TimeVarying Channel Estimation and Symbol Detection for Underwater Acoustic FBMCOQAM Communications
https://doi.org/10.1007/s11804023003582

Abstract
Filter bank multicarrier (FBMC) systems with offset quadrature amplitude modulation (OQAM) need long data blocks to achieve high spectral efficiency. However, the transmission of long data blocks in underwater acoustic (UWA) communication systems often encounters the challenge of timevarying channels. This paper proposes a timevarying channel tracking method for shortrange highrate UWA FBMCOQAM communication applications. First, a known preamble is used to initialize the channel estimation at the initial time of the signal block. Next, the estimated channel is applied to detect data symbols at several symbol periods. The detected data symbols are then reused as new pilots to estimate the next time channel. In the above steps, the unified transmission matrix model is extended to describe the timevarying channel input–output model in this paper and is used for symbol detection. Simulation results show that the channel tracking error can be reduced to less than −20 dB when the channel temporal coherence coefficient exceeds 0.75 within one block period of FBMCOQAM signals. Compared with conventional knownpilotbased methods, the proposed method needs lower system overhead while exhibiting similar timevarying channel tracking performance. The sea trial results further proved the practicability of the proposed method.Article Highlights• Different from the previous methods that use scatteredpilots to track the channel, the proposed method considers the coherence time of the UWA channel and makes use of the characteristics of adjacent symbols passing through the similar channel.• The proposed iterative verification method can improve channel estimation accuracy and reduce the required number of known pilots. 
1 Introduction
Underwater acoustic (UWA) communications constantly suffer from long delay spread, severe Doppler effects, and time variation (Eggen et al., 2000). These factors are serious obstacles, particularly for highrate transmission, and must be overcome.
Orthogonal frequency division multiplexing (OFDM) technology has been widely studied in highrate UWA communication systems because of its high bandwidth efficiency and robustness to multipath delay. However, the rectangular timedomain window used for each subcarrier in OFDM leads to significant outofband emission in the frequency domain. Moreover, OFDM systems are susceptible to synchronization errors (Pollet et al., 1995), especially in scenarios with severe Doppler frequency shifts (Morelli et al., 2007), due to the frequency limitations of rectangular timedomain pulses.
Filter bank multicarrier (FBMC) systems with offset quadrature amplitude modulation (OQAM) are highly robust to carrier frequency offsets due to the pulseforming filters. These systems reduce outofband energy leakage and can easily satisfy system synchronization requirements (PérezNeira et al., 2016). Furthermore, sharp frequency–time filters (Siohan et al., 2002) used in FBMCOQAM can minimize intersymbol interference without cyclic prefixing, which is required in OFDM. Thus, FBMC has higher spectral efficiency than OFDM (SaeediSourck et al., 2011). Overall, FBMCOQAM has a lower outofband interference rate (Nam et al., 2016), a flexible bandwidth (He et al., 2019a), better time–frequency focusing characteristics, and higher spectral efficiency than OFDM, making it an excellent candidate modulation method for future communication systems (Nissel and Rupp, 2018).
In OFDM or FBMCOQAM systems, an accurate channel estimator (CE) is conducive to accurate symbol detection. Especially in timevarying UWA channel environments, channel estimation and tracking are usually necessary. For OFDM systems, multicarrier symbol blocks transmitted at different times do not overlap with each other. Achieving channel tracking by inserting combtype pilots is easy due to the aforementioned feature. While for FBMCOQAM systems, pilots are interfered with by unknown symbols around them due to the overlapped multicarrier symbols. In other words, the received pilots in OFDM systems are only affected by the channel, while the received pilots in FBMC systems are further affected by many unknown data symbols. Therefore, the combtype pilot structure in OFDM systems cannot be directly used for channel tracking in FBMC systems, but the inherent interference of the received pilots must be eliminated or reduced first. The inherent interference lies in the effect of energy leaked by symbols in the time and frequency domains on other symbols after the orthogonality between symbols is lost. Therefore, the traditional channel estimation methods for FBMCOQAM systems can be divided into two categories: preamblebased methods (Lélé et al., 2008) and scattered pilotbased methods (Javaudin et al., 2003). When all pilots used for channel estimation are placed at the beginning of a signal block, it is called the preamblebased method. When pilots are placed at isolated time–frequency points throughout the frame, it is called the scattered pilotsbased method (Kofidis et al., 2013).
The first type of method is preamblebased and aims to set the symbols around the pilot symbols to zero or set the pilot to a specific value to reduce or cancel directly the interference received by the pilot. A twostep preamblebased design is proposed to estimate the carrier frequency offset and frequencyselective channels in MIMOFBMC/OQAM systems simultaneously (Singh et al., 2018; 2019). A simple and effective method is proposed on the basis of the continuous preamble sequences. Combined with a lowcomplexity channel estimation method, the scheme considers adequately designed pilot sequences and selfinterference. This scheme provides superior performance considering mean square error (MSE) and symbol error rate (ChenHu et al., 2018). An extended preamble structure, which exploits symmetric patterns to cancel interference and consider the weight of interference in the symmetric structure as well as the additional interference from the middle of the leading structure, is proposed (Wang et al., 2018). However, the system overhead of such methods is often large.
The second type, which is based on scattered pilots, usually needs to insert auxiliary pilots around the known pilots. The energy leaked to the pilot by the auxiliary pilot and that leaked by other unknown symbols can cancel each other out by setting different values for the auxiliary pilots. A pilotaided channel estimation method and a pilot design scheme based on a tilebased subchannel mode are proposed. This method reduces interference and improves the channel estimation performance of FBMCOQAM (He et al., 2012). A threedimensional pilotaided channel estimation scheme is proposed for MIMOOQAM systems with spatially correlated channels. The scheme is based on assigning pilots in three dimensions: time, frequency, and space. The scheme handles the inherent interference of MIMOOQAM and maximizes spatial channel correlation to reduce the total pilot overhead without degrading the channel estimation accuracy (Kalil et al., 2013). An auxiliary pilotbased method is proposed to reduce the inherent interference on the scattered pilots to improve the channel estimation accuracy (Fuhrwerk et al., 2017). When used for continuous tracking of timevarying channels, the pilots and auxiliary pilots can be arranged into a combtype structure, which is similar to the combtype pilot in OFDM systems. The overhead of the second type is less than that of the first type. However, the overhead is still relatively large for the UWA channel with a long delay.
Preamble and scattered pilotbased methods constantly assume that the symbol duration is substantially longer than the maximum channel delay. The performance of these channel estimation schemes degrades significantly when the assumptions are unsatisfied (Cheng et al., 2019a).
Deep learning (DL), known for its powerful learning and recognition and prediction capabilities, has also been introduced into communication systems to obtain superior performance in timevarying channel estimation (O'Shea and Hoydis, 2017). Support vector machines are extended from the earliest classification problems to regression estimation problems (Vapnik et al., 1996). A nonlinear complex support vector regression approach is proposed and applied to the auxiliary pilotbased FBMCOQAM channel estimation problem. The algorithm first obtains a learning model from the least square (LS) pilot estimates and then predicts the channel estimates of other subcarriers by this model (He et al., 2019b). A channel estimation and equalization scheme based on deep bidirectional long shortterm memory networks was proposed in 2019 (Cheng et al., 2019b). In the same year, they also took the channel estimation and equalization problem in FBMC systems as a DL task and proposed a DLbased channel estimation and equalization scheme for FBMC systems. Channel estimation and equalization and QAM demapping are treated as black boxes in their model, and their roles are continuously approximated by a DNN model. However, DLbased methods often require high computational complexity.
The sparsity of the UWA channel can reduce the complexity of channel estimation (Li and Preisig, 2007). Shallow water surface scattering channels have a sparse structure at sufficiently wide bandwidths. This structure is formed by multipath arrivals resolved in delayed form (Li, 2006). A sparse adaptive subspace tracking (SASP) method is proposed to improve the accuracy of LS channel estimation. Two channel frequency response estimation algorithms, namely auxiliary pilot SASP and codeSASP, are proposed for FBMCOQAM systems (He et al., 2019b). A Bayesian learningbased sparse channel estimation technique is proposed for quasistatic and doublyselective mmWave hybrid MIMOFBMC systems. In contrast to the interference approximation method model, the proposed timedomain sparse Bayesian learning (TDSBL) scheme can successfully estimate frequencyselective channels strongly (Srivastava et al., 2020; Singh et al., 2022). A lowcomplexity algorithm, namely a dynamic thresholdbased inner product operation optimization strategy for MIMO sparse channel reconfiguration, is proposed for MIMOFBMC sparse channels to reduce complexity (Wang et al., 2020). The block SBL algorithm is introduced to improve the channel estimation performance using the blocksparse structure of the sparse multipath channel model (Wang et al., 2021). A sparse channel estimation method for MIMO FBMCOQAM systems, which can achieve accurate channel estimation by adaptively selecting the support set, is introduced (Wang, 2018).
In shortrange highrate UWA communication scenarios, the sparsity of the UWA channels may not be observed, and reducing the system complexity and overhead is difficult. However, obtaining high signaltonoise ratio (SNR) signal blocks is easy for the receiver in the case of short distances. This paper studies the channel tracking algorithm based on the data reuse method to meet the requirements of highspeed UWA in this scenario.
The data reusebased timevarying channel tracking method is a mature technology in conventional OFDM and single carrier systems. Compared with pilotbased methods, the data reusebased method needs low system overhead. The basic idea is to detect the current data symbols according to the historical channels and then use the detected data symbols as new pilots to update the current channel to achieve channel tracking. For conventional OFDM systems, these data symbols can be used for channel tracking as long as the length of the cyclic prefix or guard interval is longer than the channel delay and the Doppler and residual frequency offset are well compensated. The problem of inherent interference must be overcome to apply the data reuse technology to FBMCOQAM systems. This problem can be divided into two subproblems: detection of the interference source symbols and iterative updating of the channel after interference elimination from the reused symbols.
The innovative contributions in this paper are described as follows to solve the two aforementioned problems.
1) A timevarying UWA channel input–output model for FBMCOQAM systems based on the unified transmission matrix (UTM) model is established. The timeinvariant channelbased UTM model was proposed in a previous work (Lu et al., 2022) to describe the input–output channel relationship of the UWA FBMCOQAM system. The UTM model in this paper is extended to timevarying channel scenarios without changing the model framework. An efficient symbol detection model based on timevarying UTM is derived. The characteristic of the UTM model lies in its comprehensive and independent description of the weight value of each inherent interference. Therefore, the symbol detector based on the UTM model can realize the unified detection of all interference source symbols.
2) A data reusebased timevarying channel tracking algorithm for UWA FBMCOQAM systems with long signal blocks is proposed. This algorithm utilizes an iterative verification method to reduce the impact of data detection errors on channel tracking accuracy. The conventional FBMCOQAM detector cannot detect all interference source symbols uniformly. Therefore, suppressing the inherent interference of symbols to be reused is difficult, and the data symbols cannot be selected as new pilots for channel tracking. This paper maximizes the advantage of the UTM model, which can detect all interference symbols uniformly. Therefore, the data reuse method can be applied in UWA FBMCOQAM systems.
The relationship between channel tracking accuracy and channel time coherence of the proposed method is discussed by simulation in this paper. The influence of data detection error on channel tracking error is analyzed. Compared with traditional methods based on known pilots, the proposed method has similar channel estimation accuracy but requires low overhead. The sea trial results further prove the practicability of this method in the application of highspeed UWA communication in shortrange and high SNR scenarios. Finally, the results confirmed that the proposed timevarying channel tracking algorithm in this paper is suitable for the highrate FBMCOQAM UWA communication applications under the condition of shortrange and high SNR.
2 System model
This section first introduces the FBMCOQAM transmission signal model. Next, the input–output channel model over UWA timevarying channels is derived. Afterward, the extended UTM model is presented to describe the derived input–output channel model. Simplifying the expression of the timevarying channel model in a matrix form is easy by using the UTM model. Therefore, computational complexity is reduced by dividing and solving the model in the form of subUTM models. The derivation in this section lays a theoretical basis for channel estimation and symbol detection in Section 3.
2.1 FBMCOQAM transmitting signal model
An FBMCOQAM communication system with M subcarriers and N multicarrier symbols is considered in this study. The real and imaginary parts of a complex data symbol are transmitted at staggered T/2 time. The baseband transmission signal can be expressed as
$$ x(t)=\sum\limits_{m=0}^{M1} \sum\limits_{n=0}^{N1} s_{m, n} \mathrm{e}^{\mathrm{j} \phi_{m, n}} \mathrm{e}^{\mathrm{j} 2 \pi \frac{m}{T}\left(tn \frac{T}{2}\right)} g\left(tn \frac{T}{2}\right) $$ (1) where s_{m, n} is the (m, n)th transmitted real value symbol, T is the transmission interval of adjacent complex data symbols, and g (t) is the prototype filter function. The additional phase rotation of each real value symbol is expressed as ϕ_{m, n} = π (m + n)/2 + mnπ to maintain the real orthogonality between adjacent symbols. The PHYDYAS filter designed by Bellanger et al. (2010) is adopted in the current study. The timedomain expression of g (t) is defined as
$$ g(t)=1+2 \sum\limits_{\beta=1}^{\mathcal{B}1} G_\beta \cos \left(2 \pi \frac{\beta t}{\mathcal{B} T}\right) $$ (2) where $\mathcal{B}=4, G_1=0.97196, G_2=\sqrt{2} / 2$, and G_{3} = 0.235 147. Eq. (2) shows that the period of the prototype filter g (t) is 4T, but the transmission interval of adjacent multicarrier symbols is T/2. Therefore, the FBMCOQAM multicarrier symbols overlap each other in the time domain, as shown in Figure 1.
The passband transmission signal can be expressed as
$$ \tilde{x}(t)=\operatorname{Re}\left\{x(t) \mathrm{e}^{\mathrm{j} 2 \pi f_c t}\right\} $$ (3) where Re{·} represents the real part extraction operation and f_{c} is the center frequency.
2.2 Underwater acoustic channel input – output model for FBMCOQAM
The timevarying underwater acoustic channel impulse response (CIR) is defined as
$$ c(t, \tau)=\sum\limits_{k=0}^{K1} \tilde{h}_k(t) \delta\left(\tau\tau_k(t)\right) $$ (4) where $\tilde{h}_k(t) $ and τ_{k} (t) are the timevarying amplitude gain and the delay of the kth path of the underwater acoustic channel, respectively. K is the number of channel multipaths. The current study provides the following two approximations to c (t, τ).
Assumption 1: In shallow water environments, the Doppler spread factor of different multipath signals is approximately equal when the communication distance is far larger than the water depth.
Assumption 2: The transmitter and receiver are moving at a relatively uniform speed within the duration of a signal block; that is, the Doppler factor is timeinvariant within a signal block.
Then,
$$ \tau_k(t) \approx \tau_k\alpha t $$ (5) where α = v/v_{a} is the timeinvariant Doppler factor. v is the relative speed between the transmitter and the receiver. v_{a} is the acoustic speed. The received signal $\tilde{y}(t) $ is the convolution of the transmitted signal $\tilde{x}(t) $ and the channel c (t, τ).
$$ \begin{aligned} \tilde{y}(t)= & \sum\limits_{k=0}^{K1} \tilde{h}_k(t) \tilde{x}\left(t\tau_k(t)\right)+\tilde{w}(t) \\ = & \mathcal{R}\left\{\sum\limits_{k=0}^{K1} \tilde{h}_k(t) \sum\limits_{m=0}^{M1} \sum\limits_{n=0}^{N1} s_{m, n} \mathrm{e}^{\mathrm{j} \phi_{m, n}} \mathrm{e}^{\mathrm{j} 2 \pi \frac{m}{T}\left((1+\alpha) tn \frac{\pi}{2}\tau_k\right)}\right. \\ & \left.\cdot g\left((1+\alpha) tn \frac{T}{2}\tau_k\right) \mathrm{e}^{\mathrm{j} 2 \pi f_c\left((1+\alpha) t\tau_k\right)}\right\}+\tilde{w}(t) \end{aligned} $$ (6) where $\tilde{w}(t) $ is the passband additive noise.
The resampling method is employed for Doppler compensation:
$$ \tilde{y}\left(\frac{t}{1+\hat{\alpha}}\right)=\operatorname{Re}\left\{y(t) \mathrm{e}^{\mathrm{j} 2 \pi f_c t}\right\} $$ (7) where $\hat{\alpha} $ is the estimated Doppler factor. The baseband received signal y (t) when $\hat{\alpha} $ = α can be written as
$$ \begin{aligned} y(t) & =\sum\limits_{k=0}^{K1} \tilde{h}_k\left(\frac{t}{1+\hat{\alpha}}\right) \sum\limits_{m=0}^{M1} \sum\limits_{n=0}^{N1} s_{m, n} \mathrm{e}^{\mathrm{j} \phi_{m, n}} \mathrm{e}^{\mathrm{j} 2 \pi \frac{m}{T}\left(tn \frac{T}{2}\tau_k\right)} \\ & \cdot g\left(tn \frac{T}{2}\tau_k\right) \mathrm{e}^{\mathrm{j} 2 \pi f_c \tau_k}+w(t) \end{aligned} $$ (8) where w (t) is the baseband noise corresponding to $\tilde{w}(t) $. Eq. (8) shows that the equivalent baseband channel is the sampling of the actual passband CIR after resampling in the presence of a Doppler in the system. The demodulated baseband symbol is expressed as
$$ z_{m^{\prime}, n^{\prime}}=\int_{n^{\prime} \frac{T}{2}}^{n^{\prime} \frac{T}{2}+4 T} y(t) g\left(tn^{\prime} \frac{T}{2}\right) \mathrm{e}^{\mathrm{j} 2 \pi \frac{m^{\prime}}{T}\left(tn^{\prime} \frac{T}{2}\right)} \mathrm{e}^{\mathrm{j} \phi_{m^{\prime}, n^{\prime}}} \mathrm{d} t $$ (9) 2.3 UTM model
Equation (9) is the continuous time expression of the input and output model of FBMCOQAM signal over timevarying UWA channels. The discrete expression of (9) must be further derived to facilitate computer simulation.
Assume that the signal bandwidth is B, and the time sampling interval before upsampling is Δt = 1/B. The relationship between the transmission interval T of adjacent complex symbols and the number of subcarriers M is M = TB. Therefore, t and τ_{k} in (9) can be replaced by discrete index l and k, respectively. The integral of time can be replaced by the summation of l. As described in Section 2.2, the equivalent baseband channel is the sampling of the actual passband CIR after resampling in the presence of Doppler in the system. Assume that the baseband equivalent channel is $h_k(t)=\tilde{h}_k(t /(1+\alpha)) $. Let $ L=\mathcal{B} M=4 M$, representing the baseband sampling points of the prototype filter.
The discrete expression of z_{m′, n′} can then be written as
$$ z_{m^{\prime}, n^{\prime}}=\sum\limits_{m=0}^{M1} \sum\limits_{n=0}^{N1} S_{m, n} \Gamma_{m, n}^{m^{\prime}, n^{\prime}}+w_{m^{\prime}, n^{\prime}} $$ (10) where w_{m′, n′} is the resulting additive noise on the (m, n)th received symbol. The contribution weight Γ_{m, m}^{m′, n′} is specifically expressed as
$$ \begin{aligned} \Gamma_{m, n}^{m^{\prime}, n^{\prime}}= & \sum\limits_{l=\frac{M}{2} n^{\prime}}^{\frac{M}{2} n^{\prime}+L1} \sum\limits_{k=0}^{K1} h_k(l) \mathrm{e}^{\mathrm{j} 2 \pi \frac{\left(mm^{\prime}\right) l}{M}} \mathrm{e}^{\mathrm{j} 2 \pi \frac{m k}{M}} g\left(l\frac{M}{2} n^{\prime}\right) \\ & \cdot g\left(l\frac{M}{2} nk\right) \mathrm{e}^{\mathrm{j} \phi_{\Delta}} \mathrm{e}^{\mathrm{j} 2 \pi f_c k} \end{aligned} $$ (11) where $\phi_{\Delta}=\phi_{m, n}\phi_{m^{\prime}, n^{\prime}}$, and $h_k(l)$ represents the value of the $k$ th tap of the channel at time $l$. Herein, variable $K$ is continuously used to represent the number of channel taps. Some values of $h_k(l)$ are close to zero when the channel is sparse.
The sending symbol $s_{m, n}$, receiving symbol $z_{m^{\prime}, n^{\prime}}$, contribution weight $\Gamma_{m, n}^{m^{\prime}, n^{\prime}}$, and noise $w_{m^{\prime}, n^{\prime}}$ are recombined as shown below to describe (10) in matrix form.
$$ \boldsymbol{Z}=\boldsymbol{T S}+\boldsymbol{W} $$ (12) where
$$ \begin{array}{l} \boldsymbol{Z}=\left[\begin{array}{c} \boldsymbol{z}_0 \\ \vdots \\ \boldsymbol{z}_{M1} \end{array}\right], \boldsymbol{S}=\left[\begin{array}{c} \boldsymbol{s}_0 \\ \vdots \\ \boldsymbol{s}_{M1} \end{array}\right], \boldsymbol{W}=\left[\begin{array}{c} \boldsymbol{w}_0 \\ \vdots \\ \boldsymbol{w}_{M1} \end{array}\right], \\ \boldsymbol{T}=\left[\begin{array}{ccc} \boldsymbol{\varLambda}_0^0 & \cdots & \boldsymbol{\varLambda}_{M1}^0 \\ \vdots & \ddots & \vdots \\ \boldsymbol{\varLambda}_0^{M1} & \cdots & \boldsymbol{\varLambda}_{M1}^{M1} \end{array}\right] \end{array} $$ (13) Matrix T is the UTM, and (12) is the UTM model. T comprises M × M submatrices Λ_{m}^{m'}, which are composed of N × N scalar elements, that is,
$$ \boldsymbol{\varLambda}_m^{m^{\prime}}=\left[\begin{array}{ccc} \Gamma_{m, 0}^{m^{\prime}, 0} & \cdots & \Gamma_{m, 0}^{m^{\prime}, N1} \\ \vdots & \ddots & \vdots \\ \Gamma_{m, N1}^{m^{\prime}, 0} & \cdots & \Gamma_{m, N1}^{m^{\prime}, N1} \end{array}\right] $$ (14) 2.3.1 UTM model for timeinvariant channel
Submatrix $\boldsymbol{\varLambda}_m^{m^{\prime}}$ describes the contribution weight of $N$ symbols at different times on the $m$ th subcarrier to the symbols on the $m^{\prime}$ th subcarrier. The values of $m$ and $m^{\prime}$ are fixed when the elements in $\boldsymbol{\varLambda}_m^{m^{\prime}}$ are calculated by Eq. (11). Furthermore, the values of $n$ and $n^{\prime}$ are equal when calculating the elements on the main diagonal of $\boldsymbol{\varLambda}_m^{m^{\prime}}$. If the channel is timeinvariant, then $h_k(l)=h_k$. Thus, the element values on the main diagonal of $\boldsymbol{\varLambda}_m^{m^{\prime}}$ are equal. However, the element values on the main diagonal of different submatrices $\varLambda_m^{m^{\prime}}$ are different due to varying values of $m$ and $m^{\prime}$. That is to say, the modulus of the elements on the main diagonal of matrix $\boldsymbol{T}$ presents a ladderlike change, and its envelope follows the frequency domain response characteristics of the channel.
2.3.2 UTM model for timevariant channel
Different from the case of timeinvariant channels, $h_k(l)$ changes with $n$ and $n^{\prime}$ under timevarying channel conditions. Therefore, the value of the main diagonal elements of $\boldsymbol{\varLambda}_m^{m^{\prime}}$ also changes with $n$ and $n^{\prime}$. The channel temporal coherence function (Yang, 2012) is defined to describe the correlation between channels at different times.
$$ \Theta(\tau) \equiv \mathrm{E}\left[\frac{\left\langle c^*(t) c(t+\tau)\right\rangle}{\sqrt{\left\langle c^*(t) c(t)\right\rangle\left\langle c^*(t+\tau) c(t+\tau)\right\rangle}}\right] $$ (15) where the angular brackets $ \langle a b\rangle$ denote the maximum value of the correlation between the a and b series. E[·] denotes ensemble average over the geotime t.
2.3.3 MMSEbased symbol detector for FBMCOQAM under UTM model
In an FBMCOQAM system, the symbol on one subcarrier only interferes with itself and the symbol on two adjacent subcarriers. Therefore, the UTM presents block diagonalization. A sliding window with a dimension of 3N × 5N is used to segment the UTM and solve the model. This segmentation solution can reduce the overall computational complexity. According to the model described by (12) and (13), the subUTM model is expressed as
$$ \boldsymbol{Z}_p=\boldsymbol{T}_p \boldsymbol{S}_p+\boldsymbol{W}_p $$ (16) where
$$\begin{array}{l} \boldsymbol{Z}_p=\left[\begin{array}{c} \boldsymbol{z}_{p1} \\ \boldsymbol{z}_p \\ \boldsymbol{z}_{p+1} \end{array}\right], \quad \boldsymbol{S}_p=\left[\begin{array}{c} \boldsymbol{s}_{p2} \\ \boldsymbol{s}_{p1} \\ \boldsymbol{s}_p \\ \boldsymbol{s}_{p+1} \\ \boldsymbol{s}_{p+2} \end{array}\right], \quad \boldsymbol{W}_p=\left[\begin{array}{c} \boldsymbol{w}_{p1} \\ \boldsymbol{w}_p \\ \boldsymbol{w}_{p+1} \end{array}\right]\\ \boldsymbol{T}_p=\left[\begin{array}{ccccc} \boldsymbol{\varLambda}_{p2}^{p1} & \boldsymbol{\varLambda}_{p1}^{p1} & \boldsymbol{\varLambda}_p^{p1} & \bf{0} & \bf{0} \\ \bf{0} & \boldsymbol{\varLambda}_{p1}^p & \boldsymbol{\varLambda}_p^p & \boldsymbol{\varLambda}_{p+1}^p & \bf{0} \\ \bf{0} & \bf{0} & \boldsymbol{\varLambda}_p^{p+1} & \boldsymbol{\varLambda}_{p+1}^{p+1} & \boldsymbol{\varLambda}_{p+2}^{p+1} \end{array}\right] \end{array} $$ (17) The value range of $p$ is $[1, M1]$. In particular, when $p=1$, the first $N$ columns of $\boldsymbol{T}_p$ and $\boldsymbol{s}_{p2}$ are removed in $\boldsymbol{S}_p$; when $p=M1$, the last $N$ columns of $\boldsymbol{T}_p$ and $\boldsymbol{s}_{p+2}$ are removed in $\boldsymbol{S}_p$.
Next, the real and imaginary parts of the elements in (16) are separated and rearranged. $\boldsymbol{S}_p$ is a real value vector; thus, its imaginary part is zero, and the following can be obtained:
$$ \overline{\boldsymbol{Z}}_p=\overline{\boldsymbol{T}}_p \overline{\boldsymbol{S}}_p+\overline{\boldsymbol{W}}_p $$ (18) where
$$ \overline{\boldsymbol{Z}}_p=\left[\begin{array}{l} \mathcal{R}\left\{\boldsymbol{Z}_p\right\} \\ \mathcal{I}\left\{\boldsymbol{Z}_p\right\} \end{array}\right] \quad \overline{\boldsymbol{S}}_p=\mathcal{R}\left\{\boldsymbol{S}_p\right\} \quad \overline{\boldsymbol{W}}_p=\left[\begin{array}{l} \mathcal{R}\left\{\boldsymbol{W}_p\right\} \\ \mathcal{I}\left\{\boldsymbol{W}_p\right\} \end{array}\right] $$ (19) and
$$ \overline{\boldsymbol{T}}_p=\left[\begin{array}{l} \mathcal{R}\left\{\boldsymbol{T}_p\right\} \\ \mathcal{I}\left\{\boldsymbol{T}_p\right\} \end{array}\right] $$ (20) The dimension of matrix $ \overline{\boldsymbol{T}}_p$ is 6N × 5N when p is 2–(M−2). When p = 1 or p = M − 1, the dimension of matrix $ \overline{\boldsymbol{T}}_p$ is 6N × 4N. MMSEbased detectors can then be applied for symbol detection.
$$ \hat{\overline{\boldsymbol{S}}}_p=\left(\overline{\boldsymbol{T}}_p{ }^H \cdot \overline{\boldsymbol{T}}_p+\boldsymbol{C}_{\overline{\boldsymbol{W}}_p} \bf{I}\right)^{1} \overline{\boldsymbol{T}}_p{ }^H \overline{\boldsymbol{Z}}_p $$ (21) where $\bf{I}$ is the unit matrix. When $p=1$ or $p=M1$, the dimension of $\bf{I}$ is $4 N \times 4 N$; otherwise, the dimension of $\bf{I}$ is $5 N \times 5 N . \boldsymbol{C}_{\bar{\boldsymbol{W}}_p}$ is the covariance matrix of the resulting noise $\overline{\boldsymbol{W}}_p$.
Finally, all detection results of $\hat{\boldsymbol{S}}$ are obtained by updating the value of $p$.
3 Timevarying channel estimation method for FBMCOQAM
The analysis in Section 2 shows that the UTM model can adapt to the cases of timevarying UWA channels. That is, the value of the CIR h_{k} (l) is taken at each time sampling point l into (11) to calculate $ \Gamma_{m, n}^{m^{\prime}, n^{\prime}}$. However, obtaining a considerable amount of unknown channel information remains a challenge. An effective approximate method is necessary to estimate the CIR when n^{′} = {0, M/2, M, ⋯, (N−1) M/2} and obtain the channel responses at other times through interpolation.
This section describes the proposed timevarying CIR estimation method for FBMCOQAM systems. The method is divided into two steps: initial channel estimation and channel tracking. Figure 2 shows the block diagram of the proposed timevarying channel estimation method for FBMCOQAM systems. The rest of this section will elaborate on this method.
3.1 Initialize channel estimation
Half of the symbols in the preamble are used as pilot symbols to initialize the channel estimation. Figure 3 shows the symbol distribution structure in one FBMCOQAM signal block. Each circle represents a baseband symbol. Hollow, solid, and crossed hollow circles denote zero, pilot, and data symbols, respectively. The crossed hollow circles with gray background denote the new pilots selected for channel tracking. The dotted line box with blue background is the preamble.
According to the timefrequency characteristics of the PHYDYAS prototype filter (Bellanger et al., 2010), the received symbol $z_{m^{\prime}, n^{\prime}}$ is interfered with by the surrounding symbol $s_{m, n}$, where the subscript $m \in\left\{m^{\prime}1, m^{\prime}, m^{\prime}+1\right\}$ and $n \in\left\{n^{\prime}3, n^{\prime}2, \cdots, n^{\prime}+3\right\}$. A protection gap of $3 T / 2$ is set after the preamble to avoid unknown interference with pilot symbols.
Consequently, the UWA channel is the only source of interference with the pilot symbols. Assuming that the channel is approximately timeinvariant during the preamble period, (10) can be simplified as
$$ z_{m^{\prime}, 0}=\sum\limits_{l=\frac{M}{2} n^{\prime}}^{\frac{M}{2} n^{\prime}+L1} \sum\limits_{k=0}^{K1} h_k S_{m^{\prime}, 0} \mathrm{e}^{\mathrm{j} 2 \pi \frac{m^{\prime} k}{M}} g(l) g(lk) \mathrm{e}^{\mathrm{j} 2 \pi f_c k} $$ (22) where $ m^{\prime} \in\left\{p_1, p_2, \cdots, p_{M_p}\right\}$, and M_{p} is the number of pilots. p_{1} represents the index of the subcarrier corresponding to the first pilot. The matrix form of the channel estimation model is
$$ \boldsymbol{z}_0^{\text {pilots }}=\boldsymbol{\varPsi} \boldsymbol{h}^0+\boldsymbol{w}_0^{\text {pilots }} $$ (23) where $\boldsymbol{z}_0^{\text {pilots }}=\left[z_{p_1, 0}, z_{p_2, 0}, \cdots, z_{p_{M_p}, 0}\right]^{\mathrm{T}}, \boldsymbol{h}^0=\left[h_0, h_1, \cdots, h_{K1}\right]^{\mathrm{T}}$, and $\boldsymbol{\psi}=\left[\boldsymbol{\psi}_0, \boldsymbol{\psi}_1, \cdots, \boldsymbol{\psi}_{K1}\right]$. The superscript 0 of $\boldsymbol{h}^0$ represents the CIR vector when $n^{\prime}=0 . M_p$dimensional column vector $\boldsymbol{\psi}_k=\left[\psi_{p_1}^k, \cdots, \psi_{p_{M_p}}^k\right]^{\mathrm{T}}$. The calculation expression of the element of $\boldsymbol{\psi}_k$ is
$$ \psi_{m^{\prime}}^k=\sum\limits_{l=\frac{M}{2} n^{\prime}}^{\frac{M}{2} n^{\prime}+L1} \sum\limits_{k=0}^{K1} S_{m^{\prime}, 0}\mathrm{e}^{\mathrm{j} 2 \pi \frac{m^{\prime} k}{M}} g(l) g(lk) \mathrm{e}^{\mathrm{j} 2 \pi f_c k} $$ (24) The weighted least squares (WLS) channel estimation can then be expressed as
$$ \hat{\boldsymbol{h}}^0=\left(\boldsymbol{\psi}^{\mathrm{H}} \boldsymbol{C}_{\boldsymbol{w}^{\text {piots }}}^{1} \boldsymbol{\psi}\right)^{1} \boldsymbol{\psi}^{\mathrm{H}} \boldsymbol{C}_{\boldsymbol{w}^{\text {piots }}}^{1} \boldsymbol{z}^{\text {pilots }} $$ (25) $\boldsymbol{C}_{\boldsymbol{w}^{\text {piots }}} $ represents the covariance matrix of noise w^{pilots}. The superscript [·]^{T} and [·]^{H} denote transpose and conjugate transpose, respectively.
3.2 Timevarying channel tracking
The channel is timevarying; thus, the accuracy of $\hat{s}_{m^{\prime}, n^{\prime}}$ estimated by $\hat{\boldsymbol{h}}^0$ gradually decreases with the increase of $n^{\prime}$. Therefore, the proposed channel tracking method only uses $\hat{\boldsymbol{h}}^0$ to estimate the data symbols $\hat{s}_{m^{\prime}, n^{\prime}}$ with $n^{\prime}=$ $\{4, 5, 6, 7\}$ and discards the symbols when $n^{\prime}>7$. Symbols with $n^{\prime}=\{1, 2, 3\}$ are zeros. The estimated data symbols are then reused as new pilots to estimate the CIR when $n^{\prime}=4$, i.e., $\hat{\boldsymbol{h}}^4$. Next, $\hat{s}_{m^{\prime} n^{\prime}}$, with $n^{\prime}=\{4, 5, 6, 7, 8\}$, can be estimated by updating $\hat{\boldsymbol{h}}^4$ to (11) and (12). Verifying the detection symbols is necessary due to the presence of possible detection errors of data symbols. The specific method is to redetect $\hat{s}_{m^{\prime}, n^{\prime}}$ with $n^{\prime}=\{4, 5, 6, 7\}$ after updating the estimated $\hat{\boldsymbol{h}}^4$ to (11) and then check the presence of detection errors in $\hat{s}_{m^{\prime}, n^{\prime}}$ If an error exists, then the wrong reused symbol is replaced and $\hat{\boldsymbol{h}}^4$ is reestimated, iterating several times until the estimated value of $\hat{\boldsymbol{h}}^4$ stabilizes. The channel and data symbols at each time can be estimated step by step with this method. The algorithm flowchart is shown in Figure 2.
The temporal coherence coefficient Θ(τ) defined in (15) is used as a measure of channel time variability. A small coherence coefficient indicates a low estimation accuracy. Assuming that when the channel has not excessively changed, that is, the symbols received at the time within the yellow background dotted box, as shown in Figure 3, can be correctly estimated, utilizing these correctly estimated symbols as new pilots for CIR tracking is proposed. The channel error correction code is used to reduce the impact of the timevarying channel. Another important assumption is presented in this paper.
Assumption 3: In the $ {}^{7}\!\!\diagup\!\!{}_{2}\;T$ time range, the unknown data symbols in this range can still be demodulated correctly when the channel is regarded as timeinvariant.
UWA channels are often sparse; thus, only a few pilots are necessary to achieve channel tracking and estimation. The sparsity of the channel can be determined in accordance with the estimation results of the initialized channel. K_{c} is defined as the number of nonzero channel taps, and M_{c} new pilots are selected for updating the estimated UWA CIR, where K_{c} < K and M_{c} > K_{c}.
According to the block structure shown in Figure 3, the symbols with n^{′} = {1, 2, 3} are all zero symbols, which are used to reduce the inherent interference on pilot symbols. Therefore, $\hat{\boldsymbol{h}}^4 $ is the next CIR that must be updated. If symbol s_{m′, 4} is selected as one of the new pilots to estimate $\hat{\boldsymbol{h}}^4 $, then the subcarrier index of these new pilots can be expressed as a set $ m^{\prime} \in \boldsymbol{\mathcal{S}}_c=\left\{p_1, p_2, \cdots, p_{M_c}\right\}$. Interference cancellation is then performed on the received new pilot symbols $Z_{m^{\prime}, 4}^{\text {pilot }} $; that is,
$$ z_{m^{\prime}, 4}^{\text {pilot }}=z_{m^{\prime}, 4}\sum\limits_{(m, n) \in \Omega} \Gamma_{m, n}^{m^{\prime}, n^{\prime}} \hat{s}_{m, n} $$ (26) where $m \in\left\{m^{\prime}1, m^{\prime}, m^{\prime}+1\right\}, \quad n=\left\{n^{\prime}4, \cdots, n^{\prime}+4\right\}$, and $(m, n) \neq\left(m^{\prime}, n^{\prime}\right)$. Take $\hat{\boldsymbol{h}}^0$ into (11) to calculate $\Gamma_{m, n}^{m^{\prime}, n^{\prime}}$ in (26). The updated channel pulse response $\hat{\boldsymbol{h}}^4$ is then estimated in accordance with the steps of (23) to (25). Compressed sensing methods, such as orthogonal matching pursuit (Berger et al., 2010), can be used to estimate $\hat{\boldsymbol{h}}^4$ due to the sparsity of UWA channels.
Afterward, $\hat{\boldsymbol{h}}^4$ is taken into (11) and (24) to calculate $\Gamma_{m, n}^{m^{\prime}, n^{\prime}}$ and $\boldsymbol{\varPsi}$, respectively, and $\hat{\boldsymbol{h}}^5$ can be estimated thereafter. By contrast, the CIR at each time is estimated.
4 Simulation and experiment
The analysis in Section 3.2 assumed that the unknown data symbols in this range can still be detected correctly when the timevarying channel is regarded as timeinvariant in several FBMC symbol periods. Assumption 3 holds due to the minimum channel temporal coherence coefficients obtained through computer simulation analysis in this section. The performance of timevarying channel tracking in a long FBMCOQAM signal block is also verified in this paper through simulation and sea trial.
4.1 Simulation
4.1.1 Simulation parameter configuration
The UWA channel simulation model proposed by Qarabaqi (Qarabaqi and Stojanovic, 2013) is adopted. The model of Qarabaqi controls the time variation of the channel by adjusting different channel parameters, such as the position fluctuation of the transceiver affected by the current, the fluctuation of the surface wave, and the change of the scattering. The main parameters and their values set in the simulations are listed in Table 1.
Type Channel parameters Value Channel geometry Water depth (m) 80 Transmitter depth (m) 25 Receiver depth (m) 20 Channel distance (m) 1 000 Spreading factor 2.7 Bandwidth (Hz) 3 000 Amplitude ratio of the strongest diameter to the weakest diameter 10 Speed of sound in water (m/s) 1 540 Speed of sound in the bottom (m/s) 1 580 Smallscale parameters Coherence time of the smallscale variations (s) 20 Number of intrapaths 20 Mean of intrapath amplitudes 0.025 Variance of intrapath amplitudes 1×10^{6} Largescale parameters Range of surface height variation (m) −0.5–0.5 Range of transmitter height variation (m) −0.2–0.2 Range of receiver height variation (m) −0.1–0.1 Range of channel distance variation (m) −2–2 Doppler Movement speed range of the transceivers (m/s) 0–0.4 or 1–3 Movement angle range of the transceivers (°) −25–25 Among the parameters listed in Table 1, the values of Doppler and four largescale parameters have different effects on channel time variation. Largescale parameters have considerable influence on the time variation of multipath amplitude and phase, while Doppler parameters have a remarkably significant influence on the time variation of multipath delay. A set speed range of 0–0.4 m/s indicates lowspeed movement, and the generated timevarying channel multipath delay does not change within an FBMC signal block period. Meanwhile, a set speed range of 1–3 m/s indicates highspeed movement, and the generated timevarying channel multipath delay will change within a signal block period.
An FBMCOQAM signal block with M = 128, N = 30, and 3 000 Hz bandwidth is generated in this section for simulation and analysis. The number of symbols N is set as 30, which indicates that the signal is a long signal block. The signal bandwidth is consistent with the simulated channel bandwidth. According to the above parameters, the calculated time period of one FBMCOQAM multicarrier symbol is 4M/3 000 ≈ 170.67 ms, and the duration of the entire FBMCOQAM signal block is (0.5MN + 3.5M)/3 000 ≈ 789.33 ms.
4.1.2 Simulation results and analysis
In addition to the main parameters listed in Table 1, the model of Qarabaqi also contains some random parameters, such as the phase of the sea surface wave. Therefore, each generated channel will have slight differences in multipath amplitude, phase, and delay, but their statistical values are unchanged.
An example of the simulated timevarying UWA CIR is shown in Figure 4(a), which demonstrates four main nonzero taps. According to their delay index from small to large, the four main nonzero taps correspond to the direct path, the sea bottom reflection path, the two overlapped bottom surface reflection path, and the twice surface reflection path. The propagation delay of the single surface reflection path is approximately equal to that of the direct path, overlapping to form the first main nonzero tap due to the shallow deployment depth of the transceivers. Moreover, the channel is approximately symmetric due to the depth of the transmitter, and the receiver is similar. Therefore, the channel has two paths with similar delays, and their energy is superposed, corresponding to "Path 2" in Figure 4(a). The relative positions of the transmitter and receiver slightly change in a signal block time period; the delay of the four paths hardly changes. Accordingly, the positions of the four main nonzero taps remain unchanged, while their amplitudes change slightly and continuously. Figure 4(b) shows the timevarying phase of four main nonzero taps. The sea surface is fluctuating, and the sea bottom is stationary. Thus, the phase and amplitude of the twice surface reflection paths (path 3) are highly volatile with time.
Figure 5 shows the temporal coherence function of the channels in Figure 4 to measure the time variation of the channel quantitatively. Figure 5 reveals that the channel coherence coefficient decreases to approximately 0.825 within the duration of the entire FBMCOQAM signal block. The SNR under this channel is set to 20 dB. If only $\hat{\boldsymbol{h}}^0 $ is used for symbol detection of the entire signal block, then the detection result is obtained as shown in Figure 6(a). With the increase of n, the difference between channels at different times rises, which leads to a decrease in symbol detection accuracy. If the method in Section 3.2 is used for channel tracking, then the tracked timevarying channel is taken into the UTM model (18), and the symbol detection results can be improved, as shown in Figure 6(b). The scattered points in Figure 6(b) do not diverge with the increase of n. This result shows that tracking estimation of timevarying channels reduces the bit error rate (BER) of FBMCOQAM systems with long signal blocks.
The tracking accuracy of the timevarying channel is described by the MSE of channel estimation. The MSE is defined as
$$ \text { MSE }=10 \lg \frac{\left(\hat{\boldsymbol{h}}^{n^{\prime}}\boldsymbol{h}^{n^{\prime}}\right)^{\mathrm{H}}\left(\hat{\boldsymbol{h}}^{n^{\prime}}\boldsymbol{h}^{n^{\prime}}\right)}{\left(\boldsymbol{h}^{n^{\prime}}\right)^{\mathrm{H}} \boldsymbol{h}^{n^{\prime}}} $$ (27) The CIR is mainly related to the following factors: the position and velocity of the transmitter and receiver, water depth, sea surface wave, interface reflection coefficient, sound speed, and water reflector or scatterer. In most cases, changes in the position and velocity of the transmitter and receiver, as well as the sea surface wave, significantly impact the channel pulse response within a period of the FBMCOQAM signal block. Therefore, the attenuation rate of the channel temporal coherence function is adjusted by modifying the simulation parameters, and the performance of the proposed channel tracking method is then analyzed. The MSE results of channel estimation under different SNRs are displayed in Figure 7.
The solid lines with star markers in Figure 7 show that only the amplitude and phase of the tap are timevarying while the tap delay remains unchanged, which corresponds to the relative static or lowspeed movement of the transceiver. The dotted lines with star markers indicate that the time delay of the tap is also timevarying, corresponding to highspeed motion. These results show that the channel tracking error decreases with the increase in temporal coherence. Under high SNR conditions, the change in multipath tap delay will not cause substantial performance loss on the proposed method. However, the change in tap delay will markedly reduce the channel tracking accuracy under low SNR. This phenomenon is attributed to the increase in symbol detection error and the difficulty in convergence under low SNR. In addition, Figure 7 shows that the channel tracking performance based on the traditional scattered pilot structure is compared. The conventional method based on known scattered pilot shows high channel tracking accuracy, especially in the case of low SNR, using the WLSbased CEs. The conventional method is unaffected by the data detection error. The CE based on the SBL method shows the best performance, but its computational complexity is also the highest. The CE based on orthogonal matching pursuit (OMP) with low computational complexity is suitable for sparse channels and shows good performance in the case of low channel time coherence. In the case of high SNR and a larger channel time coherence coefficient than 0.8, the channel estimation performance of the proposed method is generally similar to that of conventional methods. However, the system overhead is substantially small.
Figure 8 shows the change of data symbol detection error with SNR. The curves with different markers in Figure 8 correspond to different times of iterative verification. The movement speed of the transceivers in the simulation is set to a random value between 1 and 3 m/s. The tap delay of multipath is timevarying. The WLSbased CE is adopted. The analysis of the results in Figure 8 reveals that the data detection error decreases as the number of iterations increases. The first three iterative verifications can significantly reduce the detection error. However, the rate of data detection error gradually decreases as the number of iterations increases. The effect of eight iterations is remarkably close to that of four iterations. Moreover, the error curves demonstrate error floors when the number of iterations is 1 and 2. Therefore, in the case of high SNR, the impact of channel tracking error on symbol detection accuracy is larger than that of noise. The curve error floor disappears after additional iterative verifications, which indicates that the channel tracking accuracy has been improved, and the channel tracking error is no longer the main limitation of data detection accuracy. In the case of low SNR, the data detection error remains high and does not decrease with the increase in iterations, which indicates that the proposed method induces uncontrollable cumulative error. Therefore, the proposed channel tracking method is highly suitable for high SNR situations.
Figure 9 shows the effect of data detection error on channel estimation accuracy. The symbol detection error refers to the average error of the data symbols in the range of n ∈ {n^{′} − 4, ⋯, n^{′} + 4} because only the data symbols in this range have an impact on the channel estimation at time n. The curves in Figure 9 are the results of eight iterative verifications. The three CEs in Figure 9 are based on the proposed data reuse method in this paper. Therefore, the performance of the three CEs is affected by symbol detection error. The estimation error of the three CEs generally increases with the data symbol detection error rate. The SBLbased CE shows the best performance, and the performance of the three CEs is relatively close.
4.2 Experiment results
The sea trial was conducted at sea near Zhoushan Archipelago, China, on December 6, 2022. Figure 10 shows the configuration and sound speed profile of the sea trial. The average water depth of the experimental sea area is approximately 40 m, the transceivers were placed beneath the water surface approximately 10 m, and the distance between the transmitter and the receiver is approximately 1 000 m.
The frame structure of the transmitted signal is shown in Figure 11, where the hyperbolic frequency modulation pulse is used for signal synchronization and Doppler estimation. 511order msequence (Mseq) is used for fine synchronization and carrier frequency offset (CFO) estimation. The modulation parameters of the signal block, namely the average SNR of the signal received, is 21.5 dB. The parameters of signal block "FBMC 2" are consistent with the simulation in Section 4.1; that is, M = 128 and N = 30. The bandwidth is 3 000 Hz, and the center frequency f_{c} = 15 kHz. The number of the signal block subcarriers "FBMC 1" is 256, and other modulation parameters are the same as "FBMC 2."
The source and receiver vessels slowly drifted with the current during the experiment. The estimated average CFO is 3.4 Hz. Channel estimation is performed on the received signal after frequency offset compensation, and the results are shown in Figure 12. Figure 12(a) shows the estimated modulus of the timevarying CIR. Results showed that the actual sea trial channels are similar to the computer simulation results. That is, the index of the main nonzero taps in the channels almost remains unchanged within an FBMC signal block period, but their amplitudes change. Figure 12(b) shows the timevarying phase of the main nonzero taps. The bold solid line is the phase change of the direct path. In addition to the direct path, the main nonzero taps can be divided into two groups. The taps of "group 2" correspond to the paths which reflect more times at the sea surface and bottom than the taps of "group 1." The taps of "group 2" are affected by more timevarying factors than "group 1" due to the long propagation time and additional reflection times in the channel, resulting in strong timevarying amplitude and phase. The four dotted lines in Figure 12(b) represent the phase changes of the four nonzero taps in group 2, and the four solid lines are the phase changes of the nonzero taps in group 1. Each phase curve continuously and slowly changes without mutations, indicating a strong correlation between channels at adjacent times.
Figure 13 shows the temporal coherence function of the sea trial channel to further describe the correlation between channels quantitatively. The blue solid line corresponds to the long signal block named "FBMC 1, " and the orange solid line with a circle mark corresponds to the short signal block named "FBMC 2." The declining slope of the orange curve is lower than that of the blue curve. The value of parameter N in the two signal blocks is the same. However, the correlation between channels in signal block "FBMC 2" is strong due to its short signal period.
Figure 14 displays the demodulation results of block "FBMC 1" and block "FBMC 2" using a timeinvariant channel (TIC) and timevarying channel (TVC), respectively. If the TICs were used, then the demodulation results of the long FBMCOQAM symbol block would become worse with the increase of n. However, if the TVCs were tracked, then the BER does not increase with the n. In addition, the BER is low when M is small because the symbol period shortens with the decrease in M, and the short symbol period indicates a weak influence of TVC. For the FBMCOQAM signal with a received SNR of approximately 21.5 dB, the average BER of symbol block "FBMC 1" generally decreases by 25.5 dB, and the average BER of symbol block "FBMC 2" decreases by 28.8 dB when using the proposed TVC tracking method.
Table 2 and Figure 15 show the demodulation results of sea trial data with different channel tracking methods. These results show that the BER of short data block "FBMC 2" is low because the short data packets are only slightly affected by the TVC and the highly accurate channel tracking. However, the data block cannot be excessively short in practical applications. Ensuring that one FBMC multicarrier symbol is generally substantially longer than the channel delay is necessary; otherwise, insufficient alternatives will emerge when selecting new pilots. This phenomenon will lead to poor iterative verification effect on improving channel tracking accuracy.
Method Block BER Net data rate (kbit/s) Proposed CE with WLS FBMC 1 0.011 3 4.21 FBMC 2 0.002 2 Proposed CE with OMP FBMC 1 0.023 1 FBMC 2 0.003 6 Proposed CE with SBL FBMC 1 0.009 7 FBMC 2 0.001 6 Pilotbased CE with WLS FBMC 1 0.010 9 3.16 FBMC 2 0.001 9 2.11 Pilotbased CE with OMP FBMC 1 0.023 6 3.97 FBMC 2 0.003 2 3.72 Pilotbased CE with SBL FBMC 1 0.008 1 3.16 FBMC 2 0.001 1 2.11 Considering system overhead, the proposed estimator based on data reuse in this paper has the minimum overhead and ensures the maximum data rate. For conventional channel estimation methods based on known pilots, the number of pilots for WLS, OMP, and SBLbased CEs is 64, 32, and 64, respectively.
5 Conclusions
A lowoverhead channel tracking algorithm is proposed in this paper for shortrange highrate UWA FBMCOQAM communication applications. By extending the UTM model to TVCs, the UTM symbol detector can approximately detect all interference symbols that are under the influence of TVC; then, utilize estimated interference symbols to eliminate interference for the selected new pilots. Multiple iterative verifications can improve the accuracy of channel tracking and data detection. The simulation results show that the channel tracking error can be reduced to less than −20 dB when the channel temporal coherence coefficient exceeds 0.75 within one block period of FBMCOQAM signals. The proposed method shows that compared with conventional knownpilotsbased methods, our method requires lower overhead without reducing the CE performance. The sea trial results further proved the practicability of the proposed method under the actual UWA communication scenarios.
Competing interest Fengzhong Qu is an editorial board member for the Journal of Marine Science and Application and was not involved in the editorial review, or the decision to publish this article. All authors declare that there are no other competing interests. 
Figure 4 Simulated channel based on the proposed model by Qarabaqi and Stojanovic (2013)
Table 1 Main parameters of the UWA channel simulation model of Qarabaqi
Type Channel parameters Value Channel geometry Water depth (m) 80 Transmitter depth (m) 25 Receiver depth (m) 20 Channel distance (m) 1 000 Spreading factor 2.7 Bandwidth (Hz) 3 000 Amplitude ratio of the strongest diameter to the weakest diameter 10 Speed of sound in water (m/s) 1 540 Speed of sound in the bottom (m/s) 1 580 Smallscale parameters Coherence time of the smallscale variations (s) 20 Number of intrapaths 20 Mean of intrapath amplitudes 0.025 Variance of intrapath amplitudes 1×10^{6} Largescale parameters Range of surface height variation (m) −0.5–0.5 Range of transmitter height variation (m) −0.2–0.2 Range of receiver height variation (m) −0.1–0.1 Range of channel distance variation (m) −2–2 Doppler Movement speed range of the transceivers (m/s) 0–0.4 or 1–3 Movement angle range of the transceivers (°) −25–25 Table 2 Statistics of sea trial data demodulation results
Method Block BER Net data rate (kbit/s) Proposed CE with WLS FBMC 1 0.011 3 4.21 FBMC 2 0.002 2 Proposed CE with OMP FBMC 1 0.023 1 FBMC 2 0.003 6 Proposed CE with SBL FBMC 1 0.009 7 FBMC 2 0.001 6 Pilotbased CE with WLS FBMC 1 0.010 9 3.16 FBMC 2 0.001 9 2.11 Pilotbased CE with OMP FBMC 1 0.023 6 3.97 FBMC 2 0.003 2 3.72 Pilotbased CE with SBL FBMC 1 0.008 1 3.16 FBMC 2 0.001 1 2.11 
Bellanger M, Le Ruyet D, Roviras D, Terre M (2010) FBMC physical layer: a primer. PHYDYAS, 25(4): 710 Berger CR, Zhou S, Preisig JC, Willett P (2010) Sparse channel estimation for multicarrier underwater acoustic communication: From subspace methods to compressed sensing. IEEE Transactions on Signal Processing 58(3): 17081721. DOI: 10.1109/TSP.2009.2038424 Cheng X, Liu D, Wang C, Yan S, Zhu Z (2019a) Deep learningbased channel estimation and equalization scheme for FBMC/OQAM systems. IEEE Wireless Communications Letters 8(3): 881884. DOI: 10.1109/LWC.2019.2898437 Cheng X, Liu D, Yan S, Shi W, Zhao Y (2019b) Channel estimation and equalization based on deep BLSTM for FBMCOQAM systems. 2019 IEEE International Conference on Communications, 16. DOI: 10.1109/ICC.2019.8761647 ChenHu K, EstradaJiménez JC, García MJFG, Armada AG (2018) Continuous and burst pilot sequences for channel estimation in FBMCOQAM. IEEE Transactions on Vehicular Technology 67(10): 97119720. DOI: 10.1109/TVT.2018.2861995 Eggen TH, Baggeroer AB, Preisig JC (2000) Communication over Doppler spread channels. Part Ⅰ: Channel and receiver presentation. IEEE Journal of Oceanic Engineering 25(1): 6271. DOI: 10.1109/48.820737 Fuhrwerk M, Moghaddamnia S, Peissig J (2017) Scattered pilotbased channel estimation for channel adaptive FBMCOQAM systems. IEEE Transactions on Wireless Communications 16(3): 16871702. DOI: 10.1109/TWC.2017.2651806 He X, Zhao Z, Zhang H (2012) A pilotaided channel estimation method for FBMC/OQAM communications system. 2012 International Symposium on Communications and Information Technologies (ISCIT), 175180. DOI: 10.1109/ISCIT.2012.6380885 He Z, Zhou L, Chen Y, Ling X (2019a) Nonlinear complex support vector regression for fading channel estimation in FBMCOQAM system. IEEE Wireless Communications Letters 8(3): 753756. DOI: 10.1109/LWC.2018.2890647 He Z, Zhou L, Yang Y, Chen Y, Ling X, Liu C (2019b) Compressive sensingbased channel estimation for FBMCOQAM system under doubly selective channels. IEEE Access 7: 5115051158. DOI: 10.1109/ACCESS.2019.2898896 Javaudin JP, Lacroix D, Rouxel A (2003) Pilotaided channel estimation for OFDM/OQAM. The 57th IEEE Semiannual Vehicular Technology Conference, 15811585. DOI: 10.1109/VETECS.2003.1207088 Kalil M, Banat MM, Bader F (2013) Three dimensional pilot aided channel estimation for filter bank multicarrier MIMO systems with spatial channel correlation. The 8th Jordanian International Electrical and Electronic Engineering Conference (JIEEEC'2013), Amman, Jordan Kofidis E, Katselis D, Rontogiannis A, Theodoridis S (2013) Preamblebased channel estimation in OFDM/OQAM systems: A review. Signal Processing 93(7): 20382054. DOI: 10.1016/j.sigpro.2013.01.013 Lélé C, Javaudin JP, Legouable R, Skrzypczak A, Siohan P (2008) Channel estimation methods for preamblebased OFDM/OQAM modulations. European Transactions on Telecommunications 19(7): 741750. DOI: 10.1002/ett.1332 Li W (2006) Estimation and tracking of rapidly timevarying broadband acoustic communication channels. PhD thesis, Massachusetts Institute of Technology, Cambridge, 6063 Li W, Preisig JC (2007) Estimation of rapidly timevarying sparse channels. IEEE Journal of Oceanic Engineering 32(4): 927939. DOI: 10.1109/JOE.2007.906409 Lu X, Jiang Y, Wei Y, Tu X, Qu F (2022) A Latticereductionaided sphere decoder for underwater acoustic FBMC/OQAM communications. IEEE Wireless Communications Letters 12(3): 466470. DOI: 10.1109/LWC.2022.3230857 Morelli M, Kuo CCJ, Pun MO (2007) Synchronization techniques for orthogonal frequency division multiple access (OFDMA): A tutorial review. Proceedings of the IEEE 95(7): 13941427. DOI: 10.1109/JPROC.2007.897979 Nam H, Choi M, Han S, Kim C, Choi S, Hong D (2016) A new filterbank multicarrier system with two prototype filters for QAM symbols transmission and reception. IEEE Transactions on Wireless Communications 15(9): 59986009. DOI: 10.1109/TWC.2016.2575839 Nissel R, Rupp M (2018) Pruned DFTspread FBMC: Low PAPR, low latency, high spectral efficiency. IEEE Transactions on Communications 66(10): 48114825. DOI: 10.1109/TCOMM.2018.2837130 O'shea T, Hoydis J (2017) An introduction to deep learning for the physical layer. IEEE Transactions on Cognitive Communications and Networking 3(4): 563575. DOI: 10.1109/TCCN.2017.2758370 PérezNeira AI, Caus M, Zakaria R, Le Ruyet D, Kofidis E, Haardt M, Mestre X, Cheng Y (2016) MIMO signal processing in offsetQAM based filter bank multicarrier systems. IEEE Transactions on Signal Processing 64(21): 57335762. DOI: 10.1109/TSP.2016.2580535 Pollet T, Van Bladel M, Moeneclaey M (1995) BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise. IEEE Transactions on Communications 43(2/3/4): 191193. DOI: 10.1109/26.380034 Qarabaqi P, Stojanovic M (2013) Statistical characterization and computationally efficient modeling of a class of underwater acoustic communication channels. IEEE Journal of Oceanic Engineering 38(4): 701717. DOI: 10.1109/JOE.2013.2278787 SaeediSourck H, Wu Y, Bergmans JW, Sadri S, FarhangBoroujeny B (2011) Complexity and performance comparison of filter bank multicarrier and OFDM in uplink of multicarrier multiple access networks. IEEE Transactions on Signal Processing 59(4): 19071912. DOI: 10.1109/TSP.2010.2104148 Singh P, Mishra HB, Jagannatham AK, Vasudevan K (2019) Semiblind, training, and dataaided channel estimation schemes for MIMOFBMCOQAM systems. IEEE Transactions on Signal Processing 67(18): 46684682. DOI: 10.1109/TSP.2019.2925607 Singh P, Sharma E, Vasudevan K, Budhiraja R (2018) CFO and channel estimation for frequency selective MIMOFBMC/OQAM systems. IEEE Wireless Communications Letters 7(5): 844847. DOI: 10.1109/LWC.2018.2830777 Singh P, Srivastava S, Mishra A, Jagannatham AK, Hanzo L (2022) Sparse Bayesian learning aided estimation of doublyselective MIMO channels for filter bank multicarrier systems. IEEE Transactions on Communications 70(6): 42364249. DOI: 10.1109/TCOMM.2022.3171815 Siohan P, Siclet C, Lacaille N (2002) Analysis and design of OFDM/OQAM systems based on filterbank theory. IEEE Transactions on Signal Processing 50(5): 11701183. DOI: 10.1109/78.995073 Srivastava S, Singh P, Jagannatham AK, Karandikar A, Hanzo L (2020) Bayesian learningbased doublyselective sparse channel estimation for millimeter wave hybrid MIMOFBMCOQAM systems. IEEE Transactions on Communications 69(1): 529543. DOI: 10.1109/TCOMM.2020.3029568 Vapnik V, Golowich S, Smola A (1996) Support vector method for function approximation, regression estimation and signal processing. Advances in Neural Information Processing Systems 9: 281287. DOI: 10.5555/2998981.2999021 Wang H (2018) Sparse channel estimation for MIMOFBMC/OQAM wireless communications in smart city applications. IEEE Access 6: 6066660672. DOI: 10.1109/ACCESS.2018.2875245 Wang H, Li X, Jhaveri RH, Gadekallu, TR, Zhu M, Ahanger TA, Khowaja SA (2021) Sparse Bayesian learning based channel estimation in FBMC/OQAM industrial IoT networks. Computer Communications 176: 4045. DOI: 10.1016/j.comcom.2021.05.020 Wang H, Xu L, Wang X, Taheri S (2018) Preamble design with interference cancellation for channel estimation in MIMOFBMC/OQAM systems. IEEE Access 6: 4407244081. DOI: 10.1109/ACCESS.2018.2864221 Wang H, Xu L, Yan Z, Gulliver TA (2020) Lowcomplexity MIMOFBMC sparse channel parameter estimation for industrial big data communications. IEEE Transactions on Industrial Informatics 17(5): 34223430. DOI: 10.1109/TII.2020.2995598 Yang TC (2012) Properties of underwater acoustic communication channels in shallow water. The Journal of the Acoustical Society of America 131(1): 129145. DOI: 10.1121/1.3664053