Maximum-likelihood estimation of frequency and time offsets in OFDM systems with multiple sets of identical data

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Fig. 8. Recovered source signal^x (n).

IV. CONCLUSION

In this correspondence, a theorem that sates the condition for the existence of MIMO FIR inverses, which are also FIR inverses, is pro-posed and proved. The use of the theorem for equalization of a MIMO FIR channel is illustrated through a numerical example.

REFERENCES

[1] G. Giannakis et al., Signal Processing Advances in Wireless Commu-nications: vol I Trends in Channel Estimation and Equalization, vol II Trends in Single and Multi-User Systems. Englewood Cliffs, NJ: PTR Prentice-Hall, 2001.

[2] L. Tong, G. Xu, and T. Kailath, “Blind identiﬁcation and equalization based on second-order statistics: A time domain approach,” IEEE Trans. Inf. Theory, vol. 40, no. 2, pp. 340–349, Mar. 1994.

[3] J. K. Tugnait, “Blind estimation of digital communication channel impulse response,” IEEE Trans. Commun., vol. 42, no. 2/3/4, pp. 1606–1616, Feb./Mar./Apr. 1994.

[4] E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, “Subspace methods for the blind identiﬁcation of multichannel FIR ﬁlters,” IEEE Trans. Signal Process., vol. 43, no. 2, pp. 516–525, Feb. 1995. [5] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, 1980. [6] D. A. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and

Algo-rithms. Berlin, Germany: Springer-Verlag, 1992.

Maximum-Likelihood Estimation of Frequency and Time Offsets in OFDM Systems With Multiple

Sets of Identical Data

Mu-Huo Cheng, Member, IEEE, and Chi-Chan Chou

Abstract—This paper generalizes the existing algorithm for the

max-imum-likelihood (ML) estimation of frequency and time offsets in orthogonal frequency-division multiplexing (OFDM) systems using from two sets to multiple sets of identical data. The algorithm is derived by the application of the matrix inversion lemma; the Cramér–Rao bound for estimation of the frequency offset is also obtained. For reducingrealization complexity, a simpliﬁed algorithm is developed. Simulations using the ten sets of identical data in the preamble of IEEE 802.11a for estimation of frequency and time offsets have been performed to verify the effectiveness of the proposed algorithms.

Index Terms—IEEE 802.11a, maximum-likelihood (ML) estimation,

or-thogonal frequency-division multiplexing (OFDM).

I. INTRODUCTION

The orthogonal frequency-division-multiplexing (OFDM) scheme has been adopted in many applications such as the digital broadcast television, the wireless communications, and the high-bit-rate commu-nications over the existing copper networks [1]–[3]. The OFDM sys-tems, however, are highly sensitive to the frequency offset [4], [5]. Therefore, an accurate estimation of the frequency offset is critical. Existing approaches for the frequency-offset estimation using the pre-amble data [6], [7], the cyclic prefix data [8], [9], or the cyclostationary property [10] of the received signals have been proposed. Extensive coverage of techniques for digital synchronization is also provided in textbooks [11]–[13]. Here, we focus on the data-aided maximum-like-lihood (ML) estimation in OFDM systems. The ML estimation of fre-quency and time offsets in OFDM systems using the two sets of iden-tical cyclic prefix data has been derived in [8]. In the IEEE 802.11a [14] standard for wireless LAN communications, the preamble con-tains multiple sets of identical data for channel estimation and syn-chronization. Hence, an extension for the ML estimation algorithm to include for multiple sets of identical data is practically useful and worth studying. Therefore, in this paper, by using the matrix inversion lemma [15], we generalize the ML algorithm for the estimation of frequency and time offsets to include for the number of the identical data set more than two. Moreover, we also derive the Cramér–Rao bound for the fre-quency-offset estimate. Since the resulting ML algorithm requires high realization complexity, we further develop a simplified algorithm that can reduce significantly the realization complexity but at the cost of modest performance degradation. Simulations are then carried out to evaluate the performance of all proposed algorithms using the ten short identical symbols in the preamble of IEEE 802.11a standard.

II. ALGORITHMDERIVATION

The algorithm is derived under the assumption of the nondispersive channel and the additive white Gaussian noise (AWGN)n(k) with the

Manuscript received January 17, 2005; revised September 2, 2005. This work was supported by National Science Council, Taiwan, under NSC93-2213-E-009-040. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Mats Viberg.

The authors are with the Department of Electrical and Control Engi-neering, National Chiao Tung University, Hsinchu 30010, Taiwan (e-mail: mhcheng@cc.nctu.edu.tw; chichanchou@yahoo.com.tw).

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Fig. 1. Observed data-encompassing contiguousN sets of identical data.

transmitted complex signals(k), which is also assumed to be a white Gaussian random process. The received complex sample data for a fre-quency offset and an integer time offset are given by

r(k) = s(k 0 ) exp(j2k=N) + n(k) (1) whereN represents the number of subcarriers of a complex trans-mitted OFDM symbol. Notice that here the effect of the offset in sam-pling frequency is assumed negligible such that the time offset is an integer. The effect of nonsynchronized sampling has been discussed in [16]. Suppose that we have observedL consecutive samples of r(k), k = 1; 2; . . . ; L, shown in Fig. 1, encompassing contiguous Nssets of identical data, indexed byIi,i = 0; 1; . . . ; Ns 0 1, where each

set containsL_sdata samples. Note that in theory, it is not necessary to assume that theNs sets of identical data are contiguous. The as-sumption is made here for the simplicity in derivation and for the use of the preamble data in IEEE 802.11a standard. Then, the index sets areIi= f + iL_s; . . . ; + (i + 1)L_s0 1g for i = 0; 1; . . . ; N_s0 1. Collect the observed samples in the L 2 1 vector r = [r(1); 1 1 1 ; r(L)]T_{, where the superscript}_{T denotes the transpose}

oper-ation. Notice that the samples in theNssets, i.e.,r(k); k 2 N 01_i=0 Ii, are pairwise correlated. Hence, form; n 2 [0; Ns0 1], and m n

8k 2 I0_{: E [r(k + nL} s)r3(k + mLs)] = 2 s+ 2n; m = n 2 se0j(m0n); (m 0 n) = 1; . . . ; Ns0 1 0; otherwise (2)

whereE[1] denotes the expectation, 2_s = E[s2(k)] the signal power, 2

n= E[n2(k)] the noise power, and 0 = 2Ls=N.

The log-likelihood function3(; ) is the logarithm of f(rj; ), the probability density function (pdf) of theL observed samples r condi-tioned by the given time offset and the frequency offset . This func-tion, by (2), can be written as

3(; ) =ln f(rj; ) =ln k2I f (r(k); r(k+Ls); . . . ; r (k+(Ns01)Ls)) 2 k62 I f (r(k)) =ln k2I f(r(k); r(k+Ls); . . . ; r(k+(Ns01)Ls)) f(r(k)) f(r(k+Ls)) 1 1 1f(r (k+(Ns01)Ls))) 2 L k=1 f (r(k)) (3)

where the conditioned variables and are omitted for sim-plicity. Since f(r(k)) for each k is a one-dimensional (1-D)

zero-mean complex Gaussian distribution function, it equals exp[0jr(k)j2₌₍2

s+ 2n)]=[(s2+ n2)]; the product Lk=1f(r(k))

in (3), therefore, is independent of and . Also the denominator in (3) equals the multiplication ofNs1-D complex Gaussian pdf’s, yielding

N 01 m=0 f (r(k + mLs)) = ₍₂1 s+ 2n) N 2 exp 0 N 01m=0 jr(k + mLs)j2 _s2₊2_n = ₍₂1 s+ 2n) N 2 exp 0 zH₂(k)z(k) s+ n2 (4)

wherez(k) = [r(k); r(k + L_s); . . . ; r(k + (N_s0 1)L_s)]T and the superscriptH represents the transpose conjugate (hermitian) operation. Thenf(r(k); r(k + L_s); . . . ; r(k + (N_s0 1)L_s)) = f(zzz(k)) and its pdf is anNs0 D joint complex Gaussian function [17], given by

f (z(k)) = _N _det(R)1 exp 0zH_(k)R01_z(k) ₍₅₎

whereR denotes the correlation matrix of z(k), which, by using (2) and by observation, can be written in the form

R = E z(k)zH_(k) = 2 s+ 2n s2e0j 1 1 1 2se0j(N 01) 2 sej 2s+ n2 1 1 1 2se0j(N 02) .. . ... . .. ... 2 sej(N 01) 2sej(N 02) 1 1 1 2s+ n2 = 2 nI + 2sqqH (6)

where q = [1; ej; . . . ; ej(N 01)]T and I is the identity matrix having the same dimension as the matrixR. Then, we can directly ob-tain the matrix determinantdet(R) = (_n2)N 01(2_n+ N_s2_s) and R01_{using the matrix inversion lemma [15], yielding}

R01= 1₂

nI 0

2 s

_n2₍_n2_{+ N}_s2_s₎qqH: (7)

Equation (5), therefore, becomes

f (z(k)) = 1 N ₍_n2₎N 01₍_n2_{+ N}_s2_s₎exp 2 0 1₂ nz H_(k)z(k)+ 2s 2 n(n2+Nss2)z H_(k)qqH_{z(k) : (8)}

By simple manipulation, we obtain zH_(k)qqH_z(k) = N 01 m=0 jr(k + mLs)j2 + N 01 m=1 N 01 p=m 2Re r (k+(p0m)Ls) r3(k+pLs)ejm = zH_{(k)z(k) + 2}N 01 m=1 Re m(k)ejm (9)

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where the superscript3denotes the complex conjugate operation and m(k) = N 01_p=m r(k+(p0m)Ls)r3(k+pLs). Substituting (9), (8),

and (4) into (3) and rearranging, we obtain the following log-likelihood function 3(; ) = +L 01 k= ln c1+ 2c2 N 01 m=1 Re m(k)ejm 0 c3zH(k)z(k) + L k=1 ln f (r(k)) (10) where c₁ = ((_n2 + _s2)N )=((_n2)N 01(_n2 + N_s2_s)), c₂ = 2 s=(n2(2n+ Nss2)), c3= ((Ns0 1)4s)=(n2(2n+ Ns2s)(n2+ 2

s)). The ﬁrst term ln c1 and the last term L_k=1ln f(r(k)) are independent of and ; the ML estimate (_ML; _ML), therefore, is given by ML; ML = arg max ; +L 01 k= N 01 m=1 Re m(k)ejm 0 Ns₂0 1zH_(k)z(k) ₍₁₁₎

where = _s2=(_s2+ _n2). Note that for Ns = 2, (11) is equal to the

(5) in [8], as expected.

A. Cramér–Rao Bound for Frequency-Offset Estimate

If the time offset has been given a priori and assume the estimation of frequency offset is unbiased, then its Cramér–Rao bound can be derived directly using the result [18]

E ((r) 0 )2 _{0E @}2ln f(rj; )

@2

01

: (12) Note that extensive simulations have shown that the frequency esti-mate via the ML algorithm (11) is unbiased, but this property has not been analytically proven, and hence the assumption is made. Taking the second derivative of the log-likelihood function with respect to using (10), we obtain @2_{ln f(rj; )} @2 = +L 01 k= 02c2 2L_Ns 2 N 01 m=1 N 01 p=m m2_Re 2 r (k + (p 0 m)Ls) r3(k + pLs)ejm : (13)

We then apply the expectation, use results in (2), and replacec2by its value, yielding 0 E @2ln f(rj; )_@₂ = 2c2 2L_Ns 2 +L 01 k= N 01 m=1 N 01 p=m m2 2 Re E [r (k + (p 0 m)Ls) r3(k + pLs)] ejm = 82L3s4s N22 n(2n+ Nss2) N 01 m=1 m2_(N s0 m) = 2_3N2L₂3sN₂s2 Ns20 1 s4 n(n2+ Ns2s) : (14)

The Cramér–Rao bound, therefore, is obtained

E ((r) 0 )2 3N2n2 n2+ Ns2s

22_L3

sNs2(Ns20 1) s4:

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This bound is intuitive; when the number of identical data setNsor the number of data samples in each setLsincreases, the bound decreases. The bound also shows the advantage of using multiple sets instead of two sets of identical data for frequency-offset estimation.

III. SIMPLIFIEDALGORITHM

The computational complexity to realize (11) for estimating ; is intensive because a two-dimensional searching is required. Here, we present a simpliﬁed algorithm for reducing the realization complexity. Let x denote the angle of x. Then, the likelihood function (11) is maximized if the frequency offset given _MLmakes

[ +L 01

k= m(k)] + m0 = 0 for all m = 1; 1 1 1 ; Ns0 1. The

simpliﬁed algorithm presumes ﬁrst that the frequency-offset estimate makes [ _k=+L 01 m(k)] + m0 = 0 for m = 1; . . . ; Ns0 1.

Note that this presumption can be true forNs = 2 but is generally

impossible to attain forN_s > 2. This discrepancy, therefore, makes the simpliﬁed algorithm result in a performance loss forNs> 2. The

time offset, using (11) under the presumption, can then be estimated by the following equation:

ML;sim = arg max +L 01 k= N 01 m=1 Re m(k)ejm 0 Ns₂0 1zH_(k)z(k) = arg max N 01 m=1 +L 01 k= m(k) 0+L 01 k= Ns0 1 2 zH(k)z(k) : (16) The obtained time-offset estimate, via the above equation, is used to ﬁnd the frequency offset. For each indexm, we obtain, from the pre-sumed condition, one estimate0 = 0 [ +L 01

k= m(k)]=m.

TheseNs0 1 frequency-offset estimates are then averaged to yield the

ﬁnal estimate ML;sim=_2L0N s(Ns0 1) N 01 m=1 1 m +L 01 k= m(k) : (17) The complexity to realize the algorithm (16), (17) takes approxi-matelyN_s2LsK operations (including multiplication, addition, and

arc-tangent operations), whereKdenotes the range of for searching while to realize the ML algorithm (11) requires approximately N2

sLsKK operations, where K is the number to partition the

frequency offset for searching. Therefore, the realization complexity of the simpliﬁed algorithm is greatly reduced. The frequency-offset estimator in (17) can be analytically shown to be consistent if the time-offsetML;simis properly chosen. For brevity, we only describe the proof procedure. For the received data in the preamble set, the product r(k)r3(k + mLs) is an addition of a constant multiplying

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Fig. 2. Structure of the IEEE 802.11a preamble.

of freedom and three zero-mean Gaussian random variables, as seen from the following equation:

r(k)r3_{(k + mL}

s) = js(k 0 )j2e0jm + n(k)s3(k 0 )

+n3_{(k + mL}

s)s(k 0 ) + n(k)n3(k + mLs): (18)

Since _m(k) = N 01_p=m r(k + (p 0 m)L_s)r3(k + pL_s), the random variable m(k), therefore, is an addition of a random variable with

the chi-square distribution of higher degrees of freedom and more Gaussian random variables. Therefore, as the number of preamble data is increased, both the averaged imaginary part and the averaged real part of _m(k), by the statistical properties of both the chi-square and the Gaussian distributions, converge to a constant multiplying sin(2m0_{) and cos(2m}0_{), respectively. The variance of each}

also converges to zero. The ratio then converges to the desired value tan(2m0_{) with its variance also converging to zero; hence, the}

consistency of the time-offset estimator is obtained.

Note that forNs= 2, this simpliﬁed algorithm is just the exact ML

algorithm, which has been derived in [8]. ForN_s> 2, however, the so-lutions using the simpliﬁed algorithm are no longer the ML estimates. The simpliﬁed algorithm, therefore, reduces the realization complexity at the cost of modest performance degradation, as illustrated by the simulation results in the following section.

IV. SIMULATIONS

The IEEE 802.11a standard deﬁnes the preamble as illustrated in Fig. 2. Each preamble consists of ten identical short symbols and two identical long symbols. Note that the ten short symbols and the two long symbols can be combined together by using the method similar to the second approach proposed in [9] for performance improvement. For demonstration and for simplicity, here we only use the ten repeated short symbols for the estimation of time and frequency offsets. The number of subcarriers in IEEE 802.11a isN = 64, and the sampling frequency is 20 MHz. One short symbol is of duration 0.8s and hence has 16 samples.

Monte Carlo simulations are used to evaluate the performance of the proposed algorithms in both AWGN and dispersive channels. The dis-persive channel, similar to the model used in [8], has 15 independent Rayleigh-fading taps with an exponentially decaying power delay pro-ﬁle of the root mean-squared width equal to 0.1s (two samples) and the maximum delay spread of 0.75s (15 samples). The signal-to-noise ratio (SNR) is measured at the received signal. The squared estimation errors of the frequency offset and the time offset, averaged over 1000 trials, versus SNRs at 0, 1, to 10 dB, are shown in Figs. 3 and 4, re-spectively. These results show that the performance of the proposed algorithms, as expected, degrades for the dispersive channel because the transmitted signals passed through the channel result in signals

Fig. 3. Mean-squared error of the frequency-offset estimate versus SNRs.

Fig. 4. Mean-squared error of the time-offset estimate versus SNRs.

which are no longer uncorrelated. The performance of the simplified algorithm at low SNRs, as shown in the figures, is slightly worse than that of the ML method but its difference is decreasing as the SNR is increasing. The simplified algorithm, although inferior to the ML al-gorithm in performance, is more practical when the realization com-plexity is concerned.

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Fig. 5. Comparison of the frequency-offset estimator between the approach [8] and the proposed method.

Fig. 6. Comparison of the time-offset estimator between the approach [8] and the proposed method.

Simulations are also performed for performance comparison be-tween the proposed approach and that in [8]. For simplicity, we only demonstrate the simulation results in AWGN channels even the ten-dency in dispersive channels is also similar. The proposed algorithm uses ten sets of identical data(Ns = 10), with each set having 16

samples(L_s= 16), while the approach in [8] uses two sets of identical data( Ns = 2), with each set having Ls= MLssamples, whereM is an integer. It has been shown in [9] that the estimator using two sets of identical data with each set having Ls samples is equivalent in performance to the estimator usingM symbols, with each symbol having the cyclic prefix of lengthLs. The estimates of the frequency offset and the time offset, averaged over 1000 trials versus SNRs are shown in Figs. 5 and 6, respectively. These results demonstrate that the symbol numberM has little effect on the time-offset estimator but in-fluences significantly the performance of frequency-offset estimation. Note that the proposed approach and the method in [8] are closely comparable in the performance of frequency-offset estimation when M = 6. In fact, this number can be derived because the performance of each algorithm can be approximated by the obtained Cramér–Rao bound (15). In terms of (15), under the assumption of the same SNR

and the same number of OFDM subcarriers, two approaches attain the same Cramér–Rao bound if the following equation is satisﬁed:

N2 s Ns20 1 L3s NSR+ Ns = N2 s Ns20 1 L3s NSR+ Ns (19)

where NSR= 2_n=_s2. For example, if NSR=1 (0 dB SNR), N_s= 10 and Ns = 2, then M = Ls=Ls= 6.0822; this solution matches well

with the simulation result. Hence, the length of the cyclic preﬁx should be about six times the length of one identical set such that the approach [8] can attain the same performance as our method.

V. CONCLUSION

We have derived an ML algorithm for the estimation of frequency and time offsets in OFDM systems using multiple sets of identical data; the Cramér–Rao bound for the frequency offset is also obtained. We have further developed a simpliﬁed algorithm for reducing the realiza-tion complexity. Simularealiza-tions using the preamble data in IEEE 802.11a standard have been performed to verify the effectiveness of the pro-posed algorithms.

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