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Carrier Frequency Recovery for Oversampled Perfect Reconstruction Filter Bank Transceivers ICWMC 2013 : The Ninth International Conference on Wireless and Mobile Communications Carrier Frequency Recovery for Oversampled Perfect Reconstruction Filter Bank Transceivers 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) ap- proach 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 recon- struction 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 2 f0;:::;M − 1g, n 2 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 1 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 j:j 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[jν[m]j ] = 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.
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