DOI 10.1007/s11277-007-9295-y
Frequency estimators for MIMO-OFDM systems
David C.-H. Chiang · Yu T. SuReceived: 11 July 2006 / Accepted: 11 March 2007 / Published online: 22 May 2007 © Springer Science+Business Media, LLC 2007
Abstract Orthogonal frequency division multiplex-ing (OFDM) systems are known to be sensitive to car-rier frequency offset (CFO). This paper is concerned with the CFO estimation for multiple input multiple output (MIMO) systems employing OFDM waveforms. We present two approaches to derive maximum likeli-hood (ML) pilot-assisted frequency estimators that use either two or multiple identical training symbols. It is shown that the resulting ML frequency estimators are similar to maximum ratio combining versions of Moose estimator and Yu–Su solution, respectively. Numerical examples demonstrate that the proposed frequency esti-mators are robust against spatial Signal-to-noise ratio (SNR) variation and they yield performance superior to that of the corresponding single-antenna system.
Keywords Orthogonal frequency division multiplexing (OFDM)· Multiple input multiple output (MIMO)· Frequency estimation
D. C.-H. Chiang (
B
)Afa Technologies, 233-1 Baociao Rd., Sindian, 23145, Taiwan
e-mail: david-jiang@yahoo.com.tw Y. T. Su
Department of Communications Engineering, National Chiao Tung University, Hsinchu, 30056, Taiwan e-mail: ytsu@mail.nctu.edu.tw
1 Introduction
It is well known that the performance of an orthogo-nal frequency division multiplexing (OFDM) system is sensitive to the carrier frequency offset (CFO) caused by Doppler shifts or instabilities of and mismatch be-tween transmitter and receiver oscillators [1]. Amongst the numerous proposals for CFO compensation, two are directly related to our work. Moose’ [2] maximum likelihood (ML) estimator is based on the observation of two consecutive and identical symbols. Its maxi-mum frequency acquisition range is only±1/2 subcar-rier spacing because of the modulo 2π ambiguity. Yu and Su (YS) [6] used a transform domain approach to derive an ML estimator based on multiple (>2) identi-cal pilot symbols and presented very efficient alterna-tives that give mean squared error (MSE) performance close to the Cramér–Rao bound (CRB).
Combining the advantages of OFDM and multiple input multiple output (MIMO) techniques, a variety of (MIMO-OFDM) system architectures and the associ-ated detection techniques have been studied. The related CFO estimation issue, however, does not received much attention so far. Mody and Stüber pro-posed a CFO estimator based on the autocorrelation of the cyclic prefix [3,4]. The residual integer part of CFO is obtained by increasing the range of CFO estima-tor [3] or using the cyclic cross-correlation in the fre-quency domain [4]. The solution of Asai et al. [5] is based on the average of the autocorrelation values derived from two receiving antennas. Both approaches
are of heuristic nature and no optimality was claimed. This paper presents two ML fractional CFO estimators for MIMO-OFDM systems. We employ two different approaches to derive the ML solutions for systems using two-symbol and multiple-symbol preamble struc-tures, respectively. The resulting estimators are found to be extended versions of the corresponding ML esti-mators for the single-antenna scenario, namely, the Moose and the YS algorithms.
The rest of the paper is organized as follows. The following section gives a description of the signal and system model. In Sect.3, we derive the ML frequency estimator and show that they are extended versions of Moose and YS ML CFO estimators. Section 4 pro-vides some numerical examples and related discus-sions. Conclusions are then given in Sect.5.
2 System and signal model
Consider a frequency selective fading channel associ-ated with a MIMO system of MTtransmit and MR re-ceive antennas. If the duration of the cyclic prefix (CP),
Ng samples, is greater than or equal to the maximum relative delay that includes users’ timing ambiguities and the maximum multipath delay, we can express the equivalent received time-domain baseband sequence at the output of the ith receive antenna, after removing the CP part, as yi[n] = MT j=1 rij[n] + wi[n], n = 1, 2, . . . , N, i = 1, 2, . . . , MR, (1) where{wi[n]} is a zero-mean complex additive white Gaussian noise (AWGN) sequence and
rij[n] = N1 k∈Dj Es MT Sj[k]Hij[k]ej2πn(k+)/N (2) is the part of the received waveform contributed by the OFDM signal sent by the j th transmit antenna and received by the ith receive antenna. Moreover,
• Sj[k] represents the symbol carried by the kth
sub-carrier at the j th transmit antenna.
• Hij[k] is the frequency response at the kth
subcar-rier for the channel between the ith receive and the
jth transmit antennas. The corresponding impulse
response is given byhij[n], n = 0, 1, . . . , L − 1, i.e.Hij[k] = L−1n=0hij[n]e− j2πkn N .
• L is the maximum channel memory of all MTMR
SISO component channels. Thus not all hij[n] have nonzero values.
• denotes the relative CFO of the channel (the ratio
of the actual CFO to the intercarrier spacing).
• Dj is the set of modulated subcarrier for the j th transmit antenna.
• Es is the average energy allocated to the kth sub-carrier evenly divided across the transmit antennas. Using the short form notations
y[n] =y1[n] y2[n] · · · yMR[n] T , (3a) S[k] =S1[k] S2[k] · · · SMT[k] T , (3b) H[k] = Hij[k], (3c) w[n] =w1[n] w2[n] · · · wMR[n] T (3d) we rewrite (1) in matrix form
y[n] = 1 N k∈Dj Es MT H[k]S[k]ej2πn(k+)/N+ w[n]. (4) Taking N -point DFT on both sides of the above equa-tion leads to Y[k] = Es MT m∈Dj H[m]S[m] × 1 N N−1 n=0 ej2π(m+)n/Ne−j2πkn/N + W[k], (5) where Y[k] =Y1[k] Y2[k] · · · YMR[k] T, (6a) W[k] =W1[k] W2[k] · · · WMT[k] T, (6b) Yi[k] = N−1 n=0 yi[n]e− j2πkn N , (6c) Wi[k] = N−1 n=0 wi[n]e− j2πkn N . (6d)
Let fand ibe, respectively, the fractional and integer parts of the CFO so that = f+ i. We rewrite (5) as
Y[k] = Es MT H[k − i]S[k − i] × 1 Nsin(π f) sin(π f/N)e jπf(N−1)/N + Es MT m∈Dj m=k−i H[m]S[m] × 1 Nsin(π(f + i)) sin(π(m− k + f + i)/N) ×ejπ(f+i)(N−1)/Ne−jπ(m−k)/N+ W[k]. (7) The first term on the right hand side of the above equa-tion indicates that, besides suffering from multiplica-tive distortion caused by the channel response, the kth subcarrier signal is circularly shifted and amplitude-reduced. The second term represents the inter-carrier interference (ICI). It is clear that the presence of a fractional CFO f causes reduction of the desired sub-carrier’s amplitude and induces ICI. If f is perfectly compensated for, the integer CFO i, if exists, will re-sult in a circular shift of the desired output, causing decision errors.
3 Maximum likelihood frequency estimator
3.1 Estimator based on two pilot symbols
We consider a pilot structure similar to that proposed by Moose [2] and letD be the set of modulated sub-carrier (indexes) that bear a training sequence on the even frequencies and zeros on the odd frequencies. The resulting received time-domain training sequence has two identical halves
r[n] = 1 N k∈De Es MT H[k]S[k]ej2πn(k+)/N,(8a) r[n + N/2] = 1 N k∈De Es MT H[k]S[k]ej2π(n+N2)(k+)/N = r[n]ej2π/2, (8b) where n = 1, 2, . . . , N/2, r[n] = (r1[n] r2[n] · · ·
rM [n]T andDeis the subset of even numbers inD.
Taking into account AWGN, we obtain
y[n] = r[n] + w[n], (9a) y[n + N/2] = r[n]ej2π/2+ w[n + N/2], (9b) where w[n] =w1[n] w2[n] · · · wMR[n] T . Define the abbreviated notations y1[i] = (yi[1] yi[2], · · · yi[N/2]) (10a) y2[i] = (yi[N/2 + 1] yi[N/2 + 2] · · · yi[N]) , (10b) r1[i] = (ri[1] ri[2] · · · ri[N/2]) , (10c) r2[i] = (ri[N/2 + 1] ri[N/2 + 2] · · · ri[N]) (10d) and w1[i] = (wi[1] wi[2], · · · wi[N/2]) (11a) w2[i] = (wi[N/2 + 1] wi[N/2 + 2] · · · wi[N]) , (11b) where the subscripts on the right-hand sides of the above equations indicate either the first or the second half of a time-domain OFDM frame and the indexes within the brackets denote from which receive antenna the time domain sample is derived. Equation9aand9b
then have the equivalent expressions
y1[i] = r1[i] + w1[i], (12a)
y2[i] = r1[i]ej2π/2+ w2[i], i = 1, 2, · · · , MR. (12b) The ML estimator of the parameter , given the received vectory1[i], y2[i], is obtained by maximizing the likelihood function
f (y1[i], y2[i]|) = f (y2[i]|y1[i], )f (y1[i]|), (13) where, for simplicity, we have denoted various condi-tional probability density functions by the generic func-tional expression, f (·|·). As one can see from (8) and (12) that provides no information about y1[i] unless H[k] and S[k] is also given, i.e., f (y1[i]|) = f (y1[i]), the ML estimator of is given by
= arg max f (y2[i]|y1[i], )f
y1[i]|
= arg max fy2[i]|y1[i],
. (14) Since y2[i] = y1[i] − w1[i] ej2π/2+ w2[i] = y1[i]ej2π/2+ w2[i] − w1[i]ej2π/2 (15) and w1[i], w2[i] are independent temporally white Gaussian vector with zero mean and variance σ2I,
where I is the identity matrix, the multivariate Gauss-ian vector y2[i] has conditional mean y1[i]ej2π/2and covariance matrix Ew2[i] − w1[i]ej2π/2 w2[i] − w1[i]ej2π/2 H = 2σ2 niI. (16)
Then, given the received vectors yi[n], i = 1, 2, n = 1, 2, . . . , MR, the likelihood function becomes
() = fy1[1] · · · y1[MR], y2[1] · · · y2[MR]y1[1] · · · y1[MR], ∝ exp M R i=1 1 σ2 ni
{y2[i]yH1[i]e−j2π/2}
∝ exp ⎡ ⎣ ⎧ ⎨ ⎩ ⎛ ⎝MR i=1 γi N/2 n=1 yi∗[n]yi[n + N/2] ⎞ ⎠ × e−j2π/2 ⎫ ⎬ ⎭ ⎤ ⎦ , (17)
where in the last two equations we have dropped terms that are not dependent on , γi = (σn2i)−1 and{x} denotes the real part of the complex number x. the ML estimator of is given by = arg max () =π1Arg ⎛ ⎝MR i=1 γi N/2 n=1 yi∗[n]yi[n + N/2] ⎞ ⎠ , (18)
where Arg(x) denotes the principal argument of the complex number x. In general, the ML estimator for two identical halves pilot symbols of length NW and
ND-spaced is given by = N 2πNDArg ⎛ ⎝MR i=1 γi NW n=1 y∗ i[n]yi[n + ND] ⎞ ⎠ . (19)
Note that the correlationN/2n=1yi∗[n]yi[n + N/2] =
Cyi is weighted by the reciprocal of the noise
sam-ple’s variance and the product γiCyi is unitless and is
proportional to the signal-to-noise ratio (SNR) at the
ith receive antenna. This weighting strategy is similar
to the maximum ratio combining (MRC) rule used in a diversity receiver for combating frequency-selective fading. When equal gain weighting γi = 1 is used, the estimator is just the multiple antenna extension of the original Moose estimator, averaging all two symbol
correlation values over all receiving antennas. We refer to this estimator as the weighted Moose (WM) algo-rithm. The acquisition range of the WM algorithm is
±2NND subcarrier spacings. For a system that uses
arbi-trary even number (say 2k) of identical training sym-bols, we can also apply the WM algorithm by regarding the first half (k symbols) as the first training symbol and the second half as the second one.
3.2 Estimator based on multiple short symbols Now let us consider a MIMO-OFDM system that uses multiple (K+ 1) identical pilot symbols. For this case, we can use a procedure similar to that presented in the previous section to derive the corresponding ML esti-mator. An alternative but more efficient approach for the case of multiple pilot symbols is to apply the trans-form domain method of [6]. The mth sample of the kth (time-domain) pilot symbol received by the ith receive antenna, yi(k, m), can be represented as
yi(k, m) = xi(k, m) + wi(k, m), k = 1, . . . K,
m = 1, . . . , . . . M, (20)
where xi(k, m) and wi(k, m) are the corresponding sig-nal and noise components. The latter are uncorrelated circularly symmetric Gaussian random variables with zero mean and variance σw,i2 = E{|wi(k, m)|2}. Define
Yi(m) = [yi(1, m) · · · yi(K, m)]T , (21a)
A() =%1 ej2πM/N · · · ej2π(K−1)M/N
T
, (21b) Wi(m) = [wi(1, m) · · · wi(K, m)]T , (21c) where (·)T denote the matrix transpose.
Following an approach similar to [6] we obtain the joint log-likelihood function
(, xi(1, m)) = MR i=1 1/σn2i M m=1 ||Yi(m) −A()xi(1, m)||2, (22)
where we have used the fact
xi(k, m) = xi(1, m)ej2π(k−1)M/N. (23) For a given A(), setting ∇xi(1,m)||Yi(m) − A()
xi(1, m)||2 = 0, we obtain the conditional ML esti-mator, ˆxi(1, m) = xLSi(1, m) = A+()Yi(m), where
A+() = A()H/K and H denotes the Hermitian oper-ation. By substituting the least-square solution, xLSi
(1, m), for xi(1, m), we obtain () = MR i=1 1 σ2 ni M m=1 ||P⊥ AYi(m)||2 = (MRM)tr(PA⊥ˆRY Y), (24) where tr(·) denotes the trace of a matrix, PA⊥ def= I −
A()A+() and ˆRY Y def= M1 RM MR i=1 γi M m=1 Yi(m)YiH(m) (25) The above equation indicates that, like the 2-pilot case, the elements in the correlation matrix are also max-imum ratio combinations of correlation values com-puted in each component antenna. The desired frequency estimator is then given by
ˆ = arg{min tr(PA⊥ ˆRY Y)} = arg{max tr(PA ˆRY Y)}
= arg{max AH ˆR
Y YA}. (26)
This ML solution is similar to that derived in [6] with an
extended definition (MR combined) of the correlation
matrix ˆRY Y. It can be proved that using this extended correlation matrix and the transform domain approach of [6] we obtain an ML algorithm similar to that of [6]. For convenience of reference, we summarize the resulting ML estimation procedure as following. 1. Collect K received symbols from all receive
anten-nas and construct the sample correlation matrix ˆRY Y according to (25).
2. Calculate the coefficients of F (z) based on ˆRY Y, where z= ej2πM/N and F (z) =K−1 n=1 ns(n)zn, (27a) s(n) = (i,j)∈n ˆRY Y(i, j), (27b) n= {(i, j)|1 ≤ i, j ≤ K, j − i = n}, (27c)
3. Find the nonzero unit-magnitude roots of F (z)−
F∗(z) = 0.
4. The CFO estimator is obtained by
ˆ = j2πMN lnˆz, (28a)
ˆz = arg{max(z)}, (28b)
where
(z) = A(z)H ˆRY YA(z), (29a)
A(z) =%1 z z2 · · · zK−1
T
. (29b)
This algorithm will be referred to as the extended YS (EYS) algorithm whose frequency acquisition range is
±N
2M subcarrier spacings.
4 Simulation results and discussion
The computer simulation results reported in this sec-tion are obtained by using a pilot format the same as the IEEE 802.11a standard with a sample interval of 50 ns. The frequency-selective fading channel has a exponen-tially decaying power delay profile whose 16 paths and a rms delay spread of 50ns. The complex Gaussian dis-tributed path amplitudes are normalized such that the sum of the average power is unity. The DFT size is
N = 64. The SNR, defined as the ratio of the received
signal power (from all MT transmitters) to the noise power at the ith receive antenna, is assumed to be the same for each receive antenna for Figs.1–2. For the WM algorithm, the training part consists of two identi-cal halves with length NW = 32. Fig.1shows the MSE,
E[(ˆ − )2], performance of the WM CFO estimator for different number of transmit and receive antennas. Obviously, the MSE performance improves as the num-ber of receive antennas, MR, increases. In fact if one follows the analysis of [7], one can show that the cor-responding CRB has a similar trend and is given by CRBMIMO=
3(N/M)3(SNR)−1
2π2NK(K2− 1)M
R
. (30)
Figure2shows the performance of EYS and WM estimators when four short training symbols are used. The WM estimator regards the first two training sym-bols as one period so that two training symsym-bols with length Nw = 32 are used in computing the CFO esti-mator. The acquisition range of the WM estimator is±1 subcarrier spacings while that of the EYS estimator is
±2 subcarrier spacings. Because the EYS estimator use
all second-order information of the training symbols, it outperforms the WM estimator.
In the remaining two figures we examine the effect of the weighting schemes in combining time-correla-tions from various receive antennas; see (18), (19) or (25), (26). Figure3shows the performance of the WM CFO estimator using two weighting schemes. We plot
-2 0 2 4 6 8 10 12 14 16 18 20 -50 -45 -40 -35 -30 -25 -20 -15 -10 MT=1, MR=4 M T=2, MR=4 M T=4, MR=4 CRB (MR=4) M T=1, MR=1 MT=2, MR=1 M T=4, MR=1 CRB (M R=1) MSE (dB) Average SNR (dB) M T=1, MR=2 MT=2, MR=2 M T=4, MR=2 CRB (M R=2)
Fig. 1 MSE performance of the weighted Moose estimator for
systems using two identical training symbols; true CFO = 0.7 sub-carrier spacings -2 0 2 4 6 8 10 12 14 16 18 20 -50 -40 -30 -20 -10 0 MSE (dB) Average SNR (dB) MT=4,MR=2
Weighted Moose estimate rms delay spread 50 ns 100 ns 150 ns 200 ns Extended YS estimate rms delay spread 50 ns 100 ns 150 ns 200 ns CRB (MR=2)
Fig. 2 MSE performance of the EYS algorithm for systems
us-ing four short trainus-ing symbols; true CFO = 0.93 subcarrier spac-ings
the frequency estimator’s MSE performance when the SNR for each receive antenna is assumed to be per-fectly known. The importance of a proper weighting is clearly demonstrated. Performance of both frequency estimators using the optimal weighting is insensitive to the SNR variation across antennas. Similar observation is obtained in Fig.4which shows the performance of the EYS and WM estimators using either equal gain or optimal combining for different SNRs.
For reference purpose, we also present the CRB curves in each figures. It is found that, for a fixed num-ber of receive antennas NR, when the numnum-ber of trans-mit antennas increases the corresponding MSE perfor-mance becomes closer to the CRB for MR, especially at high SNRs. -2 0 2 4 6 8 10 12 14 16 18 20 -45 -40 -35 -30 -25 -20 -15 -10 -5 MSE (dB) Average SNR (dB)
Weighted Moose estimate M
T=2,MR=2
Equal gain weighting: SNR 1:SNR2=1:10 SNR1:SNR2=1:5 SNR 1:SNR2=1:1 SNR weighting: SNR 1:SNR2=1:10 SNR 1:SNR2=1:5 SNR 1:SNR2=1:1 CRB (M R=2)
Fig. 3 The effect of the weighting strategy on the MSE
per-formance of the WM algorithm for systems using two identical training symbols; true CFO = 0.7 subcarrier spacings
-2 0 2 4 6 8 10 12 14 16 18 20 -50 -40 -30 -20 -10 0 Extended YS estimate Equal gain weighting: SNR1:SNR2=1:10 SNR1:SNR2=1:5 SNR1:SNR2=1:1 SNR weighting: SNR1:SNR2=1:10 SNR1:SNR2=1:5 SNR1:SNR2=1:1 MT=2,MR=2 CRB (MR=2) MSE (dB) Average SNR (dB)
Weighted Moose estimate Equal gain weighting:
SNR1:SNR2=1:10 SNR1:SNR2=1:5 SNR1:SNR2=1:1 SNR weighting: SNR1:SNR2=1:10 SNR1:SNR2=1:5 SNR1:SNR2=1:1
Fig. 4 The effect of the weighting strategy on the MSE
per-formance of the EYS algorithm for systems using four identical training symbols; true CFO = 0.93 subcarrier spacings
5 Conclusion
We have shown that the ML frequency estimator for MIMO-OFDM systems with multiple (> 2) identical training symbols is equivalent to an extended version of the YS algorithm. For the special case when only two (short) training symbols are available the ML solu-tion is a weighted combinasolu-tion of the Moose estima-tors for every receive antennas. Our derivation is based on the assumption that the length of cyclic prefix is greater than or equal to the maximum delay that ac-counts for the all subchannels of the MIMO channel. When the SNRs across different receive antennas are not identical, the frequency estimators use weighted combinations of time correlations values from various
antennas. With an MRC-like combining scheme, the performance of both ML frequency estimators is insen-sitive to spatial SNR variation and improves as the number of transmit/receive antennas increases. In other words, the presence of multiple antennas not only prom-ises great capacity enhancement but entails performance improvement for the associated frequency synchroni-zation subsystem.
Acknowledgements This work was supported in part by the National Science Council of Taiwan and in part by the MediaTek Research Center of National Chiao Tung University.
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David C.-H. Chiang
re-ceived B.S. degree in me-chanical engineering and M.S. degree in electrical en-gineering from National Tai-wan University and National Chiao Tung University in 2002 and 2004, respectively. He was with Sunplus Tech-nology Co., Hsinchu, Tai-wan, from 2004 to 2006. Since May 2006 he has been a senior system engineer of Afa Technologies, Sindian, Taiwan.
Yu T. Su received the
Ph.D. degree in electri-cal engineering from the University of Southern California, Los Ange-les, USA, in 1983. From 1983 to 1989, he was with LinCom Corpora-tion, Los Angeles, USA, where he was a Corporate Scientist involved in the design of various measure-ment and digital satellite communication systems. Since Septem-ber 1989, he has been with the National Chiao Tung University, Hsinchu, Taiwan, where he is Associative Dean of the College of Electrical and Computer Engineering and was the Head of the Communications Engineering Department from 2001 and 2003. He is also affiliated with the Microelectronic and Informa-tion Systems Research Center, NaInforma-tional Chiao Tung University and served as a Deputy Director from 1997 to 2000. His main research interests include communication theory and statistical signal processing.
