Adaptive Sparse Regular Split Gaussian Kernel Least Mean Square Algorithm for Super-Low-Frequency Motion-Induced Noise Cancellation

Abstract

In super-low-frequency (SLF) submarine communication, the motion-induced noise of the towed antenna is the primary noise source, and below 500 Hz, it increases with increasing speed. We propose an improved quadratic Approximate Forward–Backward Split Gaussian Kernel Least Mean Square Algorithm (ASRSG–KLMS) based on the forward–backward split criterion using noise approximation of the nonlinear kernel least mean square, which introduces an L2-paradigm regularization term and has good sparsity while maintaining optimization stability. The ASRSG–KLMS algorithm could improve the narrowband signal-to-noise ratio by approximately 6.93 dB in the frequency range of 45–55 Hz, making it suitable for motion-induced noise cancellation in the SLF band.

1. Introduction

Super-low-frequency (SLF) underwater communications help to secure maritime communications [1,2,3,4]. In the long-term development of shore–sea communications, towed antennas of appropriate length are usually used to ensure the reception of shore-based signals. According to the organization of submarine-to-submarine communication, ultra-low-frequency (ULF) communication requires that the time and frequency are agreed upon in advance, and the submarine obtains the characteristics of the received signals by reading the pole-distance potentials of the towed antenna at a certain depth, in a certain direction, at a certain speed, etc. [5,6,7]. In particular, the low-frequency background noise in the ocean is a key factor affecting SLF communications. Due to signal attenuation in the ocean, the electric field signal becomes weak and significant as the reception speed increases. In addition, the electric field signal can easily be drowned in motion-induced noise. The key to improving the stability of towed antenna underwater communication is to reduce the motion noise and improve the working signal-to-noise ratio.

At present, the motion noise is mainly generated by the motion of the towed antenna constantly cutting the geomagnetic field, which seriously pollutes the onboard electromagnetic signal [8,9,10]. Secondly, underwater towing speed variations or turbulent motions cause electromagnetic noise to be generated inside the electric field strength sensor [11]. The resulting motion noise constrains underwater communication.

Undoubtedly, these motion-induced noises that occur in different scenarios are generated by Faraday’s law of electromagnetic induction under the action of a geomagnetic field. However, different antenna types and interference sources are used in different scenarios, and therefore different requirements are imposed on the means of attenuating and suppressing this type of noise. For airborne electromagnetic signals, wavelet neural network noise cancellation [8], denoising selfencoder [9], and linear homomorphic filtering algorithms [10] have been proposed, respectively. In the underwater context, many research works have used the least mean square (LMS) algorithm and the motion-induced noise projection algorithm to deal with electromagnetic noise caused by seawater motion [12,13]. We note that motion-induced noise is pervasive.

      Classical adaptive filters, including least mean square (LMS), affine projection (APA), and recursive least squares (RLSs) algorithms, can be formulated in terms of their inner products [14,15,16,17]. In kernel adaptive filters (KAFs), the inner products in the RKHS are efficiently computed using kernel evaluation, which can avoids direct computation in the high-dimensional feature space. The most representative KAFs include kernel least mean square (KLMS) [18,19], kernel APA (KAPA) [20,21], and kernel recursive least squares (KRLSs) [22,23].

The main bottleneck of KAFs is the linearly growing structure of each training sample, which increases the computational and memory burdens for large amounts of training data [24,25]. For practical applications of KAFs, various online sparsification methods have been proposed to curb the growth of networks, where only important input data are chosen as members of the dictionary. In particular, the sparsification methods in KAFs frequently used for data selection include the approximate linear dependency (ALD) [22], novelty criterion (NC) [26], coherence criterion (CC) [27], and quantization criterion (QC) [28].

In addition to the work described above, many of the existing KAFS algorithms mostly forget obsolete elements through forgetting factors or operate using all dictionary elements, imposing a severe burden on computation and memory, which is unacceptable for special environments such as underwater. In recent years, sparse regularization has been investigated in the context of linear adaptive filtering, and these works have used, for example, L1 paradigms or other related regularizers to constrain the deviation in unconstrained solutions [29,30]. Surprisingly, this idea is rarely used in KAFs-based adaptive filtering.

We consider a subgradient approach to this task that can automatically discard obsolete kernel functions and add appropriate elements. To the best of our knowledge, in the context of underwater ultra-low-frequency adaptive filtering, motion-induced noise manifests itself as low-frequency impulse signals, and the L2-paradigm regularization term reduces the low-frequency impulse signals and improves the smoothness of the second iteration. We conducted experiments on a tank platform and demonstrated that the algorithm has a high signal-to-noise ratio and robustness for processing underwater low-frequency signals.

2. Proposed Method

2.1. ASRSG–KLMS Algorithm

Based on the advantages of KAFs in reproducing kernel Hilbert space in non-smooth signal processing, we introduce sparsity into the correlation filter algorithm for the non-smooth time-varying characteristics of SLF motion-include noise. Moreover, we propose a new adaptive sparse regularized correlation filtering algorithm (ASRSG–KLMS) for filtering the antenna background noise. The following optimization problem is considered [30,31]:

$${\alpha }^{*}=min\left\{\sum _{\mathrm{i}=1}^{\mathrm{N}}{\left(\mathrm{d}\left(\mathrm{i}\right)-\alpha \left(\mathrm{i}\right)\kappa \left(·,{\mathrm{u}}_{{\omega }_i}\right)\right)}^2+\zeta \mathsf{\Omega }\left(\alpha \right)\right\}$$

(1)

where $\mathrm{d}\left(\mathrm{i}\right)-\alpha \left(\mathrm{i}\right)\kappa \left(·,{\mathrm{u}}_{{\omega }_i}\right)$ is a convex bounded empirical loss function with Lipschitz continuous gradient and constant $1/\eta$. $\zeta$ is a positive regularization constant. $\mathsf{\Omega }\left(\alpha \right)$ is a convex, continuous, but not necessarily a differentiable regularization term that penalizes over-complicated vectors. However, the order of the filters increases linearly with the amount of input data. This aspect dramatically increases the computational burden and memory requirement of the KAFs. To overcome this drawback, several authors have focused on a fixed-size model of the form [32,33]. We call $\kappa \left(·,{\mathrm{u}}_{{\omega }_i}\right)$ a dictionary similar to a linear transversal filter that learns updates from the input data. The subscript ${\omega }_i$ is used to distinguish between the dictionary elements from the input data. KAFs typically rely on a two-step process at each iteration: a model control step for updating the dictionary and a parameter update step.

(1) Dictionary update: In linear sparse approximation problems, coherence is a fundamental parameter that characterizes a dictionary, which can be described as follows:

$$\mu =\underset{i\ne j}{max}\left|\kappa \left(u_{{\omega }_j},u_{{\omega }_i}\right)\right|$$

(2)

$\left|\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)\right|$ is a Gaussian paradigm kernel [30]. The input data continuously replace the dictionary if its coherence remains below a particular threshold value.

$$\underset{\mathrm{i}=1,\dots ,\mathrm{M}.\mathrm{j}=1,\dots ,\mathrm{N}}{max}\left|\kappa \left({\mathrm{u}}_{{\omega }_{\mathrm{j}}},{\mathrm{u}}_{{\omega }_{\mathrm{i}}}\right)\right|\le {\mu }_0$$

(3)

where ${\mu }_0$ is a parameter that determines the coherence of the dictionary.

(2) Filter parameter update: Online sparsification is usually obtained by constructing from an empty set and gradually adding samples to a center set called the dictionary, according to the consistency criterion. The consistency criterion first calculates the distance from ${\mathrm{u}}_{{\omega }_j}$ to the current dictionary. If  this distance is smaller than some pre-set old ones, ${\mathrm{u}}_{{\omega }_j}$ is not added to the dictionary. Otherwise, ${\mathrm{u}}_{{\omega }_j}$ replaces the minimum value in the dictionary.

Namely:

$${\mu }_{i=1,\dots ,M,j=1,\dots ,N}\left\{\begin{array}{c}>max\left|\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)\right|\hfill \\ \le max\left|\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)\right|\hfill \end{array}\right.$$

(4)

$$\alpha (\mathrm{n}+1)=\left\{\begin{array}{c}\alpha \left(\mathrm{n}\right)+\eta {\mathrm{e}}_{\mathrm{n}}\sum _{i=1}^{\mathrm{M}}\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)\hfill \\ \alpha \left(\mathrm{n}\right)\hfill \end{array}\right.$$

(5)

where $e_n$ is the estimation error regarding $\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)$. Moreover, the coherence criterion can maintain the maximum finite integrity of the dictionary concerning the overall signal under constant data input.

2.2. ASRSG–KLMS Model Optimization

Based on the Kernel Least Mean Square Algorithm with forward–backward splitting (FOBOS-KLMS) [18], for the non-smooth time-varying nature of the motion-induced noise, we added an L2 paradigm regularization term-constrained filter to the objective function and sparse the weight matrix using a soft threshold operator. Figure 1 illustrates a block diagram of the ASRSG–KLMS algorithm model.

Informally, ASRSG–KLMS can be viewed as analogous to the projected subgradient method, where the forward–backward splitting method is motivated by the desire to iterate ${\alpha }^{*}$ to reach a point of nontriviality of the function. The method mitigates the problem of integrability in cases such as L2 regularization by employing an analytic minimization step that is interleaved with the subgradient step. ASRSG–KLMS is very succinct since each iteration consists of the following two steps:

$${{\alpha }_{(i+{\displaystyle \frac{1}{2}})}}^{*}={{\alpha }_{\left(i\right)}}^{*}+r_i{g}_i$$

(6)

$${\alpha }^{*}=min\left\{{∥\mathrm{d}\left(\mathrm{i}\right)-\alpha \left(\mathrm{i}\right)\kappa \left(·,{\mathrm{u}}_{{\omega }_i}\right)∥}_2^2+\zeta \mathsf{\Omega }\left(\alpha \right)\right\}$$

(7)

$g_i$ is the gradient vector and $r_i$ is the step size of the algorithm at time step i. The actual value of $r_i$ depends on the specific setup and analysis. Thus, the first step is simply equivalent to an unconstrained subgradient step about the function. In the second step, we need to consider two regularization terms to complete the parsing step interleaved with the subgradient step.

The objective function considers two regularization terms. If $\mathrm{d}\left(\mathrm{i}\right)-\alpha \left(\mathrm{i}\right)\kappa \left(·,{\mathrm{u}}_{{\omega }_i}\right)$ and $\zeta$ are fixed, the  optimal objective function ${\alpha }^{*}$ could be computed. However, the computational complexity of directly optimizing the objective function is high; therefore, we performed a quadratic process on all filtered channels for each signal element.

Second, the motion-induced noise manifests itself as an impulse signal at low frequencies. The L2-paradigm regularization term reduces the low-frequency impulse signal and improves smoothing in the secondary iteration. Moreover, we considered an adaptive L2-paradigm function where the weights were dynamically adjusted according to the signal. Sparse solutions were obtained using a soft thresholding operator, eliminating irrelevant or redundant features.

By contrast, a limitation of the soft thresholding operator is that it produces biased predictions with local optimal solutions and signal distortions. We assume that the dictionary values, regularization parameters, and Lipschitz constant $1/\eta$ are provided, and an additional auxiliary variable is introduced to aid computing and dynamically adjusting according to the filter signal weights. The approximation operator of this regularization function is expressed as follows:

$$\zeta \mathsf{\Omega }\left(\alpha \right)=max\left\{\left|\alpha \left(\mathrm{n}\right)\right|-\lambda \eta \mathrm{L}2\left|{\mathrm{w}}_{\mathrm{m}}\right|,0\right\}sign\left\{\alpha \left(\mathrm{n}\right)\right\}$$

(8)

The second term in the above minimization problem is a regularization term that controls the number of paradigms of the solution. The regularization parameter $\lambda$ provides a trade-off between measurement fidelity and noise sensitivity [18].

The ASRSG–KLMS algorithm with sparsity-induced regularization can be written as follows:

$${\alpha }_{\mathrm{n}}={\alpha }_{\mathrm{n}-1}+\eta {\mathrm{e}}_{\mathrm{n}}\left|\kappa \left({\mathrm{u}}_{{\omega }_{\mathrm{j}}},{\mathrm{u}}_{{\omega }_{\mathrm{i}}}\right)\right|-f_{\mathrm{n}-1}$$

(9)

where

$$f_{n-1}\left(n\right)=\left\{\begin{array}{cc}\lambda \eta sign\left({\alpha }_{n-1}\left(n\right)\right)\hfill & \mathrm{if}\left|{\alpha }_{n-1}\left(n\right)\right|\ge \lambda \eta \hfill \\ {\alpha }_{n-1}\left(n\right)\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(10)

The proposed ASRSG–KLMS algorithm is described in Algorithm 1.

Note that the quadratic operator is applied after the gradient descent step. When $\lambda$ = 0, this method reduces to a general KLMS algorithm.

Algorithm 1 ASRSG–KLMS.

Select the step size and the parameters of the kernel $\zeta$; Insert $\kappa \left(·,u_{{\omega }_1}\right)$ into the dictionary $i=1,\dots ,M,{\alpha }_1=0$     for n = 1,2,…, do        if ${max}_{m=1,\dots ,M}\left|\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)\right|>{\mu }_0$            Compute $\left|\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)\right|$ and ${\alpha }_n$        else if ${max}_{m=1,\dots ,M}\left|\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)\right|\le {\mu }_0$            incorporate $\kappa \left(·,{\mathrm{u}}_{\mathrm{n}}\right)$ into the dictionary;            Compute $\left|\kappa \left({\mathrm{u}}_{{\omega }_j},{\mathrm{u}}_{{\omega }_i}\right)\right|$ and ${\alpha }_n$        end if        Compute ${\alpha }_n$ using (7)     end for

3. Experiments

We conducted a water tank experiment to validate the proposed ASRSG–KLMS algorithm. The experimental setup is shown in Figure 2. The towed antenna was suspended in the water tank, with the acceleration sensor connected to the far end and the proximal output ports connected to the LNA and data collector. An oscillating device was connected to the end of the antenna to simulate the motion-induced noise. The 50 Hz frequency signal was used as the received signal, and the antenna signal and motion-induced noise offsetting operator were collected through three channels: The antenna, helix coil sensors, and acceleration sensor.

3.1. Parameters Setting

Based on [34,35,36], we generated two independent sequences of 1000 samples with initial standard deviations of 0.15 and 0.35 and a variance of 1 × ${10}^{-4}$. The results are averaged over 200 Monte Carlo loops. In order to further verify the feasibility of nonlinear algorithms in adaptive noise reduction, six adaptive noise reduction algorithms are given in this paper to make comparisons between the convergence in a non-stationary environment, which are LMS, RLS, AP, KLMS, FOBOS-KLMS, and ASRSG–KLMS calculations. As shown in Figure 3 LMS and RLS start to reach the mean squared error (MSE (dB) = 10log10(en)) gradually after nearly 1000 iterations. The AP and KLMS calculations were not robust under the simulation conditions although they converged faster. The ASRSG–KLMS, on the other hand, had basically the same robustness at a 5 dB higher steady-state MSE than FOBOS-KLMS, about which we will continue to verify and analyze by collecting real-time data from the water tank experiments.

Figure 4 shows the signal power spectrum under different regularization parameter conditions. The regularization parameter ranges from 0.0001 to 1. The SLF noise presents a higher power spectrum at low frequencies. The power spectrum is in a low position overall when the regularization parameter is 1, and the signal power spectrum is −26.89 dB. In contrast, the signal power spectrum is −20.76 dB at the peak when the regularization parameter is 0.01, and the relative background noise is lower. Thus, the filtering is more effective. Consequently, the optimal regularization parameter for the underwater low-frequency ASRSG–KLMS algorithm is $\lambda =0.01$.

3.2. Quantitative Analysis

In [6], the correlation between acceleration and noise signal collected by the spiral sensor is analyzed in detail. In order to improve the reliability of the algorithm, we use the four-channel signal collected in Figure 2 for noise cancellation. The cancellation results of different algorithms for band-limited noise sources were simulated using the SNR of the towed antenna at different noise bandwidths [18].

Table 1. Noise cancellation effect of linear model algorithm under acceleration and helix coil sensors.

Algorithm	Acceleration Sensor/Helix Coil Sensors ($\mathbf{\Delta }$SNR (dB))
	45–55 (Hz)	25–75 (Hz)	0–100 (Hz)	0–200 (Hz)
LMS	5.58/5.79	12.86/12.64	7.72/8.24	7.76/8.30
RLS	5.52/4.85	11.74/13.32	12.34/5.91	12.27/5.94
AP	5.50/5.50	12.90/12.64	19.50/15.27	19.08/15.16

lists the filtering results for the different bandwidths and linear algorithms. Note that the LMS outperforms in low-frequency narrowband 45–55 Hz filtering, and the noise-canceled $\Delta$SNR of both sensors is improved by 5.58 dB and 5.79 dB, respectively.

The introduction of KAFs processing based on the LMS algorithm modeling was considered.

Table 2. Noise cancellation effect of nonlinear model algorithm under acceleration and helix coil sensors.

Algorithm	Acceleration Sensor/Helix Coil Sensors ($\mathbf{\Delta }$SNR (dB))
	45–55 (Hz)	25–75 (Hz)	0–100 (Hz)	0–200 (Hz)
KLMS	5.45/5.48	12.75/12.48	8.10/2.26	8.09/2.32
FOBOS-KLMS	4.27/6.67	11.00/13.16	12.11/11.83	11.89/11.79
ASRSG–KLMS	4.28/6.93	11.03/13.50	12.19/12.89	11.97/12.84

lists the noise cancellation effects of the KAFs algorithms for the acceleration and helix coil sensors. Figure 5 illustrates the noise cancellation power spectra of six algorithms for the acceleration and helix coil sensors.The SNR gains of the FOBOS–KLMS algorithms are 4.27 dB and 6.67 dB. In contrast, after the L2-paradigm regularization constraints are applied, the highest SNR improvement is achieved by ASRSG–KLMS for noise cancellation under helix coil sensors, with an improvement of 6.93 dB for a narrow band of low frequency at 45–55 Hz SNR.

In addition to improving the SNR, we considered background noise to verify the noise-canceling effect.

Table 3. Noise elimination results of the spiral sensor noise source Algorithm.

Algorithm	Helix Coil Sensors ($\mathbf{\Delta }$SNR (dB))
	45–55 (Hz)	25–75 (Hz)	0–100 (Hz)	0–200 (Hz)
LMS	5.97	2.92	7.07	3.60
RLS	0.64	3.92	5.74	3.54
AP	2.22	2.14	2.72	1.93
KLMS	0.04	0.21	0.78	0.36
FOBOS-KLMS	3.40	7.46	7.89	7.34
ASRSG–KLMS	2.39	5.89	6.58	6.03

lists the background noise differences regarding the six algorithms under noise cancellation for the helix coil sensors. The results are generally consistent; noise cancellation reduces the background noise to different degrees. The ASRSG–KLMS algorithm not only has a higher SNR, but also reduces the offset of FOBOS-KLMS by 1.01 dB in the low-frequency background noise after 45–55 Hz filtering.

4. Conclusions

This study proposed the ASRSG–KLMS algorithm and analyzed its convergence behavior under motion-induced noise interference from the SLF electrodes and the antenna. The algorithm handles the L2-paradigm regularization based on forward and backward splitting to adapt the dictionary to the transient characteristics of the input signal automatically. We conducted several experiments on a range of platforms and demonstrated that the proposed ASRSG–KLMS model exhibited a high SNR robustness and faster convergence when a large number of digital signal sequences were acquired underwater in real time. The proposed algorithm can be generalized for motion-induced noise cancellation of submarine underwater antennas.