A Chirp-Based Frequency Synchronization Approach for Flat Fading Channels

Ana Belen Martinez∗, Atul Kumar∗, Marwa Chafii†, Gerhard Fettweis∗ ∗Vodafone Chair Mobile Communications Systems, Technische Universitat¨ Dresden, Germany {ana-belen.martinez,atul.kumar,fettweis}@ifn.et.tu-dresden.de †ETIS, CY Universite,´ ENSEA, CNRS, F-95000, Cergy, France [email protected]

Abstract—In different new and emerging technologies, where which requires the center value above a certain threshold at high frequency offsets (FOs) are expected due to the low-cost the output of each matched filter (MF). nature of the local oscillators, and low sampling rates are chosen A dual-chirp, which consists of a linear up-chirp transmitted for the sake of power efficiency, accurate FO estimation becomes a challenging task. This work proposes a frequency synchro- simultaneously with its complex conjugate down-chirp, acts nization approach based on a dual-chirp training sequence. Its as a reference for synchronization in AWGN channels performance is evaluated by of simulations and validated in [3]. The outputs of the MFs matched to each of the with the derived Cramer-Rao lower bound. It is shown that the linear chirps are exploited for the estimation of the FO in proposed method achieves near-optimum performance with lower frequency domain. However, its performance is linked to a complexity than the state-of-the-art approach. Index Terms—Frequency offset estimation, synchronization, high computational complexity due to the calculation of a up-down-chirp, dual-chirp, reference sequences, CRLB. discrete Fourier transform (DFT) with an factor and a subsequent polynomial interpolation. This work proposes a frequency synchronization approach I.I NTRODUCTION with a dual-chirp as a reference signal, suitable for flat fading The cost-effectiveness sought by different new and emerging channels under high FOs. Unlike in [3], the FO estimation technologies entails the choice of low-cost local oscillators. is accomplished in time domain. The FFO, obtained through Therefore, due to their low accuracy, high frequency offsets AC, is compensated in the reference signal prior to the IFO (FOs) may appear between the reference frequency at the estimation. The IFO can thus be easily estimated using the receiver used for down-conversion and the carrier frequency positions of the global maxima at the outputs of the MFs. of the received signal. Moreover, in order to achieve low Under the assumption of perfect time synchronization, the complexity at the end device, a low sampling rate for initial proposed approach is derived for a dual-chirp and for a linear synchronization is commonly selected. Thus, the low accuracy up-down-chirp. Theoretical limits in terms of the Cramer associated to a low sampling rate, together with the potentially Rao lower bounds (CRLBs) are derived for each reference high FOs, poses an important challenge in the FO estimation, a sequence, which serve as benchmarks for results evaluation. crucial task that has to be appropriately accomplished to avoid Comprehensive simulations have been conducted to verify the performance degradation at the receiver. validity of the presented technique. For the purpose of com- Aiming at high of detection in the low signal- parisons, the DFT-based approach described in [3] has been to- ratio (SNR) regime and moderate latency, a detection implemented with different upsampling factors and evaluated approach based on matched filtering is widely used in the liter- with both reference sequences. It is shown that the proposed ature. The robustness of chirp sequences against high FOs in a approach achieves a performance close to the CRLB with low matched filter-based detection, as well as their special time and computational complexity, and significantly outperforms the frequency properties, have motivated their application in the state-of-the-art algorithm for moderate to high SNR regimes. detection process and in the estimation of the synchronization The remainder of this paper is structured as follows. In [1] [2]. In [1], a training sequence consisting of a Section II, the system model and the reference sequences linear up-chirp followed by its conjugate, known as linear up- are introduced. The proposed synchronization algorithm is down-chirp, assists the synchronization process under additive presented in Section III. In Section IV, the CRLBs are derived. white Gaussian noise (AWGN) channel conditions. For the Simulation results are provided and discussed in Section V. sake of low latency, fractional and integer frequency offsets Finally, conclusions are drawn in Section VI. are estimated independently. The periodicity obtained after downsampling a linear chirp is utilized for the estimation of II.S YSTEM MODEL the fractional frequency offset (FFO) with an approach based Under perfect time synchronization, the received baseband on autocorrelation (AC). For the calculation of the integer signal in discrete time domain can be described as frequency offset (IFO), a laborious algorithm is implemented, r[n] = hx[n]ej2πn + w[n], (1) where h represents the complex channel gain, unknown but synchronization with an AC-based approach. Specifically, the constant over the transmission duration of the reference signal phase of the AC of a periodic signal at its maximum provides x[n],  indicates the FO, normalized by the sampling frequency an estimation of the FFO [4]. fs, and w[n] denotes complex AWGN with zero and At the entrance of the synchronization block, a demulti- σ2. plexer extracts samples from the received signal r[n] alter- A. Linear Up- and Down-Chirp natively, generating two downsampled signals, r1[n] = r[2n], and r2[n] = r[2n + 1]. The linear up- and down-chirp sequences can be expressed Considering the linear up-down-chirp as reference, the first as Nl/2 samples of each demultiplexed sequence correspond to  n2   n2  jπ N −n −jπ N −n xu[n] = e l , and xd[n] = e l , (2) a downsampled version of the received up-chirp, whilst the second Nl/2 samples refer to the down-chirp as: where n = 0, 1, ..., Nl − 1. The length of the sequence, Nl, known as compression factor in systems, is given by the ru,1[n] = r1[n], rd,1[n] = r1[n + Nl/2], time-bandwidth product Nl = BTl, where B is the frequency ru,2[n] = r2[n], rd,2[n] = r2[n + Nl/2], (6) band swept during the time span T . The reference sequence l for n = 0, 1, ..., N /2 − 1. Consequently, four different signals built with a linear up-chirp followed by its complex conjugate, l of length N /2 are available to perform four independent ACs. or down-chirp, is thus given by l W = N /4 ( The AC with sliding and averaging windows of l l xu[n], n = 0, 1, ..., Nl − 1, samples can be calculated according to xl[n] = ∗ (3) xu[n − Nl], n = Nl,Nl + 1, ..., 2Nl − 1. Wl−1 1 X ∗ pj,q[n] = rj,q[n + k + Wl]rj,q[n + k], (7) B. Dual-Chirp Signal Wl k=0 The dual-chirp, also referred to as composite chirp, consists where j ∈ {u, d} and q ∈ {1, 2}. Neglecting the noise terms of one linear up-chirp and its corresponding down-chirp, which for simplicity, the outputs of the autocorrelators at the maxi- are transmitted simultaneously. Since the down-chirp is the mum of their absolute values can be expressed as complex conjugate of the up-chirp, the resulting signal is 2 b jπNl purely real-valued, and can be formulated as pj,1[nj,1,max] = |h| (−1) e , (8) 2 b+1 jπNl   n2  pj,2[nj,2,max] = |h| (−1) e , (9) xc[n] = α cos π − n , (4) Nc where b = mod(Wl, 2), with mod(.) being the modulo opera- where n = 0, 1, ..., Nc − 1, being Nc the length in samples tor. The corresponding phases of the dual-chirp, and α represents a normalization factor to φj,1 =∠pj,1[nj,1,max] = πNl + bπ, (10) ensure unitary power. φj,2 =∠pj,2[nj,2,max] = πNl + (1 − b)π, (11) In terms of the underlying linear chirps, xc[n] can be expressed as can be used to estimate the individual FFOs as α   xc[n] = (xu[n] + xd[n]) , (5) 1 1 1 2 ˆj,1 = φj,1 − b , ˆj,1 ∈ [−(1 + b), (1 − b)] , Nl π Nl being xu[n] and xd[n] the sequences given in (2), of length   (12) Nc. 1 1 1 ˆj,2 = φj,2 − (1 − b) , ˆj,2 ∈ [(b − 2), b] . (13) III.F REQUENCY OFFSET ESTIMATION Nl π Nl In order to achieve an estimation with high accuracy, the The estimation of the FO is accomplished in different stages. information provided by all individual phases should be used. The FFO is initially calculated and subsequently compensated However, since ˆ and ˆ lie in different definition domains, in the received reference sequence. Afterwards, using the j,1 j,2 a simple average of the phases would reduce the effective sequence with compensated FFO, the IFO is found. Without estimation range of the FFO to [−1/N , 0]. This effect can loss of generality, the detailed analysis presented in the next l be avoided with the modification of the phases obtained from subsections is based on the up-down-chirp, and thereafter the the second demultiplexed signal as φ0 = φ − π. The FFO modifications to be done when applying the dual-chirp are j,2 j,2 can thus be obtained as explained. 0 0 φu,1 + φd,1 + φu,2 + φd,2 A. Fractional Frequency Offset Estimation ˆF,l = . (14) 4πNl The method employed for the estimation of the FFO takes Demultiplexing the dual-chirp enables the computation of advantage of the periodicity generated after downsampling a two ACs, where the sliding and averaging windows consist of chirp sequence. In particular, downsampling a linear chirp with Wc = Nc/4 samples. The maxima of the ACs are given by N 2 compression factor i by results in another linear chirp 2 2 b jπNc pc,1[n1,max] = |h| α (−1) e , (15) sequence with compression factor Ni/4. Thus, Ni should be 2 2 b+1 jπNc an integer multiple of 4. The downsampled sequence exhibits pc,2[n2,max] = |h| α (−1) e . (16) a periodicity of Ni/4 samples that can be exploited for 00 The same reasoning as in the previous case leads to an nˆi,d = arg max |mi,d[ni ]| = NiI,i + γiNi − 1, (28) estimation of the FFO as n 0 φ1 + φ2 which can, ideally, be combined to estimate the IFO. However, ˆF,c = , (17) 2πNc the presence of uncompensated FFOs of magnitude close to 0 half of the IFO spacing (fs/Ni), distorts the ideal behavior where φ1 corresponds to the phase of pc,1[n1, ] and φ refers max 2 of the MF. Its output is not characterized by a clearly defined to the phase of pc,2[n2, ] after phase compensation. max maximum, but by two values of similar amplitude. In this case, B. Integer Frequency Offset Estimation choosing the index of the highest value can lead to ambiguities in the estimation of the IFO. An accurate estimation should After FFO compensation, the IFO can be found using the consider the position of the first value exceeding a certain information provided by the outputs of the MFs. threshold at the output of each MF. The IFO can thus be estimated with an accuracy of f /(2N ), avoiding potential The impulse response gi,j[n] of the MF matched to the s i ambiguities due to uncompensated FFOs. The new indexes, signal xj[n] can be formulated as ∗ after ambiguity resolution, provide an estimation of the IFO gi,j[n] = βixj [Ni − 1 − n], (18) as nˆi,d − nˆi,u − (γi − 1)Ni where i ∈ {l, c}, j ∈ {u, d}, and ˆI,i = . (29) ( 2Ni 1, i = l, Hence, the overall FO estimated with each sequence consists βi = (19) 1/α, i = c. of two terms, as ˆi = ˆI,i + ˆF,i. [s] The convolution between the signal rf [n] and the matched filter gi,j[n] is defined as C. DFT-Based Frequency Offset Estimation ∞ 1 X m [n] = r[s][k]g [n − k], The approach presented in [3] exploits the properties of the i,j f i,j (20) Ni DFT for the estimation of the FO. After multiplying the time- k=−∞ synchronized received signal with the complex conjugate of [s] where r [n] is the reference signal after proper compensation ∗ f each reference chirp, βixj [n], two DFTs with an upsampling of the FFO ˆF , and it can be described as factor are performed. The maxima obtained at the outputs of [s] −j2πˆF,i(n+si,j Ni) the DFTs are used for the estimation of the FO, similarly to r [n] = r[n + si,jNi]e , (21) f the approach described in time domain, making use of the where ( time-frequency duality of the DFT. 0, (i = l ∧ j = u) ∨ i = c, si,j = (22) 1, (i = l ∧ j = d), IV.CRLB S FOR FREQUENCY OFFSET ESTIMATION with ∧ and ∨ as the logical operators ”and” and ”or”, respec- tively. The (LF) of the received vector r, After some algebraic manipulations, the magnitudes of the with Ni independent observations in complex AWGN, can be outputs of the MFs can be written as expressed as [5] sinc (N (n0 +  )(1 − |n0 |)) 0 0 i i I,i i 1 −(r−s(ξ))H C )−1(r−s(ξ)) |mi,u[ni]| = |h|(1 − |ni|) , p(r; ξ) = e[ r(ξ) ], (30) sinc(n0 +  ) N i I,i π det(Cr(ξ)) (23) where C = σ2I is the matrix of the received sinc (N (n00 −  )(1 − |n00|)) r(ξ) 00 00 i i I,i i signal r[n], and the components of s(ξ) are given by |mi,d[ni ]| = |h|(1 − |ni |) 00 , sinc(ni − I,i) j2πn jφh (24) s[n] = h0x[n]e e , n = 0, 1, ..., Ni − 1, (31) 0 00 h φ where the new discrete time indexes, ni and ni , are given by where 0 and h are, respectively, the magnitude and the argument of the complex channel gain h, and x[n] corresponds n − Ni + 1 n − γiNi + 1 n0 = , and n00 = , (25) to the transmitted reference sequence of length N samples, i N i N i i i which will be properly specified in the next subsections. with ( 2, i = l, The second derivative of the log-likelihood function (LLF) γi = (26) 1, i = c, of r with respect to the parameters p and q, provides the {p, q}-element of the matrix, which can and the frequency offset  represents the IFO (plus some I,i be calculated as [5] residual FFO, which will be neglected for this analysis), "N −1 # normalized by the sampling frequency. 2 Xl ∂s∗[n] ∂s[n] [I] = Re , (32) The arguments of the outputs of the MFs at their maximum p,q σ2 ∂ξ ∂ξ n=0 p q correspond to where p, q ∈ {1, 2, 3}, refer to the components of the vector nˆ = arg max |m [n0 ]| = −N  + N − 1, (27) T i,u i,u i i I,i i of unknown real parameters ξ = [h0  φh] . n A. CRLB for Linear Up-Down-Chirp to the theoretical references provided by the derived CRLBs. The sweeping frequency B of the reference signal is set to For the derivation of the CRLB of the FO attainable 180 kHz. The system is assumed to work at the . with a linear up-chirp, the signal x[n] equals x [n], for u For a fair comparison, both reference signals are transmitted n = 0, 1, ..., N − 1. The Fisher information matrix is given l with the same power and have the same length, with N = 64 by l and N = 128 samples. N and N are chosen to ensure 1 0 0  c l c integer compression factors after downsampling the sequences. 2Nl 2π2h2(N −1)(2N −1) I =  0 l l 2  . (33) 2 0 3 πh0(Nl − 1) During the simulations, a uniformly distributed FO in the σ 2 2 0 πh0(Nl − 1) h0 range [−18, 18] kHz has been generated, corresponding to a 900 With SNR = h2/σ2, and after some algebraic manipula- transmission in the MHz band and an accuracy of the local 0 20 tions, the minimum variance associated to the up-chirp can oscillators of ppm. be expressed as In order to compare the performance of the proposed ap- proach with the state-of-the-art algorithm [3], the latter has 3 −1 been implemented and simulated with upsampling factors 4 V ar(ˆl,u) ≥ [I ]22 = 2 2 , (34) 2π Nl(Nl − 1)SNR and 8. Additionally, least-squares (LS) based second-order- which is the same as the variance attainable with the down- polynomial interpolation with an upsampling factor 4 has chirp. Therefore, due to the additivity property of the infor- been incorporated to increase the accuracy of the estimation. mation for independent observations [6], the variance obtained The coefficients of the second-order-polynomial are calculated with the training sequence consisting of one up-chirp followed according to the least-squares principle, based on three samples by one down-chirp results in selected at the output of the DFT, namely the highest one, the 1 preceding, and the following. V ar(ˆ ) = V ar(ˆ ). (35) l 2 l,u The metric selected for the performance evaluation is the B. CRLB for Dual-Chirp mean squared error (MSE) of the estimation, defined as MSE = E ( − ˆ)2, where  and ˆ are normalized by the In this case, x[n] = xc[n], for n = 0, 1, ..., Nc − 1. The same sampling frequency. procedure as in the preceding case yields the new Fisher The simulation results obtained with the up-down-chirp information matrix and with the dual-chirp are depicted in Fig. 1(a) and 1(b), α2U 0 0  respectively. Even though they show similar behavior, there 2 2 2 2 2 2 is a clear difference in the magnitude of the MSE. This effect I = 2  0 4π h0α W 2πh0α V  . (36) σ 2 2 2 2 0 2πh0α V α h0U is in accordance with the theoretical ratio between After applying the following approximations, given in (41), which in this case has a value of 5.65 dB.√ It can be shown that, for ρ = 2 and long sequences, α ≈ 1/ 2, and Nc−1   2  X n Nc the ratio R tends to 6 dB. The main reason for the superiority of U = cos2 π − n ≈ , (37) N 2 the dual-chirp is the higher compression factor of its underlying n=0 c linear reference sequences. Nc−1   2  X n Nc(Nc − 1) V = n cos2 π − n ≈ , (38) The estimation obtained with the proposed approach N 4 n=0 c achieves an MSE close to the CRLB for SNR values above 5 dB. In the low SNR regime, the MSE is significantly im- Nc−1   2  X n Nc(Nc − 1)(2Nc − 1) W = n2 cos2 π − n ≈ , proved through the phase compensation described in III-A N 12 n=0 c (Proposed mod). The performance achievable with the DFT- (39) approach highly depends on the upsampling factor and the LS the lower bound for the variance of the FO obtainable with a interpolation. Especially the latter operation can substantially improve the accuracy of the estimation, and provide, in a dual-chirp of length Nc samples can be written as limited SNR range, an MSE lower than the proposed approach. −1 3 V ar(ˆc) ≥ [I ]22 ≈ . (40) Nevertheless, the MSE obtained with the DFT-approach shows 4π2α2N (N 2 − 1)SNR c c an error floor, which can only be reduced through the use of With Nc = ρNl, the ratio between the CRLBs achievable with higher upsampling factors and interpolation. Consequently, a both training sequences can be expressed as performance improvement leads to an increase in latency and V ar(ˆ ) α2ρ(ρ2N 2 − 1) computational complexity. R = l = l . (41) 2 Table I lists the computational complexity of each approach V ar(ˆc) (Nl − 1) in terms of number of complex multiplications (CMs). The proposed approach considers the CMs associated to the ACs V.S IMULATION RESULTS and MFs. In case of the up-down-chirp, four ACs of length This section presents the performance evaluation of the pro- Nl/4 are calculated, whilst with the dual-chirp, two ACs posed synchronization approach described in Section III. For of length Nc/4 are computed. For the sake of efficiency, this purpose, simulations have been conducted and compared matched filtering has been implemented in frequency domain. 10-4 10-4

3 10-8

-6 -6 10 10 5 6 7

10-8 10-8

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5 6 7 10-10 10-10 -5 0 5 10 15 20 25 30 -5 0 5 10 15 20 25 30

(a) Linear up-down-chirp (b) Dual-chirp Fig. 1: MSE of the frequency offset estimation obtained with (a) a linear up-down-chirp and (b) a dual-chirp.

Appropriate zero padding has been considered to achieve has been evaluated by means of simulations, compared to equivalence between linear and circular convolution, and en- the state-of-the-art approach, and validated with theoretical able a DFT operation based on Radix-2 fast Fourier transform limits provided by CRLBs. Aiming at low complexity, the (FFT). These assumptions lead to the new extended lengths, synchronization process can be undertaken at a low sampling Nl,ext = 128, and Nc,ext = 256 samples. rate, avoiding upsampling and interpolation. Nevertheless, a The DFT-based approach includes the initial multiplication high accuracy is achieved, and the approach exhibits near- with the complex conjugate of each linear reference signal, optimum performance for SNR values above 5 dB. the complexity due to the DFT operation (assuming Radix-2 Two training sequences have been considered for the evalua- FFT implementation), and the complexity required for the LS tion. Under the assumption of identical length and transmit interpolation (CLS). The latter includes the operations needed to power, it is shown that the dual-chirp achieves a lower MSE find the coefficients of the second-order polynomial, evaluate than the linear up-down-chirp, at the cost of higher computa- the polynomial at the newly defined positions, and find the tional complexity. maximum. The ratio between the derived CRLBs and the complexity Considering the simulation parameters, the DFT-based ap- analysis can be used as tools for the design and selection of the proach with upsampling factor 8 requires approximately 40% most suitable reference sequence that allows to meet specific more CMs than the proposed approach. A further improvement system requirements. of the accuracy achieved with the state-of-the-art algorithm, VII.A CKNOWLEDGMENT through the use of higher upsampling factors, would increase significantly the number of CMs, making the difference in Authors acknowledge the European Research Council under terms of complexity between both approaches more evident. the European Union’s Horizon 2020 Research and Innovation The compression factor of the reference signal, is respon- Programme No. 732174 (ORCA) and the Paris Seine Initiative sible, not only for the accuracy of the estimation, but also (ASIA Chair of Excellence Grant). for the complexity. Therefore, a higher compression factor REFERENCES selected for the dual-chirp, compared to the up-down-chirp, [1] S. Boumard and A. Mammela, “Time domain synchronization using implies a higher number of CMs, independently of the adopted Newman chirp training sequences in AWGN channels,” in Proc. IEEE algorithm. Intern. Conf. Commun. (ICC 2005), vol. 2, May 2005, pp. 1147–1151. [2] J. Zhang, M. M. Wang, and T. Xia, “Practical Synchronization Waveform TABLE I: Complexity of the presented approaches. for Massive Machine-Type Communications,” IEEE Trans. Commun., Approach Complexity (CMs) vol. 67, no. 2, pp. 1467–1479, Feb. 2019. 1 [3] J.-C. Lin and Y.-T. Sun, “Estimation of timing delay and frequency offset Proposed 0.5γiNi + 2 (1.5 log2(Ni,ext) + 1) using a dual-chirp sequence,” in Proc. Wireless Commun., Veh. Technol., DFT 2Ni + fupNi log2(fupNi) + 2 CLS Inf. Theory and Aerosp. Electron. Sys. Technol. (Wireless VITAE 2009), 1 2009. γi as defined in (26) [4] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchroniza- tion for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613–1621, Dec. 1997. VI.C ONCLUSIONS [5] S. M. Kay, Fundamentals of Statistical . Volume I. Estimation Theory. Upper Saddle River (N.J.): Prentice-Hall, 1993. In this work, a chirp-based frequency synchronization ap- [6] P. C. Papaioannou, “On Statistical and Related proach, suitable for flat fading channels under high FOs, Measures of Information,” Ph.D. dissertation, Iowa State University, has been proposed and examined in detail. Its performance 1970. [Online]. Available: https://lib.dr.iastate.edu/rtd/4258