Carrier Frequency Recovery for Oversampled Perfect Reconstruction Filter Bank Transceivers

ICWMC 2013 : The Ninth International Conference on Wireless and Mobile Communications

Siavash Rahimi Benoit Champagne Dept. of Electrical and Computer Eng. Dept. of Electrical and Computer Eng. McGill University McGill University Montreal, Canada Montreal, Canada [email protected] [email protected]

Abstract—In this paper, we consider the problem of effect, limited number of frequency estimation or syn- data-aided carrier frequency offset (CFO) estimation for chronization algorithms for FBMC systems have been filter bank multicarrier systems, with emphasis on over- considered. In [5], a non data-aided (i.e., blind) CFO sampled perfect reconstruction filter banks. By exploiting estimator for FMT systems is obtained based on the statistical properties of the transmitted pilots in such sys- maximum likelihood (ML) principle. Another blind CFO tems, the maximum likelihood (ML) estimator of the CFO is derived and its performance is investigated numerically estimator based on the best linear unbiased estimation for different channel scenarios. The Cramer Rao bound (BLUE) principle, under the assumption of additive (CRB) on CFO estimator variance for the additive white white Gaussian noise (AWGN) channel, is proposed Gaussian noise (AWGN) channel is also derived as a in [6]. Alternatively, a data-aided joint symbol timing performance benchmark. Simulation results show that the and frequency synchronization scheme for FMT systems proposed ML estimator reaches the provided CRB over is presented in [7], where the synchronization metric AWGN channel, while it also exhibits a robust performance is derived by calculating the time-domain correlation in the case of frequency selective channels. between the received signal and a known pseudo-random Keywords—Data-aided estimation, carrier frequency off- training sequence. In [8], a synchronization scheme for set estimation, maximum likelihood, filter bank multicarrier data-aided symbol timing and frequency offset recovery systems. is developed by employing the least-squares (LS) approach and exploiting the known structure of a special I.INTRODUCTION training sequence whose estimation error does not reach the provided lower bound. Authors in [9], [10] propose Due to its desirable characteristics, multicarrier modu- CFO estimation schemes based on the ML criterion lation (MCM) is currently the main choice for high speed that are specifically tailored for FBMC systems. While wireless communications. For instance, one specific form [9] uses a sequence of consecutive pilots, the approach of MCM, orthogonal frequency division multiplexing in [10] employs scattered pilots instead, which helps (OFDM), has been used in many standards, including estimate and track channel variations during multicarrier WiMAX and LTE-Advanced. Recently, to overcome cer- burst transmissions. This method exhibits an improved tain limitations of OFDM, alternative forms of MCM estimation accuracy when compared to the blind ones, have been proposed, which fall into the general category while requiring a moderate pilot overhead and a low of filter bank multicarrier systems (FBMC). Filtered complexity. multitone (FMT) [1] and oversampled perfect reconstruction filter banks (OPRFB) [2], [3] are examples Although these data-aided methods perform well for of such systems. However, while FMT and OPRFB FMT systems with root raised cosine prototype filters, have been shown to be less sensitive than OFDM to they do not demonstrate the same level of accuracy for carrier frequency offset (CFO) [4], some counter measure OPRFB systems. In particular, since OPRFB transceiver techniques should be also applied to fully exploit the employs longer prototype filters, some of the essential benefits of MCM in this case. assumptions of these methods, e.g., constant CFO effect To mitigate this sensitivity and remove the CFO during the filter support in [9], [10], are not valid in prac-

Copyright (c) IARIA, 2013. ISBN: 978-1-61208-284-4 140 ICWMC 2013 : The Ninth International Conference on Wireless and Mobile Communications

x [n] μˆ x [n] 0 F (zw0) 0 0 K represent the number of subbands and the upsam- K 0 Channel H0 (zw ) K y[m] y[m] Offset  

... C (z) K > M ... Correction pling/downsampling factor, respectively, and is ...... F (zwM−1) M−1 K 0 ν H0(zw ) K x [n] xM−1[n] [m] Offset Estimation assumed. Here, i denotes the complex-valued data xM−1[n] Transmit Filter Bank Receive Filter Bank sequence transmitted on the ith subband at discrete-time −1 nTs, where i ∈ {0,...,M − 1}, n ∈ Z, Ts = Fs Fig. 1. DFT modulated OPRFB transceiver and Fs is the input sampling rate. In DFT modulated FBMC systems, the transmit and receive subband filters can be derived from common prototypes with finite tice, which in turn significantly degrades the estimation impulse responses (FIR) of length D and respective accuracy as we have been able to observe and therefore, PD−1 −n system functions F0(z) = n=0 f0[n]z and H0(z) = they can not be applied to OPRFB systems that employ PD−1 n n=0 h0[n]z . For convenience in analysis, Hi(z) is longer filters. Consequently, finding a synchronization assumed non-causal although in practice, causality can method well suited for OPRFB systems is of particular be restored simply by introducing an appropriate delay interest. in the receiver. Defining w = e−j2π/M , the transmit In this paper, we propose a data-aided, ML-based and receive filters for the ith subband are respectively CFO estimation method that is well suited to OPRFB- obtained as based MCM systems with longer prototype filters, and i i Fi(z) = F0(zw ),Hi(z) = H0(zw ). (1) investigates its performance numerically for different wireless channel scenarios. By applying judicious simpli- In this work, the filter length D is restricted to be a fications to the log-likelihood function and ignoring the multiple of M and K, i.e., D = dP P , where P denotes negligible terms, we are able to significantly reduce the the least common multiple of M and K and dP is a implementation complexity of the proposed ML estima- positive integer. As proposed in [2], [3], to enforce the tor. The Cramer Rao bound (CRB) on CFO estimator perfect reconstruction (PR) property, the paraconjugates variance for the AWGN channel is also derived as a of the transmit filters are employed as receive filters, i.e., ∗ performance benchmark. Results of simulation experi- hi[n] = fi [n]. Therefore, PR can be expressed as ments show that the proposed ML estimator reaches the ∞ X ∗ provided CRB over AWGN channel, while it exhibits fj[q − pK]fi [q − nK] = δijδnp, (2) a robust performance in the case of frequency selective q=−∞ channels, despite the fact that no channel state informa- where δij denotes the Kronecker delta function. As tion (CSI) is employed. shown in Fig. 1, the transmitter output signal at discrete- The paper organization is as follows. The OPRFB time mTs/K is given by M−1 system model is outlined in Section II along with a X X discussion of CFO in such systems. The proposed ML y[m] = xi[q]fi[m − qK], (3) estimator for CFO and the CRB are presented in Section i=0 q III. The estimator performance is assessed in Section IV. where the range of the summation over q is delimited by Finally, conclusions are drawn in Section V. the finite support of the subband FIR filter, fi[m]. Notations: Bold-faced letters indicate vectors and ma- We assume that during a time interval equal trices. Superscript (.)∗ denotes the complex conjugation, to the processing delay of the transceiver system Re[.] the real part and |.| the absolute value of a complex (i.e., 2DTs/K), the transmission channel can be mod- number. IK denotes the K × K identity matrix. More- elled as a linear time-invariant system with FIR c[l] of over, (.)T represent transpose, (.)H Hermitian, (.)−1 the length Q and corresponding system function C(z) = PQ−1 −l inverse of a matrix, while E[.] stands for the statistical l=0 c[l]z . The channel output is corrupted by an expectation. AWGN sequence ν[m], with zero-mean and variance 2 2 E[|ν[m]| ] = σν, assumed to be statistically independent II.PROBLEM FORMULATION from the input data. The input-output relationship of the noisy channel can therefore be expressed as A. OPRFB System Model Q−1 X We consider a DFT modulated OPRFB transceiver y¯[m] = c[l]y[m − l] + ν[m], (4) system, as depicted in Fig. 1, where parameters M and l=0

Copyright (c) IARIA, 2013. ISBN: 978-1-61208-284-4 141 ICWMC 2013 : The Ninth International Conference on Wireless and Mobile Communications

where y¯[m] denotes the received baseband discrete-time l 6= 0, the presence of the CFO term e2πµq in the inter- n,p signal. On the receiver side, y¯[m] is passed through a ference factors Γi,j (µ) (8) would render the transceiver n,p bank of analysis filters and downsampled by K. Accord- system non-PR. That is, the terms Γi,j (µ)xj[p] would be ingly, for each subband, the reconstructed signal x¯i[n] non-zero for j 6= i or p 6= n, and this in turn would result can be written as in a loss of performance in the data transmission process. X It is worth to mention that in previous work [9], [10], it x¯ [n] = y¯[q]f ∗[q − nK]. i i (5) is assumed that the CFO factor e2πµq can be taken out of q the summation in (8) and consequently the interference n,p B. Effect of Carrier Frequency Offset terms Γi,j (µ)xj[p] when j 6= i or p 6= n are negligible, which does not hold for the OPRFB systems. Our interest In practice, there often exists a mismatch between therefore lies in the development of a suitable, data- the carrier frequency in the receiver and the transmitter, aided ML-based approach for the estimation of the CFO denoted as CFO. In this case, the received signal y¯[m] parameter µ. can be modelled as [6]–[8] As seen from Fig. 1, once a suitable estimate of µ is Q−1 2πµm X available, say µˆ, it can be used to compensate the CFO at y¯[m] = e c[l]y[m − l] + ν[m], (6) the receiver front-end and thereby avoid its deleterious l=0 effects. In this paper, we focus on a simplified model where µ is a normalized CFO with respect to the subband of the noisy channel, i.e., AWGN for which the above spacing FsK/M. Upon substitution of (3) and (6) into condition on the channel coefficients c[l] is satisfied, but (5), the reconstructed signal for the ith subband, x¯i[n], extensions of our proposed approach to more complex can be written in terms of the input signals xj[n], for time dispersive channels with joint equalization and CFO j ∈ {0, ..., M − 1}, as recovery is possible. M−1 X X n,p III.FREQUENCY OFFSET ESTIMATION x¯i[n] = Γi,j (µ)xj[p] + νi[n], (7) p j=0 In this section, we first derive a novel CFO estima- n,p tor based on the ML principle, which employs known where Γi,j (µ) and νi[n] are defined as transmitted pilots. We then propose a number of practical Q−1 n,p X X 2πµq ∗ simplifications in the calculation of the associated log- Γi,j (µ) = e c[l]fj[q − l − pK]fi [q − nK], likelihood function (LLF) that result in a lower imple- l=0 q (8) mentation complexity for this estimator. Finally, the CRB X on the variance of unbiased CFO estimators is derived ν [n] = ν[q]f ∗[q − nK]. (9) i i as a performance benchmark. q n,p Here, the complex factor Γi,j (µ) (8) characterizes the A. ML Estimator interference level of the pth input sample from the jth subband on the nth output sample of the ith subband, As indicated above, we consider a simplified AWGN in the presence of CFO with magnitude µ. We note that channel model (i.e., C(z) = 1) in the derivation of the for |n − p| > (D + Q)/K, due to the finite support of proposed ML-based CFO estimator; consequently, the n,p resulting approach will not require the use of a priori fi[n], Γi,j (µ) = 0; accordingly, the range of the sum CSI. In this special case, (8) reduces to over p in (7) is finite. The term νi[n] (9) represents the additive noise passed through the ith subband of the n,p X 2πµq ∗ Γi,j (µ) = e fj[q − pK]fi [q − nK]. (11) receive filter bank. This term has zero-mean and, due to q the PR property imposed on fi[n], its covariance is given by In this work, we define a data frame as the set ∗ 2 E[νi[q]νj [p]] = δijδqpσν. (10) of M subband inputs xi[n] (i ∈ {0, 1,...,M − 1}) entering the transmit filter bank at time n. We assume Considering the reconstructed signal x¯i[n] in (7), that within a burst of N consecutive frames, say from it appears that even if the channel could be perfectly n = 0 to N − 1, a total of L frames, with time indices equalized, which is equivalent to c[0] = 1 and c[l] = 0 for 0 ≤ t0 < t1 < ··· < tL−1 ≤ N − 1, are selected for

Copyright (c) IARIA, 2013. ISBN: 978-1-61208-284-4 142 ICWMC 2013 : The Ninth International Conference on Wireless and Mobile Communications

the transmission of pilot tones. At any given time tn, a of µ. Take the natural logarithm of this PDF, the LLF subset of S subbands, with indices 0 ≤ s0 < s1 < ··· < [11] can be expressed (up to a constant term) in the form sM−1 ≤ M − 1, are dedicated to the transmission of a 1 H log(f (Z; µ)) = − 2 [Z − Λ(µ)] [Z − Λ(µ)] unit-energy pilot symbol psi [tn]. We therefore consider σ N = LS S−1 L−1 a rectangular lattice of P pilot tones distributed 1 X X tn 2 over the time-frequency plane. However, our approach = − zsi [tn] − γ (µ) (22). σ2 si can be applied to other distributions of pilot symbols. i=0 n=0 Without loss in generality (since the pilot symbols are Finally the ML estimator of the CFO can be written as: known to the receiver), we set p [t ] = 1 for all si n µˆ = arg max{log(f (Z; µ))}, (23) µ∈M pair (si, tn). Let zsi [tn] denote the reconstructed signal corresponding to the transmitted pilot psi [tn]. From (7), where M is the search range for µ. According to (22), it follows that maximization of the LLF attempts to find the CFO µ, such that the skewed pilots by this hypothetical µ best tn zs [tn] = γ (µ) + νs [tn], (12) i si i match (in the LS sense) the reconstructed pilot data at where we defined the output of the receive filter bank. M−1 n,p X X t ,p B. Simplifications of Γi,j (µ) γtn (µ) = Γ n (µ) (13) si si,j n,p p j=0 Here, we propose two simplifications of Γi,j (µ) to speed up the calculation of the (22). First consider (11), In order to express (12) in compact vector form, we which includes a summation over the length D (often introduce: large) of the prototype filter f0[q]. Recall that fi[q] = −iq f0[q]w , therefore, (11) can be first simplified as T zsi = [zsi [t0], zsi [t1], ··· , zsi [tL−1]] (14) n,p K(pj−in) n,p Γi,j (µ) = w ϕi−j(µ), (24) λ (µ) = [γ0 (µ), γ1 (µ), ··· , γL−1(µ)]T si si si si (15) where n,p X 2πµq ∗ q∆ T ϕ (µ) = e f0[q − pK]f0 [q − nK]w . (25) νsi = [νsi [t0], νsi [t1], ··· , νsi [tL−1]] (16) ∆ q Therefore, we can write (12) as n,p By this implementation, instead of calculating Γi,j (µ) for all the M 2 possible pairs (i, j), it is sufficient to zsi = λsi (µ) + νsi . (17) n,p compute ϕ∆ (µ) for 2M − 1 possible different values Moreover, by arranging , we can write of i − j = ∆ ∈ {−M + 1, ··· ,M − 1} and find n,p the corresponding Γi,j (µ) by a multiplication as in Z = Λ(µ) + W (18) (24). Therefore, we can roughly reduce the number of Γn,p(µ) where operations needed to compute the terms i,j by a factor of M/2. Z = [zT , zT , ··· , zT ]T , s0 s1 sS−1 (19) Due to the excellent spectral containment of the T T T T Λ(µ) = [λs0 (µ) , λs1 (µ) , ··· , λsS−1 (µ) ] , (20) prototype filters, we can assume that the main source of the CFO-induced interference on each target subband is W = [νT , νT , ··· , νT ]T . (21) s0 s1 sS−1 due to the first few neighbouring subbands, and that the interference from more distant subbands is negligible. As a consequence of the AWGN model assumption, it Therefore, as the second proposed simplification, to follows that W is a zero-mean Gaussian random vector derive the total interference from the other subbands ∗ with diagonal covariance matrix CW = E[WW ] = on the subband with pilot index si, it is sufficient to 2 σνI , where I is the identity matrix. Accordingly, for a only factor in the contribution from the two neighbouring given value of the unknown CFO parameter µ, the obser- subbands on each side of the sith one. As a result, (13) vation vector Z in (18) is also Gaussian with mean Λ(µ) is approximated as 2 and covariance σνI. The probability density function si+2 X X t ,p (PDF) of Z, say f (Z; µ) can therefore be formulated and γtn (µ) ≈ Γ n (µ) (26) si si,j subsequently maximized to produce the desired estimate p j=si−2

Copyright (c) IARIA, 2013. ISBN: 978-1-61208-284-4 143 ICWMC 2013 : The Ninth International Conference on Wireless and Mobile Communications

1 -1 10

CRB 0.8 AWGN Rayleigh 0.6

0.4

0.2

0 10 RMSE

5 Input time frame 60 -2 10 20 30 40 50 10 Input subband n,p Fig. 2. Interference level Γi,j (µ) of pth input sample from jth subband on the nth output sample of the ith subband (p ∈ {0, 1,..., 10}, j ∈ {0, 1,..., 63}, n = 4, i = 16 and µ = 0.08)

C. Cramer Rao Bound -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 CFO Next, we derive a compact expression for the CRB on the variance of an unbiased data-aided CFO esti- Fig. 3. RMSE of CFO estimator µˆ versus µ (SNR=10dB, L = 10) mator obtained over the AWGN channel. Considering 1. The results are reported for an OPRFB system with ∂CW/∂µ = 0, the Fisher I(µ) [11] is M = 64 subbands, up/downsampling factor K = 72 H ∂Λ(µ) −1 ∂Λ(µ) and real prototype filter of length D = 1728, designed I(µ) = 2Re CZ ∂µ ∂µ based on the method [2], [3]. Results are presented for S−1 L−1 2 ∂γtn (µ) different values of the SNR, defined as Es/N0, where 2 X X si = , (27) 2 σ2 ∂µ Es = E |xi[n] | is the input symbol energy and i=0 n=0 2 N0 = σ is the variance of the channel induced Gaussian where Re[.] represents the real part of its argument and noise. Moreover, L known pilots are inserted at the start M−1 of the transmitted burst on all the available subbands ∂γtn (µ) si X X X 2πµq ∗ = 2π qe fj[q−l−pK]f [q−tnK]. (S = M). ∂µ si p j=0 q Figure 2 justifies the assumptions made in Section (28) III-B about the cross-channel interference. In this figure, Therefore, we can obtain the CRB on the variance of an the level of interference Γn,p(µ) from the pth input unbiased CFO estimator µˆ as i,j sample of the jth subband on the nth output sample S−1 L−1 2!−1 ∂γtn (µ) of the ith subband is plotted for p ∈ {0, 1,..., 10}, 1 2 X X si Var(ˆµ) ≥ = . (29) j ∈ {0, 1,..., 63}, n = 4, i = 16 and µ = 0.08. It is I(µ) σ2 ∂µ i=0 n=0 evident that only a few subbands surrounding the target It can be seen that this CRB is inversely proportional subband are contributing as interference sources. to the signal-to-noise ratio (SNR), or proportional to the The root mean square error (RMSE) of the proposed noise variance σ2. Moreover, it generally depends on the estimator, i.e., pE[|µˆ − µ|2], is shown in Figure 3 versus number of the observed pilots N = LS. Specifically, it p the true CFO µ for SNR=10dB and L = 10 pilot frames. is a decreasing function of both L and S. Here, the AWGN acquisition range (i.e., the CFO values where the algorithm’s RMSE coincides with the CRB) is IV. RESULTS AND DISCUSSION observed to be |µ| < 0.2 (i.e., 20% of subband spacing). In this section, the performance of the proposed ML Clearly, in this interval, the RMSE of the proposed estimator of the CFO is assessed and compared with estimator over the AWGN channel is almost independent the CRB (29). In addition to the AWGN channel (where of the CFO. It is worth mentioning that the results Q = 1 and c[0] = 1), we consider a frequency selective reported in [10] show an acquisition range |µ| < 0.1 channel consisting of Q = 5 independent Rayleigh- when implemented for FMT transceivers, whereas in the fading taps with an exponentially decaying power delay OPRFB context, it leads to unsatisfactory results (i.e., profile, i.e., E[|c[l]|2] = βe−αl for l ∈ {0, ··· ,Q − 1}, RMSE level of the order of 10%) due to its underly- PQ−1 2 α = 0.5, and β is a constant such that l=0 E[|c[l]| ] = ing assumptions. Moreover, the proposed method also

Copyright (c) IARIA, 2013. ISBN: 978-1-61208-284-4 144 ICWMC 2013 : The Ninth International Conference on Wireless and Mobile Communications

-1 -1 10 10 CRB CRB AWGN AWGN Rayleigh Rayleigh

-2 10

-2 10 RMSE RMSE

-3 10

-3 -4 10 10 2 4 6 8 10 12 14 0 5 10 15 20 25 30 35 40 Number of Pilots SNR

Fig. 4. RMSE of CFO estimator µˆ versus L (SNR=10dB, µ = 0.08) Fig. 5. RMSE of CFO estimator µˆ versus SNR (CFO=0.08, L = 10) exhibits a robust performance over frequency selective REFERENCES Rayleigh channel, with an RMSE of about 1% within [1] G. Cherubini, E. Eleftheriou, and S. Olcer, “Filtered multitone the range |µ| < 0.1. modulation for very high-speed digital subscriber lines,” IEEE J. Sel. Areas Commun., vol. 20, pp. 1016–1028, Jun. 2002. Sensitivity of the proposed estimator to L, the number [2] S. Rahimi and B. Champagne, “Perfect reconstruction DFT of the pilots being used, is depicted in Figure 4 with modulated oversampled filter bank transceivers,” in Proc. EU- SNR= 10dB and CFO µ = 0.08. As expected, by in- SIPCO, Barcelona, Spain, Aug. 2011, pp. 1588–1592. creasing L, a better estimation accuracy can be achieved. [3] ——, “Oversampled perfect reconstruction DFT modulated filter banks for multi-carrier transceiver systems,” Signal Pro- In addition, over AWGN channel, the CRB is always cessing, in press. attainable. Figure 5 exhibits the RMSE performance [4] ——, “On the robustness of oversampled filter bank multi of the proposed estimator as a function of SNR with carrier systems against frequency offset,” in Proc. ISWCS, true CFO µ = 0.08 and L = 10. Interestingly, in Paris, France, Aug. 2012, pp. 944–948. contrast to the result for the CFO synchronization method [5] V. Lottici, M. Luise, C. Saccomando, and F. Spalla, “Blind carrier frequency tracking for filterbank multicarrier wireless reported in [8], the proposed ML estimator can achieve communications,” IEEE Trans. Commun., vol. 53, pp. 1762 – a performance very close to the CRB over a wide range 1772, Oct. 2005. of SNR values.. [6] T. Fusco, A. Petrella, and M. Tanda, “Blind CFO estimation for noncritically sampled FMT systems,” IEEE Trans. Signal V. CONCLUSION Process., vol. 56, pp. 2603 –2608, June 2008. [7] A. Tonello and F. Rossi, “Synchronization and channel estima- In this paper, we considered the problem of data-aided tion for filtered multitone modulation,” in Proc. WPMC, Abano CFO estimation and recovery for FBMC systems with Terme, Italy, Sep. 2004, pp. 590–594. emphasis on OPRFBs. By exploiting statistical properties [8] T. Fusco, A. Petrella, and M. Tanda, “Data-aided symbol timing of inserted pilots transmitted by such systems over an and CFO synchronization for filter bank multicarrier systems,” IEEE Trans. Wireless Commun., vol. 8, pp. 2705–2715, May AWGN channel, the ML estimator for the CFO was 2009. derived. The complexity of the proposed estimator is [9] M. Carta, V. Lottici, and R. Reggiannini, “Frequency recovery considerably reduced by identification of the insignificant for filter-bank multicarrier transmission on doubly-selective LLF terms and consequently neglecting them in the fading channels,” in Proc. ICC, Glasgow, Scotland, June 2007, estimation. This method was tested over AWGN and pp. 5212–5217. [10] V. Lottici, R. Reggiannini, and M. Carta, “Pilot-aided carrier frequency selective channels with various CFOs, SNRs frequency estimation for filter-bank multicarrier wireless com- and number of pilots. The results show that over the munications on doubly-selective channels,” IEEE Trans. Signal AWGN channel, the proposed estimator exhibits a per- Process., vol. 58, pp. 2783 –2794, May 2010. formance close to the CRB with wider acquisition range [11] S. Kay, Fundamentals of Statistical Signal Processing, Vol. I: compared to other methods, whereas its performance Estimation Theory. Upper Saddle River, NJ, USA: Prentice Hall, 2001. remains satisfactory over frequency selective channels.