Abstract
Applying the orthogonal matching pursuit (OMP) to estimate the underwater acoustic (UWA) orthogonal frequency division multiplexing (OFDM) channels is attractive because of its high estimation accuracy and low computational cost. However, most existing OMP-based algorithms suffer the limited estimation accuracy in impulsive noise (IN) cases. Through the studies can be found, only part of channels’ estimation is affected due to the random IN which appears transient and intermittent in time and frequency. Based on this observation, joint time-frequency OMP (JTF-OMP) method is proposed, where the estimation of the affected channels benefits adaptively from that of adjacent channels in time or frequency. It is well known that preliminary Doppler scale estimation is key to the subsequent OMP algorithm, which is difficult to deal with due to the IN. To solve this problem, an adaptive Doppler scale estimation (ADSE) method is proposed. It involves generating two shorter identical cyclic prefixes (CPs) for each OFDM symbol, placed before two adjacent OFDM symbols. The repetition pattern can adaptively defend the IN which appears randomly and shortly in time. Simulation results show that the proposed algorithms integrating JTF-OMP with ADSE can achieve much higher estimation accuracy and better system reliability than the OMP in the IN environment.
1. Introduction
The increasing marine service requirements urgently need to enhance the data rate of underwater acoustic (UWA) communication [1, 2]. However, the UWA channels at the physical layer are facing many challenges for high speed communication, such as long delay spread, significant Doppler scale, and severe channel attenuation [3]. Orthogonal frequency division multiplexing (OFDM) is recognized as an effective technique to meet the challenges, due to its robustness against frequency selectivity and interference [4]. To improve UWA OFDM system, instantaneous channel state information (CSI) through channel estimation is required, which is the basis of adaptive resource allocation at the transmitter and stable signal detection at the receiver [5].
By modeling and estimating the UWA channels as a basis expansion model (BEM), the time-varying nature of the channels can be modeled arbitrarily well [6, 7]. However, the BEM method alone will become more inefficient with much longer delay spread or larger Doppler scale of the UWA channels. The effective way to solve this problem is to exploit the fact that UWA channels are naturally sparse, i.e., most channel energy is concentrated in a few delay and/or Doppler values [8]. In recent years, to make full use of the sparsity of the UWA channels, compressed sensing (CS) algorithms have been applied to the UWA channel estimation, such as basis pursuit (BP), matching pursuit (MP), and orthogonal matching pursuit (OMP) [9, 10]. Among them, OMP algorithm is the most representative one, which can achieve high estimation accuracy with low computational complexity [11]. To fully exploit its potential in the UWA channel estimation, lots of related research is in progress. By adaptive iteration and gradient descent with sparse signal, an adaptive gradient descent OMP algorithm is proposed in [12], where the process of signal reconstruction is transformed to the unconstrained optimization problem. A look-ahead backtracking OMP-based sparse channel estimation technique is developed in [13], where the Doppler scale is estimated and compensated by PN sequence in time domain. A novel and computationally efficient channel estimation algorithm is proposed in [14]; the OMP algorithm is combined with the maximum a posteriori probability (MAP) technique to estimate the UWA channel gains. A greedy algorithm technique of the OMP implemented with the BEM is proposed in [15], where no prior information about channel statistics is required and the spectral leakage can be avoided. To make the expansion coefficients uncorrelated with the increasement of the OFDM frame length, time-varying complex-valued channel path gains are estimated using BEM based on the orthonormal discrete Legendre polynomials [16]. To determine an appropriate iteration termination condition, [17] derives an adaptive OMP (A-OMP) algorithm based on the power ratio of the residual vector, where the recovery sparsity of the UWA channels is exploited.
Most of the aforementioned works assume that impulsive noise (IN) can be ignored in the UWA system; hence, they suffer the limited estimation accuracy in the IN cases. Unfortunately, due to human activities as well as natural sources, IN is frequently encountered, and it cannot be completely removed at the receiver by using blanking or clipping techniques [18]. Accordingly, joint channel and IN estimation methods have attract attention. The proposed method in [19] adopts the generalized approximate message passing (GAMP) to jointly estimate the channel and the IN, which assumes that a priori statistics about the IN can be acquired. Two pilot-subcarrier-based joint algorithms are proposed in [20] to improve the performance of channel estimation and IN mitigation, but the accurate positions of the IN are required in the received OFDM signal. A recursive CS algorithm incorporating Kalman filter and Smoother (KFS) is developed in [21], which adopts the sparse Bayesian learning (SBL) to construct the CS framework. However, these methods either require extra bandwidth for null subcarriers or use the statistical feature of the IN as the basis for instantaneous estimation.
In fact, the IN in the UWA system is very complex and difficult to be classified into a specific statistical characteristic. Furthermore, the statistical value of the IN may be quite different from the instantaneous value. Through the studies can be found, only part of the UWA channels’ estimation is affected due to the random IN which appears transient and intermittent in time and frequency. According to literature [17], the IN will lead to huge channel gain variations on the elements of the estimated channels, and the iteration should be terminated early in the process of the OMP algorithm. In view of these, joint time-frequency OMP (JTF-OMP) method is proposed in this paper; the pilot subcarriers are divided into groups at the same interval and estimated separately. If the iteration corresponding to a group is terminated early due to the IN, meaning that this part of channels’ estimation is not accuracy, then the estimation is adaptively weighted by the estimation of adjacent channels in time or frequency.
In addition, Doppler scale is also difficult to be estimated due to the IN, which is key to the subsequent OMP algorithm. Based on the guard interval insertion way, OFDM schemes can be categorized into three types: cyclic-prefix (CP), zero-padding (ZP), and time domain synchronization (TDS) OFDM [22–24]. In this paper, a CP-based adaptive Doppler scale estimation (ADSE) method is proposed; two shorter CPs for each OFDM symbol are generated and placed before two adjacent OFDM symbols, and a bank of parallel and self-correlator matched branches as [25] is used. Doppler scale estimation is performed by searching the branch with the largest self-correlator value, and two CPs can ensure the effective search in the IN cases. The main contributions can be summarized as follows: (i)The proposed JTF-OMP method makes full use of the transient and intermittent feature of the IN occurring in time and frequency. The pilot subcarriers are grouped at the same interval and estimated separately; hence, some groups are not affected by the IN which is frequency selective, while the estimation of the affected channels can benefit adaptively from that of adjacent channels in time or frequency(ii)The proposed ADSE method takes advantage of the randomness and transience of the IN occurring in time. On the premise of maintaining the same spectral and energy efficiency as traditional single CP structure, two shorter CPs with reasonable distance can ensure the self-correlation of the received signal in the IN cases, which is key to the Doppler scale estimation(iii)The proposed algorithms do not require null subcarriers for channel estimation; hence, channel resource is saved. In addition, the proposed JTF-OMP method and ADSE method can be used in combination, alone, or embedded into other existing algorithms; they have good compatibility
The rest of this paper is structured as follows. The system model is discussed in Section 2, and the proposed ADSE method is presented in Section 3. Then, the JTF-OMP algorithm is deduced and realized in Section 4. The numerical results and analysis are illustrated in Section 5. Finally, the conclusion is drawn in Section 6.
Notation. The following notations will be used throughout this paper: Bold lowercase letters and uppercase letters denote the vectors and matrices, respectively. , , and denote the transpose, conjugate transpose, and inverse of , respectively.
2. System Model
In this paper, a novel OFDM transmission structure is proposed matching to the ADSE method. As shown in Figure 1, it involves generating two identical CPs for each OFDM symbol, placed before two adjacent OFDM symbols. To maintain the same spectral and energy efficiency as traditional single CP structure, the CP length is set to half of the traditional.
Let denote the OFDM symbol duration and the length of CP. Let denote the number of subcarriers, and define the system bandwidth. The th subcarrier is at frequency where is the carrier center frequency and is the subcarrier set.
Let denote the symbol transmitted on the th subcarrier. The transmitted signal in baseband can be given as where denotes the segmented rectangular windows of total length as
where .
Then, the corresponding passband signal can be written as
The channel impulse response for the time-varying UWA channel is often defined as
where and are the gain and time-varying delay of the th path, respectively.
As in [25], the UWA OFDM symbol duration is around 100 ms to 200 ms, while the channel coherence time is usually on the order of seconds. Hence, for the duration shown in formula (3), the path gain can be assumed constant , and the time-varying delay can be reasonably approximated by a Doppler rate as
where and are the initial delay and Doppler scale of the th path, respectively.
Hence, the channel model in (5) can be simplified to
When the passband signal in (4) passes through the UWA channel described in (7), the received passband signal is
where and is the composite noise at passband.
By converting the passband signal to baseband , satisfying , the following can be given
where , , and are the composite noise, background noise, and IN at baseband, respectively.
3. Adaptive Preliminary Doppler Scale Estimation
According to literature [26], all the paths have approximately the same Doppler scale, i.e., , and the small difference can be treated as additive noise. Then, the received baseband signal in (9) can be simplified as
for corresponding to the rectangular window , where .
Because two identical CPs are generated for each OFDM symbol as shown in Figure 1, some of the transmitted signal has a repetition pattern as
where .
As can be verified from (10), when the IN can be ignored, i.e., , the received baseband signal still retains such a repetition pattern even after the UWA multipath propagation as
for .
In fact, the IN cannot be completely removed at the receiver. But it has the randomness and transience feature occurring in time. As in formula (12), the received baseband signal retains the repetition pattern in two different time periods, because two identical CPs are designed in this paper, i.e., includes and ; the distance is . There is a great possibility that one of the time intervals is not affected by the IN. The problem is then to detect the repetition period to find the Doppler scale.
An ADSE method is proposed in this paper to find the Doppler scale based on the correlation metrics from the bank of self-correlators as shown in [25]. Assuming the oversampling time is , where the oversampling factor is an integer, then the oversampled baseband signal is . The self-correlation metric can be given as
where , , and are the window size, delay, and CP size, respectively.
As can be seen from (12), Doppler scale changes the period to ; the optimal estimated window size shall be , where is the estimated Doppler scale; hence, it can be estimated as
The CP size can be obtained by satisfying , where means the integer rounding. The delay is used to find the starting position of the two CPs.
The proposed ADSE algorithm is shown in Algorithm 1.
When the correlation metric of any branch exceeds the threshold in step 8, it means that a repetition pattern is detected. However, the repetition pattern occurs in two cases as shown in Figure 2. In case A, the correlation metric shown by marks 1 and 2 in Figure 2 is calculated at step 10. In case B, the correlation metric shown by marks 3 and 4 is calculated at step 11. Note that the positions and may be negative, which are determined by the reference position. By obtaining the maximum value in step 12, the influence of the IN can be greatly weakened because it rarely appears in all the positions shown in Figure 2 at the same time, and the Doppler Scale can be estimated more accurately.
4. Joint Time-Frequency OMP Channel Estimation Algorithm
4.1. The Processing of the Received Signal
After the preliminary Doppler scale estimation, the received bandpass signal in (8) needs to be resampled according to the estimated Doppler Scale to mitigate the Doppler effect, leading to
where and is the composite noise.
By converting the resampled passband signal to the baseband , such that , the following can be obtained
where is the residual Doppler scale and , , and are the composite noise, background noise, and IN at baseband, respectively.
Performing CP-OFDM demodulation, the output on the th subchannel can be calculated as
where is the transfer coefficient and is the composite noise on the th subchannel.
Collecting the into the transfer matrix , and the , , and into vectors , , and , respectively. Then, formula (17) can be denoted as
Due to the delay spread and Doppler scale of the UWA channel, OFDM subcarriers lose their orthogonality, where coefficient denotes the contribution of the th subcarrier to the th FFT output. According to (16) and (17), the following equation is satisfied
where the value of the rectangular windows is one in the integral expression.
Set and , leading to
where .
Defining the mixing matrix with th element , and the diagonal matrix with th element , then the transfer matrix can be expressed as
4.2. Sparse Channel Estimation
According to the sparsity of the UWA channel, applying compressed sensing theory to channel estimation can reduce the complexity of the algorithm. As in [27], an overcomplete dictionary of the delay and Doppler scale parameters is established in this paper, which can be expressed as
where the delay-resolution index is an integer; and are the resolutions of the delay and Doppler scale, respectively; and is the maximum value of . A number of delay steps and Doppler steps are and , respectively. Hence, the gain of the th path with delay and Doppler scale can be denoted as ; the matrix can be rewritten as
Note that the channel model in (24) is not a BEM itself. However, considering that the UWA signal is a narrowband signal, the model reduces to the complex exponential BEM as shown in [27].
Then, formula (24) can be expressed equivalently as the following linear model
where is the original sample set composed of the delay and residual Doppler scale and is the set of the corresponding channel gains.
The UWA channels are naturally sparse, namely, most channel energy is concentrated in a few delay and/or Doppler values. Hence, in (27) is sparse, and only some elements are not zero, which can be estimated by substituting the known pilot signal for in (25). In this paper, OMP-based algorithm is considered, which is the most representative one among the CS algorithms. Since the composite noise in (25) includes the IN which appears transient and intermittent in time and frequency, part of the channels’ estimation will be affected.
To resolve this problem, the pilot subcarriers are divided into groups at the same interval and estimated separately. Let denote the number of pilot subcarriers, and denotes the number of groups on the premise that is an integer. Then, formulas (25) and (26) can be rewritten as
where , , , and are the received signal, part of the transfer matrix, part of the noise, and transmitted pilots corresponding to the th group pilots, respectively. is the selection matrix that selects the elements in that correspond to nonzero entries in the th group pilots.
The proposed OMP-based algorithm is shown in Algorithm 2. It is named JTF-OMP because the estimation of the affected channels can benefit adaptively from that of adjacent channels in time or frequency.
| ||||||||||||||||||||||||||||||||||||||||||||||
When is not satisfied in step 11, it means that the residual vector is small enough and the channel estimation is accurate. When is not satisfied, it indicates that the remaining estimated taps are resulted by the IN according to literature [17], where is decided by the power of transmitted signals, the power of the IN, and the number of pilots. Then, this part of channel estimation needs to be multiplied by a coefficient in step 19. The overall channel estimation is obtained by the weighted addition of each in step 21. Hence, the estimation of the affected channels benefits adaptively from that of adjacent channels in frequency. Finally, the judgement is done in step 22. If the partial estimation is interrupted many times due to the IN, meaning that the IN is serious, then the channel estimation is corrected adaptively by the value in the previous OFDM-period. In a word, the proposed JTF-OMP algorithm makes full use of the transient and intermittent feature of the IN occurring in time and frequency.
5. Simulation and Analysis
In this section, the performance of the proposed algorithms is presented and compared through Monte Carlo simulations, in terms of the normalized mean square error (NMSE) and the bit-error-rate (BER). The NMSE is defined as
where and denote the real and the estimated channel impulse response, respectively.
The simulation parameters are depicted as follows. The CP-OFDM specifications are consistent with [28] except for the CP structure. The carrier center frequency, bandwidth, number of subcarriers, number of pilots, symbol duration, subcarrier spacing, and CP length are summarized in Table 1. The pilot subcarriers are distributed on every fourth subcarrier. There are no null subcarriers; the remaining 768 subcarriers are used for data symbols. All of the symbols are encoded with a rate 1/2 nonbinary LDPC code and modulated by QPSK constellations.
The number of discrete paths is 15, and the delay difference between adjacent paths follows an exponential distribution with the average value . To reflect different scenarios over distances, is within the range of ms. The amplitude follows a Rayleigh distribution with the average power decreasing exponentially with the delay. The Doppler scale is Gaussian distributed with the mean and standard deviation . The resolution of the Doppler scale in the overcomplete dictionary is set to be 0.01 m/s.
The composite noise in (25) is modeled as two-component Gaussian mixture (GM) model, which is widely used to study the IN environment [20]. The probability density function of the noise on the th subchannel is modelled as
where is the complex Gaussian distribution function and is the IN occurrence probability (INOP). and are the variances of the background noise and IN, respectively. In the following simulations, is within the range of ; accordingly, about IN samples are generated within one OFDM block. In addition, and are varied to generate different signal-to-noise ratio (SNR).
In Figure 3, the NMSE performance of the proposed algorithms versus SNR is evaluated under different and . As can be seen, when the is smaller, the smaller has little loss of performance. However, when the gets larger enough, the smaller makes the performance worse obviously. The proposed ADSE method involves generating two shorter identical CPs for each OFDM symbol, as shown in Figure 1. In order to maintain the same spectral and energy efficiency as traditional single CP structure, the CP length is set to be 12.3 ms, i.e., half of the traditional value which is 24.6 ms. Herein, the length of the guard interval has not been changed. The difference is that only half of the OFDM tail data has been copied to the CP. Therefore, the proposed two CPs structure can also well mitigate the effect of the intersymbol interference (ISI) and intercarrier interference (ICI). Meanwhile, high estimation accuracy is maintained when the is not large enough, corresponding to the small or medium distance UWA environment.
Figure 4 shows the BER performance versus INOP under different number of groups , where and are kept at the values so that the SNR is 16 dB when the INOP is 0.02. When the INOP is larger, the BER performance is improved with the increase of . However, when increases further, the BER performance deteriorates gradually. It is because that appropriate grouping can make good use of the estimation of the channels which are not affected by the random IN. But, too many groups will lead to too few pilot subcarriers, which degrades the channel estimation. In fact, needs to be adjusted adaptively according to the actual number of the pilot subcarriers. In the current simulation configuration, setting to be or is a relatively good choice.
Figure 5 shows the BER performance versus SNR under different standard deviation . The notation in the figure means that the Doppler scale is estimated by the proposed ADSE algorithm. The notation no PDSE means that the Doppler scale is not preliminarily estimated, i.e., in (15) and in (16). As can be seen, BER is decreased significantly through the ADSE algorithm when the SNR is higher, and the improvement is more obvious with the increase of Doppler scale. Because the Doppler effect is mitigated through the ADSE algorithm, i.e., is much closer to 0 than ; accordingly, the overcomplete dictionary of Doppler scale in (23) gets much smaller. This shows that the ADSE algorithm can improve the accuracy and convergence speed of the JTF-OMP algorithm.
In Figure 6, the BER performance under different and is evaluated. It can be observed that the performance is improved with the increase of , but the improvement saturates quickly. It is because that the increase of the dictionary size of the delay can better reflect the discrete nature of the channel in continuous time. In addition, the oversampling factor increases the performance slightly. Clearly, the Doppler scale can be better estimated through finer self-correlation metric in (13). However, larger and increase the computational complexity. To achieve better comprehensive performance, and have both adaptively set to be in the current simulation configuration.
In Figure 7, the NMSE performance of the proposed algorithms is compared with the A-OMP [17], the OMP [14], and the least-square (LS) algorithm, respectively. The notation in the figure means that the Doppler scale is preliminarily estimated by the DSE algorithm [25]. As can be observed, the CS-based algorithms significantly outperform the LS-based algorithm. This can be explained by the fact that the former better exploit the sparsity of the UWA channels. Meanwhile, the proposed algorithms integrating JTF-OMP with ADSE can achieve NMSE performance advantages under all SNR cases, because the interference problem of the IN has been better considered and dealt with.
Figures 8 and 9 show the BER performance of these algorithms. As can be seen, the proposed algorithms always maintain the best BER performance, while the OMP algorithm with no PDSE has always been the worst. There are performance gaps between the JTF-OMP and the two others and gaps between the ADSE and the two others. The results also show that the JTF-OMP and the ADSE can be adaptively embedded into other existing algorithms, and some performance gain can be achieved.
In Figures 10 and 11, the BER performance of these algorithms versus INOP is evaluated, where the values of and are consistent with Figure 4. It can be found that the proposed algorithms integrating JTF-OMP with ADSE remain the best BER performance with the increase of INOP. The performance gaps between these algorithms are similar to Figures 8 and 9. Obvious, the proposed algorithms can achieve much higher estimation accuracy and better system reliability than others in the IN environment. It can be explained that the JTF-OMP and the ADSE have made full use of the transient and intermittent feature of the IN occurring in time and frequency, and the impact of the IN has been greatly mitigated.
6. Conclusions
Considering the UWA communication in impulsive noise environment, adaptive algorithms integrating JTF-OMP with ADSE are proposed in this paper. In the ADSE method, on the premise of maintaining the same spectral and energy efficiency as traditional single CP structure, two shorter identical CPs for each OFDM symbol are generated and placed before two adjacent OFDM symbols. By searching the branch with the largest self-correlator value related to the two CPs, the Doppler scale can be estimated effectively in the IN cases. In the JTF-OMP algorithm, the pilot subcarriers are divided into groups at the same interval, and the estimation is adaptively weighted by the estimation of adjacent channels in time or frequency according to the severity of IN. The proposed algorithms do not require null subcarriers for channel estimation; hence, channel resource is saved. Simulation results verify the validity of the proposed algorithms; much higher estimation accuracy and better system reliability can be achieved in the IN environment. Our future research work will focus on the cross-layer channel estimation techniques via acknowledge/not-acknowledge (ACK/NAK) feedback for UWA communication in the IN environment.
Data Availability
The MATLAB Source Code data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Zhejiang Provincial Natural Science Foundation of China under Grant LY18F010001, the Ningbo Natural Science Foundation of China under Grant 2019A610076, the Humanities and Social Science Research Project of the Ministry of Education of China under Grant 20YJAZH130, the Public Technology Research Project of Zhejiang under Grant LGF20F020004, and the Natural Science Foundation of China under Grant 61571250.