Hopping Pilots for Estimation of Frequency-Offset and Multi-Antenna Channels in MIMO OFDM

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1 Hopping Pilots for Estimation of Frequency-Offset and Multi-Antenna Channels in MIMO OFDM Mi-Kyung Oh 1,XiaoliMa 2, Georgios B. Giannakis 2 and Dong-Jo Park 1 1 Dept. of EECS, KAIST; Guseong-dong, Yuseong-gu, Daejon 35-71, Republic of Korea 2 Dept. of ECE, Univ. of Minnesota; 2 Union Str. SE., Minneapolis, M 55455, USA Abstract We design pilot symbol assisted modulation for carrier frequency offset (CFO) and channel estimation in orthogonal frequency division multiplexing (OFDM) transmissions over multi-input multi-output (MIMO) frequency-selective fading channels. By separating CFO and channel estimation from symbol detection, the novel training patterns lead to low-complexity CFO and channel estimators. The performance of our algorithms is investigated analytically, and then compared with an existing approach by simulations. I. ITRODUCTIO Orthogonal frequency division multiplexing (OFDM) has been adopted by many standards (e.g., IEEE82.11a, IEEE82.11g in the US and DAB/DVB, HiperLA/2 in Europe), because it offers high data-rates and low decoding complexity [6]. On the other hand, space-time multiplexing and multi-antenna transmissions over multi-input multi-output (MIMO) channels, has been recently proved effective in combating fading, and enhancing data rates; see [1], [5] and references therein. Therefore, MIMO-OFDM has got much attention. Implementing MIMO-OFDM however, faces two major challenges: i) with the number of antennas increasing, channel estimation becomes more challenging as the number of unknowns to be estimated increases accordingly; and ii) similar to single-antenna OFDM, MIMO-OFDM exhibits great sensitivity to CFO. Many existing approaches have dealt with CFO and channel estimation in a single-input single-output (SISO) OFDM setup (e.g., [2] [4], [8]). Some rely on training-blocks [4], while others just take advantage of the standardized transmission format (e.g., [2], [8] exploits presence of null sub-carriers). Recently, optimal training for MIMO channel estimation has been considered in [9]. However, CFO estimation was not taken into account. We incorporate it in this paper using sparse-training sequences. Unlike [9], which estimates the channel on a per block basis, we collect several blocks for estimating CFO and MIMO channels. Thus, we can afford fewer number of training symbols per block than [9]. Specifically, we design training patterns for estimating CFO and MIMO frequency-selective channels in block transmission systems. For MIMO-OFDM, we design training symbols that enable decoupling of CFO and channel estimation from symbol decoding, which in turn leads to a low-complexity receiver compared to blind alternatives [7]. Moreover, our MIMO setup enlarges the acquisition range of CFO estimators compared to existing SISO algorithm, e.g., [4]. Our scheme is also flexible to accommodate any space-time coded transmission. otation: Upper (lower) bold face letters will indicate matrices (column vectors). Superscript ( ) H will denote Hermitian, ( ) T transpose, and will stand for the nearest integer. The real and imaginary parts are denoted as R[ ] and I[ ]; E[ ] will stand for expectation and diag[x] for a diagonal matrix with x on its main diagonal. Matrix D (h) with a vector argument will denote an diagonal matrix with D (h) = diag[h]. For a vector, denotes the Euclidean norm. We will use [A] k,m to denote the (k, m)th entry of a matrix A, and [x] m for the mth entry of the column vector x; I will denote the identity matrix; e i the (i+1)-st column of I ; [F ] m,n = (1/2) exp( j2πmn/) the fast fourier transform (FFT) matrix. We define f (ω) :=[1, exp(jω),...,exp(j( 1)ω)] T. II. SYSTEM MODEL Let us consider the discrete-time equivalent baseband model of a block transmission system communicating over MIMO frequency-selective channels in the presence of CFO, shown in Figure 1. Every information symbol block, [s(k)] n = s(k s + n), is drawn from a finite alphabet. At the transmitter, each block s(k) is first encoded and/or multiplexed in space and time, to yield blocks {c µ (k)} t of length c, which are forwarded to each transmit antenna. Training symbols (either zero or non-zero), which are known to both the transmitter and the receiver, are then inserted into c µ (k) to form a vector ū µ (k) with length, fortheµth antenna. Following the insertion of training symbols, we just perform MIMO-OFDM. In detail, we implement inverse FFT (IFFT) (via left multiplication with F H ) and cyclic prefix insertion (via left multiplication with the appropriate matrix operator T cp := [I T L IT ]T, where I L denotes the last L columns of I ). Define the resulting vectors as {u µ } t. After parallel to serial (P/S) multiplexing, the blocks {u µ } t of size P 1 are transmitted through the frequency-selective channels, which in discrete-time equivalent form have a finite impulse response h (ν,µ) (l), l [,L]. Letf o be the frequency offset (in Hz), which could be due to Doppler, or, mismatch between transmit-receive oscillators. If the sampling period is T, then the samples at the ν-th receive-antenna filter output can be written as: L x ν (n) =e jωon h (ν,µ) (l)u µ (n l)+η ν (n), (1) l= GLOBECOM /3/$ IEEE

2 where ν [1, r ], ω o := 2πf o T is the normalized CFO, and η ν (n) is zero-mean, white, complex Gaussian distributed noise with variance ση. 2 The sequence x ν (n) is then serial to parallel () converted into P 1 blocks with entries [x ν (k)] p := x ν (kp +p). Selection of the block size P greater than the channel order L implies that each received block x ν (k) depends only on two consecutive transmitted blocks, u µ (k) and u µ (k 1), which is referred to as interblock interference (IBI). In order to remove IBI at the receiver, we discard the cyclic prefix by left multiplying x ν (k) with the matrix R cp := [ L I ]. Denoting the resulting IBI-free block as y ν (k) :=R cp x ν (k), we obtain the following vectormatrix input-output relationship: y ν (k) =e jωokp R cp D P (ω o )H (ν,µ) T cp F H ū µ (k) + R cp η ν (k), where η ν (k) := [η ν (kp),η ν (kp +1),...,η ν (kp + P 1)] T ; H (ν,µ) is a P P lower triangular Toeplitz matrix with first column [h (ν,µ) (),...,h (ν,µ) (L),,...,] T ; and D P (ω o ) is a diagonal matrix defined as D P (ω o ) := diag[1,e jωo,...,e jωo(p 1) ]. Based on the structure of the matrices involved, it can be readily verified that R cp D P (ω o ) = e jωol D (ω o )R cp, where D (ω o ) := diag[1,e jωo,...,e jωo( 1) ]. Following (ν,µ) this identity, let us define H := R cp H (ν,µ) T cp, where the matrix H (ν,µ) is circulant with first column [h (ν,µ) (),...,h (ν,µ) (L),,..., ]. Letting also v ν (k) := R cp η ν (k), we can re-write (2) as: y ν (k) =e jωo(kp +L) D (ω o ) H (ν,µ) F H ū µ (k)+v ν (k). (3) Because H (ν,µ) in (3) is circulant, it follows that F H (ν,µ) F H is a diagonal matrix D ( h (ν,µ) ), where h := [ h (ν,µ) (),, h (ν,µ) (2π( 1)/ )] T, with h (ν,µ) (2πn/) := L l= h(ν,µ) (l)exp( j2πln/) denoting the channel s frequency response values on the FFT grid. ext, we insert F H F = I between D (ω ν ) and H (ν,µ) in (3), and using T sc (k), we re-express (3) as y ν (k) =e jωo(kp +L) D (ω o ) F H D ( h (ν,µ) )ū µ (k) + v ν (k), ν [1, r ]. We deduce from (4) that estimating CFO and the multiple channels based on {y ν (k)} r ν=1 is a nonlinear problem. Given {y ν (k)} r ν=1, our goal is to design training pilots for estimating the CFO ω o and the t r channels h (ν,µ) := [h (ν,µ) (),...,h (ν,µ) (L)] in MIMO-OFDM systems. III. CFO AD CHAEL ESTIMATIO FOR MIMO-OFDM Although ū µ (k) contains both information bearing symbols and training symbols, their separation is challenging due to the presence of CFO. In the following, we specify how to (2) (4) insert pilot symbols (both zero and non-zero ones) so that CFO estimation can be separated from MIMO channel estimation. The insertion of pilot symbols will be performed in two steps. In the first step, we insert the pilot block b µ (k) into the information bearing block c µ (k), per transmit-antenna, as follows: ũ µ (k) =P A c µ (k)+p B b µ (k), (5) where the two permutation matrices P A, P B have sizes K c and K b, respectively, and are selected to be mutually orthogonal: P T A P B = c b. ote that c + b = K. One example of such matrices is to form P A with the last c column of I b + c, and P B with the first b columns of I b + c given as: P A =[e b... e K 1 ], and P B =[e... e b 1]. (6) The structure of ũ µ (k) in (5) is shown in Figure 2. When ũ µ (k) is left multiplied by F H, the permutation matrices in (6) assign OFDM subcarriers to information and training symbols (pilot tones). We will specify the structure of the training block b µ (k) later. In the second step, we insert zeros per block ũ µ (k) to obtain ū µ (k). This insertion can be implemented by left-multiplying ũ µ (k) with the null subcarrier insertion matrix defined as [ ] T sc (k) := e qk (mod ),...,e q k +K 1(mod ), (7) where q k := k /(L +1). We call each subcarrier corresponding to a zero symbol as null subcarrier. Dependence of the null subcarrier insertion matrix T sc (k) on the block index k, implies that the position of the inserted zero is changing from block to block. In other words, (7) implements a null subcarrier hopping operation from block to block. Plugging (7) and (5) into (4), we deduce that the resulting signal at the νth receive antenna, takes the following form: y ν (k) =e jωo(kp +L) D (ω o )F H D ( h (ν,µ) )T sc (k) (8) [P A c µ (k)+p B b µ (k)] + v ν (k). We have described the insertion of two types of training symbols: zero and non-zero ones. In the following, we will show that this idea of hopping pilots is instrumental in establishing identifiability of our CFO estimator. A. CFO Estimation If the CFO was absent (ω o in (8)), then similar to [9], we could isolate from the received block the part corresponding to the training symbols, and by collecting several blocks (enough number of pilots), we could eventually estimate the channels. However, CFO destroys the orthogonality among subcarriers, and the training information is mingled with the unknown symbols, and channels. This motivates acquiring the CFO first, and estimating the channels afterwards. Our CFO estimation algorithm will rely on a de-hopping operation, implemented on a per block basis using the dehopping matrix D H (k) :=diag[1,e j 2π q k,...,e j 2π q k( 1) ]. (9) GLOBECOM /3/$ IEEE

3 Because T sc (k) is a permutation matrix and D ( h (ν,µ) ) is a diagonal matrix, it is not difficult to verify that D ( h (ν,µ) )T sc (k) =T sc (k)d K ( h (ν,µ) (k)), where h (ν,µ) (k) is formed by permuting the entries of h (ν,µ) as dictated by T sc (k). Using the well-designed de-hopping matrix in (9), it is easy to establish the identity D H (k)f H T sc (k) =F H T zp, (1) where T zp := [I K K ( K) ] T is a zero-padding operator. Multiplying (8) by the de-hopping matrix, and using (1), we obtain ȳ ν (k) =D H (k)y ν (k) (11) = e jωo(kp +L) D (ω o )F H T zp g ν (k)+ v ν (k), where g ν (k) := t D K( h (ν,µ) (k))ũ µ (k), and v ν (k) := D H (k)v ν(k). Eq. (11) shows that after dehopping, null subcarriers in different blocks are at the same location, because T zp does not depend on the block index k. The system model (11) per receive antenna, is similar to the one used in [2], [8] for the SISO-OFDM case. This observation suggests that we can generalize the method of [2], [8] to estimate the CFO for MIMO-OFDM systems. To this end, we consider the covariance matrix of ȳ ν (k) Rȳν = D (ω o )F H T zp R (ν) gg T H zpf D H (ω o )+σ 2 I, (12) where R (ν) gg := E [ g ν (k)gν H (k) ], and the noise v ν (k) has covariance matrix σ 2 I. In practice, the ensemble correlation matrix Rȳν is replaced by its sample estimate formed by averaging across M blocks (M > K): ˆRȳν = 1 M M 1 k= ȳ ν (k)ȳ H ν (k). (13) It has been shown in [8] that the column space of Rȳν consists of two parts: the signal subspace and the null subspace. In the absence of CFO, if R (ν) gg has full rank, the null space of Rȳν is spanned by the missing columns (the location of the null subcarriers) of the FFT matrix. The presence of CFO introduces a shift in the null space. Similar to [8], a cost function can be built to measure this CFO-induced shift for our MIMO-OFDM setup. With ω denoting the candidate CFO, this cost function can be written as 1 ( ) { r } 2πn J(ω):= f H D 1 (ω) ( ) 2πn Rȳν D (ω)f, n=k ν=1 (14) where { r ν=1 R ȳ ν =D (ω o )F H T r } zp ν=1 R(ν) gg T H zpf D H (ω o). Clearly, if ω = ω o, then D (ω o ω) =I. ext, recall that the matrix F H T zp is orthogonal to {f (2πn/)} 1 n=k. Hence, if ω = ω o, the cost function J(ω o ) is zero in the absence of noise. However, we have to confirm that if r ν=1 R(ν) gg has full rank, then ω o is indeed the unique zero of J(ω). Here we omit the proof for limitation of space. Therefore, CFO estimates can be found by minimizing J(ω) as ˆω o = arg min J(ω). (15) ω Thanks to subcarrier hopping, J(ω) has a unique minimum in [ π, π) regardless of the position of channel nulls. This establishes consistency of ˆω o, and shows that the acquisition range of our CFO estimator in (15) is [ π, π), which is the full range. B. Estimating MIMO Channels Based on the estimated CFO in (15), we can remove the terms that depend on ω o from {ȳ ν (k)} M 1 k=, and proceed with channel estimation. To derive our MIMO channel estimator, we temporarily assume that the CFO estimate is perfect; i.e., ˆω o = ω o. At the receiver, after removing the CFO related terms from (11), we first take the FFT and then remove the null-subcarriers by multiplying the blocks with T T zp to obtain z ν (k) =e j ˆωo(kP +L) T T zpf D 1 (ˆω o)ȳ ν (k) = D K ( h (ν,µ) (k)) [P A c µ (k)+p B b µ (k)] + ξ ν (k), (16) where ξ ν (k) :=e j ˆωo(kP +L) T T zpf D 1 (ˆω o) v ν (k). By the definitions of P B and the de-hopping matrix in (1), we have D K ( h (ν,µ) (k))p B = P B diag[f(k)h (ν,µ) ], (17) where the b (L +1) matrix F(k) contains the first L +1 columns and q k related b rows of the F, and h (ν,µ) := [h (ν,µ) (),,h (ν,µ) (L)] T. Since P T B P B = I b,wehave z ν,b (k) =P T Bz ν (k) = B µ (k)f(k)h (ν,µ) + ξ ν,b (k), (18) where B µ (k) := diag[b µ (k)] and ξ ν,b (k) := P T B ξ ν(k). Stacking z ν,b (k) for M blocks, we can write the input-output relationship for training symbols as z ν,b = Bh ν + ξ ν,b, (19) where h ν consists of {h (ν,µ) } t, and B 1 ()F() B t ()F() B = B 1 (M-1)F(M-1) B t (M-1)F(M-1) otice that B is the same ν [1, r ]. Collecting z ν,b s from all receive antennas into z b := [ z T 1,b,, zt r,b ]T, the LMMSE channel estimator is now given by ĥ LMMSE := ( σ 2 R 1 h + I r (B H B) ) 1 (Ir B H ) z b, (2) where R h := E[hh H ] with h := [h T 1,, h T r ] T is the channel covariance matrix, and σ 2 denotes the noise variance. These two quantities are assumed to be available. If R h is unknown, the least-squares (LS) estimator can be used instead. GLOBECOM /3/$ IEEE

4 To guarantee that LS estimation can be performed, we need to ensure that the minimum number of blocks is L +1. ow our problem has been reduced to the one in [9]. Due to page limitations, we will not derive the optimum training for our current scheme. Based on results in [9], however, the training block length b for each block can be selected equal to t, and the training block can be designed as: b µ (k) =[ 1 (µ 1),b, 1 (t µ)] T. (21) C. Phase Estimation So far, we have estimated the CFO and the t r channels. However, we will show by simulations that the residual CFO will degrade the bit-error rate (BER) severely as the number of blocks increases. Therefore, one additional step is necessary to deal with the residual CFO. After CFO compensation using the estimate in (15), the received block can be written as [c.f. (11)] ỹ ν (k) =e j(ωo ˆωo)(kP +L) D (ω o ˆω o )F H T zp g ν (k)+ζ ν (k), (22) where ˆω o ω o is the residual CFO, and ζ ν (k) := e j ˆωo(kP +L) D 1 (ˆω o) v ν (k). We observe from (22) that when the CFO estimate is accurate enough, the matrix D (ˆω o ω o ) can be approximated well by an identity matrix. However, the phase term (ˆω o ω o )(kp + L) becomes increasingly large as the block index k increases. Without mitigating it, the phase distortion degrades not only the performance of the channel estimator, but also the BER performance over time. To enhance BER performance, we will use the non-zero training symbols to estimate the phase per block, which was originally designed for channel estimation. Suppose that for the kth block, we obtain the estimated channel from (2). Let us adopt the training sequence in (21) and suppose that the channel estimation step is perfect. After removing the channel, for the νth antenna and the µth entry of z ν,b (k), the equivalent input-output relationship, provided that D (ˆω o ω o ) I, becomes φ ν (k) =e j(ˆωo ωo)(kp +L) b + w ν, (23) where φ ν (k) :=[z ν,b (k)] µ /[ h (ν,µ) b (k)] µ, and w ν is the equivalent noise term after removing the channel. Since b is known, the phase (ˆω o ω o )(kp + L) can be estimated based on the observations from r receive-antennas on a per block basis. To perform this phase estimation step, we do not need to insert any additional pilot symbol and the extra complexity is negligible. IV. PERFORMACE AALYSIS To benchmark the performance of our estimators, we derive the Cramér-Rao lower bounds (CRLB) for the CFO. Starting from the system model in (11), the CRLB for ω o is: ( 2 r M 1 [ ] ) 1 CRLB ω = σ 2 v tr D(k)F H T zp R (ν) gg T T zpf D(k), ν=1k= (24) where D(k) :=diag[pk + L,...,P(k +1) 1]. It follows from (24), that as the number of blocks increases, the CRLB for CFO decreases. Similar comments apply for the signal-tonoise ratio (SR) versus CRLB. If 1, we have that T zp I. Assuming that R (ν) gg = EI, and P, M are sufficiently large, we obtain CRLB ω σ2 v 3 E 2(P L)P 2 M 3 r 3. (25) As expected, the CRLB of CFO is independent of the channel and the number of transmit-antennas, inversely proportional to the SR, and the cube of the number of space-time data. V. SIMULATIOS We conduct simulations to verify the performance of our MIMO OFDM designs. In all experiments, we consider channels with exponential power profile, Rayleigh fading independent taps, and additive white Gaussian noise, with zero-mean, and variance ση. 2 We define SR = E 2 /ση 2 with E 2 denoting energy per symbol. To verify the acquisition range of our CFO estimator, we compare against the algorithm in [4] with t =1and r =1. Fig. 3 illustrates true versus estimated CFO. The ideal line is also shown for comparison. Our CFO estimate is identifiable over the entire range, which is larger than [4]. Figure 4 shows average normalized mean square error (MSE) of ω o for ( t, r ) = (2, 2), = 64, L = 3, and CFO randomly selected in the range [.5π,.5π]. As a figure of merit, we depict the CFO MSE defined as: E[ ˆω o ω o 2 ]/ ω o 2. The CRB we derived in Section IV is also shown as a benchmark. ext, we test the performance of MIMO channel estimation and BER. The simulation parameters are the same as the ones for CFO MSE. Figure 5 shows channel estimation performance. As a performance measure of channel estimation quality, we compute the average channel MSE E[ ĥ h 2 ]/ h 2, where ĥ is obtained using the LS method. We compare with the case where CFO is absent. As expected, we can see that increasing the number of OFDM blocks does not improve the channel MSE accordingly. We also plot BER versus SR in Figure 6. Zero-forcing equalization is used to estimate the information symbols. The ideal case corresponding to perfect channel estimation is plotted for comparison. As discussed in Section III-C, BER performance degrades with time due to residual CFO errors. The BER performance after mitigating for the phase distortion are also shown, which corroborates our claim that the phase estimation improves BER performance considerably. VI. COCLUSIOS In this paper, we derived algorithms to estimate the carrier frequency offset and the channels in multi-antenna OFDM transmissions. We have shown that at least one zero pilot per OFDM block for CFO estimation, and orthogonal training blocks of size t are sufficient for MIMO channel estimation. Moreover, we proved that hopping pilots from block to block GLOBECOM /3/$ IEEE

5 s Space-Time Coder x 1 c 1 c t R cp Insertion of null & training Insertion of null & training y 1 FFT z 1 u 1 u t IFFT IFFT T cp T cp Space-Time Decoder u 1 u t P/S P/S ŝ Estimated CFO Ideal Proposed Standard [11] x r R cp y r FFT z r CFO & Channel Estimator True CFO Fig. 1. Discrete-time equivalent baseband model of MIMO system Fig. 3. CFO acquisition range comparison (( t, r)=(1, 1), L =3) ull Training Information k = 1 2 MSE of CFO 1 k = Fig. 2. One example of ū µ(k) structure k = 2 Average MSE for CFO M=L+1 M=K M=2K M=3K CRLB (M=K) 1 1 enlarges the CFO acquisition range to the full range, while inserting training symbols orthogonally per transmission block leads to low-complexity high performance channel estimation. Our training pattern shows flexibility to be adjusted according to different standards, and is capable of achieving desirable complexity-performance tradeoffs. The performance of our estimators was benchmarked with CRLB, investigated by simulations, and compared favorably with existing alternatives. REFERECES [1] A. F. aguib,. Seshadri and R. Calderbank, Increasing data rate over wireless channels, IEEE Signal Processing Magazine., vol. 17, pp , May 2. [2] H. Liu and U. Tureli, A high efficiency carrier estimator for OFDM communications, IEEE Commun. Letters, vol. 2, pp , Apr [3] M. Morelli and U. Mengali, Carrier-frequency estimation for transmissions over selective channels, IEEE Trans. on Commun., vol. 48, no. 9 pp , Sep. 2. [4] P. H. Moose, A technique for orthogonal frequency division multiplexing frequency offset correction, IEEE Trans. on Commun., vol. 42, pp , Oct [5] Q. Sun, D. C. Cox and H. C. Huang, Estimation of continuous flat fading MIMO channel, IEEE Trans. on Wireless Commun., vol. 1, no. 4, pp , Oct. 22. [6] R. egi and J. Cioffi, Pilot tone selection for channel estimation in a mobile OFDM system, IEEE Trans. on Consumer Electronics, vol. 44, no. 3, pp , Aug [7] U. Tureli, H. Liu and M. Zoltowski, OFDM blind carrier offset estimation: ESPRIT, IEEE Trans. on Commun., vol. 48, pp , Sept. 2. [8] X. Ma, C. Tepedelenlioglu, G. B. Giannakis and S. Barbarossa, ondata-aided carrier offset estimations for OFDM with null subcarriers: Identifiability, Algorithms, and Performance, IEEE Trans. on Commun., vol. 19, no. 12, pp , Dec. 21. [9] X. Ma, L. Yang and G. B. Giannakis, Optimal training for MIMO frequency-selective fading channels, Proc. of 36th Asilomar Conf. on Signals, Systems, and Computers, Pacific Grove, CA, ov. 3-6, SR Fig. 4. Average CFO MSE (( t, r)=(2, 2), L =3) Channel MMSE MSE of Channel 1 1 M=L+1 M=K M=2K M=3K Ideal(w/perfect CFO) SR Fig. 5. Average Channel MSE (( t, r)=(2, 2), L =3) Average BER M=L+1 (w/ comp) M=K (w/ comp) 1 2 M=2K (w/ comp) M=3K (w/ comp) M=L+1 (w/o comp) M=K (w/o comp) M=2K (w/o comp) M=3K (w/o comp) Ideal (perfect channel and CFO) SR Fig. 6. Average BER performance (( t, r)=(2, 2), L =3) GLOBECOM /3/$ IEEE