Joint Frequency Offset, Time Offset and Channel Estimation for OFDM/OQAM Systems

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Baghaki and Champagne EURASIP Journal on Advances in Signal EURASIP Journal on Advances Processing (2018) 2018:4 DOI 10.1186/s13634-017-0526-4 in Signal Processing

RESEARCH Open Access Joint frequency offset, time offset, and channel estimation for OFDM/OQAM systems Ali Baghaki* and Benoit Champagne

Abstract Among the multicarrier modulation techniques considered as an alternative to orthogonal frequency division multiplexing (OFDM) for future wireless networks, a derivative of OFDM based on offset quadrature amplitude modulation (OFDM/OQAM) has received considerable attention. In this paper, we propose an improved joint estimation method for carrier frequency offset, sampling time offset, and channel impulse response, needed for the practical application of OFDM/OQAM. The proposed joint ML estimator instruments a pilot-based maximum-likelihood (ML) estimation of the unknown parameters, as derived under the assumptions of Gaussian noise and independent input symbols. The ML estimator formulation relies on the splitting of each received pilot symbol into contributions from surrounding pilot symbols, non-pilot symbols and additive noise. Within the ML framework, the Cramer-Rao bound on the covariance matrix of unbiased estimators of the joint parameter vector under consideration is derived as a performance benchmark. The proposed method is compared with a highly cited previous work. The improvements in the results point to the superiority of the proposed method, which also performs close to the Cramer-Rao bound. Keywords: OFDM/OQAM, Joint estimation, Filter bank multicarrier, Carrier frequency offset, Sampling time offset, Channel impulse response, Cramer-Rao bound

1 Introduction paramount importance for the successful application of Due to the several benefits of multicarrier modulation OFDM/OQAM in practical systems. (MCM) over single carrier modulation, the former has There exist three main categories of synchronization been considered as the primary choice in the physical and channel estimation methods for OFDM/OQAM sys- layer implementation of telecommunication systems for tems: blind, semi-blind and pilot-based methods. While quite a long time. Among the MCM family, orthogonal blind methods, e.g., [4–7], provide higher spectral effi- frequency division multiplexing (OFDM) has been largely ciency by avoiding the overhead of training sequences, the studied and adopted in many wireless and wireline stan- requirement of a longer observation window for accurate dards [1, 2]. Still, as an alternative and promising form of estimation limits their tracking ability, rendering them MCM for future generations of wireless networks, a vari- less popular in most practical applications. In contrast, ant of OFDM based on offset quadrature amplitude mod- the semi-blind methods only require the transmission of ulation (OFDM/OQAM) has attracted much research a small number of parameters to resolve an estimation interest in recent years, due to its many advantages over ambiguity, e.g., [8] and as such, they offer a useful trade- classical OFDM, including higher spectral efficiency and off between spectral efficiency and estimation accuracy. reduced sensitivity to timing and frequency mismatch However, since in practical scenarios, the training symbol [3]. In spite of these advantages, accurate carrier fre- overhead needed to obtain a better estimation perfor- quency and timing synchronization along with channel mance is usually tolerated, our focus here is on pilot-based estimation (for the purpose of equalization) remain of synchronization and channel estimation. Compared to channel estimation, pilot-based carrier frequency offset (CFO), and sampling time off- *Correspondence: [email protected] set (STO) estimation has received less attention in the Department of Electrical and Computer Engineering, McGill University, 3480 OFDM/OQAM literature. In [9], a maximum likelihood University Street, H3A 0E9 Montreal, Canada (ML) symbol timing estimator is derived by using two

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training symbols per burst transmission. In [10], using the Although the aforementioned synchronization and same preamble structure, the authors extend this work equalization problems have been separately addressed by proposing a joint ML-based estimator of the CFO and throughout the literature on OFDM/OQAM systems, STO. Then, to avoid the computational complexity of a only a few research papers can be found, e.g., [29], that are two dimensional ML search, a feasible joint estimation devoted to STO, CFO and channel estimation at the same method, called approximate maximum-likelihood (AML), time, let alone a joint estimation approach based on a uni- is developed by assuming a small CFO and using only one fied criterion. In fact, to the best of our knowledge, a joint OQAM preamble symbol per burst. The same authors, estimation method for OFDM/OQAM systems, account- in [11], propose a joint least-squares (LS) CFO and STO ing for all the three error sources, i.e., CFO, STO, and CIR, estimation method by using two identical OFDM/OQAM has not yet been developed. Hence, our focus in this paper pilot symbols per burst transmission. Therein, as a time- is to develop and investigate a general estimation method domain method, the estimation is performed before the to fill this need. analysis filter bank (AFB) at the receiver. In [12], by using Specifically, a new formulation of the joint parame- a polyphase network implementation of OFDM/OQAM, ter estimation problem in OFDM/OQAM system is first the preloading technique, and a conjugate-symmetric introduced, which is based on splitting the interference preamble, the CFO and STO are separately estimated. The term on the desired received pilot into adjacent pilot, non- CFO estimation exploits the phase difference between pilot and noise contributions. Then, by assuming Gaus- the adjacent pilots while the frame detection and STO sian noise and independent input symbols, a pilot-based estimation are derived based on the conjugate-symmetry joint ML estimator of CFO, STO, and CIR is derived. property. Moreover, in [13, 14], a joint CFO and STO Such a general approach offers many advantages, includ- estimation method is proposed by using a four-column ing a unified framework for the estimation of multiple preamble per burst transmission, which contains zeros parameters using a common preamble/burst structure in every other subcarrier and every other symbol time and the proper treatment of different types of interfer- index. ence in the estimator derivation. More importantly, a In contrast to the CFO and STO estimation, during significant performance improvement is expected in the the past decade, many pilot-based channel estimation joint estimation of the aforementioned error sources as schemes have been proposed for OFDM/OQAM systems, opposed to their separate estimation. Through numeri- which can be broadly classified into frequency domain cal simulations of wireless OFDM/OQAM transmission and time domain methods. Frequency domain methods, over multipath fading channels, the proposed estimator e.g., [15–23], rely on the assumption that the symbol dura- is evaluated by comparing the accuracy of the result- tion is much longer than the maximum channel delay ing parameter estimates to that obtained with a selected spread. While these methods are generally characterized benchmark approach among a few existing works where by a lower computational complexity, when the above all the three error sources of our focus are estimated1,as condition is not satisfied, they will suffer from a perfor- well as to the CRB. The simulation results show that, the mance degradation. Time domain methods, e.g., [24–26], proposed method is capable of significant improvements attempt to estimate the channel impulse response (CIR) in parameter estimation accuracy, performing close to the by using sequences of pilot tones. In [24], a time domain CRB. In turn, this improved performance leads to a lower CIR estimator is proposed based on the multiple sig- bit error rate (BER) of the compensated (i.e., synchronized nal classification (MUSIC) and LS algorithms. In [25], a and equalized) transceiver system. per-subchannel estimator is proposed in which the CIR This paper is a more developed and improved version on each subcarrier is estimated separately. In [26], the of our previous work [30] addressing the joint estima- authors exploit pilot tone structures in OFDM/OQAM tion problem under a more restrictive set of assumptions. systems to derive two new CIR estimators, namely the lin- Specifically, the new contributions of the current work ear minimum mean square error (LMMSE) and weighted include the following: least-square (WLS) estimators. The former exploits a pri- ori knowledge of the CIR covariance matrix while the • In [30], orthogonality of the OFDM/OQAM latter only requires the knowledge of the channel length; analysis/synthesis filters in the complex domain is both methods are benchmarked against the Cramer-Rao assumed, as opposed to orthogonality in the real bound (CRB). In a recent paper [27], based on a combina- domain only. The former condition leads to tion of the ideas of [15] and [28], a coded auxiliary pilot important simplifications in the derivation of the ML scheme is proposed for frequency domain channel esti- estimator, especially in the statistical properties of the mation. The coded auxiliary pilots are carefully designed subband noise and data interference. to compensate for the inherent imaginary interference of As a consequence of such simplifications, the OFDM/OQAM. resulting estimator in [30] only qualifies as an Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 3 of 19

approximate ML estimator, although it shows Notations: Bold-faced letters indicate vectors and matri- performance improvements compared to earlier ces, e.g., A.The(i, j)th entry of a matrix is represented work. In contrast, herein, by invoking the true by [ A]i,j. The superscripts T and H stand for the trans- orthogonality condition of OFDM/OQAM in the real pose and Hermitian transpose of a vector or matrix, domain, we can strive for an exact ML-based respectively. The operator ∗ represents a linear convolu- estimator, which achieve even better estimation tion while the superscript ∗ denotes complex conjugation. accuracy. The identity and zero matrices are denoted by I and 0, • Another important contribution of this paper is the respectively. The paraconjugate operation on a matrix analysis of the distributions of the subband noise and function E(z) is defined by E˜ (z) = E(1/z∗)H . The opera- data interference terms in the general OFDM/OQAM tors E{.}, R[.]and I[ .] stand for the expected value, real framework, where the pilot tones can be scattered or part and imaginary part of their arguments, respectively. appended as preamble to the data. In particular, we The floor operation is denoted by . while ||.|| represents show that the subband noise contributions, after the the second norm operation. real operation, are uncorrelated along the time and frequency axes, as a consequence of the exact 2 Problem formulation orthogonality relation. Furthermore, we show In this section, the OFDM/OQAM system model is pre- through analysis and numerical simulations that the sented along with its input-output relation over a fre- data interference terms are well modeled by a quency selective fading channel. The effects of the CFO Gaussian distribution for which we derive the second and STO on the reconstructed signal are discussed and order statistics. We further show that the data finally, the joint estimation problem for the CFO, STO, interference terms are only weakly correlated along and CIR is stated. the time and frequency axes. We conclude that with a carefully designed pilot distribution, their correlation 2.1 OFDM/OQAM System Model can be confidently approximated as zero. The OFDM/OQAM system model, as implemented in • Based on this reformulation of the problem and this work and commonly used in the literature, e.g., [31], subsequent derivation of an accurate log-likelihood is illustrated in Fig. 1. OFDM/OQAM makes use of a function for the received pilot tones, we derive in specific filter bank structure where the upsampling and detail the CRB for the joint parameter estimation downsampling factor equals half the number of subcar- problem under consideration. The former plays a key riers, denoted by M. At each input symbol time, with role in demonstrating the near optimality of the symbol duration Ts, a vector of discrete input symbols newly derived joint ML-based estimator, whose is loaded on the M available subcarriers. The latter are performance (estimation error) comes within 1 dB or separated in frequency by Fs = 1/Ts, so that the system less from the bound. occupies a total bandwidth of W = MFs. • In addition to the above new theoretical On the transmitter side, let xk,n ∈ A denote the complex contributions, the paper contains a number of valued symbols at the input, where k ∈{0, 1, ..., M − 1} improvements, including a computational complexity is the frequency index, n ∈ Z is the symbol time index, analysis and discussion of practical approaches to and A is the digital constellation from which the symbols reduce implementation complexity. are drawn. In the first stage of pre-processing, each xk,n is converted to a pair of real symbols, dk,n, according to the The paper is organized as follows. Section 2 is dedi- following equations, cated to reviewing the OFDM/OQAM system model as implemented in this work. In Section 3, the joint ML esti- R I mator of the CFO, STO, and CIR is developed in details xk,n , k even xk,n , k even dk,2n = dk,2n+1 = . based on a new formulation. Several related aspects are I xk,n , k odd R xk,n , k odd also discussed, including: computational simplifications (1) for efficient implementation; evaluation of computational complexity; CFO and STO compensation and channel equalization. The CRB on the unbiased estimator of the This complex-to-real (C2R) operation doubles the sam- aforementioned parameters is derived in Section 4. The pling rate of the subcarrier signals. The second stage of pre-processing involves multiplication of theπ real OQAM methodology used in our simulations and the results are j (k+n) d θ = e 2 provided in Section 5, while Section 6 concludes the paper. symbols, k,n, by the sequence k,n , which Appendices A and B provide important developments results in complex symbols about statistical properties of the subband noise and data π j (k+n) interference terms. sk,n = dk,nθk,n = dk,ne 2 .(2) Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 4 of 19

Fig. 1 OFDM/OQAM system model

∞ It is notable that the OQAM symbol duration is T /2, i.e., nM s s¯ = y¯[ m] g − m ,(4) one half of the input symbol duration. Next, in the syn- k,n k 2 m=−∞ thesis filter bank (SFB), the input subcarrier signals sk,n are first upsampled by M/2, and then passed through syn- where the range of summation over m is determined by thesis filters with finite impulse responses (FIR) fk[ m]of thefinitesupportofthesubbandFIRfilters. ( ) = length Lp and corresponding system functions Fk z The symbols s¯k,n then pass through the first post- − Lp 1 −m = fk[ m] z . Finally, the individual filter outputs are processing stage, which involves multiplication by the m 0 θ ∗ added together to form the baseband output y[ m]as sequence k,n followed by taking the real part, i.e., follows, ∗ − π ( + ) ¯ = ¯ θ = ¯ j 2 k n dk,n R sk,n k,n R sk,n e .(5) − M1 nM y[ m] = s f m − .(3)The second post-processing stage is the real-to-complex k,n k 2 k=0 n∈Z (R2C) conversion, where two consecutive real valued symbols are combined into a complex one as follows, In a practical implementation of OFDM/OQAM, the ¯ + ¯ output signal y[ m] is passed through a pulse shaping fil- ¯ = dk,2n jdk,2n+1, k even , xk,n ¯ ¯ (6) ter and up-converted to an appropriate frequency band dk,2n+1 + jdk,2n, k odd . for transmission over the physical medium. In this work, however, we consider an equivalent baseband channel We consider a complex-valued, uniform modulated fil- model for simplicity. Specifically, the channel is mod- ter bank, where the subchannel filters are all generated eled as a linear time-invariant system with FIR h[ l]of from a common low-pass prototype filter, p[ m], by means ( ) = of exponential modulation as follows, length Q, and corresponding system function H z Q−1 −l l=0 h[ l] z . The filter length Q is proportional to the τ = τ / + 2πkm ∗ channel delay spread ds,thatis,Q M ds Ts 1. The = j M = − fk[ m] p[ m] e , gk[ m] fk [ m], (7) channel coefficients h[ l] are assumed to remain constant during the transmission time of one data block of N sym- where k ∈{0, 1, ..., M − 1}. bols, i.e., block duration plus overall processing delay of The prototype filter used in this work is a near perfect the transceiver system. Finally, the channel output is cor- reconstruction (NPR), real-valued linear phase (symmet- rupted by additive white Gaussian noise (AWGN) η[ m], ric) FIR low-pass filter with length Lp and support region assumed to be circularly complex with zero mean and m ∈{0, 1, ..., Lp − 1}. It is derived by using the frequency 2 2 variance E[ |η[ m] | ] = ση . A more detailed discussion of samplingtechnique,asin[31],withoverlapfactorK,so the effects of fading channels, CFO and STO is provided that its non-zero coefficients can be represented in closed in Subsection 2.2. form as On the receiver side, let y¯[ m] denote the received base- K− α 1 2πl band signal after transmission through the noisy chan- p[ m] = 1 + 2 (−1)lA[ l]cos (m + 1) , nel. In the analysis filter bank (AFB), signal y¯[ m]is KM KM l=1 passed through the analysis filters with FIR gk[ m]of (8) 2 ( ) = length Lp and corresponding system functions Gk z 0 g [ m] z−m, whose outputs are downsampled 2 + − 2 = m=−Lp+1 k where the coefficients A[ l]satisfyA[ l] A[ K l] 1 by M/2 afterwards. The resulting symbols at the output of for l = 1, 2, ..., K/2 and α is a normalization factor 2 = the AFB can be represented as such that m p[ m] 1. Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 5 of 19

−j2πμ0m/M In particular, for the adopted value of the overlap factor, y¯[ m] = (h[ m] ∗y[ m − τ0] ) e + η[ m] (11) = ⎛ ⎞ i.e., K 4, we have Q−1 ⎝ ⎠ −j2πμ0m/M = h[ l] y[ m − l − τ0] e + η[ m], √ l=0 = = / = − 2 A[1] x, A[2] 1 2, A[3] 1 x ,(9) (12)

4 where τ0 is the normalized STO with respect to the sam- = where A[1] x can be determined by using various opti- pling period at the baseband transmitter output, Ts/M, mization criteria. Since the LS criterion is used in this and μ0 is the normalized CFO with respect to Fs,thesub- work (i.e., minimizing stopband energy), we set A[1]= carrier frequency spacing. It is worth mentioning that, 0.97741677 according to Table 1 in [31]. Since the pro- similar to previous works on this subject (e.g., [12, 13]) totype filter is linear-phase (symmetric), the overall pro- the second-order effects, i.e., those of CFO, STO, and CIR cessing delay of the complete OFDM/OQAM transceiver on one another, have been neglected in this model. It has / system will be LpTs M. been observed through simulations that these effects are, By using the paraconjugates of the synthesis filters as indeed, negligible. the analysis filters in the receiver, as specified in (7), the From (12), (4), and (5), useful expressions can be ¯ orthogonality condition of the transceiver system can be obtained for the real-valued output symbols dk,n that expressed as appear in the OQAM post-processing module on the receiver side in Fig. 25. Specifically, ∞ ¯ = ζ + η ∗ ∗ nM n M dk,n k,n k,n , (13) R θ θ f m − f m − ≈ δ δ , k,n k ,n k 2 k 2 kk nn m=−∞ where ζk,n represents the contribution from the transmit- (10) ted data (pilot and information symbols) as given by

∞ δ 3 where kk denotes the Kronecker delta function [3, 32, 33] . ζ = θ ∗ ( ∗ − τ ) k,n R k,n h[ m] y[ m 0] m=−∞ 2.2 Effects of fading channel, carrier frequency offset, nM and sampling time offset −j2πμ0m/M e gk − m , (14) In addition to channel fading and additive noise, as illus- 2 trated in Fig. 1, the received signal y¯[ m] at the front-end of thereceiverwillbeaffectedbyCFOduetooscillatormis- and ηk,n represents the additive noise (i.e., the contribu- match or Doppler effect, as well as STO due to imperfect tion from η[ m]) passed through analysis filter bank and sampling. These effects can be mathematically modeled as first post-processing stage, as given by

Fig. 2 Block diagram of the OFDM/OQAM receiver with ML-based compensation Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 6 of 19

∞ ∗ nM is critical to achieve the low level of bit error rate (BER) η = R θ η − k,n k,n [ m] gk m . (15) required for the practical operation of multi-carrier mod- =−∞ 2 m ulation in broadband communication systems. ζ Substituting (3) into (14) and using (7), k,n can be The main focus of this work, therefore, lies in the joint further developed as follows, estimation and compensation of above imperfections for ⎧ ⎫ ⎨Q−1 ⎬ the OFDM/OQAM system. To this end, the use of pilot- ζ = λ ( μ τ ) based estimation is preferred over the blind approach, k,n R ⎩ h[ l] k,n l, 0, 0 ⎭ , (16) l=0 since the latter generally requires a longer data record to achieve a desired level of accuracy, which in turns where we define increases the computational complexity and limits appli- M−1 ∞ cations to static or very slowly time-varying channels. λ ( μ τ ) = γ k ,n ( μ τ ) k,n l, 0, 0 dk ,n k,n l, 0, 0 , (17) Furthermore, the framework of point estimation theory k=0 n=−∞ is employed here, where the parameters under estimation are modeled as unknown, yet deterministic quantities, ∞ k ,n ∗ n M i.e., no prior distribution is assumed. By transmitting a γ (l, μ , τ ) = θ θ f m − l − τ − k,n 0 0 k,n k ,n k 0 2 sequence of known pilot tones, and observing the received m=−∞ sequence over a given time interval, our specific interest ∗ nM − πμ / f m − e j2 0m M . (18) lies in developing and investigating the properties of the k 2 joint ML estimator of the CFO μ0,STOτ0 and CIR h[ l] γ k ,n ( μ τ ) for the generic OFDM/OQAM transceiver system illus- The term k,n l, 0, 0 in (18), known as ambiguity function [34], characterizes the level of the ‘intrinsic trated in Fig. 1 and described in mathematical terms in Section 2.1. We shall denote the resulting ML estimates by interference’ of the n th real input sample from the k th ˆ μˆ , τˆ and h[ l], respectively. subband, dk,n ,onthenth output sample from the kth subband, through the lth channel tap, h[ l], in the presence of 3 Joint estimation CFO, μ0,andSTO,τ0.Inthespecialcasel = μ0 = τ0 = In this section, we first introduce our proposed approach 0, the quantity γ k ,n (0, 0, 0) describes the level of com- k,n to the estimation problem by splitting a received pilot plex orthogonality of the analysis/synthesis filters of the symbol into pilot, data, and noise contributions. Next, OFDM/OQAM transceiver system up to a multiplicative ∗ we formulate and develop a pilot-based joint ML esti- θ θ γ k ,n ( ) factor k ,n k,n. The values of k,n 0, 0, 0 for the filter matoroftheCFO,STOandCIR.Wethenpresent bank adopted in this work, with the prototype filter p[ m] possible simplifications to reduce the implementation and its parameters as described in Section 2.1, are given complexity of the resulting joint ML estimator and dis- in Table 1. We note that due to the finite length of the cuss its computational requirements. Finally, we explain subband filters fk[ m], the summation in (18) is, in fact, how the jointly estimated parameters can be used performed over a finite range. to compensate the detrimental effects of STO, CTO, For the same reason, the range of summation over the and CIR. symbol time index n in (17) is also finite. 3.1 Structure of received pilots 2.3 Problem statement In this work, for convenience in analysis, a real OQAM If estimates of the CFO and STO are available, they can symbol at time n is defined as the ordered set of M be compensated at the receiver front-end to avoid their subband symbols dk,n for k ∈{0, 1, ..., M − 1},asthey degrading effects. Likewise, if estimates of the CIR coeffi- appear at the output of the C2R modules in the pre- cients are available, they can be used on the receiver side processing stage of the SFB in Fig. 1 To allow for flexibility to design a set of subband equalizers to compensate for in the application of the derived pilot-based estimation the distortion caused by the multipath fading channel6. method, we consider a general framework for the alloca- The estimation and compensation of these imperfections tion of pilots. Specifically, within a burst of N consecutive

γ k ,n ( ) Table 1 Transmultiplexer response k,n 0, 0, 0 of the OFDM/OQAM system using Bellanger’s (also known as PHYDYAS) filter [41, 42] with K = 4 (numerical values truncated to 4 digits) n = n − 4 n = n − 3 n = n − 2 n = n − 1 n = nn = n + 1 n = n + 2 n = n + 3 n = n + 4 k = k − 1 0.0107 j 0.0506 j 0.1246 j 0.1980 j 0.2283 j 0.1980 j 0.1246 j 0.0506 j 0.0107 j k = k -0.0002 0.0765 j -0.0005 0.5720 j 1 -0.5720 j -0.0005 -0.0765 j -0.0002 k = k + 1 -0.0107 j 0.0506 j -0.1246 j 0.1980 j -0.2283 j 0.1980 j -0.1246 j 0.0506 j -0.0107 j Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 7 of 19

= − p symbols (e.g., from time n 0toN 1), a total of i.e. , d for (k , n ) ∈ P and (k , n ) = (k, n) .Sincethe ≡ p k ,n Np symbols, denoted as dk,n dk,n,aretransmittedas pilots are known symbols, this part can be accounted for ( ) ∈ P pilots with time and frequency indexes k, n ,where as a deterministic component (albeit dependent on the P ⊆{0, ..., M − 1}×{0, ..., N − 1}.Inordertosim- unknown parameters μ0, τ0 and h[ l]) in the derivation of plify the mathematical developments, we may consider a the ML estimator. To further proceed with this derivation, P = S × T rectangular distribution of pilots, as in where we therefore need to characterize the statistical prop- S ⊆{0, ..., M − 1} and T ⊆{0, ..., N − 1}; however, erties of the noise term ηk,n and the data contribution the extension of the resulting estimator to an arbitrary ζ d term k,n in the decomposition (19) of the received pilot time-frequency grid P is straightforward. By definition, ¯ p symbol d . the pilot symbols d p are deterministic quantities, known k,n k,n From the AWGN assumption made earlier on the addi- to the receiver side. tive channel noise η[ m], and the linearity of the processing For clarity in the presentation, the data or information operations involved in the OFDM/OQAM receiver, it fol- symbols (i.e., non-pilot) are denoted as d ≡ d d where k,n k,n lows that the subband noise contribution η is a jointly (k, n)/∈ P. These symbols, unknown to the receiver, are k,n Gaussian (real) random process in the variables (k, n).Fur- modeled as independent and identically distributed (i.i.d.) thermore, on account of the orthogonality property of the random variables with zero-mean and variance 1 σ 2. 2 x analysis and synthesis filters, as stated in (10), it follows As mentioned earlier, a demodulated pilot symbol on that the various random variables η (for different pairs the receiver side of the OFDM/OQAM system, assuming k,n (k, n)) are uncorrelated and therefore statistically inde- a general pilot distribution P which is scattered among pendent. Specifically, it can be shown (see Appendix A) the data symbols, consists of additive contributions from that surrounding pilot symbols, surrounding data symbols and σ 2 noise. Specifically, for the case (k, n) ∈ P, the real-valued η E {η η }= δ δ . (23) ¯ ≡ ¯ p n,k n ,k kk nn output symbol dk,n dk,n in (13) can be written as 2 ζ d ¯ p p The data contribution term in (22) is unknown to d = ζ + ζ d + η , (19) k,n k,n k,n k,n k,n the receiver and therefore should be modeled as a (real- ζ p ζ d valued) random process. Based on the assumptions made where k,n and k,n, respectively, denote the contributions d from surrounding pilots and data. The pilot contribution above on dk,n, we note that the expression (22) involves the weighted sum of a large number of statistically indepen- can be expressed as d ⎧ ⎫ dent, zero-mean terms dk,n .Hence,invokingthecentral ⎨ − ⎬ Q1 limit theorem, we shall assume that ζ d in (22) can be con- ζ p = R λ¯ ( μ τ ) k,n k,n ⎩ h[ l] k,n l, 0, 0 ⎭ , (20) veniently modeled as zero-mean (real-valued) Gaussian = l 0 random process. This assumption is further investigated where in Appendix B. In addition, under mild assumptions usually satisfied in applications, it can be shown that the λ¯ ( μ τ ) = p γ k ,n ( μ τ ) k,n l, 0, 0 dk,n k,n l, 0, 0 . (21) random variables ζ d (for different pairs (k, n))arenearly k,n (k ,n )∈P uncorrelated. Specifically (see Appendix B), we have that The data contribution, which can be interpreted as “data σ 2 d d ζ d interference”,can be expressed as ζ ζ ≈ δ δ E k,n k ,n kk nn , (24) ⎧ ⎫ 2 ⎨Q−1 ⎬ 2 σ 2 d d k ,n k ,n ζ = d R h[ l] γ (l, μ , τ ) . σ 2 = x γ ( μ τ ) k,n k ,n ⎩ k,n 0 0 ⎭ ζ d R h[ l] k,n l, 0, 0 . 2 (k ,n )/∈P l=0 (k,n)/∈P l (22) (25) In the sequel, we use the described splitting of a = ζ d +η From (19), (23), and (24) by introducing vk,n k,n k,n received symbol into pilot, data and noise contribu- and assuming that the data and the additive noise are tions to develop the joint ML estimator of CFO, STO statistically independent, it follows that the terms vk,n and CIR for a general pilot-data distribution. When are independent Gaussian random variables with variance ¯ ¯ p 2 the received symbol d = d corresponds to a ση k,n k,n σ 2 = σ 2 + . Finally, by substituting v in (19) we will transmitted pilot, in the aforementioned formulation in v ζ d 2 k,n p have (20) and (21), the various terms dk,n represent the ¯ p = ζ p + contribution from the corresponding transmitted pilot dk,n k,n vk,n , (26) p i.e. dk,n , as well as, depending on the pilot distribution, which provides a convenient basis for the derivation of the possible contributions from surrounding pilot symbols covetedMLestimator. Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 8 of 19

! !! ¯ p ¯ p 3.2 Pilot-based joint ML estimator L D ; μ0, τ0, h = log f D ; μ0, τ0, h Focusing on the pilot contribution in (20), we have 1 T =− D¯ p − (μ τ )h D¯ p − (μ τ )h 2 0, 0 0, 0 . ⎧ ⎫ 2σv − ⎨Q1 ⎬ (31) ζ p = R λ¯ ( μ τ ) k,n ⎩ h[ l] k,n l, 0, 0 ⎭ (27) l=0 The joint ML estimators of the CFO, CIR and STO can be obtained by maximizing the derived LLF with respect Q−1 ! to the parameters μ , τ and h . Let the unknown search = hR[ l] λ¯ R (l, μ , τ ) − hI [ l] λ¯ I (l, μ , τ ) , 0 0 k,n 0 0 k,n 0 0 parameters for CFO and STO be denoted by μ and τ.By l=0 fixing μ and τ and varying h in C2Q, the LLF achieves its (28) maximum at where the superscripts R and I are used in the sequel to h˜(μ, τ) = (μ, τ)†D¯ p , (32) identify the real and imaginary parts of the underlying (μ τ)† = ((μ τ)H (μ τ))−1(μ τ)H quantity. Hence, by letting where , , , , is the (μ τ) pseudo-inverse of , . By substituting the resulting λR (μ τ ) = λ¯ R ( μ τ ) λ¯ R ( μ τ ) ... λ¯ R ( − μ τ ) channel guess of (32) into the LLF we can obtain the k,n 0, 0 k,n 0, 0, 0 , k,n 1, 0, 0 , , k,n Q 1, 0, 0 , CFO and STO estimates using a two-dimensional search λI (μ τ ) = λ¯ I ( μ τ ) λ¯ I ( μ τ ) ... λ¯ I ( − μ τ ) k,n 0, 0 k,n 0, 0, 0 , k,n 1, 0, 0 , , k,n Q 1, 0, 0 , according to R R R R T h =[ h [0],h [1],..., h [ Q − 1] ] , (μˆ , τ)ˆ = arg max L(D¯ p; μ, τ, h˜) . (33) (μ τ) hI =[ hI [0],hI [1],..., hI [ Q − 1] ]T , , The ML estimate of the CIR can be obtained by substi- we can obtain the relationship between the received sym- tuting the estimates (μˆ , τ)ˆ , resulting in bol and the channel taps as hˆ = h˜(μˆ , τ)ˆ = (μˆ , τ)ˆ †D¯ p . (34) hR ζ p = λR (μ , τ ) λI (μ , τ ) . (29) k,n k,n 0 0 k,n 0 0 × − I 3.3 Computational simplifications 1 2Q h 2Q×1 Herein, three simplifications are introduced in comput- ¯ p ¯ λR (μ τ ) λI (μ τ ) ing λk,n(l, μ0, τ0) in (21) to speed up the calculation of By stacking dk,n, k,n 0, 0 and k,n 0, 0 ,andvk,n the LLF (31) significantly. To this end, we first consider over the time index n and then over the frequency index γ k ,n ( μ τ ) k, we arrive at the following matrix-vector equation, the term k,n l, 0, 0 in (18), whose definition includes a summation over the length (pretty large) of the proto- R = ¯ p = R(μ τ ) I (μ τ ) h type filter p[ m]. Since for the filter banks, we have fk[ m] D × 0, 0 0, 0 I π / Np 1 " #$ % −h p[ m] ej2 km M,wecanwrite " #$ % (μ ,τ ) 0 0 Np×2Q k ,n ∗ n,n h 2Q×1 γ ( μ τ ) = θ θ ϕ ( μ τ ) k,n l, 0, 0 k,n k ,n k−k l, 0, 0 + ' [V]Np×1 . j2π M exp − k (l + τ )+ (n k − nk) , M 0 2 As a result of the AWGN assumption and the ensu- (35) ing assumptions on the noise and data interference terms η ζ d α = − k,n and k,n, V will be a real Gaussian random vector where letting k k with zero mean and a nearly diagonal covariance matrix ∞ = T ≈ σ 2 μ n,n n M CV E[ VV ] v I. Similarly, for given values of 0, ϕα (l, μ0, τ0) = p m − l − τ0 − ¯ p 2 τ0 and h, the observation D is also a Gaussian random m=−∞ 2 vector with mean (μ , τ )h and covariance C ¯ p ≈ σ I. −j2π 0 0 D v nM m(μ +α) ¯ p p m − e M 0 . (36) Hence, the probability density function (PDF) of D can 2 be presented as, k ,n In this way, instead of calculating γ (l, μ0, τ0) for all k,n 1 the possible pairs of (k , k), it is sufficient to compute ( ¯ p μ τ ) = & f D ; 0, 0, h ϕn,n ( μ τ ) − = α ( π)Np ( ) α l, 0, 0 for only possible values of k k and 2 det CD¯ p k ,n ! ! find the corresponding γ (l, μ , τ ) by multiplication 1 T − k,n 0 0 exp − D¯ p − (μ , τ )h C 1 D¯ p−(μ , τ )h . 2 0 0 D¯ p 0 0 with a discrete phase factor as in (35). The number of pos- α (30) sible values of depends on the distribution of the pilots over the frequency axis. Thus, the log-likelihood function (LLF) is written, up to In the second simplification, regarding the calcula- a constant, as, tion of the interference from the surrounding pilots, Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 9 of 19

¯ λk,n(l, μ0, τ0), we might assume that, owing to the excel- in the burst. This complexity evaluation is based on the lent spectral containment of the prototype filters, the case where none of the aforementioned practical simplifi- main source of the CFO-induced interference on each cations above in Subsection (3.3) are applied. Employing subband is due to its first few neighboring subbands; i.e, these simplifications reduces the complexity ! of the practi- the interference from more distant subbands is negligi- cal implementation by a factor of O QM2 , where the first ble. Hence, to derive the total interference from subbands two foregoing simplifications each reduce the complexity k on the subband with index k in (21), it suffices to take by a factor O(M) and the third one by a factor O(Q). into the account the interference from a few neighboring pilot-carrying subbands on each side of the kth subband. 3.4 Compensation of the estimated parameters Consequently, (21) can be approximated as Figure 2 illustrates the block diagram of the OFDM/OQAM receiver as implemented in the proposed k+β λ¯ ( μ τ ) ≈ p γ k ,n ( μ τ ) method for estimation and compensation of CFO, STO k,n l, 0, 0 dk,n k,n l, 0, 0 , (37) and CIR. First, after passing through the AFB and mul- k =k−β ∗ θ (k ,n )∈(P) tiplication by k,n and real taking, the received symbols areusedtoobtaintheMLestimatesoftheCFO,STOand where in practice, the value of β can be set7to 2. CIR. The CFO and STO estimates are then fed back to To reduce the complexity of the estimator even further correct the signal at the front-end of the receiver. The and make the two-dimensional search for CFO-STO more CFO-STO compensated signal can be written as practical, the third compromise is to only consider the πμˆ c j2 m first few channel taps in computing pilot contribution in y¯ [ m] = WI[ m; τˆ] ∗ y¯[ m] e M , (38) (27). This is due to the fact that these taps contribute the most to the entire power of the channel. By considering where WI[ m; τˆ] represents the hamming-windowed sinc the implemented channel as described in Section 5.1, this fractional-delay interpolation filter used for STO simu- reduces the running time of the proposed method approx- lation [35]. Next, the CFO-STO corrected symbols pass imately by three times. As our experiments confirm, this through the AFB again where, this time, a single-tap per simplification only introduces a marginal degradation to subcarrier equalization is performed based on the DFT of the performance of the estimator, while maintaining it still the estimated CIR according to the following equation, significantly superior to that of the benchmark. ( ( It is notable that, from (32) to (34), a second iteration = 1 ( ek ˆ ( , (39) of this estimation method can be performed by running H(z) z=ej2πk/M a two-dimensional search over (μ, τ) using the obtained ∈{ ... − } channel estimate and continuing to obtain a new set of where ek for k 0, , M 1 is the coefficient of the ˆ ( ) = Q−1 ˆ −l ˆ estimates. However, our experiments indicate that the equalizer for subband k and H z l=0 h[ l] z , h[ l] improvement gained by performing a second iteration is being the estimated CIR coefficients. It is notable that not significant enough to justify the additional complex- although a single-tap per subcarrier equalizer is used here, ity. Indeed, as it will be seen from the results, performing generalizations to other, more advanced types of equaliz- close to the CRB, a single iteration suffices to provide a ers are possible. This simple equalization scheme inverts significant improvement over the benchmark method. the channel at the center frequency of the corresponding As the LLF in (31) provides a closed form solution to subcarrier and it works well in mildly selective channels as the estimation problem, the multi-dimensional ML esti- long as the number of subcarriers is sufficiently large [36]. mation is reduced to a two-dimensional search over the The equalized symbols can be represented as, unknown CFO and STO, μ and τ.Thesearchisper- ∞ formed in two stages, namely, a coarse search followed nM s¯c = y¯c[ m] g − m e . (40) by a fine one in proximity of the coarse estimate. The k,n k 2 k m=−∞ decisive factor in the complexity of the proposed ML estimator is the number of operations required for each The final received symbols are then obtained by under- evaluation of the LLF. This is approximately calculated as going the OQAM post-processing stage. 2 + 2 + O( ) C 8MQNp 8Q Np QNp complex-valued oper- It is notable that the formulation of the problem ations where the first term is the cost of forming (μ, τ), through the linear equations obtained from the LLF of the second term is the cost of QR decomposition to solve the received pilots, leads to performing the channel esti- p D¯ = (μ, τ)h˜ (μ, τ) and the third term is the cost mation in the time domain. Furthermore, as mentioned of forming the LLF. Since the first term is dominant for earlier in Section 1, the channel estimation methods in typical Np and Q, we conclude that the overall complex- time domain do not make assumption on the subchannels ity is proportional to the squared number of pilot tones being almost flat. In contrast, the channel equalization has Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 10 of 19

been performed in the frequency domain by using one- where l = a − 2. In addition, for Q + 2 ≤ a ≤ 2Q + 2 tap-per-subcarrier scheme, which is a common equaliza- ∂A ∂A ∂hI tion technique in MCM systems including OFDM and = =−λI (μ, τ) =−λ¯ I (l, μ, τ). (47) ∂θ ∂hI[ l] k,n ∂hI[ l] k,n OFDM/OQAM. In addition to simplicity, this technique a Thus, J (θ) can be written as allows for flexibility of equalization in a multi-user sce- ⎡ ⎤ nario where different subchannels need to be equalized J1,1 J1,2 κ separately. ⎢ φρ⎥ J (θ) = ⎢ J2,1 J2,2 ⎥ ⎣ ()T (φ)T χψ⎦ , (48) 4 Joint Cramer-Rao bound analysis based on the (κ)T (ρ)T (ψ)T ζ Gaussian assumption for data interference ζ d k,n where according to (42)–(47) we have, In this section, assuming known transmitted symbols, i.e., ⎡ ( ( ⎤ pilots, the CRB on the covariance matrix of unbiased esti- ( R I (2 1 (∂(λ (μ, τ)) ∂(λ (μ, τ)) ( mators of the CFO, STO and CIR are derived [37]8. Letting J = ⎣ ( k,n hR − k,n hI ( ⎦ , 1,1 σ 2 ( ∂μ ∂μ ( θ be the vector including the unknown (real) parameters v (k,n)∈P we have, (49)

!T T ⎡ ( ( ⎤ θ = μ, τ, (hR)T , hI . (41) ( (2 1 (∂(λR (μ, τ)) ∂(λI (μ, τ)) ( J = ⎣ ( k,n hR − k,n hI ( ⎦ , θ ( + ) 2,2 σ 2 ( ∂τ ∂τ ( Therefore, holds 2Q 2 real entries with indexes v (k,n)∈P ∈{ ... + } denoted by a and b 1, 2, ,2Q 2 . The Fisher infor- (50) mation matrix (FIM), J ,isthen(2Q + 2) × (2Q + 2). ⎡ ⎛ ⎞ For the covariance matrix C ,sinceforalla in the afore- V ∂ λR (μ, τ) ∂ λI (μ, τ) mentioned interval ∂CV /∂θa = 0, the entries of FIM are 1 ⎣ ⎝ k,n R k,n I ⎠ J1,2 = h − h given by, σ 2 ∂μ ∂μ v (k,n)∈P ⎛ ⎞⎤ 2 ¯ p ∂ L(D ; θ) ∂ λR (μ, τ) ∂ λI (μ, τ) [ J (θ)] =−E k,n k,n a,b ∂θ ∂θ ⎝ hR − hI⎠⎦ a b ∂τ ∂τ , T ∂( (μ, τ)h ) − ∂( (μ, τ)h ) = C 1 , (42) (51) D¯ p ∂θa ∂θb = which results in, J2,1 J1,2 . (52) ⎡ × ∂ λR (μ τ) R − λI (μ τ) I Also, is a 1 Q vector whose entries can be written as 1 k,n , h k,n , h ⎛ ⎞ J (θ) = ⎣ R I [ ]a,b 2 ∂ λ (μ, τ) ∂ λ (μ, τ) σ ∂θa 1 k,n k,n v (k,n)∈P = ⎝ hR − hI⎠ ⎤ 1,b σ 2 ∂μ ∂μ ∂ λR (μ τ) R − λI (μ τ) I v (k,n)∈P k,n , h k,n , h ⎦ λ¯ R ( μ τ) . k,n b, , , (53) ∂θb (43) 1 ∂(λR (μ, τ)) ∂(λI (μ, τ)) κ =− k,n hR − k,n hI 1,b σ 2 ∂μ ∂μ λR (μ τ) R − λI (μ τ) I v ( )∈P Letting A k,n , h k,n , h ,thepartial k,n θ derivative of A with respect to a for four different ranges λ¯ I (b, μ, τ). (54) of the indexes a and b,namelya = 1, a = 2, 3 ≤ a ≤ k,n φ × Q + 2andQ + 3 ≤ a ≤ 2Q + 2canbewrittenas, Moreover, is a 1 Q vector whose entries can be represented as ∂ ∂ ∂λR (μ τ) ∂λI (μ τ) A A k,n , k,n , = = hR − hI , (44) 1 ∂(λR (μ, τ)) ∂(λI (μ, τ)) ∂θ ∂μ ∂μ ∂μ φ = k,n hR − k,n hI 1 1,b σ 2 ∂τ ∂τ v (k,n)∈P R I ∂A ∂A ∂λ (μ, τ) ∂λ (μ, τ) λ¯ R (b, μ, τ), (55) = = k,n hR − k,n hI . (45) k,n ∂θ2 ∂τ ∂τ ∂τ 1 ∂(λR (μ, τ)) ∂(λI (μ, τ)) Also, for 3 ≤ a ≤ Q + 2, ρ =− k,n hR − k,n hI 1,b σ 2 ∂τ ∂τ ∂ ∂ ∂ R v (k,n)∈P A A R h ¯ R = = λ (μ τ) = λ ( μ τ) I R k,n , R k,n l, , , (46) λ¯ ( μ τ) ∂θa ∂h [ l] ∂h [ l] k,n b, , . (56) Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 11 of 19

In addition, χ, ψ and ζ are Q × Q matrices with the output symbol, the impact of all the subbands are taken following entries, into account. 1 χ = λ¯ R (a, μ, τ)λ¯ R (b, μ, τ), (57) 5 Performance evaluation a,b σ 2 k,n k,n This section begins with the simulation setup and param- v (k,n)∈P eter settings for performance evaluation of the proposed method compared to the existing one, followed by pre- 1 sentation and discussion on their estimation results and ψ =− λ¯ R (a, μ, τ)λ¯ I (b, μ, τ), (58) a,b σ 2 k,n k,n complexity evaluation. v (k,n)∈P 5.1 Methodology 1 The prototype filter of the transceiver system is obtained ζ = λ¯ I (a, μ, τ)λ¯ I (b, μ, τ). (59) a,b σ 2 k,n k,n using the frequency sampling technique with overlap fac- v (k,n)∈P tor K = 4 as described in [31] and used in [29]. The data are modulated to a 4-QAM constellation. The input The CRB of an unbiased estimator of θ, denoted as θˆ, sampling frequency is Fs = 175 kHz corresponding to a is expressed as Cov(θˆ) ≥ J (θ)−1. As a result, we channel bandwidth of MFs = 11.2 MHz. can compute the variance of the unbiased CFO estimator, To obtain BER that are more representative of a μˆ ,as practical digital communications system, a punctured − convolutional channel coding scheme is applied to the Var(μ)ˆ ≥[ (J ) 1] = CRBμ . (60) 1,1 information sequence with the overall rate of 2/3byusing Similarly, the variance of the unbiased STO estimator, τˆ, constraint lengths vector [5 4] and vector of function is derived as generators [23 35 0; 0 5 13]. A frequency selective wireless channel is used with Q = 8 randomly gen- −1 Var(τ)ˆ ≥[ (J ) ]2,2 = CRBτ . (61) erated coefficients h[ l] based on the ITU-Vehicular A channel guidelines [39]. The channel is assumed constant Finally, for the lth tap of the channel, the lower bound of during the transmission of a burst but changes over differ- the unbiased estimator can be obtained as ent transmissions910. During each transmission, the STO and CFO obey a uniform distribution within the intervals ˆ ˆ ˆ −1 Var(h[ l] ) =Var(h [ l] ) + Var(h [ l] ) ≥[ (J ) ] + + R I l 2,l 2 − Ts Ts and − Fs Fs . The root mean squared error =+ (J )−1 4 4 4 4 [ ]Q+l+2,Q+l+2 CRBh[l] . (62) (RMSE) and BER results are obtained by running 500 independent Monte-Carlo simulations for given values of The lower bound on the average variance of the CIR the SNR per bit. The latter is expressed as E /N ,where estimator over different taps can be obtained by assuming b 0 E denotes the bit energy and N is the noise power spec- that the tap estimates are independent. Then, we can write b 0 tral density level. Regarding the implementation results ! 1 − − − of [29], we follow the estimation and equalization algo- CRB = tr[(J ) 1(θ)]−[(J ) 1(θ)] −[(J ) 1(θ)] . h Q 1,1 2,2 rithms and structure precisely as described in the paper. (63) This method is referred to as “Stitz” in Figs. 4, 5, 6 and 7. We compare the proposed method to [29], one of a few It is worth emphasizing that in general the entries of works that estimate all the three aforementioned parame- the vectors κ, , φ, ρ and χ are not negligible, i.e., there ters in their paper. To this end, two different distributions is a coupling between the estimation errors that can be of pilots are considered in the implementation of the pro- achieved for μ, τ,andh. This means that, for example, the posed method. In the first distribution, the pilots are scat- CRB on μ with no channel knowledge will be greater than tered within each burst as mentioned in [29]. Specifically, the one obtained with a known channel, which could be the size of the transmitted bursts in time and frequency directly computed as the inverse of the first entry of the are according to DL-PUSC configuration as illustrated in −1 FIM, i.e. (J1,1) . This also applies to the STO and CIR Fig. 3, with M = 64 subcarriers and N = 54 input sym- estimators with or without knowledge of other parame- bols. The other distribution, which also uses M = 64 ters. Furthermore, as it has been mentioned in [38], the subcarriers, adopts a full preamble of pilot tones of length derivations imply that the CRB is a function of the par- T = 4, followed by N − T = 50 data symbols. In this ticular channel realization. This has also been observed way, the two bursts share the same data rate and hence are through simulations. Also, it should be noted that in the comparable. The consideration of these two different dis- derivation and implementation of the CRB, the simplifi- tributions of pilot tones is useful to assess the performance cations of Section 3.3 are not introduced, i.e., on a given of different transmission modes. Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 12 of 19

Fig. 3 Pilot distribution in WiMAX, DL-PUSC configuration [29]

5.2 Results and discussion performance of the proposed estimator is very close to the The performance and the complexity comparison of the average CRB as a lower bound. proposed estimator vis-a-vis the benchmark are, respec- Comparison of the two methods and the average tively, presented in this subsection. CRB in terms of RMSE of STO is depicted in Fig. 5, where a fixed CFO of 5% and the multi-path chan- 5.2.1 Estimation results nel are used. The superior performance of the pro- Figure 4 compares the proposed method with [29] in posed method implemented in both configurations can termsofRMSEofCFOasafunctionofEb/N0.Theaver- be seen in the figure. Again, the best result belongs age CRB of the CFO estimation is also presented in the to the one based on a full grid of pilot tones fol- figure. The estimation is jointly performed in the presence lowed by the data; although, the proposed method of other sources of error, namely, a fixed STO of 2.5% with remarkably reduces the estimation error with the same respect to Ts, and Rayleigh fading channel as described burst structure as in [29]. Similar to the previous earlier. The figure indicates that the proposed method not figure, the average CRB runs closest to the full-preamble only outperforms the other method as implemented with case. a preamble of full 64 × 4 pilot tones, but also, is capa- InFig.6,theRMSEofCIR,inthepresenceoffixedCFO ble of significant improvement when adopting the burst and STO, is depicted for the three implementations along structure of DL-PUSC. Especially in the former case, the with the average CRB. Similar to the previous figures,

−1 10

Stitz Proposed, DL−PUSC distribution Proposed, preamble−data distribution CRB

−2 10 RMSE CFO error (percent of subcarrier spacing)

0 2 4 6 8 10 12 14 16 18 20 Eb/N0 (dB)

Fig. 4 Comparative RMSE error of CFO estimation versus input Eb/N0 with STO = 2.5% and Rayleigh channel Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 13 of 19

Stitz Proposed, DL−PUSC distribution Proposed, preamble−data distribution /2) s −2 CRB 10 RMSE STO error (half a symbol duration T

−3 10 0 2 4 6 8 10 12 14 16 18 20 Eb/N0 (dB)

Fig. 5 Comparative RMSE error of STO estimation versus input Eb/N0 with CFO = 5% and Rayleigh channel

σ 2 the proposed method achieves a lower estimation error interference, ζ d rather than the noise power. However, in both configurations. The figure indicates that the CIR our experiments show that the noise power effect begins estimator performs very close to the average CRB over to rise when applying lower SNR values with an abrupt different channels. increase in the estimation error around Eb/N0 =−10 dB. It is worth mentioning that, in this work, all the CRB σ 2 The coded BER performance of the methods, after esti- σ 2 = σ 2 + η terms are inversely proportional to v ζ d 2 .This mation and compensation, are compared in Fig. 7. The explains the mild slope of the estimation error graphs with figure indicates that, in both configurations, the proposed respect to Eb/N0. In other words, in our range of inter- method is capable of a significant decrease in BER of the est of Eb/N0, the dominant term is the power of the data system especially at high input Eb/N0.

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Stitz Proposed, DL−PUSC distribution Proposed, preamble−data distribution CRB

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Fig. 6 Comparative RMSE error of channel estimation versus input Eb/N0 with CFO = 5% and STO = 2.5% Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 14 of 19

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Fig. 7 Comparative coded BER versus input Eb/N0 in the presence of CFO, STO and channel estimation

5.2.2 Complexity evaluation and discussion performance gain of the proposed estimator presented in The complexity of the two methods is compared in Table 2 Section 5.2, the complexity compromise seems justifiable. in terms of the running time needed for the processing From a theoretical perspective, a key advantage of the of a burst transmission of size 64 × 54. For the proposed proposed ML-based approach is to offer a unified treat- method, three different implementations are considered ment leading to a compact solution format (i.e., near as follows: closed form) for the joint estimation of CFO, STO and CIR • in OFDM/OQAM systems, in contrast to making use of Prop 0: the proposed method with the basic different signal processing techniques (correlation, phase simplification in (35)-(36); estimation, etc.) for the treatment of different impair- • Prop 1: as above with additional simplification (37); • ment sources. Another important motivation behind the Prop 2: as Prop 1, but by considering only the first ML-based approach lies in its asymptotic optimality, as three channel taps in computing the pilot observed in many practical situations of interest, under contribution in (27). conditions of high SNR or long observation time [37]. In addition, Table 2 includes the RMSE figures of CFO, From a practical perspective, the proposed joint ML- STOandCIR(forEb/N0 = 20 dB), to illustrate based estimator leads to significant improvements in the achievable trade-off between estimation accuracy estimation accuracy compared to existing methods, as and computational complexity. The RMSE figures are demonstrated by the simulation results in this section. obtained based on the average of 500 burst transmissions Specifically, over the complete range of SNR consid- of size 64 × 54 with the scattered pilot distribution as ered (from 0 to 20dB), the proposed estimator achieves described in Section 5.1, on an ordinary quad-core PC the best performance for the three types of parameters, running MATLAB 2016 b. The table indicates that the i.e., CFO, STO and CIR coefficients. For each parameter performance degradation due to the suggested simplifi- type, the resulting RMSE obtained with the joint-ML cations is rather small, i.e., on the order of 10% for Prop estimators comes within 1dB of the CRB. In turn, the 1 and 25% for Prop 2. Furthermore, by employing all the foregoing simplifications, the obtained running time of Table 2 Comparative running time and RMSE estimation errors the proposed method comes within about two times that of the simplified versions of the proposed method vs. Stitz for / = of the benchmark, which implies that, by using state-of- Eb N0 20 dB [29] the-art DSP technology, the running time of the proposed Prop 0 Prop 1 Prop 2 Stitz method remains practical. It should be noted that, as can Running time (s) 4617 65 28 15 be seen from the results in Subsection 5.2.1, in mid and − − − − RMSE CFO 1.1 × 10 2 1.2 × 10 2 1.4 × 10 2 2.0 × 10 2 specially low SNR regime, the performance gap between × −3 × −3 × −3 × −3 the two methods is much larger than that presented in RMSE STO 2.2 10 2.4 10 2.7 10 2.8 10 RMSE CIR 2.9 × 10−2 3.1 × 10−2 3.4 × 10−2 4.0 × 10−2 Table 2 for Eb/N0 = 20 dB.Hence,consideringthe Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 15 of 19

improved estimation accuracy results in lower BER for Due to the linearity of this expression, it follows that ηk,n the OFDM/OQAM transceiver with ML-based compen- is a (real) Gaussian random variable with zero mean. For sation. the second order moments we have, The main limitation of the proposed method is the addi- {η η } tional computational burden. Indeed, the calculation of E k,n k ,n the joint ML estimator involves several matrix operations = +1 +2 +3 4 , (67) and a two-dimensional search over the CFO and STO =1θ ∗ θ ∗ {η η } ∗ − nM space. However, by allowing a number of possible sim- 1 k,n k,n "E [ m#$] [ m ]% fk m 4 m 2 plifications to reduce the processing time as discussed m 0 in Section 3.3, the proposed method offers a trade-off ∗ n M f m − = 0 , (68) between complexity and performance. k 2 1 ∗ ∗ ∗ nM = θ θ {η η } − 6Conclusions 2 k,n k ,n "E [ m]#$ [ m ]% fk m 4 m 2 A new general pilot-based ML joint estimation method m σ 2δ η mm for OFDM/OQAM systems has been developed and evaluated. The CFO, STO, and CIR effects have been jointly n M fk m − (69) estimated and compensated. The CRB on the joint esti- 2 mator variance was also derived and implemented as a 1 ∗ ∗ nM n M = σ 2θ θ f m − f m − , reference to evaluate the performance of the proposed η k,n k ,n k k 4 m 2 2 algorithm. The comparison has been made with a highly- (70) cited method among the few research papers of the same = focus. The results have shown the significant improve- 3 complex conjugate of 2 , (71) ment that the proposed method offers in both transmis- =4 complex conjugate of 1 . (72) sion modes, i.e., as scattered pilots in data and as a full preamble of pilot tones followed by the data. As it was Therefore, observed on the figures, the proposed estimator, espe- σ 2 η ∗ ∗ nM cially when used in a full-preamble setup, performs close E {η η }= θ θ f m − k,n k ,n 4 k,n k ,n k 2 to the CRB and can robustly estimate the CFO, STO, and m CIR. Furthermore, by comparing the performance and nM f m − + 3 (73) running times of the simplified versions of the proposed k 2 method to those of the benchmark, we conclude that σ 2 η ∗ nM the former provides a significant improvement over the = θ θ − R k,n k ,n fk m estimation result while maintaining computational com- 2 m 2 ' plexity in a feasible range. This, in turn, offers a useful σ 2 ∗ n M η − = δ δ trade-off between performance and complexity. f m kk nn . (74) k 2 2 Appendix A: Statistical properties of the subband where we have used the orthonormality relation (10) for noise ηk,n OFDM/OQAM systems. Let η[ m] be a zero-mean complex circular AWGN process 2 with variance ση . By definition, we have, Appendix B: Statistical properties of the data ζ d interference term k,n ∗ 2 E {η[ m] η[ m ] }=0, E{η[ m] η [ m ] }=δmm ση , ∀m, m . To further investigate the Gaussian assumption of the data (64) interference, the histogram of data contribution terms at pilot locations is obtained and depicted in Fig. 8. To According to (15) and (7), we can write acquire pure data interference, a data burst of the same structure and size as described in Section 5.1 is trans- ∗ ∗ nM mitted and received where pilots are replaced by zero η = R θ η[ m] f m − (65) k,n k,n k 2 tones. No noise was added to the system. The histogram m of data interference terms at all the 512 pilot locations is 1 ∗ ∗ nM = θ η[ m] f m − illustrated. A Gaussian distribution with zero mean and 2 k,n k 2 a standard deviation of σ = 0.45 is also depicted for m comparison. The figure suggests that the statistical distri- ∗ nM +θk,n η [ m] fk m − . (66) bution of the data interference terms tends towards the m 2 PDF of a Gaussian distribution. Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 16 of 19

Histogram of data interference at pilot locations Zero-mean Gaussian pdf with standard deviation =0.45

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0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Data interference at pilot locations ζ d Fig. 8 Histogram of k,n at pilot locations in presence of fading channel, STO and CFO

( ) γ k ,n ( μ τ ) In addition, to examine the covariance of these data Therefore, for a given k , n either k,n l, 0, 0 or interference terms at pilot locations, let the data symbols k ,n γ¯ (l, μ , τ ) tend to be zero. Hence, d k,n¯ 0 0 dk,n be i.i.d. (real) random variables with zero mean and variance 1 σ 2. By definition, we have (¯ ¯) = ( ) =⇒ ζ d ζ d ≈ 2 x If k, n k, n E ¯ ¯ 0 , (77) k,n k,n 1 (¯ ¯) = ( ) =⇒ ζ d ζ d = (ζ d )2 d d 2 If k, n k, n E k,n ¯ ¯ E k,n , (78) E d d¯ = δ ¯ δ ¯ σ . (75) k,n k ,n k ,n¯ 2 k k n n x for which we can write ⎛ ⎧ ⎫⎞ According to (22) we can write 2 ' ⎨Q−1 ⎬ 2 σ 2 ⎧ ⎫ k ,n E ζ d = x ⎝R h[ l] γ (l, μ , τ ) ⎠ . ⎨Q−1 ⎬ k,n ⎩ k,n 0 0 ⎭ 2 d d k ,n (k ,n )/∈P l=0 ζ = d R h[ l] γ (l, μ0, τ0) . k,n k ,n ⎩ k,n ⎭ (79) (k,n)/∈P l=0 Itcanbeseenfrom(79)thatthevarianceofthedata Following the observation above in the opening of this interference term depends on the particular channel real- appendix, by invoking the central limit theorem and the ization and the unknown CFO and STO parameters. d ζ d k ,n assumptions made on dk,n, it follows from (75) that k,n γ ( μ τ ) However, since the filter bank response k,n l, 0, 0 can be modeled as a zero-mean (real-valued) Gaussian does not profoundly vary with the aforementioned param- random process. For the second order moments, it follows eters, the effect of the channel has been approximated by from (75) that a fixed gain. This approximation can be further inspected ⎧ ⎫ by the simulation result in Fig. 9. The figure illustrates the ⎨Q−1 ⎬ σ 2 d d x k ,n covariance of data interference contributions with respect E ζ ζ¯ = R h[ l] γ (l, μ , τ ) k,n k,n¯ ⎩ k,n 0 0 ⎭ to one another in the simulated OFDM/OQAM system 2 ( )/∈P = ⎧ k ,n l 0 ⎫ with the scattered pilot-data distribution considered in ⎨Q−1 ⎬ this work. More specifically, it depicts the covariance of a ¯ k ,n ¯ R h[ l] γ¯ (l, μ , τ ) . (76) ⎩ k,n¯ 0 0 ⎭ data interference contribution at a pilot location with data ¯ l=0 interference contributions to other pilot locations in pres-

ence of 8-tap Rayleigh fading channel, 20% CFO, 25% STO γ k ,n We note that the filter-bank response term k,n with no noise. For the considered pilot-data distribution (l, μ0, τ0) tend to be non-zero only in the vicinity of (k, n). with length of 160 OQAM symbols, the second largest Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 17 of 19

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60 70 40 60 Subcarrier index 50 40 20 30 20 10 Symbol index Fig. 9 Covariance of data interference at a pilot location with data interference in other pilot locations covariance is less than 10% of the peak value. Although advanced types of equalizers are possible. This simple only one sample is presented here, we were able to verify equalization scheme inverts the channel at the center the consistency of this result for different values of CFO frequency of the corresponding subcarrier and it works and STO with various fading channels, indicating that the well in mildly selective channels as long as the number of observed maximum off-center covariance value does not exceed 14% of the peak value. This result is consistent with subcarriers is sufficiently large [36]. our approximation above in (77) and (79). 7 This is only valid for the PHYDYAS filter. There are other prototype filter functions in discussion for Endnotes OFDM/OQAM systems which may require a different 1 The authors’ intention in the performance compari- value for β [34]. son is not to include the existing estimators in which 8 It is worth emphasizing that the CRB derived in this only one or two of the aforementioned error sources are section is based on the approximation of Gaussian dis- ζ d considered. tribution for the data interference term k,n;hence,it 2 For convenience in our analysis, Gk(z) is assumed non- provides a benchmark for the performance of the esti- causal; although, in practice, causality can be restored mators developed on such an assumption. Nevertheless, simply by introducing an appropriate delay in the receiver. as it can be seen later, the proposed estimator obtains 3 The precise orthogonality condition corresponding satisfactory results by a significant improvement over the to the specific configuration of OFDM/OQAM system existing method. adopted in this work does not exist in the literature. How- 9 The authors have also evaluated the performance of ever, by applying appropriate modifications pertained to the proposed and the existing method by adopting a time- the changes in the configuration, the presented orthogo- varying fading channel in contrast to the static channel nality condition can be derived. used in the following simulations. However, the results 4 In practice, non-integer STO can be modeled with show the same relative trend among the curves corre- the help of an ideal low-pass interpolation filter (see sponding to the two methods and the two pilot distribu- Section 3.4). tion schemes described in this section. 5 Due to the inherent property of OFDM/OQAM, as 10 Although, in the proposed method, the channel implied in (10), which only sustains real orthogonal- length is supposed to be known, the authors have inves- ity, estimation is performed on the received pilots after tigated a mismatch in the presumed channel length with extracting the real part. the actual one and did not find a significant performance 6 In this work, we develop a single-tap-per-subcarrier degradation in CFO and STO estimation. Nevertheless, equalizer, although generalizations to other, more the effect of such a mismatch on channel estimation and Baghaki and Champagne EURASIP Journal on Advances in Signal Processing (2018) 2018:4 Page 18 of 19

BER performance is considerable. In a practical system, to 16. C Lélé, P Siohan, R Legouable, J-P Javaudin, in Proc. IEEE Int. Symp. Power Line Commun. Its Appl. (ISPLC). Preamble-based channel estimation mitigate this problem, as a conventional way to make the techniques for OFDM/OQAM over the powerline (IEEE eXpress channel estimation task more robust, the channel length Conference Publishing, New York, 2007), pp. 59–64 17. C Lélé, P Siohan, R Legouable, in Proc. IEEE Int. Conf. Commun. (ICC).2dB can be matched to a worst-case scenario and may be better than CP-OFDM with OFDM/OQAM for preamble-based channel modified adaptively [40]. estimation (IEEE eXpress Conference Publishing, New York, 2008), pp. 1302–1306 18. J Du, S Signell, in Proc. IEEE Int. Conf. Commun. (ICC).Novel Funding preamble-based channel estimation for OFDM/OQAM systems (IEEE This work was supported by a grant from the Natural Sciences and Engineering eXpress Conference Publishing, New York, 2009), pp. 1–6 Research Council of Canada, under sponsorship of InterDigital Canada. 19. S Hu, G Wu, S Li, Preamble design and iterative channel estimation for OFDM/Offset QAM system. J. Netw. 4(10), 1050–1057 (2009) Authors’ contributions 20. D Katselis, E Kofidis, A Rontogiannis, S Theodoridis, Preamble-based AB has contributed to the conception and design of this research, channel estimation for CP-OFDM and OFDM/OQAM systems: a mathematical developments, performing the simulations, analysis and comparative study. IEEE Trans. Signal Process. 58(5), 2911–2916 interpretation of the results, and drafting the manuscript. BC has participated (2010) in the conception and design of this research, mathematical developments, 21. H Lin, P Siohan, Robust channel estimation for OFDM/OQAM. 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