Blind Estimation of OFDM Carrier Frequency Offset ...

160

ECTI TRANSACTIONS ON ELECTRICAL ENG., ELECTRONICS, AND COMMUNICATIONS VOL.5, NO.1 February 2007

Blind Estimation of OFDM Carrier Frequency Offset Using Shift-invariance Properties Aifeng Ren1 and Qinye Yin2 , Non-members ABSTRACT We address the problem of carrier frequency offset (CFO) synchronization in Orthogonal Frequency Division Multiplexing (OFDM) communications systems in the context of frequency-selective fading channels. The mathematic model of OFDM systems with cyclic prefix (CP) and virtual subcarrier is derived. And we propose the shift-invariance CFO estimation algorithm based on DOA-MATRIX method. After the rough estimation, it can acquire the frequency offset and matrix including channel information simultaneously, which are favorable for offset compensation and demodulation of received signals. Simulation results illustrate performance of this algorithm. Keywords: Carrier frequency offset, invariance, OFDM, DOA-Matrix

Shift-

1. INTRODUCTION The basic principle of Orthogonal Frequency Division Multiplexing (OFDM) is to split a high-rate data stream into a number of lower rate streams that are transmitted simultaneously over a number of subcarriers. Although the total channel has the frequency selectivity, for each subchannel to say that is flat (frequency nonselective) fading. One of the principal advantages of OFDM is its utility for transmission at very nearly optimum performance in unequalized channels and in multipath channels [1]. To account for Inter-Block Interference (IBI), OFDM systems rely on the so called cyclic prefix (CP) inserted at the transmitter, after IFFT modulation. To eliminate IBI, the length of CP is chosen larger than the FIR channel memory [2][3]. But one of the principal disadvantages of OFDM is more sensitivity to carrier frequency offset (CFO) than single carrier modulations. There are two destructive effects caused by Manuscript received on August 6, 2006 ; revised on October 11, 2006. This work was supported in part by the National Natural Sciences Foundation (No. 60572046 and No. 60502022) of China. 1 The author is with School of Electronic and Engineering, XiDian University, Xi’an 710071, P. R. China Email: [email protected] 1,2 The authors are with School of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, P.R. China Email: [email protected], [email protected]

CFO in OFDM systems. One is the reduction of signal amplitude and the other is the introduction of intercarrier interference (ICI) from the other carriers. The resulting ICI degrades Bit Error Rate (BER) performance severely [4]. Thus, accurate carrier offset estimation and compensation is more critical in OFDM systems. Recently, the available frequency offset estimation algorithm can be divided into two categories: the non-blind estimation methods and blind estimation methods. The non-blind estimators are based on pilots[5][6] or on the cyclic prefix, such as Maximum Likelihood (ML) carrier-frequency offset estimator proposed by Jan-Jaap van de Beek[7], which makes use of the redundant information introduced by CP availably. In [8] and [9], the channel impulse response (CIR) is first removed from the likelihood function and then by summing over all the possible signals, the frequency domain transmitted signals are removed. The blind estimators contain MUSIC method[10] proposed by H. Liu and ESPRIT method[12] by U. Tureli, which are based on subspace technique and have the super distinguishing performance. On analyzing the signal structure of CP-based OFDM modulation, using virtual carrier technique and shift-invariance properties in DOA-MATRIX method, we present a new CFO estimation algorithm. 2. SIGNAL MODEL In OFDM systems, the information stream bn drawn randomly from a finite-alphabet is parsed into T blocks s(k) = [b0 (k), b1 (k), . . . , bp−1 (k)] , where [•]T denotes transpose. By applying an G-point inverse discrete Fourier transform (IDFT), P parallel data bits are modulated onto G(G > P ) subcarriers and the whole bandwidth is evenly divided for G narrowband subcarriers such that the bandwidth for each subcarrier is relatively small compared to the coherence bandwidth of the channel. The remaining G − P subcarriers are not used for data modulation in order to avoid aliasing at the receiver, which are often referred to as the virtual carriers (VC) [10]. We model the frequency-selective channel as a finite impulse response (FIR) filter with channel CIR T h = [h(0), h(1), . . . , h(L)] , where L is the channel order and [•]T denotes the transpose. The channel has L independent Rayleigh-fading taps and the power gain of each tap is randomly generated with the total

Blind Estimation of OFDM Carrier Frequency Offset Using Shift-invariance Properties

PL power of all taps, l=0 |h(l)|2 , normalized to 1. We assume that the CIR is time invariant over K(≥ 1) consecutive OFDM symbol blocks, but could vary from one set of K blocks to the next. After inserting the cyclic prefix (CP) with the length Ng (Ng ≥ L) in OFDM symbols at the transmitter and removing CP at the receiver, we can make a practical assumption that there is no synchronization error and no intersymbol interference (ISI) from the previous OFDM symbol block. At the receiver end, the removal of the CP makes the received sequence the circular convolution of the transmitted sequence with the channel impulse response {h(l)}L l=0 . In the absence of the carrier frequency offset between the receiver and the transmitter, i.e., ∆f = 0 , we can describe the IBIfree received data blocks as y(k) in the receiver [3]. 

T

˜ H s(k) y(k)=WDH b0 (k), . . . , bp−1 (k), 0, · · · , 0 =WP D | {z } G−P

(1) where DH = diag [H(0, · · · , H(P − 1)]), and H(n) = PL −j 2π G nl denote the complex channel frel=0 h(l)e quency response at the n-th subcarrier frequency. ˜ H = diag [H(0), · · · , H(P − 1)]. Wp comAnd D prises the first P columns of the G × G IDFT matrix W = [w0 w1 · · · wp−1 wp · · · wG−1 ] . Then, the effect of the frequency-selective channel on the OFDM signal is completely captured by scalar multiplications of the data symbols by the frequency responses of the channel at the subcarrier frequencies[11]. In radio communications, because of the local oscillators between the transmitter and receiver inaccuracy and Doppler shift, there is typically a frequency offset after the subcarrier is removed from the received signal, i.e.∆f 6= 0. The received data block y(k)becomes [12][13] ˜ H s(k)ej2π(k−1)(G+Ng )∆f y(k) = EWp D

˜ HS ˜ +V Y = [y(1), · · · , y(K)] = EWp D

3. SHIFT-INVARIANCE CFO ESTIMATION In this section, we estimate the CFO using DOAMATRIX method presented by Q. Yin in 1989 for direction-of-arrival (DOA) estimation. Compared with the well-known ESPRIT method, DOAMATRIX is simpler and more generalized [14]. From (3), we have two received data matrices with (G−1)× K dimensions, X and Z, which are the first G−1 rows and the last rows of Y respectively ˜ HS ˜ + Vhead X = EG−1 W(G−1)×P D ˜ + Vhead = AS

(3)

˜ = [s(1), · · · , s(K)]diag[1, ej2π(G+Ng )∆f , where S j2π(K−1)(G+Ng )∆f ··· ,e ] . Each entry of the matrix V with dimensions G × K is the independent identically distributed (i.i.d.) complex zero-mean Gaussian noise with variance σv2 .

(4)

˜ H ΦS ˜ + Vtail Z = EG−1 W(G−1)×P D ˜ + Vtail = AΦS

(5)

˜ H with dimensions where A = EG−1 W(G−1)×P D (G − 1) × P . EG−1 and W(G−1)×P can be expressed as E(1 : G − 1; 1 : G) and W(1 : G − 1; 1 : P ) , respectively. and are the first rows and the last G − 1 rows of the noise matrix V , respectively.Φ is a P × P diagonal matrix including the information of carrier frequency offset, iand h P −1 1 j2π∆f j2π( G +∆f ) Φ = diag e ,e , · · · , ej2π( G +∆f ) . Note that Φ is an unitary matrix that relates the measurements from subarray X to those from subarray Y. From above deduction we can find (4) and (5) have the ESPRIT structure, in which Φ and A correspond to the shift operator and channel frequency attenuation matrix respectively. As the shift operator and the array manifold matrix can be achieved from eigenvalue-eigenvector pairs using TLS-ESPRIT [15] or DOA matrix method [14], we can also estimate Φ and A similar to [14] or [15] by using shift-invariance property. The auto-correlation matrix of subarray X and the cross-correlation matrix of the two subarrays Z and X can be respectively written as ˜S ˜ H ]AH + σ 2 IG−1 RXX = E[XXH ] = AE[S v

(2)

where E = diag[1, ej2π∆f , · · · , ej2π(G−1)∆f ] is CFO matrix. The presence of the CFO makes the energy loss of the desired signal and introduces intercarrier interference (ICI) resulting from other subcarriers. Without loss of generality, considering the thermal noise and stacking K IBI-free received OFDM symbol blocks, the received data matrix can be expressed as

161

= ARS˜S˜ AH + σv2 IG−1

(6)

= RXX0 + σv2 IG−1 and ˜S ˜ H ]AH + σ 2 JG−1 RZX = E[ZXH ] = AΦE[S v = AΦRS˜S˜ AH + σv2 JG−1 = RZX0 +

(7)

σv2 JG−1

where [•]H denotes conjugate transpose and E[•] ˜ H ] denotes the expectation operator. RS˜S˜ = E[SS the auto-correlation matrix S of and is an P × P Hermitian matrix.IG−1 is an G − 1identity matrix. JG−1 is an square G − 1 matrix, where all the first minor diagonal elements on upper right of the main diagonal are set to 1 and all others to 0.

162


Obviously, the rank of RXX0 in (6) is equal to P , the number of bits in one OFDM symbol block. By doing eigen decomposition on the noisy autocorrelation matrix RXX , we can obtain the eigenvalues λi P and the corresponding eigenvectors ηi , and G−1 RXX = i=0 λi ηi ηiH . We sort the eigenvalues as λ1 ≥ λ2 ≥ · · · ≥ λP > λP +1 = · · · = λG−1 ≈ σv2 . Then, the variance of noise, σv2 , can be estimated PG−1 1 as σ ˆv2 = G−P ˜S ˜ is nonsingui=P +1 λi .When RS −1 lar and G − 1 > P , RXX0 = RXX − σv2 IG−1 = PP ARS˜S˜ AH = i=1 (λi − σv2 ηi ηiH ). Then, we define an auxiliary matrix R referred to as the DOA-MATRIX shift-invariance estimator in [14]. R = RZX0 [RXX0 ]+

(8)

4. SIMULATION RESULTS Numerical simulations are carried out to demonstrate the performance of the proposed shiftinvariance CFO estimation algorithm. In this simulation, we choose BPSK for modulation schemes, and we use Nt = 50 Monte Carlo simulations to evaluate the performance of the estimation of CFO in which we consider OFDM system with G = 64 subcarriers, Ng = 10 and coping with the FIR composite channel of length L, (L < Ng ) for each subcarrier channel. We use the following multipath model to hold for the channel impulse response within the block of data[16]. h(t) =

L X

Al δ(t − τl ), Al = αl e−jθ1

(12)

l=1

where [RXX0 ]+ is the Penrose-Moore pseudoinverse of RXX0 . RXX0 , [RXX0 ]+ and RZX0 can be calculated by RXX0 =

P X

(λi − σv2 )ηi ηiH

i=1

[RXX0 ]+ =

P X i=1

1 ηi η H λi − σv2 i

(9)

RZX0 = RZX0 − σv2 JG−1 It is shown in [14] that if the matrix A in (4) and (5) is column-rank, RS˜S˜ is nonsingular and there are not identical terms in the main diagonal of Φ, the eigenvalues and corresponding eigenvectors of the auxiliary matrix R are the diagonal elements of Φ and columns of the matrix A, respectively. That is RA = AΦ. After the eigen decomposition on matrix ˆ . Then ˆ and A R , we have the estimated matrix Φ the estimated CFO can be calculated from ˆ tr(Φ) ˜ ej2π∆f = PP −1 2π j Gm m=0 e

(10)

where tr(•) denotes trace of the matrix, or the sum of the elements of the principal diagonal of the matrix. Similar to [10], the carrier frequency offset, 2π∆fˆ , calculated from (10) may be at all possible values of ˆ [0, 2π] on the unit circle ej2π∆f . Eq.(10) shows that the proposed algorithm can provide the estimation of CFOs up to integer times the channel spacing. In (10), the CFO estimation is in closed-form, and thus is also better than the MUSIC-like searching method [10] that minimizes the cost function as (11) for CFO estimation.

P (e

j2π∆fˆ

) = min

(G−1 X i=P

) ˆ −1 YYH Ew ˆ i wH i E

(11)

where αl and θl are the amplitude and phase of the channel path associated with the delay τl . The complex-valued path amplitudes {Al }are modelled as zero-mean complex-valued Gaussian random variables and are mutually independent. The L different delays τl = l/B l = 1,2,· · · ,L and B is the bandwidth occupied by the real bandpass signal, are mutually independent and generated according to a truncated exponential distribution[17], with an average delay τmean and a maximum delay τmax . In the following simulations, the carrier frequency offset ∆f is fixed at 0.2ω except where otherwise noted, where ω = 2π G is the subcarrier frequency spacing. We evaluate the performance of the estimator by means of the Mean-Square Error (MSE) as Nt 1 X M SE = Nt i=1

Ã

∆fˆ − ∆f 1G

!2 (13)

Fig. 1 shows the performance of three CFO estimations with the variety of the signal to noise ratio (SNR) in the case of L = 5 and G − P = 10. The bit error rate (BER) performance of an OFDM system, which is retrieved and compensated for the CFO by using the proposed algorithm based on L = 5 and K = 100, is shown in Fig.2. For comparison, we also give the BER performance of the same system with no CFO compensation and with CFO compensation by using ML CP-based estimation and ESPRIT-like estimation. Fig. 3 shows the performance of the proposed estimator at different value K. We assume that the channel fading is slow enough so that the channel does not vary rapidly while CFO estimation is performed. The MSE of the proposed CFO estimator decreases as the OFDM blocks is increased. Fig. 4 shows the performance of the estimated carrier frequency offset estimation algorithm proposed over the range of 1/2 the subcarrier spacing at the SNR of 15dB. In order to comparison, we plot the actual CFO curve.

Blind Estimation of OFDM Carrier Frequency Offset Using Shift-invariance Properties

163

Fig.5 shows the results for estimating the CFO with the number of virtual carriers (G − P ) and the channel length L. In Fig.5, the channel has 5, 10 and 15 independent Rayleigh-fading taps respectively, and the power gain of each tap is randomly generated with the total power of all taps normalized to 1. The results show that the DOA-MATRIX CFO estimator is robust to frequency-selective fading.

Fig.4: Estimated CFO vs. actual CFO

Fig.1: Comparison of MSE vs. SNR

Fig.5: MSE versus the number of virtual carriers (G − P ) 5. CONCLUSIONS

Fig.2: BER Performance of CFO Estimations

In this letter we have presented a DOA-MATRIX based blind CFO estimation for OFDM system with virtual carriers. The method can effectively estimate CFO by exploiting the inherent shift-invariance structure of received signal in time-domain. Based on these simulation results, we conclude that the proposed algorithm can estimate CFOs that are both fractional and integer multiples of the carrier spacing. References [1]

[2]

[3]

Fig.3: MSE versus SNR at different value K

[4]

H. M. Paul, “A technique for Orthogonal Frequency Division Multiplexing frequency offset correction,” IEEE Trans. on communications, vol. 42, pp. 2908-2914, Oct. 1994. J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., vol. 28, pp. 5-14, May 1990. Z. Wang, and B. G. Georgios, “Wireless multicarrier communications: where Fourier meets Shannon,” IEEE Signal Processing, pp. 29-48, May 2000. T. Pollet, M. Vanbladel, and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier

164


frequency offset and Wiener phase noise,” IEEE Trans. Commun., vol. 43, pp. 191-193, Feb-MarApr. 1995. [5]

P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE. Trans. Commun., 42(10):2908-2914, Oct. 1994.

[6]

T. M. Schmidl and D. C. Cox., “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., 45(12):1613-1621, Dec. 1997.

[7]

J. J. V. D. Beek, M. Sandell, and P. O. Borjesson, “ML Estimation of Time and Frequency Offset in OFDM Systems,” IEEE Trans. Signal Processing, vol. 45, pp.1800-1805, July 1997.

[8]

M. Morelli and U. Mengali, “Carrier-frequency estimation for transmissions over frequency selective channels,” IEEE Trans. Commun., vol. 48, no. 9, pp. 1580-1589, Sep. 2000.

[9]

E. Chiavaccini and G. M. Vitetta, “Maximumlikelihood frequency recovery for OFDM signals transmitted over multipath fading channels,” IEEE Trans. Commun., vol. 52, no. 2, pp. 244 251, Feb. 2004.

[10] H. Liu, and U. Tureli, “A high-efficiency carrier estimator for OFDM communications,” IEEE Commun., vol. 2, pp. 104-106, April 1998. [11] M. Ghogho, and A. Swami, Signal Processing for Mobile Communications Handbook, CRC Press, July 29, 2004, Ch. 8. [12] U. Tureli, H. Liu, and M. D. Zoltowskl, “OFDM blind carrier offset estimation: ESPRIT,” IEEE Trans. Commun., vol. 48, pp. 1459-1461, Sept. 2000. [13] H. M. Paul, “A technique for Orthogonal Frequency Division Multiplexing frequency offset correction,” IEEE Trans. Commun., vol. 42, pp. 2908-2914, Oct. 1994. [14] Q. Y. Yin, R. W. Newcomb, and L. Zou, “Estimating 2-D angles of arrival via two parallel linear array,” Proc. IEEE ICASSP’89, Glasgow, Scotland, vol. 4, pp. 2803-2806, May 1989. [15] R. Roy and T. Kailath, “ESPRIT Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. ASSP, vol. 37, pp. 984995, July 1989. [16] J. G. Proakis, Digital Communications, Third Edition, McGraw-Hill, Inc., 1995, Ch. 14. [17] M. Luise, M. Marselli, and R. Reggiannini, “Low-Complexity Blind Carrier Frequency Recovery for OFDM Signals Over FrequencySelective Radio Channels,” IEEE Trans. on communications, vol. 50, pp. 1182-1188, July 2002.

Aifeng Ren was born in Hebei Province in China, in Nov. 1974. He received the B.S. and M.S. degrees on electronic and engineering from XiDian University in China in 1997 and 2000 respectively, and now pursuing his Ph.D. degree in electronics and information engineering in Xi’an Jiaotong University, P.R. China. His research interests lie in the areas of wireless communications, multimedia communication, and signal processing. Email: [email protected]

Qinye Yin was born in Jiangsu Province in China, in 1950. He received the B.S. and M.S. degrees from Xi’an Jiaotong University, and received Ph.D. degree from the University of Maryland, U.S.. He is now a professor and Ph.D. advisor of the School of Electronics and Information Engineering, Xi’an Jiaotong University. His research interests lie in the areas of signal processing and communications, space spectrum estimation, neural net and its applications, and time-frequency analysis. Email: [email protected]