract—We consider training-aided low complexity estima-tion of multiple carrier frequency offsets (CFOs) in OFDM-based wireless systems. Typically multiple CFOs incur a loss oforthogonality between the transmitted training sequences, eithercausing a severe error floor in the performance of an estimatoror requiring estimation algorithms of higher complexity. In thispaper we introduce design criteria for training signals that allowestimators for multiple CFOs to be decoupled nearly free ofsuch multiple access interference. We propose three differentapproaches to training signal design fulfilling the presentedcriteria. Consequently, even estimators developed for the singleCFO case may be applied. Our simulations show that theproposed approach provides excellent estimator performance andat the same time allows low complexity estimators to be used.

I. INTRODUCTION

Cooperative systems have attracted a lot of research at-tention recently. While for example cooperative transmissionschemes have been fairly thoroughly investigated, certainpractical issues such as time-frequency synchronization amongnodes are not yet fully explored. As an example, cooperativesystems involving small and cheap nodes suffer from thefact that the carrier frequencies generated by the nodes’ lowquality local oscillators (LO) may significantly deviate fromthe nominal frequency [1]. On the other hand, as describedin [2][3], carrier frequency offsets (CFOs) may become veryproblematic even with more expensive equipment and accurateLOs if the cooperative transmission scheme itself is sensitiveto phase rotations. Furthermore, in for example multi-cell localarea networks involving inexpensive access point equipment,CFOs may cause loss of orthogonality between transmissionsfrom different access points and therefore aggravate the im-pacts of co-channel interference. Moreover, mobility and theresulting Doppler shifts may cause similar effects in wirelesssystems.

Coping with the aforementioned problems requires estima-tion of multiple CFOs. This could be done either for thepurpose of mitigating the resulting impairments at the receiveror for the purpose of reporting the errors to other nodesand adjusting the frequencies accordingly, i.e. for purposesof distributed network synchronization.

Estimation of multiple frequency offsets was first addressedin [4], where MIMO flat fading channels with multiple fre-quency offsets were considered and the maximum likelihood(ML) estimator for the problem was derived. This work was

later extended to frequency-selective channels in [5], andsimilar studies have been made also for uplink OFDMA casein [6][7]. The ML solution requires a multi-dimensional searchover the parameter space and as such is computationallyprohibitively complex. To reduce the computational burden,in [6][7] it was proposed to utilize the alternating projectionmethod with certain approximations to solve the ML problem.However, the computational complexity remains fairly high.Another possibility to reduce the complexity is to approx-imate the ML solution by assuming the training signals tobe uncorrelated [4][5]. Such training signals have also beenshown to minimize the asymptotic Cramer-Rao bound (CRB)[4][8]. However, it is acknowledged already in [4] and willalso be later shown in this paper that with practical datasamples this type of an approximation does not hold, sincethe orthogonality of the signals is lost due to the CFOs.This results in multiple access interference and insufficientCFO compensation. This issue has been also pointed out in[3], and as a solution algorithms taking into account the lossof orthogonality were proposed. While such algorithms mayprovide the needed performance, complexity is increased aswell. To the authors’ knowledge, so far the only solutionproviding good performance with low receiver complexity isto orthogonalize the training signals via TDM multiplexing [1][4]. However, this leads to inefficient transmit power usage andpower amplifier inefficiency at the transmitter side. In additionto the ML solutions, several papers have proposed subspace-based methods that rely on existence of null subcarriers [9][10]. Also these can be considered to have high complexity.

Low computational complexity of multiple CFO compen-sation is a critical issue in particular in systems with in-expensive nodes and transceiver front ends. Most existingsolutions on the multiple CFO estimation problem still seemto require fairly high processing power. In this paper our maincontribution is showing that with properly designed trainingsignals, the estimators for multiple CFOs may be decoupledfrom each other, and low complexity and high performancemethods developed for single CFO case may be applied. Ourcontributions can be further listed as follows:

1) We introduce training signal design criteria that allowthe estimation of the multiple CFOs to be decoupled.In essence we aim at preserving low cross-correlation

ISSSTA2010, Taichung, Taiwan, October 17-20, 2010

978–1–4244–6015–1/10/$26.00 c© 2010 IEEE81

between the sequences even in presence of CFOs.2) We propose three different design approaches: in the first

one the signals are obtained as a result of optimizing agiven cost function whereas the two others are moreheuristic methods that fulfil the given design criteria.

3) We study the performance of the approximate MLmethod in simulation using the proposed training sig-nals. As a further example of a low complexity es-timation method made feasible by the signal design,we show the performance of the well-known repetitiveslots -based estimation method [11] [12] in case ofmultiple CFOs. Our results show superior performancecompared to signals stemming from the asymptotic CRBconsiderations [4][8].

The paper is organized as follows: In section II, we presentthe signal model for our problem. In section III we reviewthe ML estimator for estimating multiple CFOs. In sectionIV we propose three different approaches for training signaldesign. In section V we review the repetitive slots -based CFOestimator and assess its applicability to multiple CFOs usingthe optimized sequences. Section VI presents our simulationresults and finally, section VII concludes the paper.

II. SIGNAL MODEL

We assume a system consisting of 𝐾 transmitters usingOFDM waveforms and operating on common radio resources.This system could represent e.g. a local area network or a set ofcooperating transmitters as described in the previous section.Coarse slot synchronization between transmitters is assumedsuch that signals coming from different transmitters arrive tothe receiver within the cyclic prefix (CP). After discardingthe CP, the time-domain signal model for received symbol 𝑚taking into account frequency offsets as well as the frequencyselective fading channel is expressed as follows:

𝒓𝑚 =

𝐾∑𝑗=1

𝑒𝚥2𝜋𝜀𝑗𝑚𝑁𝑠+𝐺

𝑁 𝑫𝜀(𝜀𝑗)��(𝑗)

𝑚 ��(𝑗)

𝑚 + ��𝑚 (1)

where 𝚥 =√−1, 𝜀𝑗 is the normalized frequency offset

between transmitter 𝑗 and the receiver and 𝑁𝑠 = 𝑁 + 𝐺,where 𝑁 is the FFT size and 𝐺 is the CP length. Matrix

𝑫𝜀(𝜀𝑗) = diag{𝑒𝚥2𝜋𝜀𝑗

𝑛𝑁

}, 𝑛 = 0, . . . , 𝑁 − 1 and ��

(𝑗)

𝑚 isthe (𝐿 + 1) × 1 channel impulse response vector. Matrix

��(𝑗)

𝑚 is an 𝑁 × (𝐿 + 1) circular convolution matrix, i.e.its columns are cyclic shifts of the transmitted signal vectorwith elements ��(𝑗)𝑚 [⟨𝑛− 𝑙⟩𝑁 ], 𝑛 = 0, . . . , 𝑁 − 1, where ⟨⋅⟩𝑁denotes the modulo-𝑁 operation. We denote these columnvectors as ��(𝑗)

𝑚 (𝑙). Noise is assumed i.i.d. circular complexGaussian, i.e. ��𝑚 ∼ 𝒞𝒩 (0, 𝜎2𝑣𝑰𝑁 ). It is noted that the channelimpulse response is assumed to contain also the impact ofpropagation delay differences such that the propagation delaydifferences and delay spread are both still contained withinthe cyclic prefix.

Arranging the sum in (1) into matrix form, we can also

express the signal as

𝒓𝑚 = ��𝑚(𝜺)��𝑚 + ��𝑚 (2)

��𝑚(𝜺) =[𝑒𝚥𝜙1(𝑚)𝑫𝜀(𝜀1)��

(1)

𝑚 . . . 𝑒𝚥𝜙𝐾(𝑚)𝑫𝜀(𝜀𝐾)��(𝐾)

𝑚

]

𝜙𝑗(𝑚) = 2𝜋𝜀𝑗𝑚𝑁𝑠 +𝐺

𝑁

��𝑚 =[��(1)𝑇

𝑚 . . . ��(𝐾)𝑇

𝑚

]𝑇.

Next, the maximum likelihood estimator for the frequencyoffsets based on this signal model is briefly reviewed [3]-[7].

III. MAXIMUM LIKELIHOOD CFO ESTIMATOR

For maximum likelihood (ML) estimation of the unknownparameters, we assume that training signals are transmit-ted from each transmitter simultaneously and known by thereceiver. Then, following [3]- [7], the (deterministic) MLestimator for the frequency offsets can be derived by firstcalculating the ML solution for the channel vector assum-ing known frequency offsets, and plugging it into the log-likelihood function. From (2), the negative log-likelihood is(for notational simplicity we drop the symbol index 𝑚)

Λ𝑀𝐿(𝜺, ��) = ∣∣𝒓 − ��(𝜺)��∣∣2. (3)

From this we get the cost function to be optimized to get thefrequency offset estimates:

Λ𝑀𝐿(𝜺) = 𝒓𝐻��(𝜺)(��𝐻(𝜺)��(𝜺)

)−1

��𝐻(𝜺)𝒓. (4)

Further simplification done in the earlier studies [3]-[5] followsfrom assuming ��

𝐻(𝜺)��(𝜺) ≈ 𝑰𝐾(𝐿+1). The cost function

(4) may then be re-written as

Λ𝑀𝐿(𝜺) ≈ 𝒓𝐻��(𝜺)��𝐻(𝜺)𝒓 (5)

=

𝐾∑𝑗=1

𝐿∑𝑙=0

∣∣∣∣∣𝑁−1∑𝑛=0

𝑟∗[𝑛]��(𝑗)[⟨𝑛− 𝑙⟩𝑁 ]𝑒𝚥2𝜋𝜀𝑗𝑛𝑁

∣∣∣∣∣2

. (6)

From this it becomes obvious that the originally 𝐾-dimensional optimization problem can be reduced to 𝐾 one-dimensional optimizations, e.g. the frequency offset for 𝑗:thtransmitter follows from

𝜀𝑗 = argmax𝜀

𝐿∑𝑙=0

∣∣∣∣∣𝑁−1∑𝑛=0

𝑟∗𝑚[𝑛]��(𝑗)𝑚 [⟨𝑛− 𝑙⟩𝑁 ]𝑒𝚥2𝜋𝜀

𝑛𝑁

∣∣∣∣∣2

. (7)

Problem here is that, as pointed out in [3] [4], the usedapproximation ��

𝐻(𝜺)��(𝜺) ≈ 𝑰𝐾(𝐿+1) does not necessarily

hold in practice. Consequently, the estimator (7) experiencesperformance loss. In the next section we discuss how thetraining signals should be designed to make the approximationvalid.

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IV. TRAINING SIGNAL DESIGN

In order to decouple the estimation of different CFOs, thetraining signals should have very low cross-correlation even inpresence of frequency offsets, i.e. ��

𝐻(𝜺)��(𝜺) ≈ 𝑰𝐾(𝐿+1)

should remain valid. From this, the correlation between the𝜏𝑖:th cyclic shift of the 𝑖:th training sequence and the 𝜏𝑗 :thcyclic shift of the 𝑗:th training sequence may be written as

𝜒𝑖𝑗(𝜏𝑖, 𝜀𝑖; 𝜏𝑗 , 𝜀𝑗) = ��(𝑖)𝐻(𝜏𝑖)𝑫𝜀(𝜀𝑗 − 𝜀𝑖)��(𝑗)(𝜏𝑗) (8)

= 𝒔(𝑖)𝐻𝑫𝐻𝑑 (𝜏𝑖)Φ𝜀(𝜀𝑗 − 𝜀𝑖)𝑫𝑑(𝜏𝑗)𝒔

(𝑗) (9)

where Φ𝜀(𝜀𝑗 − 𝜀𝑖) = 𝑭𝑫𝜀(𝜀𝑗 − 𝜀𝑖)𝑭𝐻 is the ICI matrix

presenting the leakage from a subcarrier to other subcarriersdue to frequency offsets. Matrix 𝑭 is the Fourier transformmatrix and 𝑫𝑑(𝜏) = diag

{𝑒−𝚥2𝜋

𝑘𝜏𝑁

}, 𝑘 = 0, . . . , 𝑁 − 1.

Vector 𝒔(𝑖) is the frequency-domain training sequence of the𝑖:th transmitter. It is noted here that (8) can be recognized asthe time-frequency cross-correlation i.e. the cross-ambiguityfunction of signals ��(𝑖) and ��(𝑗).

Considering the design of training signals for CFO estima-tion, based on above discussion it is obvious that the signalsshould be designed to have their cross-ambiguity functionas close to ideal as possible. While sequences with idealambiguity function do not exist, it is important to note thatlow cross-correlation is required only within a specific regionof the 𝜏 − 𝜀 plane. In particular, we have assumed that allsignals arrive within the cyclic prefix, hence it is enough thatthe cross-correlation is low with delays up to 𝐺. Similarly,typically limits can be placed also on the maximum expectedCFO. To formalize the design criteria, we write the cross-ambiguity function in a slightly more convenient form:

𝜒𝑖𝑗(𝜏, 𝜀) = ��(𝑖)𝐻𝑫𝜀(𝜀)��(𝑗)(𝜏) (10)

= 𝒔(𝑖)𝐻Φ𝜀(𝜀)𝑫𝑑(𝜏)𝒔(𝑗). (11)

This can be done due to the symmetry properties of the ambi-guity function since we are only interested in the magnitude.The sequences should then be designed as

min𝒔(1),...,𝒔(𝐾)

∣𝜒𝑖𝑗(𝜏, 𝜀)∣2 , ∀(𝜏, 𝜀) ∈ (Ω𝜏 ,Ω𝜀) (12)

s.t. ∣𝜒11(0, 0)∣2 = . . . = ∣𝜒𝐾𝐾(0, 0)∣2 = 1.

Here Ω𝜏 and Ω𝜀 denote the desired part of the 𝜏 − 𝜀 plane,e.g. Ω𝜏 = [−𝐺,𝐺] and Ω𝜀 = [−2𝜀𝑚𝑎𝑥, 2𝜀𝑚𝑎𝑥].

An alternative design target is to design one base sequence𝒔 that has low ambiguity function sidelobes in a larger regionof the 𝜏 − 𝜀 plane, and allocate different time- or frequency-domain cyclic shifts of that base sequence to different trans-mitters. In this case the design criterion reduces to

min𝒔

∣𝜒𝒔𝒔(𝜏, 𝜀)∣2 , ∀(𝜏, 𝜀) ∈ (Ω𝜏 ,Ω𝜀) (13)

s.t. ∣𝜒𝒔𝒔(0, 0)∣2 = 1.

where 𝜒𝒔𝒔(𝜏, 𝜀) is the auto-ambiguity function of sequence𝒔, and Ω𝜏 and Ω𝜀 include now a larger range of time-frequency offsets in order to accommodate the time- and/or

frequency-domain cyclic shifts allocated to the transmitters.For example, in case of time-domain cyclic shifts one coulddefine Ω𝜏 = [−𝐾(𝐺+ 1) + 1,𝐾(𝐺+ 1)− 1] and Ω𝜀 =[−2𝜀𝑚𝑎𝑥, 2𝜀𝑚𝑎𝑥].

The above design criteria ensure that the signals have verylow cross-correlation sidelobes even in presence of time-frequency offsets. This facilitates decoupling the estimationof multiple CFOs into many single CFO estimation problems.Note that this is valid for almost any estimation method, notonly ML, since with the given criteria the overlap between thesignal subspaces of different transmitters is minimum also inpresence of CFOs.

Next, we propose three methods for finding suitable se-quences: first method is based on optimizing the ambiguityfunction within the desired part of the 𝜏 − 𝜀 plane directly,while the other two methods are more heuristic methods thatfulfil the above criteria.

A. Direct optimization of the ambiguity function

As the first method, we propose to optimize the ambiguityfunction in the desired part of the 𝜏 − 𝜀 plane directly. Onemethod for achieving this is to find a sequence that minimizesthe total energy in the ambiguity function sidelobes. This canbe expressed as following constrained optimization problem:

min𝒔

∑𝜏∈Ω𝜏

∑𝜀𝑛∈Ω𝜀

∣∣𝒔𝐻Φ𝜀(𝜀𝑛)𝑫𝑑(𝜏)𝒔∣∣2 (14)

s.t. 𝒔𝐻𝒔 = 𝒙𝐻𝒙 = 1.

Here, Ω𝜏 and Ω𝜀 are dimensioned to fit the time- or frequency-domain cyclic shifts of different transmitters as in (13).

Solving the above optimization problem will produce sig-nals that have non-constant modulus in frequency domain.Since often the same signals would be used also for otherpurposes, e.g. channel estimation, it may be desirable to designthe sequences to have constant modulus property. This isachieved by optimizing phase only, which leads to followingunconstrained problem:

min𝝋

∑𝜏∈Ω𝜏

∑𝜀𝑛∈Ω𝜀

∣∣𝒔𝐻(𝝋)Φ𝜀(𝜀𝑛)𝑫𝑑(𝜏)𝒔(𝝋)∣∣2 . (15)

Here 𝒔(𝝋) = 1√𝑁[𝑒𝚥𝜑1 , . . . , 𝑒𝚥𝜑𝑁 ]

𝑇 and 𝝋 = [𝜑1, . . . , 𝜑𝑁 ]𝑇 .

Within the desired 𝜏 − 𝜀 region, both above cost func-tions are typically highly non-convex in the set of feasiblevectors, hence convex optimization methods do not typicallyconverge to the global optimum. However, it turns out thatalmost any attained minimum yields good results in termsof residual cross-ambiguity function sidelobes. Therefore, itseems enough to run the optimization with several differentinitial values and select the local optimum that provides thesmallest value for the cost function. It should be emphasizedthat this optimization is obviously done offline for examplein system design phase. Hence, no costly optimizations arerequired during online operation. This allows even complexsearches for the optimum signal vectors. See section VI forfurther discussion on the optimization procedure used in thispaper.

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B. IFDMA with guard subcarriers

Interleaved FDMA (IFDMA) has been used in contextof uplink OFDMA for estimation of multiple CFOs sinceit allows utilization of null subcarriers for the estimation[9]. While IFDMA-based sequences are fully orthogonal inabsence of frequency offsets, their cross-correlation suffersgreatly even from small CFOs. However, equation (9) suggestsan enhancement to IFDMA which makes it more robust againstloss of orthogonality due to CFOs.

From the ICI matrix Φ𝜀(𝜀), the power leaked from subcar-rier 𝑘 to subcarrier 𝑙 due to frequency offset 𝜀 is [2]

∣Φ𝑘𝑙(𝜀)∣2 =1

𝑁2

sin2(𝜋(𝜀− 𝑘 + 𝑙))sin2( 𝜋𝑁 (𝜀− 𝑘 + 𝑙)) . (16)

This decays very quickly as ∣𝑘 − 𝑙∣ increases. Hence, most ofthe multiple access interference can be avoided by allowingadditional guard (null) subcarriers between the interleavedsequences. The proposed frequency-domain training signal oftransmitter 𝑖 would thus be constructed from base signal 𝑠(𝑖)𝑏 [𝑘]as

𝑠(𝑖)[𝑘𝑃+(𝑖−1)𝑃

𝐾+𝑚] =

{ √𝑃𝑠

(𝑖)𝑏 [𝑘], 𝑚 = 0

0, 𝑚 = 1, . . . , 𝑃 − 1

where 𝑃 is the spacing in frequency between two consecutivesubcarriers allocated to the same transmitter. For simplicity weassume here that 𝑁 mod 𝑃 = 0.

Typically, 𝑃 should be chosen as an integer multiple of𝐾. This leaves then a minimum of 𝑃/𝐾 − 1 null subcarriersbetween signals from different transmitters. It is noted thatany IFDMA-based signals are repetitive in time domain whichrestricts the choice of 𝑃 and hence the number of guardsubcarriers: The signal repetition period in time domain is𝑁/𝑃 . Therefore, to avoid ambiguous correlation peaks in timewithin the cyclic prefix, 𝑃 should be kept small enough tosatisfy 𝑁/𝑃 > 𝐺. Thus, there is a tradeoff between multipleaccess interference suppression capability and unambiguouscorrelation range in time. Still in typical cases there are enoughdegrees of freedom to suppress cross-ambiguity function side-lobes in the desired 𝜏 − 𝜀 region. Note that normal IFDMAcorresponds to 𝑃 = 𝐾, i.e. without guard subcarriers.

Obviously, further optimization is possible by designingthe base sequences allocated to the transmitters using theprocedures outlined in the previous section.

C. Sequence selection

Sequences that have desirable time-frequency correlationproperties have been widely studied in context of radarwaveform design [13]. There exists sequences that have lowsidelobes at least within certain subregion of the 𝜏 − 𝜀 plane.For example, sequences based on Costas arrays are known tohave very low sidelobes in the ambiguity function over thewhole 𝜏 − 𝜀 plane. However, such sequences are based onfrequency hopping and hence are not suitable for transmissionin one OFDM training symbol. Since we concentrate espe-cially on OFDM systems in this paper, we have chosen fordemonstration purposes to use properly selected Zadoff-Chu

sequences [14]. Zadoff-Chu base sequences of length 𝑁𝑍𝐶 aregiven as

𝑠𝑢[𝑘] =

⎧⎨⎩

exp(𝚥𝜋𝑢𝑘2

𝑁𝑍𝐶

), 𝑁𝑍𝐶 even

exp(𝚥𝜋𝑢𝑘(𝑘+1)𝑁𝑍𝐶

), 𝑁𝑍𝐶 odd

(17)

where 𝑢 is the root index that is relatively prime to 𝑁𝑍𝐶 .Zadoff-Chu sequences have autocorrelation sidelobes of

zero magnitude in absence of frequency offsets. However, theambiguity function of Zadoff-Chu sequences over the whole𝜏 − 𝜀 plane in fact contains several ambiguous correlationpeaks. Hence, multiple access interference is still presentunless the sequences are carefully chosen: With a limitednumber of transmitters it is typically possible to find a basesequence that has ambiguity function sidelobes very close tozero in the desired region (𝜏, 𝜀) ∈ (Ω𝜏 ,Ω𝜀), i.e. the ambigu-ous peaks fall outside of this region. As will be shown insection VI, Zadoff-Chu sequences allow very good estimationperformance provided that the root index and cyclic shifts arechosen such that (13) is satisfied.

V. LOW COMPLEXITY ESTIMATION

With the described methods, the training signals remainnear-orthogonal even in presence of frequency offsets and de-lay spread. It is noted that actually the signals transmitted fromdifferent transmitters remain approximately in their distinctsignal subspaces. This means that the estimators for differ-ent CFOs are decoupled, and certain single CFO estimationmethods become feasible also in case of multiple CFOs.

As an example of such low complexity estimation, herewe generalize the method developed in [11][12] to case ofmultiple CFOs. The method is based on two repetitive slotsin time domain or equivalently, nulling every other subcarrierin frequency domain. Our training signal for transmitter 𝑗 is

then ��(𝑗) =[��(𝑗)𝑇1/2 ��

(𝑗)𝑇1/2

]𝑇in time domain.

We first estimate the timing of the strongest multipathcomponent for transmitter 𝑗 by correlating the received signalwith the corresponding training sequence. Since we alreadyassumed coarse slot synchronization, we get the timing simplyby taking the largest correlation peak as

𝜏𝑗 = argmax𝜏

(��(𝑗)[𝜏 ]⊙ ��(𝑗)∗[𝜏 ]

)(18)

where ��(𝑗)[𝜏 ] = ��(𝑗)𝐻

𝒓 and ⊙ denotes the Hadamardproduct. In other words, we simply run periodic correlationover the cyclic prefix length to get the optimum timing.

For frequency synchronization, we define 𝒓1[𝑛] = 𝒓[𝑛] and𝒓2[𝑛] = 𝒓[𝑛 + 𝑁/2] for 𝑛 = 0, . . . , 𝑁/2 − 1. Further, wedefine

𝑧(𝑗)𝑚 [𝜏𝑗 ] =

𝑁/2−1∑𝑛=0

𝑟𝑚[𝑛]��(𝑗)∗1/2 [⟨𝑛− 𝜏𝑗⟩𝑁 ], 𝑚 = 1, 2. (19)

Then we get the CFO estimate for the 𝑗:th transmitter as

𝜀𝑗 =1

𝜋arg

{𝑧(𝑗)∗1 [𝜏𝑗 ]𝑧

(𝑗)2 [𝜏𝑗 ]

}. (20)

84

It is noted that in this case Ω𝜏 is chosen slightly differentlyfrom ML, e.g. zero delay 𝜏 = 0 is not included in Ω𝜏 . Thisis because in this case a high correlation peak at zero delayis desirable independently of the frequency offset.

VI. SIMULATIONS

We simulated the proposed approaches using OFDM nu-merology adapted from 3GPP LTE to be more suitable forlocal area scenarios. For example, cyclic prefix as well assymbol length were reduced. The main simulation parametersare summarized in table I. We simulated 10000 drops whereon each drop the frequency offsets, channels and propagationdelays were generated randomly. Maximum frequency offsetwas half of the subcarrier spacing. At chosen carrier frequencyof 2 GHz, this corresponds to 15 ppm LO accuracy, whichmay be considered as a lower bound for the performance ofinexpensive LOs (typical values could be around 10 ppm).We assumed that the channel remained static during thetraining symbol period. We simulated both the approximateML estimator (7) as well as the repetitive slot -based estimator(20). The signal-to-noise ratio in our simulations is defined tobe equal to 𝜎−2

𝑣 and we have normalized the channel poweras

∑𝐺𝑙=0 ∣ℎ(𝑗)[𝑙]∣2 = 1 ∀𝑗. Thus, the signal power in the SNR

definition is the received signal power per transmitter. Thisway the impacts of multiple access interference are directlyvisible in the results.

TABLE ISIMULATION PARAMETERS.

Carrier frequency 2.0 GHzFFT size 256Cyclic prefix length 9Subcarrier spacing 60 kHzNumber of transmitters {1, 2, 4, 6}Channel model Extended Pedestrian A

Flat Rayleigh fading

As a baseline training sequence, we used a pseudo-randomQPSK sequence that satisfies 𝐸

{��(𝑖)∗[𝑛]��(𝑗)∗[𝑛]

}= 0, ∀𝑖 ∕=

𝑗 as suggested by the asymptotic CRB considerations [4][8].We then compared this to the proposed signal designs.

As the first case, we found training signals by optimizing(14) and (15). The optimization was done using the gradientdescent method. The unit norm constraint in (14) was takeninto account by using the geometric methods described in [15],i.e. the gradient descent updates at each iteration were donealong the sphere ∥𝒔∥ = 1 in which case no separate normal-izations at each iteration were needed in order to maintain unitnorm. We ran the optimization using 100 different randomlychosen initial vectors, and chose the signal vector that mini-mized the cost (14) or (15), respectively. We also simulatedIFDMA with guard subcarriers using 𝑃 = 16. Finally, we alsoselected an optimized set of Zadoff-Chu sequences using thedescribed ambiguity function considerations. The parametersof our optimized Zadoff-Chu sequences are listed in tableII. Finally, we benchmarked the estimators with proposedsequences also with the Cramer-Rao bound.

Figure 1 shows the performance of the proposed designscompared to PN sequences in case of approximate ML es-

TABLE IIZADOFF-CHU SEQUENCE PARAMETERS USED IN THE SIMULATIONS.

Scheme 𝑁𝑍𝐶 𝑢 Cyclic shifts

Maximum likelihood, K=1,2,4 256 51 {0, 10, 20, 30}Maximum likelihood, K=6 256 85 {0, 10, 20, 30, 40, 50}Repetitive slots 128 43 {0, 10, 20, 30}

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Fig. 1. Performance of the approximate ML estimator with the proposedtraining signal designs compared to the pseudo-random sequences with 𝐾 = 2(left) and 𝐾 = 4 (right) in Extended Pedestrian A channel. The error floor dueto multiple access interference almost disappears with the optimized signals,hence estimation of different CFOs may be decoupled.

timator (7) and 𝐾 = 2 or 𝐾 = 4 in case of frequency-selective Extended Pedestrian A channel. As expected, thePN sequences perform poorly as the large cross-ambiguityfunction sidelobes cause severe multiple access interferencethat causes an error floor in the estimation. With the proposeddesigns, the impact of the error floor is much less severe dueto low ambiguity function sidelobes, and good performance isobtained. Especially with the optimized Zadoff-Chu sequencesthe error floor almost disappears in this (most practical) SNRrange and the performance is very close to the CRB evenwith such low complexity estimator. This is because Zadoff-Chu sequences have almost zero sidelobes in the ambiguityfunction with the exception of several ambiguous correlationpeaks. When the cyclic shifts of the base sequence are properlychosen as we have described, the ambiguous peaks are avoidedand the sidelobes are kept extremely low. Other approachesperform slightly worse than Zadoff-Chu. However, still theperformance is significantly better than with PN sequencesand the error floor starts to be visible only at high SNR. Thenon-constant modulus and constant modulus designs (14) and(15) have only a small difference in performance. IFDMAwith guard subcarriers also performs clearly better than PNsequences. Figure 2 shows the performance of the approxi-mate ML estimator for different numbers of transmitters atSNR=10 dB. From this result, the impact of multiple accessinterference is clearly visible: In case of PN training sequencethe error increases with the number of transmitters, while incase of optimized sequences multiple access interference hasnegligible impact and the error is a few percent of subcarrierspacing which is enough for OFDM.

Figure 3 shows the performance of the repetitive slots -basedestimator with different training signals in case of Extended

85

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Fig. 2. Performance of the approximate ML estimator with different trainingsequences at SNR=10 dB in Extended Pedestrian A channel. With properlydesigned sequences the multiple access interference impact is negligible.

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Fig. 3. Performance of the repetitive slots -based estimator with thesynthesized sequences as well as optimized Zadoff-Chu sequences comparedto the pseudo-random sequences with 𝐾 = 2 (left) and 𝐾 = 4 (right) inExtended Pedestrian A channel. With the optimized sequences the estimationmethod designed for single CFO case becomes applicable also to multipleCFOs as the training signals remain near-orthogonal even in presence of CFOs.

Pedestrian A channel, and figure 4 shows the same in flatRayleigh fading channel. It is obvious from these resultsthat the estimation method is not suitable for estimation ofmultiple CFOs unless the training signals are optimized tominimize multiple access interference as described. In fact,with our proposed approaches to training signal design themethod performs very well. These examples illustrate that lowcomplexity single CFO estimation methods become feasibleand perform very well also in case of multiple CFOs if thesignals are designed to have low cross-ambiguity functionsidelobes with delays and CFOs of interest. It is noted thatthe additional loss compared to CRB in case of frequency-selective channels is because the algorithm inherently onlytakes one channel tap into account.

VII. CONCLUSION

We have addressed design of training signals for estima-tion of multiple frequency offsets in OFDM systems. It wasshown that by designing the signals to have minimum cross-correlation also in presence of CFOs allows one to decouplethe estimation of different CFOs. This facilitates decreasingthe complexity of the estimator significantly as one mayresort to single CFO estimation methods instead of complexalgorithms that attempt to estimate the multiple CFOs jointly.

We proposed three approaches for obtaining signals thathave the required low cross-ambiguity function sidelobes withthe delays and CFOs of interest. The proposed approacheswere shown to yield training signals that have superior es-

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Fig. 4. Performance of the repetitive slots -based estimator with thesynthesized sequences as well as optimized Zadoff-Chu sequences comparedto the pseudo-random sequences with 𝐾 = 2 (left) and 𝐾 = 4 (right) in flatRayleigh fading channel.

timation performance with practical sample sizes comparedto signals implied by asymptotic CRB minimization. Thesedesigns allow low complexity algorithms to be applied at thereceiver side without sacrificing good performance.

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