Carrier frequency offset estimation for multiuser MIMO OFDMuplink using CAZAC sequences : performance and sequenceoptimizationCitation for published version (APA):Wu, Y., Bergmans, J. W. M., & Attallah, S. (2011). Carrier frequency offset estimation for multiuser MIMO OFDMuplink using CAZAC sequences : performance and sequence optimization. Eurosip Journal on WirelessCommunication and Networking, 2011, 570680-1/11. https://doi.org/10.1155/2011/570680

DOI:10.1155/2011/570680

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Hindawi Publishing CorporationEURASIP Journal on Wireless Communications and NetworkingVolume 2011, Article ID 570680, 11 pagesdoi:10.1155/2011/570680

Research Article

Carrier Frequency Offset Estimation forMultiuser MIMO OFDM Uplink Using CAZAC Sequences:Performance and Sequence Optimization

Yan Wu,1 J. W. M. Bergmans,1 and Samir Attallah2

1 Signal Processing Systems Group, Department of Electrical Engineering, Technische Universiteit Eindhoven, P.O. Box 513,5600 MB Eindhoven, The Netherlands

2 School of Science and Technology, SIM University, Singapore 599491

Correspondence should be addressed to Yan Wu, y.w.wu@tue.nl

Received 12 November 2010; Accepted 15 February 2011

Academic Editor: Claudio Sacchi

Copyright © 2011 Yan Wu et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper studies carrier frequency offset (CFO) estimation in the uplink of multi-user multiple-input multiple-output(MIMO) orthogonal frequency division multiplexing (OFDM) systems. Conventional maximum likelihood estimator requirescomputational complexity that increases exponentially with the number of users. To reduce the complexity, we propose a sub-optimal estimation algorithm using constant amplitude zero autocorrelation (CAZAC) training sequences. The complexity ofthe proposed algorithm increases only linearly with the number of users. In this algorithm, the different CFOs from differentusers destroy the orthogonality among training sequences and introduce multiple access interference (MAI), which causes anirreducible error floor in the CFO estimation. To reduce the effect of the MAI, we find the CAZAC sequence that maximizes thesignal to interference ratio (SIR). The optimal training sequence is dependent on the CFOs of all users, which are unknown. Tosolve this problem, we propose a new cost function which closely approximates the SIR-based cost function for small CFO valuesand is independent of the actual CFOs. Computer simulations show that the error floor in the CFO estimation can be significantlyreduced by using the optimal sequences found with the new cost function compared to a randomly chosen CAZAC sequence.

1. Introduction

Compared to single-input single-output (SISO) systems,multiple-input multiple-output (MIMO) systems increasethe capacity of rich scattering wireless fading channelsenormously through employing multiple antennas at thetransmitter and the receiver [1, 2]. Orthogonal FrequencyDivision Multiplexing (OFDM) is a widely used technologyfor wireless communication in frequency selective fadingchannels due to its high spectral efficiency and its ability to“divide” a frequency selective fading channel into multipleflat fading subchannels (subcarriers). Hence, MIMO-OFDMis an ideal combination for applying MIMO technologyin frequency fading channels and has been included invarious wireless standards such as IEEE 802.11n [3] and IEEE802.16e [4]. An extension of the MIMO-OFDM system is themultiuser MIMO-OFDM system as illustrated in Figure 1.

In such a system, multiple users, each with one or multipleantennas, transmit simultaneously using the same frequencyband. The receiver is a base-station equipped with multipleantennas. It uses spatial processing techniques to separatethe signals of different users. If we view the signals fromdifferent users as signals from different transmit antennas ofa virtual transmitter, then the whole system can be viewedas a MIMO system. This system is also known as the virtualMIMO system [5].

Carrier frequency offset (CFO) is caused by the Dopplereffect of the channel and the difference between the trans-mitter and receiver local oscillator (LO) frequencies. InOFDM systems, CFO destroys the orthogonality betweensubcarriers and causes intercarrier interference (ICI). Toensure good performance of OFDM systems, the CFOmust be accurately estimated and compensated. For SISO-OFDM systems, periodic training sequences are used in

2 EURASIP Journal on Wireless Communications and Networking

User 1

User 2

User nt

· · ·

. . .

Virtual multiantennatransmitter

Base-station

Figure 1: Overview of multiuser MIMO-OFDM systems.

[6, 7] to estimate the CFO. It is shown that these CFOestimators reach the Cramer-Rao bound (CRB) with low-computational complexity. A similar idea was extended tocollocated MIMO-OFDM systems [8–10], where all thetransmit antennas are driven by a centralized LO and soare all the receive antennas. In this case, the CFO is stilla single parameter. For multiuser MIMO-OFDM systems,each user has its own LO, while the multiple antennas atthe base-station (receiver) are driven by a centralized LO.Therefore, in the uplink, the receiver needs to estimatemultiple CFO values for all the users. In [11, 12], methodswere proposed to estimate multiple CFO values for MIMOsystems in flat fading channels. In [13], a semiblind methodwas proposed to jointly estimate the CFO and channelfor the uplink of multiuser MIMO-OFDM systems infrequency selective fading channels. An asymptotic Cramer-Rao bound for joint CFO and channel estimation in theuplink of MIMO-Orthogonal Frequency Division MultipleAccess (OFDMA) system was derived in [14] and trainingstrategies that minimize the asymptotic CRB were studied. In[15], a reduced-complexity CFO and channel estimator wasproposed for the uplink of MIMO-OFDMA systems using anapproximation of the ML cost function and a Newton searchalgorithm. It was also shown that the reduced-complexitymethod is asymptotically efficient. The joint CFO andchannel estimation for multiuser MIMO-OFDM systemswas studied in [16]. Training sequences that minimize theasymptotic CRB were also designed in [16].

It is known in the literature that the computationalcomplexity for obtaining the ML CFO estimates in the uplinkof multiuser MIMO-OFDM system grows exponentially withthe number of users [15, 16]. A low-complexity algorithmwas proposed in [16] for CFO estimation in the uplinkof multiuser MIMO OFDM systems based on importancesampling. However, the complexity required to generatesufficient samples for importance sampling may still be highfor practical implementations. In this paper, we study algo-rithms that can further reduce the computational complexityof the CFO estimation. Following a similar approach as

in [17], we first derive the maximum likelihood (ML)estimator for the multiple CFO values in frequency selectivefading channels. Obtaining the ML estimates requires asearch over all possible CFO values and the computationalcomplexity is prohibitive for practical implementations. Toreduce the complexity, we propose a sub-optimal algorithmusing constant amplitude zero autocorrelation (CAZAC)training sequences, which have zero autocorrelation for anynonzero circular shifts. Using the proposed algorithm, theCFO estimates can be obtained using simple correlationoperations and the complexity of this algorithm grows onlylinearly with the number of users. However, the multipleCFO values destroy the orthogonality between the trainingsequences of different users. This introduces multiple accessinterference (MAI) and causes an irreducible error floor inthe mean square error (MSE) of the CFO estimates. Wederive an expression for the signal to interference ratio (SIR)in the presence of multiple CFO values. To reduce the MAI,we find the training sequence that maximizes the SIR. Theoptimal training sequence turns out to be dependent on theactual CFO values from different users. This is obviously notpractical as it is not possible to know the CFO values andhence select the optimal training sequence in advance. Toremove this dependency, we propose a new cost function,which is the Taylor’s series approximation of the original costfunction. The new cost function is independent of the actualCFO values and is an accurate approximation of the originalSIR-based cost function for small CFO values. Using the newcost function, we obtain the optimal training sequences forthe following three classes of CAZAC sequences:

(i) Frank and Zadoff Sequences [18],

(ii) Chu Sequences [19],

(iii) Polyphase Sequence by Sueshiro and Hatori (S&HSequences) [20].

Both Frank and Zadoff sequences and S&H sequences existfor sequence length of N = K2, where N is the length of thesequence and K is a positive integer, while Chu sequencesexist for any integer length. For both Frank and Zadoff andChu sequences, there are a finite number of sequences foreach sequence length. Therefore, the optimal sequence can beobtained using a search among these sequences. However, forS&H sequences, there are infinitely many possible sequences.As the optimization problem for S&H sequences cannotbe solved analytically, we resort to a numerical method toobtain a near-optimal solution. To this end, we use theadaptive simulated annealing (ASA) technique [21]. Forsmall sequence lengths, for example, N = 16 and N = 36,we are able to use exhaustive search to verify that the solutionobtained using ASA is globally optimal. (Because CFO valuesare continuous variables, theoretically, it is not possibleto obtain the exact optimum using exhaustive computersearch, which works in discrete variables. If we keep the stepsize in the search small enough, we can be sure that theobtained “optimum” is very close to the actual optimumand can be practically assumed to the actual optimum. Inthis way, we are able to verify the solution obtained bythe ASA is “practically” optimal.) Computer simulations

EURASIP Journal on Wireless Communications and Networking 3

were conducted to evaluate the performance of the CFOestimation using CAZAC sequences. We first compare theperformance using CAZAC sequences with the performanceusing two other sequences with good correlation properties,namely, the IEEE 802.11n short training field (STF) [3] andthe m sequences [22]. The results show that the error floorusing the CAZAC sequences is more than 10 times smallercompared to the other two sequences. Comparing the threeclasses of CAZAC sequences, we find that the performanceof the Chu sequences is better than the Frank and Zadoffsequences due to the larger degree of freedom in the sequenceconstruction. The S&H sequences have the largest numberof degree of freedom in the construction of the CAZACsequences. However, the simulation results show that theyhave only very marginal performance gain compared to theChu sequences. This makes Chu sequences a good choicefor practical implementation due to its simple constructionand flexibility in sequence lengths. By using the identifiedoptimal sequences, the error floor in the CFO estimation issignificantly lower compared to using a randomly selectedCAZAC sequence.

The rest of the paper is organized as follows. In Section 2,we present the system model and derive the ML estimator forthe multiple CFO values. The sub-optimal CFO estimationalgorithm using CAZAC sequences is proposed in Section 3.The training sequence optimization problem is formulatedin Section 4 and methods are given to obtain the optimaltraining sequence. In Section 5, we present the computersimulation results and Section 6 concludes the paper.

2. System Model

In this paper, we study a multiuser MIMO-OFDM systemwith nt users. For simplicity of illustration and analysis, weassume that each user has a single transmit antenna. Thebase-station has nr receive antennas, where nr ≥ nt . Thereceived signal at the ith receive antenna can be written as

ri(k) =nt∑

m=1

⎛⎝e jφmk

L−1∑

d=0

hi,m(d)sm(k − d)

⎞⎠ + ni(k), (1)

where φm is the CFO of the mth user, k is the time index, andL is the number of multipath components in the channel.The dth tab of the channel impulse response between themth user and the ith receive antenna is denoted as hi,m(d),sm denotes the transmitted signal from the mth user and ni isthe additive white Gaussian noise at the ith receive antenna.Here we assume the initial phase for each user is absorbed inthe channel impulse response. From (1), we can see that wehave nt different CFO values (φm’s) to estimate. We consider atraining sequence of lengthN and cyclic prefix (CP) of lengthL. The received signal after removal of CP can be written inan equivalent matrix form

ri =nt∑

m=1

E(φm)

Smhi,m + ni, (2)

where ri = [ri(0), . . . , ri(N − 1)]T and superscript T denotesvector transpose. The CFO matrix of userm is denoted E(φm)

and is a diagonal matrix with diagonal elements equal to[1, exp( jφm), . . . , exp( j(N − 1)φm)]. We use Sm to denotethe transmitted signal matrix for the mth user, which is anN × N circulant matrix with the first column defined by[sm(0), sm(1), sm(2), . . . , sm(N − 1)]T . Here we assume N > Lso the channel vector between the mth user and the ithreceive antenna hi,m is anN×1 vector by appending the L×1channel impulse response [hi,m(0), . . . ,hi,m(L − 1)]T vectorwith N − L zeros.

Using this system model, the received signals from all nrreceive antennas can be written as

R =A(φ)H + N , (3)

where

R = [r1, . . . , rnr]N×nr ,

A(φ) = [E(φ1

)S1, . . . , E

(φnt)

Snt]N×(N×nt).

(4)

For clearness of presentation, we use subscripts under thesquare bracket to denote the size of the correspondingmatrix. The vector φ = [φ1, . . . ,φnt ] is the CFO vectorcontaining the CFO values from all users, and the channelsof all users are stacked into the channel matrix H given as

H =

⎡⎢⎢⎢⎢⎣

H1

...

Hnt

⎤⎥⎥⎥⎥⎦

(N×nt)×nr

, (5)

with Hi = [h1,i, . . . , hnr ,i]N×nr being the channel matrix forthe ith user. The noise matrix is given by N = [n1, . . . , nnr ].

Because the noise is Gaussian and uncorrelated, thelikelihood function for the channel H and CFO values φ canbe written as

Λ(H , φ

)= 1(πσ2

n

)N×nr exp

{− 1σ2n

∥∥∥R −A(φ)H∥∥∥

2}

,

(6)

where H and φ are trial values for H and φ and σ2n is the

variance of the AWGN noise. Following a similar approach asin [17], we find that for a fixed CFO vector φ, the ML estimateof the channel matrix is given by

H(φ)=[AH

(φ)A(φ)]−1

AH(φ)R, (7)

where superscript H denotes matrix Hermitian. Substituting(7) into (6) and after some algebraic manipulations, weobtain that the ML estimate of the CFO vector φ is given by

φ = arg maxφ

{tr(RHB

(φ)R)}

, (8)

with

B(φ)=A

(φ)[

AH(φ)A(φ)]−1

AH(φ)

, (9)

and tr(•) denotes the trace of a matrix. To obtain the MLestimate of the CFO vector φ, a search needs to be performedover the possible ranges of CFO values of all the users.The complexity of this search grows exponentially with thenumber of users and hence the search is not practical.

4 EURASIP Journal on Wireless Communications and Networking

3. CAZAC Sequences for MultipleCFOs Estimation

To reduce the complexity of the CFO estimation for mul-tiuser MIMO-OFDM systems, in this section, we propose asub-optimal algorithm using CAZAC sequences as trainingsequences. CAZAC sequences are special sequences with con-stant amplitude elements and zero autocorrelation for anynonzero circular shifts. This means for a length-N CAZACsequence, we have s(n) = exp( jθn) and the auto-correlation

R(k) =N∑

n=1

s(n)s∗(n� k) =⎧⎨⎩N , k = 0,

0, k /= 0,(10)

for all values of k = 0, 1, . . . ,N − 1. Here we use � todenote circular subtraction. Let S be a circulant matrixwith the first column equal to [s(0), s(1), . . . , s(N − 1)]T .The autocorrelation property of CAZAC sequences can bewritten in equivalent matrix form as

SHS = NIN , (11)

where IN is the identity matrix of size N × N . This meansthat S is both a unitary (up to a normalization factor of N)and a circulant matrix.

In [23], we showed that for collocated MIMO-OFDMsystems, using CAZAC sequences as training sequencesreduces overhead for channel estimation while achievingCramer Rao Bound (CRB) performance in the CFO esti-mation. Here, we extend the idea to the estimation ofmultiple CFO values in the uplink of multiuser MIMO-OFDM systems. Let the training sequence of the first userbe s1. The training sequence of the mth user is the cyclicshifted version of the first user, that is, sm(n) = [s1(n�τm)]T ,where τm denotes the shift value. It is straightforward to showthat the training sequences between different users have thefollowing properties.

(i) The autocorrelation of the training sequence for theith user satisfies

SHi Si = NIN , (12)

for i = 1, . . . ,nt .

(ii) The cross correlation between training sequences ofthe ith and jth users satisfies

SHi S j = N�τj−τi , (13)

where �τj−τi denotes a matrix which results fromcyclically shifting the one elements of the identifymatrix to the right by τj − τi positions.

For SISO-OFDM systems, an efficient CFO estimationtechnique is to use periodic training sequences [6, 7]. Inthis paper, we extend the idea to multiuser MIMO-OFDMsystems. In this case, each user transmit two periods ofthe same training sequences and the received signal overtwo periods can be written as (We assume here timing

synchronization is perfect. We also assume a cyclic prefixwith length L is appended to the training sequence duringtransmission and removed at the receiver.)

R =⎡⎣

E(φ1)

S1 · · · E(φnt)

Snt

e jNφ1 E(φ1)

S1 · · · e jNφnt E(φnt)

Snt

⎤⎦H + N .

(14)

Without loss of generality, we show how to estimate the CFOof the first user and the same procedure is applied to all theother users to estimate the other CFO values. Since sameprocedure is applied to all the users, the complexity of thisCFO estimation method increases linearly with the numberof users.

We first consider a special case when there are no CFOsfor all the other uses except user one, that is, φm = 0 form = 2, . . . ,nt . In this case, we cross correlate the trainingsequence of the first user with the received signal as shownbelow

Y′1 =W1R

=⎡⎣

SH1 0

0 SH1

⎤⎦⎡⎣

E(φ1)

S1 · · · Snt

e jNφ1 E(φ1)

S1 · · · Snt

⎤⎦H + N

′

=

⎡⎢⎢⎢⎢⎣

SH1 E(φ1)

S1H1 +nt∑

m=2

SH1 SmHm

ejNφ1 SH1 E(φ1)

S1H1 +nt∑

m=2

SH1 SmHm

⎤⎥⎥⎥⎥⎦

+ N′

=

⎡⎢⎢⎢⎢⎣

SH1 E(φ1)

S1H1 +nt∑

m=2

�τmHm

ejNφ1 SH1 E(φ1)

S1H1 +nt∑

m=2

�τmHm

⎤⎥⎥⎥⎥⎦

+ N′.

(15)

Because �τm is a matrix resulting from cyclic shifting theidentity matrix to the right by τm elements, �τmHm producesa matrix resulting by cyclic shifting the rows of Hm by τmelements downwards.

We make sure that the cyclic shift between the m − 1thand mth users is not smaller than the length of the channelimpulse response, that is, τm − τm−1 ≥ L. Since the channelhas only L multipath components, only the first L rows inthe N × nr matrix Hm are nonzero. Therefore, �τmHm hasall zero elements in the first L rows when τm − τm−1 ≥ Lfor m = 2, . . . ,nt and N − τnt ≥ L (notice that to ensurethese conditions hold, we need to have the training sequencelength N ≥ ntL). Hence, the first L rows of Y1 will be free ofthe interference from all the other users. Let us define IL asthe first L rows of the N ×N identity matrix; we have

Y1 =⎡⎣IL 0

0 IL

⎤⎦Y′

1 =⎡⎣

ILSH1 E(φ1)

S1H1

e jNφ1 ILSH1 E(φ1)

S1H1

⎤⎦ + N ′′.

(16)

The multiplication of IL is to select the first L rows fromthe matrix SH1 E(φ1)S1H1. Because the CFOs of all the other

EURASIP Journal on Wireless Communications and Networking 5

users are 0, the shift orthogonality between their trainingsequences and user 1’s training sequence is maintained. Inthis case, Y1 is free of interferences from the other users.Following the similar approach as in [23], we can show thatthe ML estimate of user 1’s CFO given Y1 can be obtained as

φ1 = 1N�

⎧⎨⎩

L∑

k=1

nr∑

m=1

Y∗1 (k,m)Y1(k +N ,m)

⎫⎬⎭, (17)

where �(•) denotes the angle of a complex number. Thecomputational complexity of this estimator is low.

When the other users’ CFO values are not zero, Y1 isgiven by

Y1 =⎡⎣

ILSH1 E(φ1)

S1H1

e jNφ1 ILSH1 E(φ1)

S1H1

⎤⎦

+

⎡⎢⎢⎢⎢⎣

IL

nt∑

m=2

SH1 E(φm)

SmHm

IL

nt∑

m=2

e jNφm SH1 E(φm)

SmHm

⎤⎥⎥⎥⎥⎦

+ N ′′

=⎡⎣

ILSH1 E(φ1)

S1H1

e jNφ1 ILSH1 E(φ1)

S1H1

⎤⎦ + V + N ′′.

(18)

From (18), we can see that the orthogonality between thetraining sequences from different users is destroyed by thenon-zero CFO values φm. As a result, there is an extraMultiple Access Interference (MAI) term V in the correlationoutput Y1. This interference is independent of the noise andtherefore it will cause an irreducible error floor in MSE of theCFO estimator in (17). The covariance matrix of the MAI canbe expressed as

E{VVH

}= E

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

⎡⎢⎢⎢⎢⎢⎣

IL

nt∑

m=2

SH1 E(φm)

SmHm

IL

nt∑

m=2

e jNφm SH1 E(φm)

SmHm

⎤⎥⎥⎥⎥⎥⎦

×⎡⎣

nt∑

m=2

HHm SHmEH

(φm)

S1IHL ,

nt∑

m=2

e− jNφmHHm SHmEH

(φm)

S1IHL

⎤⎦

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭.

(19)

We assume the channels between different transmit andreceive antennas are uncorrelated in space and different pathsin the multipath channel are also uncorrelated. We definepi,m = [pi,m(0), . . . , pi,m(L − 1), 0, . . . 0]T(N×1) as the power

delay profile (PDP) of the channel between the mth user andthe ith receive antenna and we have

E{HmHH

n

}=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, m /= n,

diag

⎛⎝

nr∑

i=1

pi,m

⎞⎠, n = m.

(20)

Defining Pm = diag(∑nr

i=1 pi,m), we can rewrite the covariancematrix of the interference as

E{VVH

}=⎡⎣

C D

DH C

⎤⎦, (21)

where

C = IL

⎧⎨⎩

nt∑

m=2

SH1 E(φm)

SmPmSHmEH(φm)

S1

⎫⎬⎭IH

L ,

D = IL

⎧⎨⎩

nt∑

m=2

e− jN2φm SH1 E(φm)

SmPmSHmEH(φm)

S1

⎫⎬⎭IH

L .

(22)

We can see that the interference power is a function of thetraining sequence Sm, the channel delay power profile Pm,and the CFO matrices E(φm).

4. Training Sequence Optimization

In the previous section, we showed that the multipleCFO values destroy the orthogonality among the trainingsequences of different users and introduces MAI. In thissection, we study how to find the training sequence such thatthe signal to interference ratio (SIR) is maximized.

4.1. Cost Function Based on SIR. From the signal model in(18), we can define the SIR of the first user as

SIR1 =tr[IL{

SH1 E(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]

tr[IL{∑nt

m=2 SH1 E(φm)

SmPmSHmEH(φm)

S1

}IHL

] .

(23)

From the denominator of (23), we can see that the totalinterference power depends on the CFO values φm of allthe other users. As a result, the optimal training sequencethat maximizes the SIR is also dependent on φm for m =1, . . . ,m. In this case, even if we can find the optimal trainingsequences for different values of φm, we still do not knowwhich one to choose during the actual transmission as thevalues φm are not available before transmission. This makes(23) an unpractical cost function.

Let us look at user 1 again. In the absence of the CFO,the signal from user 1 is contained in the first L rowsof the received signal Y1. When the CFO is present, suchorthogonality is destroyed and some information from user1 will be “spilled” to the other rows of Y1, thus causinginterference to the other users. For user 1, therefore, to keep

6 EURASIP Journal on Wireless Communications and Networking

the interference to the other users small, such “spilled” signalpower should be minimized. On the other hand, the usefulsignal we used to estimate the CFO of user 1 is containedin the first L rows of Y1 and such signal power shouldbe maximized. Therefore, considering user 1 alone, we candefine the signal to “spilled” interference (to other users)ratio for user 1 as

SIR1 =tr[IL{

SH1 E(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]

tr[IL{

SH1 E(φ1)

S1P1SH1 EH(φ1)

S1

}IL

H] , (24)

where IL is the complement of IL, that is, IL is the lastN−Lrows of the N ×N identity matrix.

The denominator in (24) can be expressed as

tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IL

H]

= N tr[

S1P1SH1]

− tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IL

H]

= N 2 tr[P1]−tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IL

H].

(25)

Substituting this into (24), we have

SIR1 =tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]

N 2 tr[P1]− tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

] .

(26)

Now we can define the training sequence optimizationproblem as

Sopt = arg maxS1

SIR′1

= arg maxS1

tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]

�− tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]

= arg minS1

�− tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]

tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]

= arg minS1

⎧⎨⎩

�

tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]−1

⎫⎬⎭

= arg maxS1

{tr[IL

{SH1 E

(φ1)

S1P1SH1 EH(φ1)

S1

}IHL

]},

(27)

where � denotes N 2 tr[P1].From (27), we can see that the optimal training sequence

depends on the power delay profile P1 and the actual CFOvalue φ1. The channel delay profile is an environment-dependent statistical property that does not change veryfrequently. Therefore, in practice, we can store a few trainingsequences for different typical power delay profiles at thetransmitter and select the one that matches the actual

Table 1: Number of possible Frank-Zadoff and Chu sequences fordifferent sequence lengths.

N Frank-Zadoff Sequence Chu Sequence

16 2 8

36 2 12

64 4 32

channel delay profile. On the other hand, it is impossibleto know the actual CFO φ in advance to select the optimaltraining sequence. In the following, we will propose a newcost function based on SIR approximation which can removethe dependency on the actual CFO φ1 in the optimization.

4.2. CFO Independent Cost Function. Let us assume that theCFO value φ is small. In this case, we can approximate theexponential function in the original cost function by its first-order Taylor series expansion, that is, exp( jφ) ≈ 1 + jφ.Therefore, we have

E(φ1) ≈ IN + jφ1N, (28)

where N is a diagonal matrix given by N = diag[0, 1,2, . . . ,N − 1]. Using this approximation, we get

SHE(φ)

SPSHEH(φ)

S ≈ SH(

I + jφN)

SPSH(

I− jφN)

S

= P + jφSHNSP− jφPSHNS

+ φ2SHNSPSHNS.(29)

Here we omitted the subscript 1 for the clearness of thepresentation. Therefore, the optimization problem can beapproximated as

Sopt = arg maxS

{tr[IL

(P + jφSHNSP− jφPSHNS

+φ2SHNSPSHNS)IHL

]}.

(30)

Notice that the first term P in the summation is independentof S and hence can be dropped. It can be shown that thediagonal elements of the second term jφSHNSP are constantand independent of S. Therefore, tr[IL( jφSHNSP)IH

L ] isalso independent of S and hence can be dropped fromthe cost function. The same applies to the third term− jφPSHNS, which is the conjugate of the second term.Therefore, the final form of the optimization using Taylor’sseries approximation can be written as

Sopt = arg maxS

{tr[IL

(SHNSPSHNS

)IHL

]}. (31)

The advantage of (31) is that the optimization problem isindependent of the actual CFO value φ as long as the value ofφ is small enough to ensure the accuracy of the Taylor’s seriesapproximation in (28).

Now we look at how we can obtain the optimal CAZACtraining sequences for the cost function (31). In particular,

EURASIP Journal on Wireless Communications and Networking 7

we look at three classes of CAZAC sequences, namely, theFrank-Zadoff sequences [18], the Chu sequences [19], andthe S&H sequences [20]. The Frank-Zadoff sequences existfor sequence length N = K2 where K is any positive integer.For N = 16, all elements of the Frank-Zadoff sequences areBPSK symbols while for N = 64, all elements are BPSK andQPSK symbols. Therefore, the advantage of the Frank-Zadoffsequences is that they are simple for practical implemen-tation. The disadvantage is that there are limited numbersof sequences available for each sequence length as shown inTable 1. The advantage of Chu sequences is that the lengthof the sequence can be an arbitrary integer N . Compared toFrank-Zadoff sequences, there are more sequences availablefor the same sequence length as shown in Table 1. For bothFrank-Zadoff and Chu sequences, there are a finite numberof possible sequences for each N . The optimal sequence canbe found by using a computer search using the cost function(31). The S&H sequences only exist for sequence length N =K2. The sequences are constructed using a sizeK phase vector

exp( jθ) = [e jθ1 , . . . , e jθK ]T

. Therefore, the optimization oftraining sequence S is equivalent to the optimization on thephase vector θ given by

θ = arg maxθ

{J(θ)}

with

J(θ)= tr

[IL

(SH(θ)

NS(θ)

PSH(θ)

NS(θ)IHL

)].

(32)

Notice that this is an unconstrained optimization problemand each element of the phase vector can take any valuesin the interval [0, 2π). From the construction of the S&Hsequence [20], it can be easily shown that S(θ+ψ) = e jψS(θ),where θ + ψ = [θ1 + ψ, . . . , θK + ψ]T . Hence, from (32), wecan get J(θ) = J(θ + ψ). By letting ψ = −θ1, the originaloptimization problem over the K-dimension phase vectorθ = [θ1, θ2, . . . , θK ]T can be simplified to the optimizationover a (K−1)-dimension phase vector θ′ = [0, θ′1, . . . , θ′K−1]T

where θ′k = θk+1 − θ1.There are an infinite number of possible S&H sequences

for each sequence length; it is impossible to use exhaustivecomputer search to obtain the optimal sequence. We resort tonumerical methods and use the adaptive simulated annealing(ASA) method [21] to find a near-optimal sequence. To testthe near-optimality of the sequence obtained using the ASA,for smaller sequence lengths of N = 16 and N = 36, we useexhaustive computer search to obtain the globally optimalS&H sequence. The obtained sequence through computersearch is consistent with the sequence obtained using ASAand this proves the effectiveness of the ASA in approachingthe globally optimal sequence.

5. Simulation Results

In this section, we use computer simulations to studythe performance of the CFO estimation using CAZACsequences and demonstrate the performance gain achievedby using the optimal training sequences. In the simulations,we assume a multiuser MIMO-OFDM systems with twousers. (In multiuser MIMO-OFDM systems, the number

0 5 10 15 20 25 3010−5

10−4

10−3

10−2

SNR (dB)

Nor

mal

ized

MSE

802.11n STFCAZAC sequencesSingle-user CRB

Figure 2: MSE of CFO estimation usingN = 32 Chu sequences andIEEE 802.11n STF for uniform power delay profile.

CAZAC sequences

0 5 10 15 20 25 30

SNR (dB)

10−5

10−4

10−3

10−2

Nor

mal

ized

MSE

m sequences

Single-user CRB

Figure 3: Comparison of CFO estimation using N = 31 Chusequences and m sequence for uniform power delay profile.

of receive antennas has to be no less than the numberof transmit antennas from all users. Due to the practicallimitations, it is not possible to implement too many base-station antennas. Therefore, to accommodate more users, themultiuser MIMO-OFDM systems can be used in conjunctionwith other multiple access schemes such as TDMA andFDMA.) Each user has one transmit antenna and the base-station has two receive antennas. We simulate an OFDMsystem with 128 subcarriers. The CFO is normalized withrespect to the subcarrier spacing. Unless otherwise stated, theactual CFO values for the two users are modeled as random

8 EURASIP Journal on Wireless Communications and Networking

5 10 15 20 25 30 35 40 45 5010−8

10−7

10−6

10−5

10−4

10−3

10−2

Opt. Chu sequence

SNR (dB)

Nor

mal

ized

MSE

Opt. Frank-Zadoff sequence Random-selected sequenceSingle-user CRB

Opt. S & H sequence

Figure 4: Comparison of CFO estimation using different N = 36CAZAC sequences for L = 18 channel for uniform power delayprofile.

variables uniformly distributed between [−0.5, 0.5]. Themean square error (MSE) of the CFO estimation is definedas

MSE = 1Ns

Ns∑

i=1

(φ− φ2π/M

)2

, (33)

where φ and φ represent the estimated and true CFO’s,respectively, M is the number of subcarriers, and Ns denotesthe total number of Monte Carlo trials.

First we compare the performance of CFO estimationusing CAZAC sequences with the following two sequenceswhich also have good autocorrelation properties:

(1) IEEE 802.11n short training field [3],

(2) m sequences [22].

In the simulations, we use the 802.11n STF for 40 MHzoperations which has a length of 32. For the m sequence, weuse a sequence length of 31. To provide a fair comparison,we compare the performance using the 802.11n STF with alength-32 Chu (CAZAC) sequence generated by [19]

s(n) = exp

[jπ

(n− 1)2

N

], (34)

and we compare the performance with the m sequence usinga length-31 Chu sequence generated by [19]

s(n) = exp[jπ

(n− 1)nN

]. (35)

The performance of CFO estimation using the 802.11n STFand N = 32 Chu sequence is shown in Figure 2. Here we

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Nor

mal

ized

MSE

5 10 15 20 25 30 35 40 45 50

SNR (dB)

Opt. Chu sequenceOpt. Frank-Zadoff sequence Random-selected sequence

Single-user CRBOpt. S & H sequence

Figure 5: Comparison of CFO estimation using different N = 36CAZAC sequences for L = 18 channel for exponential power delayprofile.

5 10 15 20 25 30 35 40 45

SNR (dB)

010−6

10−5

10−4

10−3

10−2

Nor

mal

ized

MSE

Opt. Chu sequence N = 36Opt. Chu sequence N = 49Opt. Chu sequence N = 64

Figure 6: Comparison of CFO estimation using different length ofoptimal Chu sequences for L = 18 channel for uniform power delayprofile.

use 16-tab multipath channels and the circular shift betweenthe training sequences of the two users τ2 = 16. To gaugethe performance of the CFO estimation, we also included thesingle-user CRB in the comparison. The single-user CRB isobtained by assuming no MAI and can be shown to be [24]

CRB = M2

4π2nrN 3γ, (36)

EURASIP Journal on Wireless Communications and Networking 9

100

101

102

103A

mpl

itu

deN = 36, L = 18

User 1User 2

10 20 30

k

10−1

10−2

(a)

100

101

102

103

Am

plit

ude

User 1User 2

10 20 30 40 50 60

k

10−1

10−2

N = 64, L = 18

(b)

Figure 7: Comparison of useful signal and interference power for different sequence lengths (uniform power delay profile).

where γ is the SNR per receive antenna and M is the numberof subcarriers. From the results, we can see that the CFOestimation using the 802.11n STF has a very high errorfloor above MSE of 10−3. The performance using CAZACsequences is much better. In low to medium SNR regions, theperformance is very close to the single-user CRB. An errorfloor starts to appear at SNR of about 25 dB. The error flooris around 100 times smaller compared to the error floor usingthe 802.11n STF.

The performance of the CFO estimation using the N =31 m sequence and Chu sequence is shown in Figure 3. Hereto satisfy the condition of N ≥ ntL, we use 15-tab multipathfading channels and the circular shift between user 1 and2’s training sequence is also set to 15. Again using CAZACsequences leads to a much better performance. We can seethat in low to medium SNR regions, their performance isvery close to the single-user CRB. The error floor usingCAZAC sequences is more than 10 times smaller than thatusing the m sequence.

The performance of CFO estimation using differentCAZAC sequences is compared in Figure 4. Here we fixthe sequence length to 36 and the multipath channel hasL = 18 tabs with uniform power delay profile. Comparingthe performances of optimal Chu sequence and the optimalFrank-Zadoff sequence, we can see that the error floor ofthe Chu sequence is smaller. This is because there are morepossible Chu sequences compared to Frank-Zadoff sequencesand hence more degrees of freedom in the optimization.However, comparing the performance of optimal Chusequence with that of the optimal S&H sequence, we can seethat the additional degrees of freedom in the S&H sequence

do not lead to significant performance gain. Compared to theperformance using a randomly selected CAZAC sequence,we can see that the error floor using an optimized sequenceis significantly smaller. Simulations were also performed inmultipath channels with exponential power delay profileand root mean square delay spread equal to 2 samplingintervals. The other simulation parameters are the same asin the uniform power delay profile simulations. Simulationresults in Figure 5 show again that the error floor in CFOestimation can be significantly reduced when using theoptimized training sequence.

From both Figures 4 and 5, we can see that the gain ofusing S&H sequences compared to Chu sequences is reallysmall. Therefore, in practical implementation, it is better touse the Chu sequence because it is simple to generate andit is available for all sequence lengths. Another advantage ofthe Chu sequence is that the optimal Chu sequence obtainedusing cost function (31) is the same for the uniform powerdelay profile and some exponential power delay profiles wetested. Hence, a common optimal Chu sequence can be usedfor both channel PDP’s. This is not the case for the S&Hsequences.

Figure 6 shows the performance of CFO estimationfor different lengths of optimal Chu sequences. Here wefix the channel length to L = 18. From the previoussections, to accommodate two users, the minimum sequencelength is ntL. Therefore, we need Chu sequences of lengthat least 36. We compare the performance of the optimallength-36 sequence with that of optimal length-49 andlength-64 sequences. For the length-49 sequence, the cyclicshift between training sequence of two users is 24, while

10 EURASIP Journal on Wireless Communications and Networking

for length-64 sequence, the cyclic shift is 32. From thecomparison, we can see that there are two advantages using alonger sequence. Firstly, in the low to medium SNR regions,there is SNR gain in the CFO estimation due to the longersequences length. Secondly, in the high SNR regions, theerror floor using longer sequences is much smaller. This canbe explained using Figure 7. In Figure 7, we plotted the signalpower for user 1 and user 2 after the correlation operationin (15) for sequence length of 36 and 64. In the absenceof the CFO, user 1’s signal should be contained in the first18 samples (L = 18). However, due to CFO, some signalcomponents are leaked into the other samples and becomeinterference to user 2. For the case of L = 18 and N = 36,all the leaked signals from user 1 become interference touser 2 and vice versa. If we use a longer training sequence,there is some “guard time” between the useful signals ofthe two users as shown in Figure 7 for the N = 64 case.As we only take the useful L samples for CFO estimation(16), only part of the leaked signal becomes interference.Hence, the overall SIR is improved. The cost of using longersequences is the additional training overhead that is required.Therefore, based on the requirement on the precision ofCFO estimation, the system design should choose the bestsequence length that achieves the best compromise betweenperformance and overhead.

6. Conclusions

In this paper, we studied the CFO estimation algorithmin the uplink of the multiuser MIMO-OFDM systems. Weproposed a low-complexity sub-optimal CFO estimationmethods using CAZAC sequences. The complexity of theproposed algorithm grows only linearly with the numberof users. We showed that in this algorithm, multiple CFOvalues from multiple users cause MAI in the CFO estima-tion. To reduce such detrimental effect, we formulated anoptimization problem based on the maximization of theSIR. However, the optimization problem is dependent onthe actual CFO values which are not known in advance.To remove such dependency, we proposed a new costfunction which closely approximate the SIR for small CFOvalues. Using the new cost function, we can obtain optimaltraining sequences for a different class of CAZAC sequences.Computer simulations show that the performance of theCFO estimation using CAZAC sequence is very close to thesingle-user CRB for low to medium SNR values. For highSNR, there is an error floor due to the MAI. By using theobtained optimal CAZAC sequence, such error floor can besignificantly reduced compared to using a randomly chosenCAZAC sequence.

Acknowledgment

The work presented in this paper was supported (in part) bythe Dutch Technology Foundation STW under the projectPREMISS. Parts of this work were presented at IEEE WirelessCommunication and Networking conference (WCNC) April2009.

References

[1] G. J. Foschini, “Layered space-time architecture for wirelesscommunication in a fading environment when using multi-element antennas,” Bell Labs Technical Journal, vol. 1, no. 2,pp. 41–59, 1996.

[2] E. Telatar, “Capacity of multi-antenna Gaussian channels,”European Transactions on Telecommunications, vol. 10, no. 6,pp. 585–595, 1999.

[3] “IEEE P802.11n/D1.10 Wireless LAN Medium Access Control(MAC) and Physical Layer (PHY) specifications: Enhance-ments for HigherThroughput,” IEEE Std., February 2007.

[4] “IEEE 802.16e: Air Interface for Fixed and Mobile BroadbandWireless Access Systems Amendment 2: Physical and MediumAccessControl Layers for Combined Fixed and Mobile Opera-tion in Licensed Bands,” IEEE Std., 2006.

[5] M. Dohler, E. Lefranc, and H. Aghvami, “Virtual antennaarrays for future wireless mobile communication systems,” inProceedings of the International Conference on Telecommunica-tions (ICT ’02), pp. 501–505, Beijing, China, June 2002.

[6] P. H. Moose, “Technique for orthogonal frequency divisionmultiplexing frequency offset correction,” IEEE Transactionson Communications, vol. 42, no. 10, pp. 2908–2914, 1994.

[7] T. M. Schmidl and D. C. Cox, “Robust frequency andtiming synchronization for OFDM,” IEEE Transactions onCommunications, vol. 45, no. 12, pp. 1613–1621, 1997.

[8] A. N. Mody and G. L. Stuber, “Synchronization for MIMOOFDM systems,” in Proceedings of the IEEE Global Telecom-municatins Conference (GLOBECOM ’01), vol. 1, pp. 509–513,November 2001.

[9] G. L. Stuber, J. R. Barry, S. W. Mclaughlin, Y. E. Li, M. A.Ingram, and T. G. Pratt, “Broadband MIMO-OFDM wirelesscommunications,” Proceedings of the IEEE, vol. 92, no. 2, pp.271–293, 2004.

[10] A. Van Zelst and T. C. Schenk, “Implementation of a MIMOOFDM-based wireless LAN system,” IEEE Transactions onSignal Processing, vol. 52, no. 2, pp. 483–494, 2004.

[11] O. Besson and P. Stoica, “On parameter estimation of MIMOflat-fading channels with frequency offsets,” IEEE Transactionson Signal Processing, vol. 51, no. 3, pp. 602–613, 2003.

[12] Y. Yao and T. S. Ng, “Correlation-based frequency offsetestimation in MIMO system,” in Proceedings of the 58th IEEEVehicular Technology Conference (VTC ’03), vol. 1, pp. 438–442, October 2003.

[13] Y. Zeng, R. A. Leyman, and T. S. Ng, “Joint semiblindfrequency offset and channel estimation for multiuser MIMO-OFDM uplink,” IEEE Transactions on Communications, vol.55, no. 12, pp. 2270–2278, 2007.

[14] S. Sezginer, P. Bianchi, and W. Hachem, “Asymptotic Cramer-Rao bounds and training design for uplink MIMO-OFDMAsystems with frequency offsets,” IEEE Transactions on SignalProcessing, vol. 55, no. 7, pp. 3606–3622, 2007.

[15] S. Sezginer and P. Bianchi, “Asymptotically efficient reducedcomplexity frequency offset and channel estimators for uplinkMIMO-OFDMA systems,” IEEE Transactions on Signal Pro-cessing, vol. 56, no. 3, pp. 964–979, 2008.

[16] J. Chen, Y. C. Wu, S. Ma, and T. S. Ng, “Joint CFO and channelestimation for multiuser MIMO-OFDM systems with optimaltraining sequences,” IEEE Transactions on Signal Processing,vol. 56, no. 8, pp. 4008–4019, 2008.

[17] M. Morelli and U. Mengali, “Carrier-frequency estimation fortransmissions over selective channels,” IEEE Transactions onCommunications, vol. 48, no. 9, pp. 1580–1589, 2000.

EURASIP Journal on Wireless Communications and Networking 11

[18] R. Frank, S. Zadoff, and R. Heimiller, “Phase shift pulse codeswith good periodic correlation properties (corresp.),” IRETransactionson Information Theory, vol. 8, no. 6, pp. 381–382,1962.

[19] D. C. Chu, “Polyphase codes with good periodic correlationproperties,” IEEE Transactions on Information Theory, vol. 18,no. 4, pp. 531–532, 1972.

[20] N. Sueshiro and M. Hatori, “Modulatable orthogonalseqeuences and their application to SSMA systems,” IEEETransactions on Information Theory, vol. 34, no. 1, pp. 93–100,1988.

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[22] S. Solomon and G. Gong, Signal Design for Good Correlation:For Wireless Communication, Cryptography, and Radar, Cam-bridge Univeristy Press, Cambridge, UK, 2005.

[23] W. Yan, S. Attallaht, and J. W. M. Bergmans, “Efficient trainingsequence for joint carrier frequency offset and channelestimation for MIMO-OFDM systems,” in Proceedings of theIEEE International Conference on Communications (ICC ’07),pp. 2604–2609, June 2007.

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Photograph © Turisme de Barcelona / J. Trullàs

Preliminary call for papers

The 2011 European Signal Processing Conference (EUSIPCO 2011) is thenineteenth in a series of conferences promoted by the European Association forSignal Processing (EURASIP, www.eurasip.org). This year edition will take placein Barcelona, capital city of Catalonia (Spain), and will be jointly organized by theCentre Tecnològic de Telecomunicacions de Catalunya (CTTC) and theUniversitat Politècnica de Catalunya (UPC).EUSIPCO 2011 will focus on key aspects of signal processing theory and

li ti li t d b l A t f b i i ill b b d lit

Organizing Committee

Honorary ChairMiguel A. Lagunas (CTTC)

General ChairAna I. Pérez Neira (UPC)

General Vice ChairCarles Antón Haro (CTTC)

Technical Program ChairXavier Mestre (CTTC)

Technical Program Co Chairsapplications as listed below. Acceptance of submissions will be based on quality,relevance and originality. Accepted papers will be published in the EUSIPCOproceedings and presented during the conference. Paper submissions, proposalsfor tutorials and proposals for special sessions are invited in, but not limited to,the following areas of interest.

Areas of Interest

• Audio and electro acoustics.• Design, implementation, and applications of signal processing systems.

l d l d d

Technical Program Co ChairsJavier Hernando (UPC)Montserrat Pardàs (UPC)

Plenary TalksFerran Marqués (UPC)Yonina Eldar (Technion)

Special SessionsIgnacio Santamaría (Unversidadde Cantabria)Mats Bengtsson (KTH)

FinancesMontserrat Nájar (UPC)• Multimedia signal processing and coding.

• Image and multidimensional signal processing.• Signal detection and estimation.• Sensor array and multi channel signal processing.• Sensor fusion in networked systems.• Signal processing for communications.• Medical imaging and image analysis.• Non stationary, non linear and non Gaussian signal processing.

Submissions

Montserrat Nájar (UPC)

TutorialsDaniel P. Palomar(Hong Kong UST)Beatrice Pesquet Popescu (ENST)

PublicityStephan Pfletschinger (CTTC)Mònica Navarro (CTTC)

PublicationsAntonio Pascual (UPC)Carles Fernández (CTTC)

I d i l Li i & E hibiSubmissions

Procedures to submit a paper and proposals for special sessions and tutorials willbe detailed at www.eusipco2011.org. Submitted papers must be camera ready, nomore than 5 pages long, and conforming to the standard specified on theEUSIPCO 2011 web site. First authors who are registered students can participatein the best student paper competition.

Important Deadlines:

P l f i l i 15 D 2010

Industrial Liaison & ExhibitsAngeliki Alexiou(University of Piraeus)Albert Sitjà (CTTC)

International LiaisonJu Liu (Shandong University China)Jinhong Yuan (UNSW Australia)Tamas Sziranyi (SZTAKI Hungary)Rich Stern (CMU USA)Ricardo L. de Queiroz (UNB Brazil)

Webpage: www.eusipco2011.org

Proposals for special sessions 15 Dec 2010Proposals for tutorials 18 Feb 2011Electronic submission of full papers 21 Feb 2011Notification of acceptance 23 May 2011Submission of camera ready papers 6 Jun 2011