Remote underwater acoustic communication non-sparse channel estimation method for multi-carrier system

Abstract

The invention discloses a remote underwater acoustic communication non-sparse channel estimation method for a multi-carrier system, and relates to the technical field of remote underwater acoustic communication. For a non-sparse structure of a remote underwater acoustic communication channel, a maximum likelihood estimation method is provided to estimate dense multipath of the non-sparse channel by approximating a received signal of a multi-carrier system as a Gaussian signal. For the problem of channel indistinguishability caused by dense multipath, a received signal is modeled into Gaussian distribution based on a central limit theorem, and a novel channel estimation algorithm Novel ML is deduced. In a multi-carrier orthogonal frequency division multiplexing (OFDM) system, a received signal obeys normal distribution after modeling, and estimation of the impulse response of a dense multipath channel is realized by constructing a likelihood function and optimizing and solving the likelihood function. Results of numerical simulation and remote offshore experiments show that compared with a traditional channel estimation algorithm, the method provided by the invention can effectively perform accurate estimation of a non-sparse channel.

Description

Technical Field

[0001] The present invention relates to the technical field of long-range underwater acoustic communication, and in particular to a non-sparse channel estimation method for long-range underwater acoustic communication in a multi-carrier system. Background Art

[0002] With the development of underwater acoustic communication technology, applications in seabed mapping, ocean current monitoring, military search and investigation missions have become a reality. Today, these long-range monitoring missions are primarily accomplished by advanced equipment such as autonomous underwater vehicles, gliders, and tactical submarines. Given that these devices often need to travel long distances, powerful long-range underwater acoustic communication technology plays a vital supporting role in their control. For example, by using one-way signal transmission to a remote node, the current task of the machine can be effectively changed, thereby achieving flexible control of underwater equipment and ensuring that various underwater missions are completed efficiently and according to the intended objectives (J. Lee, J. An, H.-i. Ra, and K. Kim, “Long-range acoustic communication using differential chirp spread spectrum,” Applied Sciences, vol. 10, no. 24, p. 8835, 2020).

[0003] Because the absorption coefficient in water increases with carrier frequency, low-frequency transducers are often used for long-range underwater acoustic communications to reduce acoustic absorption. However, compared to short-range underwater acoustic communications, the channel impulse response for long-range underwater acoustic communications is more complex. Specifically, the channel impulse response for short-range underwater acoustic communications is generally sparse, while the channel impulse response for long-range underwater acoustic communications is non-sparse, and the delay spread can sometimes even exceed 1 second. Figure 1 Given a long-range underwater acoustic communication channel observed in the Mediterranean Sea, the channel has a delay spread of approximately 300ms and exhibits dense multipath and non-sparse characteristics. Therefore, to overcome the long delay spread and complex channel structure, a long-range underwater acoustic non-sparse channel estimation method for multi-carrier systems is proposed.

[0004] Researchers have proposed a variety of technical solutions for long-range underwater acoustic communications. J. Lee (J. Lee, J. An, H. -i. Ra, and K. Kim, "Long-range acoustic communication using differential chirp spread spectrum," Applied Sciences, vol. 10, no. 24, p. 8835, 2020; Z. Liu, K. Yoo, T C Yang, SEC Ho, H C Song, and DE Ensberg, "Long-range double-differentially coded spread-spectrum acoustic communications with a towed array,) proposed chirp spread spectrum modulation and double differentially coded spread spectrum systems for long-range underwater acoustic communications. Combined with beamforming, H. Song (H. Song, S. Cho, T. Kang, W. Hodgkiss, and J. Preston, "Long-range acoustic communication in deepwater using a towed array," The Journal of the Acoustical Society of America, vol. 129, no. 3, pp. EL71–EL75, 2011) proposed a quadrature phase-shift keying (QPSK) modulation method and achieved good results in long-range underwater acoustic communication experiments. However, this modulation is only suitable for specific channel types. Therefore, based on environmental information and through training for various channel types, J. Huang (J. Huang and R. Diamant, “Adaptive modulation for long-range underwater acoustic communication,” IEEE Transactions on Wireless Communications, vol. 19, no. 10, pp. 6844–6857, 2020) proposed a technology that pre-sets the modulation scheme based on the prediction of the long-range underwater acoustic channel.

[0005] For long-range underwater acoustic communications, researchers have proposed a variety of receivers. Zhang, S (Zhang, S. Xiao, H. Cui, D. Gao, and D. Sun, “Modal dispersion compensation receiver for the long range shallow water acoustic communications,” The Journal of the Acoustical Society of America, vol. 145, no. 6, pp. EL483–EL487, 2019) proposed a modal dispersion compensation receiver to compensate for the waveform expansion of high-order modes. To improve spectral efficiency, T. Shimura (T. Shimura, Y. Watanabe, H. Ochi, and H. Song, “Long-range time reversal communication in deep water: Experimental results,” The Journal of the Acoustical Society of America, vol. 132, no. 1, pp. EL49–EL53, 2012) designed a long-range underwater acoustic communication system using high-order constellations, passive time reversal, and a decision feedback equalizer (DFE). A. Zhao (A. Zhao, C. Zeng, J. Hui, L. Ma, and X. Bi, “Experimental demonstration of long-range underwater acoustic communication using a vertical sensor array,” Sensors, vol. 17, no. 7, p. 1516, 2017) proposed a time-reversal receiver for long-range underwater acoustic communication with a vertical sensor array. This receiver effectively reduces intersymbol interference (ISI) and lowers the bit error rate (BER). Based on this, a channel estimation algorithm was also applied to long-range underwater acoustic communication.For multi-carrier orthogonal frequency division modulation (OFDM) receivers in long-range underwater acoustic communications, T. Kang (T. Kang, H. Song, and W. Hodgkiss, “Long-range multi-carrier acoustic communication in deep water using a towed horizontal array,” The Journal of the Acoustical Society of America, vol. 131, no. 6, pp. 4665–4671, 2012) applied the orthogonal matching pursuit (OMP) channel estimator to the towed horizontal array signal. H. Song (H. Song and W. Hodgkiss, “Diversity combining for long-range acoustic communication in deep water,” The Journal of the Acoustical Society of America, vol. 132, no. 2, pp. EL68–EL73, 2012) proposed an OMP-based channel estimation equalizer for long-range underwater acoustic communication, combined with octal phase-shift keying modulation technology. The above solutions regard the long-range underwater acoustic communication channel as sparse and exploit the sparse characteristics of the channel through compressed sensing technology to improve the performance of channel estimation. However, the non-sparse nature of the long-range underwater acoustic channel will lead to mismatch and performance degradation of the compressed sensing algorithm.

[0006] For non-sparse channel estimation, researchers have developed adaptive channel estimation algorithms that adapt to the modulation scheme, transmission rate, channel coding scheme, and interference cancellation. These adaptive algorithms can maximize communication performance under varying system parameters. Existing solutions describe hybrid sparse / diffuse models for non-sparse channel estimation (N. Michelusi, U. Mitra, A.F. Molisch, and M. Zorzi, “Uwb sparse / diffuse channels, part i: Channel models and Bayesian estimators,” IEEE Transactions on Signal Processing, vol. 60, no. 10, pp. 5307–5319, 2012). By modeling both the non-sparse and sparse components of the channel, a hybrid sparse / non-sparse model is proposed to estimate underwater acoustic channels. However, this algorithm models the non-sparse component as following Rayleigh fading. For non-sparse, long-range underwater acoustic communication channels with dense, indistinguishable multipath, the dense multipath components may not follow Rayleigh fading. X. Geng (X. Geng and A. Zielinski, “An eigenpath underwater acoustic communication channel model,” in 'Challenges of Our Changing Global Environment'. Conference Proceedings. OCEANS'95 MTS / IEEE, vol. 2. IEEE, 1995, pp. 1189–1196) pointed out that the Ricean fading channel model is more suitable than the Rayleigh fading channel model. The aforementioned research approaches have failed to adequately model and estimate the non-sparse nature of long-range underwater acoustic channels. Summary of the Invention

[0007] The purpose of the present invention is to provide a non-sparse channel estimation method for long-range underwater acoustic communication for a multi-carrier system by approximating the received signal of the multi-carrier system as a Gaussian signal, aiming at the non-sparse structure of the long-range underwater acoustic communication channel, so as to solve the problem of inaccurate non-sparse channel estimation of long-range underwater acoustic communication in the prior art and realize effective and accurate estimation of non-sparse channels.

[0008] In order to achieve the above-mentioned object of the invention, the present invention provides the following technical solutions.

[0009] A non-sparse channel estimation method for long-range underwater acoustic communication in a multi-carrier system, comprising the following steps:

[0010] 1) Modeling the transmission and reception of multi-carrier signals: Baseband data is allocated to parallel subcarriers via OFDM modulation, generated into time-domain symbols via inverse Fourier transform, and transmitted after adding guard intervals. The receiving end removes the guard intervals and demodulates the received signal into a frequency-domain signal via Fourier transform. The received signal contains non-sparse underwater acoustic channel information composed of dense multipath and noise, and is modeled as a Gaussian distribution based on the central limit theorem, obeying a normal distribution.

[0011] 2) Constructing a maximum likelihood estimator based on a Gaussian model: This uses the Gaussian distribution of the received signal to establish a probability density function, which is converted into an optimization objective that includes the channel autocorrelation matrix and the noise variance. Through logarithmic transformation and extremization, the objective function to be minimized is formed to solve the channel impulse response that maximizes the probability of the received signal occurring.

[0012] 3) Solve the channel impulse response through iterative optimization: Take the partial derivative of the objective function to obtain the gradient vector, which is used to determine the optimization direction of the channel estimation. Use the gradient descent method to iteratively update the channel estimation value from the initial value until the difference between the estimates of two adjacent iterations is less than the preset threshold. The iteration stops when the convergence condition is met.

[0013] In step 1), the specific steps of modeling the transmission and channel reception of the multi-carrier signal may be:

[0014] 1.1) Mapping the baseband data onto the parallel subcarriers of the OFDM system, converting them into time-domain OFDM symbols through an inverse Fourier transform (IDFT), and adding a cyclic prefix (guard interval) to combat multipath interference, completing the modulation processing before signal transmission;

[0015] 1.2) The receiver removes the guard interval from the received time domain signal and demodulates it into the frequency domain using a Fourier transform (DFT) to obtain a received signal vector containing the channel impulse response and additive noise.

[0016] 1.3) The long-range underwater acoustic channel is defined as a linear time-invariant system whose characteristics are composed of dense multipath. Based on the central limit theorem, it is confirmed that the received signal approximately obeys a Gaussian distribution due to the multipath superposition effect.

[0017] In step 2), the specific steps of constructing the maximum likelihood estimator based on the Gaussian model may be:

[0018] 2.1) Based on the Gaussian distribution characteristics of the received signal, its probability density function is established. The channel impulse response, the received signal autocorrelation matrix, and the noise variance are incorporated into the model to reflect the correlation between the signal and the channel parameters.

[0019] 2.2) The probability density function is logarithmically transformed and extremized, transforming the maximum likelihood estimation into an optimization problem of minimizing the objective function. The objective function comprehensively considers the channel autocorrelation characteristics and the influence of noise to solve the channel parameters that maximize the probability of the received signal.

[0020] In step 3), the specific steps of solving the channel impulse response through iterative optimization may be:

[0021] 3.1) Taking partial derivatives of the objective function with respect to each channel component, calculating the gradient vector, and determining the optimization direction of the channel estimate. The gradient vector reflects the rate at which the objective function changes with the channel parameters.

[0022] 3.2) Using the gradient descent method, starting from the initial channel estimate, the channel estimate is iteratively updated in the reverse direction of the gradient according to a preset step size until the difference between the estimation results of two consecutive iterations is less than a preset threshold. The iteration is terminated when the convergence condition is met to obtain the final channel impulse response estimate.

[0023] Compared with the prior art, the present invention has the following beneficial effects:

[0024] 1. Existing technologies usually assume sparse channels or use simple linear models, which makes it difficult to effectively handle the non-sparse characteristics of dense multipath in long-range underwater acoustic communications. The present invention breaks through the traditional sparse channel assumption and models the received signal as a Gaussian distribution based on the central limit theorem, effectively characterizing the dense multipath superposition effect. The channel reconstruction error rate (CRER) is improved by more than 2dB compared with traditional algorithms, and the non-sparse channel modeling is more accurate and closer to the actual channel characteristics.

[0025] 2. Traditional MUSIC algorithms rely on a strict partitioning of the signal and noise subspaces, which can lead to resolution degradation when energy is concentrated in non-sparse channels. Hybrid estimation algorithms, however, face model mismatch risks when assuming Rayleigh fading for dense components. The proposed NovelML algorithm solves the maximum likelihood objective function through iterative optimization, fully leveraging the channel statistics under a Gaussian signal model. This algorithm reduces the bit error rate (BER) by 0.01-0.05 when the signal-to-noise ratio is low, outperforming MUSIC and hybrid estimation algorithms. Experiments show that the proposed algorithm significantly enhances anti-interference capabilities.

[0026] 3. The computational complexity of the present invention is lower. The single computational complexity of the iterative optimization algorithm of the present invention is linear order (O(MN)), and the total complexity is only related to the number of iterations. While ensuring the estimation accuracy, it avoids the computational overhead caused by complex matrix operations (such as the eigendecomposition of MUSIC) in traditional algorithms, and is more suitable for real-time underwater acoustic communication systems.

[0027] 4. The present invention has greater adaptability to practical scenarios, demonstrating stable performance in complex environments such as hard and soft bottom parameters and positive and negative sound velocity gradients. PE model simulations and field experiments in the Mediterranean Sea have demonstrated an average CRER gain of 3.32 dB at a distance of 100 km, under complex seabed parameters and sound velocity profiles. These experiments demonstrate that the present invention significantly outperforms existing technologies in engineering applications over long distances and in non-sparse channels.

[0028] 5. This invention establishes a Gaussian signal model for non-sparse channels for the first time, providing a new framework for long-range underwater acoustic communication channel estimation. Independent of channel sparsity priors, this method can be directly applied to long-range scenarios with dense and indistinguishable multipath paths. This expands the scope of underwater acoustic channel estimation technology and has significant implications for deep-sea exploration, long-distance underwater communications, and other fields. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 Schematic diagram of the non-sparse structure of the long-range underwater acoustic channel.

[0030] Figure 2 This is a system block diagram of the Novel ML algorithm for non-sparse long-range underwater acoustic channel estimation in the multi-carrier system of the present invention.

[0031] Figure 3 Figure 2 shows the performance results of different algorithms for non-sparse long-range underwater acoustic simulation channels at different signal-to-noise ratios, where (a) is CRER and (b) is BER.

[0032] Figure 4 CRER and BER results of different algorithms in non-sparse long-range underwater acoustic simulation channel for the case of signal-to-noise ratio SNR=5dB.

[0033] Figure 5 Diagram of the experimental transmitter and receiver setup.

[0034] Figure 6 Figure 2 shows the performance results of different algorithms for non-sparse long-range underwater acoustic communication channels at different signal-to-noise ratios, where (a) is CRER and (b) is BER.

[0035] Figure 7 The performance results of different algorithms for the case of SNR=5dB in a non-sparse long-range underwater acoustic channel, where (a) is CRER and (b) is BER. DETAILED DESCRIPTION

[0036] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the following embodiments will be further described in conjunction with the accompanying drawings. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0037] The present invention designs a maximum likelihood estimation method for long-range underwater acoustic non-sparse channels. For non-sparse long-range underwater acoustic channels with indistinguishable dense multipath, after a multi-carrier underwater acoustic communication signal passes through the channel, the received signal can be regarded as the sum of a large number of independent random variables. According to the central limit theorem, the received signal can be approximated as a Gaussian signal. Therefore, the received signal can be regarded as obeying a normal distribution (O. Bialer, D. Raphaeli, and AJ Weiss, "Efficient time of arrival estimation algorithm achieving maximum likelihood performance indense multipath," IEEE Transactions on signal processing, vol. 60, no. 3, pp. 1241–1252, 2011). Based on this model, a maximum likelihood estimator is derived to achieve estimation of non-sparse channels.

[0038] This paper proposes a novel maximum likelihood channel estimation method (Novel ML) for long-range underwater acoustic non-sparse channels with dense multipath. Secondly, numerical simulations are carried out based on the acoustic parabolic model (PE) to verify the effectiveness of the proposed algorithm. Finally, a long-range underwater acoustic communication experiment was carried out in the Mediterranean Sea. The results show that the proposed method is superior to traditional non-sparse channel estimation algorithms, such as the LS algorithm (X. Jiang, W.-J. Zeng, E. Cheng, and C.-R. Lin, “Multipath channel estimation using fast least-squares algorithm,” in 2011 Third International Conference on Communications and Mobile Computing. IEEE, 2011, pp. 433–436), the RLS algorithm (Ren, J. Li, G. Lu, and J. Ge, “Per-subcarrier rls adaptive channel estimation combined with channel equalization for fbmc / oqam systems,” IEEE Wireless Communications Letters, vol. 9, no. 7, pp. 1036–1040, 2020), and the hybrid channel estimation method (N. Michelusi, U. Mitra, A.F. Molisch, and M. Zorzi, “Uwb sparse / diffuse channels, part ii: Estimator analysis and practical channels,” IEEE transactions on signal processing, vol. 60, no. 10, pp. 5320–5333, 2012), and MUSIC algorithm (R. Berger, S. Zhou, J. C. Preisig, and P. Willett, “Sparse channel estimation for multicarrier underwater acoustic communication: From subspace methods to compressed sensing,” IEEE transactions on signal processing, vol. 58, no. 3, pp. 1708–1721, 2009), the method of the present invention shows better communication performance.

[0039] Notation: Bold capital letters denote matrices. Superscripts *, T, and H denote the conjugate operator, transpose operator, and conjugate transpose operator, respectively. The set ∪ denotes a parallel operation. x denotes the absolute value of vector x. x2 and x1 denote the l2 norm (Euclidean norm) and l1 norm of vector x, respectively. supp(a) denotes the number of nonzero elements of vector a, i.e., a0 = supp(a). Re{·} denotes the real part of a complex-valued expression. ⊙ denotes the Schur product. diag{x} denotes a diagonal matrix with the elements of vector x on the main diagonal. The M×M discrete Fourier transform (DFT) matrix is

[0040] The embodiment of the present invention includes the following steps:

[0041] 1. System modeling and signal generation

[0042] In this paper, we study the channel estimation of long-range underwater acoustic communication using multi-carrier modulation. In a multi-carrier orthogonal frequency division multiplexing (OFDM) system, baseband data is modulated on parallel subcarriers using an inverse discrete Fourier transform (IDFT). In the time domain, the transmitted OFDM symbol x(t) is:

[0043]

[0044] where χ(m) is the OFDM symbol at subcarrier m, 0≤m≤M-1, and X=[χ(0),χ(1),…,χ(M-1)] T , χ(m) is a zero-mean random variable, and |χ(m)| 2 =1.

[0045] Assuming that the channel between the transmitter and the receiver is linear time-invariant, the underwater acoustic channel can be expressed as:

[0046]

[0047] Where N is the number of multipaths in the channel, h(n) is the amplitude of the nth multipath, τ(n) is the delay associated with the nth multipath, and δ(t) is the unit impulse function. After removing the guard interval of the OFDM signal and performing DFT processing and demodulation, the received OFDM signal can be expressed as:

[0048] y(m)=χ(m)c(m)+w(m),0≤m≤M-1, (3)

[0049] Among them, w(m) means the mean is 0 and the variance is Additive Gaussian white noise, c(m) represents the Fourier transform of the channel at subcarrier m, that is:

[0050]

[0051] Therefore, we have:

[0052] H=[c(0),c(1),…,c(M-1)] T =F N h, (5)

[0053] where h=[h(0),···,h(N-1)] T is the channel impulse response, F N It is an M×N submatrix of the DFT matrix F. The matrix can be expressed as:

[0054] Y=X D F N h+W, (6)

[0055] in:

[0056] X D =diag{χ(0),χ(1),…,χ(M-1)}, (7a)

[0057] Y=[y(0),y(1),…,y(M-1)] T , (7b)

[0058] W=[w(0),w(1),…,w(M-1)] T . (7c)

[0059] For non-sparse long-range underwater acoustic communication channels, this paper proposes a new maximum likelihood channel estimator:

[0060]

[0061] Where f is the probability density function, Y∈C M×1 is the received long-range underwater acoustic communication signal, and h is the long-range underwater acoustic communication channel. Because the multipath of a non-sparse long-range underwater acoustic communication channel is dense, the received signal can be viewed as the sum of a large number of independent random variables. Therefore, according to the central limit theorem, the received signal can be modeled as a Gaussian signal, obeying a normal distribution.

[0062] 2. Non-sparse long-range underwater acoustic channel estimation:

[0063] Approximate the maximum likelihood estimate f(Y;h) as a Gaussian function of dense multipath:

[0064]

[0065] Among them, R Y =E[YY H ] is the autocorrelation matrix. Applying the natural logarithm and multiplying by -1, the estimation problem can be expressed as:

[0066]

[0067] Substitute equation (6) into the autocorrelation matrix R Y middle:

[0068]

[0069] Among them, I M ∈C M×M is the identity matrix, R h =E(hh H )∈C N×N is the autocorrelation matrix of h. Assuming that the components of h(n) are statistically independent, then:

[0070]

[0071] in,

[0072] In order to solve Equation (10), the optimization criterion is based on the differentiation of each channel component to obtain the optimal solution. Definition:

[0073]

[0074] And find the partial derivative of G with respect to h(k), k∈[0,N-1], and we get:

[0075]

[0076] The gradient descent iterative formula for minimizing ξ(h) is:

[0077] h i+1 =h i -μξ(h), (15)

[0078] Where μ is the step size and i is the number of iterations. Repeat the above steps until the following conditions are met:

[0079]

[0080] where ε is the noise factor.

[0081] Figure 2 The system block diagram of the Novel ML algorithm for non-sparse remote underwater acoustic channel estimation of the multi-carrier system of the present invention is given. The computational complexity of the proposed Novel ML algorithm is as follows: Therefore, the total complexity of the above algorithm is Where M is the received signal in the sample, N is the length of the long-range underwater acoustic communication channel, is the number of iterations.

[0082] 3. Numerical simulation

[0083] 1) Simulation settings

[0084] The numerical simulation results of the long-range underwater acoustic communication channel estimation method are as follows: The PE propagation model (D. Tappert, “The parabolic approximation method,” in Wave propagation and underwater acoustics. Springer, 1977, pp. 224–287) is used as the channel model. The ray model is a common acoustic channel model, but its accuracy is not as good as the PE model for low-frequency and long-distance transmission (R. J. Urick, Principles of Underwater Sound 3rd Edition. McGraw-Hill, 1983). In the simulation experiment, the positions of the receiver and transmitter were adjusted to generate a non-sparse channel. When the receiving and transmitting depths are lower than the first major water depth layer in the acoustic channel, the acoustic beam is more likely to scatter (J. Ogilvy, “Wave scattering from rough surfaces,” Reports on Progress in Physics, vol. 50, no. 12, p. 1553, 1987), resulting in a non-sparse channel structure. Based on the PE model, 5050 channels were generated with a transmission and reception distance of 100 km. The receivers and transmitters were uniformly and randomly placed between 30 m and 170 m. In each case, the depth difference between the receiver and transmitter was less than 10 m. Three seabed parameters were randomly selected from hard to soft bottoms: sound velocity, density, and attenuation rate, which were [1780 m / s, 2.10 g / cm 3 ,0.53dB / m], [1680m / s,1.90g / cm 3 ,0.46dB / m], [1620m / s,1.80g / cm 3 ,0.39dB / m] and [1580m / s,1.65g / cm 3 ,0.28dB / m]. At the same time, considering the sound velocity profiles with zero gradient and negative gradient, the sea surface sound velocity is uniformly and randomly distributed between 1500m / s and 1540m / s. The specific simulation parameters are shown in Table 1.

[0085] Table 1 Simulation parameters

[0086]

[0087] To validate the effectiveness of the proposed NovelML algorithm, we compared it with channel estimators such as LS, RLS, MUSIC, and hybrid channel estimation. The first two algorithms provide optimal mean square error (MSE) estimates for high signal-to-noise ratio signals, while MUSIC provides high-resolution channel estimation. The latter is a sparse / diffuse hybrid model suitable for complex channel reconstruction.

[0088] In this paper, the system simulation performance indicators are channel reconstruction error rate (CRER) and bit error rate (BER). The bit error rate is calculated from the received symbols, which are recovered by a linear minimum mean square error (MMSE) equalizer (JR Magnus and H. Neudecker, Matrix differential calculus with applications in statistics and econometrics. John Wiley & Sons, 2019). The channel reconstruction error rate CRER is defined as:

[0089]

[0090] Where h represents the actual underwater acoustic channel, represents the estimated underwater acoustic channel.

[0091] The derivation process of formula (14) is as follows:

[0092] definition:

[0093] F N =[f0,f1,…,f N-1 ],(18a)

[0094]

[0095]

[0096] In particular, |X(k)| 2 =1. Therefore, R can be derived from the following formula: Y The inverse matrix of

[0097]

[0098] Derivation of the first term of formula (14):

[0099] set up:

[0100]

[0101] R FhThe inverse derivative of h(k) is:

[0102]

[0103] in,

[0104]

[0105] Φ(k)=diag{Γ(k)(22b)because so:

[0106]

[0107] And formula (21) can be expressed as:

[0108]

[0109] Therefore, the first term of equation (14) can be expressed as:

[0110]

[0111] The derivation of the last term of formula (14) is as follows:

[0112] R Y The determinant of is:

[0113]

[0114] According to the Weinstein–Aronszajn identity (C. Pozrikidis, An introduction to grids, graphs, and networks. Oxford University Press, 2014), and It is equivalent to:

[0115]

[0116] so,

[0117]

[0118] For the last term in (14), we can get:

[0119]

[0120] Finally, adding the above two terms, the derivative of G is:

[0121]

[0122] Figure 3The CRER and BER results of different algorithms at different signal-to-noise ratios (SNRs) are presented in this paper, using a non-sparse long-range underwater acoustic simulation channel. The results show that the proposed algorithm improves CRER and BER by 2dB and 0.01, respectively, compared to traditional channel estimation algorithms. In particular, the performance of MUSIC, RLS, and LS is affected by the energy concentration of the non-sparse long-range underwater acoustic communication channel. Furthermore, due to the presence of dense and sparse components in the multipath of the channel, hybrid channel estimation algorithms offer better performance. However, hybrid channel estimation algorithms assume Rayleigh fading for the dense components, which may lead to model mismatch.

[0123] Figure 4 The CRER and BER results for different non-sparse, long-range underwater acoustic communication channels at a signal-to-noise ratio (SNR) of 5 dB are shown. It can be observed that the proposed Novel ML algorithm achieves better performance than traditional estimation methods, with an average CRER gain of 3.32 dB and an average BER gain of 0.05. The Novel ML algorithm models the received signal as a Gaussian signal, which can better exploit the dense multipath characteristics of the channel, effectively improving the estimation performance of non-sparse, long-range underwater acoustic communication channels.

[0124] 4. Sea experiment

[0125] In numerical simulation, the underwater acoustic channel model is used to model the simulation channel. However, under complex long-distance underwater acoustic communication channels, the sea trial results are of great significance for verifying the feasibility of the proposed method. This experiment gives the results of long-distance underwater acoustic communication experiments in the Mediterranean Sea. The positions of the transmitter and receiver in the experiment are as follows: Figure 5 The channel examples recorded in this experiment are as follows. Figure 1 The specific sea trial parameters are shown in Table 2.

[0126] Table 2 Sea trial parameter settings

[0127]

[0128] Based on the sea trial results, the present invention analyzes the CRER and BER of the proposed algorithm. In this experiment, nine non-sparse long-range underwater acoustic channels were collected. Different receiving signal-to-noise ratios were achieved by passing the OFDM signal through the collected underwater acoustic channels and superimposing the ocean noise recorded during the experiment.

[0129] Figure 6 (a) and (b) in the figure give a signal-to-noise ratio range of -5dB to 20dB. Figure 1 CRER and BER results of medium and long-range non-sparse underwater acoustic communication channel estimation. In view of the dense non-sparse characteristics of the channel, the proposed algorithm can effectively recover the received signal by modeling it as a Gaussian signal. Figure 1The non-sparse long-range underwater acoustic channel in the proposed method is used to obtain better estimation performance than other comparison algorithms (i.e., LS, RLS, MUSIC, and hybrid estimation algorithms), with an average CRER gain of 2.37 dB and an average BER reduction of 0.019.

[0130] also, Figure 7 The CRER and BER results of different algorithms are given in Figure 1. The signal passes through 9 non-sparse long-range underwater acoustic channels collected, and the received signal-to-noise ratio is set to 5dB. Figure 7 It can be seen that the MUSIC algorithm has the worst performance and is difficult to handle the estimation of such dense multipath. However, by modeling the received signal as a Gaussian signal, the NovelML estimator proposed in this invention has the best performance in non-sparse channel recovery.

[0131] This paper proposes a channel estimator for multi-carrier long-range non-sparse underwater acoustic communication channels. For non-sparse channels with dense multipath, a novel maximum likelihood channel estimator is designed for multi-carrier long-range underwater acoustic communication systems, aiming to effectively estimate the dense multipath of the channel. Numerical simulations and long-range offshore experiments demonstrate that the proposed Novel ML algorithm achieves superior performance in both BER and channel estimation compared to traditional channel estimation algorithms for non-sparse long-range underwater acoustic channels.

[0132] The maximum likelihood estimation method proposed in this paper models the received signal as a Gaussian signal, targeting the non-sparse structure of long-range underwater acoustic communication channels. This method effectively exploits the dense multipath characteristics of the channel and improves the accuracy of channel estimation. Numerical simulations and long-range offshore experiments demonstrate that compared to traditional channel estimation algorithms (such as the LS algorithm, RLS algorithm, hybrid channel estimation method, and MUSIC algorithm), the proposed method outperforms traditional channel reconstruction error rate (CRER) and bit error rate (BER), achieving significant improvements in both CRER and BER, enabling more efficient and accurate estimation of non-sparse channels.

[0133] The above embodiments are only preferred embodiments of the present invention and should not be considered to limit the scope of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of the patent of the present invention.

Claims

1. A non-sparse channel estimation method for long-range underwater acoustic communication in a multi-carrier system, characterized in that The following steps are involved: 1) Modeling the transmission and reception of multi-carrier signals: Baseband data is allocated to parallel subcarriers via OFDM modulation, generated into time-domain symbols via inverse Fourier transform, and transmitted after adding guard intervals. The receiving end removes the guard intervals and demodulates the received signal into a frequency-domain signal via Fourier transform. The received signal contains non-sparse underwater acoustic channel information composed of dense multipath and noise, and is modeled as a Gaussian distribution based on the central limit theorem, obeying a normal distribution. 2) Constructing a maximum likelihood estimator based on a Gaussian model: This uses the Gaussian distribution of the received signal to establish a probability density function, which is converted into an optimization objective that includes the channel autocorrelation matrix and the noise variance. Through logarithmic transformation and extremization, the objective function to be minimized is formed to solve the channel impulse response that maximizes the probability of the received signal occurring. 3) Solve the channel impulse response through iterative optimization: Take the partial derivative of the objective function to obtain the gradient vector, which is used to determine the optimization direction of the channel estimation. Use the gradient descent method to iteratively update the channel estimation value from the initial value until the difference between the estimates of two adjacent iterations is less than the preset threshold. The iteration stops when the convergence condition is met.

2. A non-sparse channel estimation method for long-range underwater acoustic communication in a multi-carrier system according to claim 1, characterized in that In step 1), the specific steps of modeling the transmission and channel reception of the multi-carrier signal are: 1.1) Mapping the baseband data onto the parallel subcarriers of the OFDM system, converting them into time-domain OFDM symbols through inverse Fourier transform, and adding a cyclic prefix to combat multipath interference, completing the modulation processing before signal transmission; 1.2) The receiving end removes the guard interval from the received time domain signal and demodulates it into the frequency domain signal through Fourier transform to obtain the received signal vector containing the channel impulse response and additive noise; 1.3) The long-range underwater acoustic channel is defined as a linear time-invariant system whose characteristics are composed of dense multipath. Based on the central limit theorem, it is confirmed that the received signal approximately obeys a Gaussian distribution due to the multipath superposition effect.

3. A non-sparse channel estimation method for long-range underwater acoustic communication in a multi-carrier system according to claim 1, characterized in that In step 2), the specific steps of constructing the maximum likelihood estimator based on the Gaussian model are: 2.1) Based on the Gaussian distribution characteristics of the received signal, its probability density function is established. The channel impulse response, the received signal autocorrelation matrix, and the noise variance are incorporated into the model to reflect the correlation between the signal and the channel parameters. 2.2) The probability density function is logarithmically transformed and extremized, transforming the maximum likelihood estimation into an optimization problem of minimizing the objective function. The objective function comprehensively considers the channel autocorrelation characteristics and the influence of noise to solve the channel parameters that maximize the probability of the received signal.

4. A non-sparse channel estimation method for long-range underwater acoustic communication in a multi-carrier system according to claim 1, characterized in that In step 3), the specific steps of solving the channel impulse response through iterative optimization are: 3.1) Taking partial derivatives of the objective function with respect to each channel component, calculating the gradient vector, and determining the optimization direction of the channel estimate. The gradient vector reflects the rate at which the objective function changes with the channel parameters. 3.2) Using the gradient descent method, starting from the initial channel estimate, the channel estimate is iteratively updated in the reverse direction of the gradient according to a preset step size until the difference between the estimation results of two consecutive iterations is less than a preset threshold. The iteration is terminated when the convergence condition is met to obtain the final channel impulse response estimate.

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