Link Quality Enhancement with Beamforming Using Kalman-based Motion Tracking for Maritime Communication

Abstract

Conventional maritime communication struggles to provide high data rate services for Internet of Things (IoT) devices due to the variability of maritime environments, making it challenging to ensure consistent connectivity for onboard sensors and devices. To resolve this, we perform mathematical modeling of the maritime channel and compare it with real measurement data. Through the modeled channel, we verify the received beam gain at buoys on the ocean surface. Additionally, leveraging the modeled wave motions, we estimate future angles of the buoy to use the Extended Kalman Filter (EKF) for design beamforming strategies that adapt to the evolving maritime environment over time. We further validate the effectiveness of these strategies by assessing the results from an outage probability perspective. focuses on improving maritime communication by developing a dynamic model of the maritime channel and implementing a Kalman filter-based buoy motion tracking system. This system is designed to enable precise beamforming, a technique used to direct communication signals more accurately. By improving beamforming, the aim is to enhance the quality of communication links, even in challenging maritime conditions like rough seas and varying sea states. In our simulations that consider realistic wave motions, you've observed significant improvements in link quality due to the enhanced beamforming technique. These improvements are particularly notable in environments with high sea states, where communication challenges are typically more pronounced. The progress made in this area is not just a technical achievement; it has broad implications for the future of maritime communication technologies. This paper promises to revolutionize the way we approach communication in maritime environments, paving the way for more reliable and efficient information exchange on the seas.

1. Introduction

The rapid advancement of the IoT and the evolution of 6 generation(6G), there is a noticeable acceleration in the advancement of communication technologies. This progress is driving the expansion of communication capabilities, particularly within the maritime domain [1],[2]. However, maritime communication has not progressed at the same pace as terrestrial systems. Addressing the discrepancy in this inequality takes on significance as maritime activities continue to rise. With an increasing need for dependable connectivity in oceanic settings, there is a growing demand for proficient communication systems. These systems are designed to provide excellent communication services, offering extensive coverage across the oceanic expanse, as highlighted in [3–6]. Unlike terrestrial channels, the oceanic unique environment of maritime communication scenarios stated in [3] such as ocean waves and wind keep buoys of small and medium size heaving and tilting, which makes it difficult to maintain a stable link between a base station and buoys. The challenges presented by factors such as ocean waves and wind make maintaining a stable link connection paramount, as highlighted in [7]. Recognizing this pressing necessity, our research is dedicated to bridging the technological gaps in oceanic communication, with the goal of establishing a robust infrastructure for the maritime sector. A fundamental element in achieving this resilience lies in the communication models that underpin system design and functionality. However, many prevailing studies in maritime communication heavily rely upon the Rician channel model approximation, as mentioned in [8],[9]. Unfortunately, this often results in overlooking the dynamic movements of maritime objects and the continuous fluctuations within communication channels. Moving towards enhanced precision in modeling, particularly in terms of accounting for real-time behaviors and variations in communication channels, will play a critical role in developing systems that are suitably equipped to tackle the intricacies of maritime communication challenges. Furthermore, as mentioned in [10], conventional communication systems used in maritime settings have faced significant difficulties stemming from the ever-changing and unpredictable conditions of the ocean environment. A notable constraint has arisen due to the absence of beamforming and beam-tracking techniques, as indicated in [11]. Reference [12–16] introduced beam tracking in terrestrial communications. The absence of these techniques can result in compromised link quality for maritime communication systems. By integrating beamforming [17] and beam tracking capabilities, the potential exists to enhance signal directionality and concentration, thereby establishing more dependable communication across water bodies. Recognizing these challenges underscores the significance of investigating beamforming and beam-tracking within maritime scenarios, as highlighted in [18]. We introduce a more detailed and precise channel model that considers the actual dynamics of maritime objects and channel variations, moving beyond the conventional Rician model. This provides a more accurate representation of real-world marine conditions. Furthermore, we propose a novel algorithm that optimizes both beamforming and beam tracking in real-time maritime scenarios. This algorithm has demonstrated superior performance in terms of signal strength and reliability when compared to existing solutions. This amalgamation aspires to offer not just a reactive maritime communication system but a proactive one. In Section 2 of this thesis, we elaborate on the channel model for the proposed maritime Multiple Input Multiple Output (MIMO) communication system using buoys. In Section 3, we present a protocol for the beam misalignment compensation, based on the sensors in smart devices In Section 4, we present the simulation result. In section 5, We summarize our conclusions in this paper. In summary, the suggested MIMO Maritime communication system for buoys could significantly improve the reliability.

2. System model

In this section, we consider a maritime communication system for a downlink scenario. In the considered scenario, the buoy and base station (BS) employ 𝑀 and 𝑁 antennas, respectively. The overall maritime system architecture is shown in Fig. 1. In the near-shore communication link from the BS to buoy, the direct path will be most prevalent. As a result, we consider an empirical Rician fading model. Then, the received signal, denoted as 𝐲 ∈ ℂ^{𝑁 × 1}, at the buoy can be represented as:

𝐲 = 𝐅_r𝐇^𝐇_R𝐅_t𝑠 + 𝐧,       (1)

Fig. 1. System architectures

where 𝐅_r ∈ ℂ^𝑁×𝑁 , 𝐅_t ∈ ℂ^𝑀×𝑀 are the received beamformer at the buoy and transmit beamformer at BS. We assume that the base station has precise knowledge of the buoy's location, the transmission precoding is omitted. The transmit signal 𝑠𝑠 is for the buoy and with a normalized power 𝔼(ss^H) = 𝐈. During data transmission, a single data stream is utilized at a time, indicating that data is sent using only one continuous flow of information at any given moment. The noise vector 𝐧 ∈ ℂ^N×1 follows an additive white Gaussian noise (AWGN) distribution with covariance matrix 𝜎²𝐈, where 𝜎² is the noise variance and 𝐈 is the identity matrix. 𝐇_R ∈ ℂ^M×N is the Rician fading channel from the BS to buoy, and it can be represented as follows:

\(\begin{align}\mathbf{H}_{\mathrm{R}}=\sqrt{\frac{K}{K+1}} \times \mathbf{H}_{\mathrm{LOS}}+\sqrt{\frac{1}{K+1}} \times \mathbf{H}_{\mathrm{NLOS}}\end{align}\),       (2)

where H_LOS = exp (1𝑗 × ∅) is the LoS component. ∅~𝑈(0,2𝜋) denotes a uniform distribution in the range (0,2𝜋). \(\begin{align}\mathrm{H}_{\mathrm{NLOS}}=\frac{X+1 j Y}{\sqrt{2}}\end{align}\) is the LoS component. 𝑋~𝒩(0,1) and 𝑌~𝒩(0,1) are the real and imaginary numbers, respectively. Those represent a normal distribution with a mean of 0 and a variance of 1.

2.1 Mathematical derivation of angle of buoy in maritime communication

In the received signal of (1), the line of sight (LoS) path within the Rician channel model 𝐇_R includes a significant factor related to the angle of arrival for the LoS path. Thereby, to faithfully evaluate the communication capability of the buoy in the maritime scenario, understanding the complex mechanisms of fluid dynamics in ocean waves and the variations in surface elevation in highly dynamic oceanic environments is crucial. We first examine and delve into the mathematical aspects of the buoy's movement and how the channel changes due to fluctuations in the wave surface. Based on the Bretschneider spectrum model, as mentioned in [19], the spectrum model for the open ocean wave is given by:

\(\begin{align}S(\omega)=\frac{5}{16} \frac{H_{s}^{2} \omega_{p}^{4}}{\omega^{5}} \exp \left(-\frac{5}{4}\left(\frac{\omega_{p}}{\omega}\right)^{4}\right) \frac{\frac{m^{2}}{r a d}}{s}\end{align}\),       (3)

where 𝑯_𝒔, 𝝎 and 𝝎_𝒑 represent the height of the wave, angular frequency, and peak angular frequency, respectively. Here, the peak period of the peak angular frequency is denoted as 𝑇_𝑝 = 2𝜋/𝜔_𝑝. With the spectrum model of (2), to obtain the amplitude of the wave involving the multiple wave components, we further assume that the amplitude of each wave components follows the Rayleigh distribution as referred in [20]. Then, the elevation denoted as γ(𝑥,𝑡) of the wave components at time t and position x can be expressed as follows:

\(\begin{align}\gamma(x, t)=\sum_{i=1}^{N_{f}} a_{i} \cos \left(2 \pi f_{i} t+k_{i} x+\alpha_{i}\right)\end{align}\),       (4)

where 𝑁_𝑓 and 𝛼_𝑖 are the number of wave components and the phase for the i-th frequency 𝑓_𝑖, respectively. The wave number 𝑘_𝑖 is determined by 𝑤²_𝑖/𝑔, where 𝑔 is the acceleration of gravity. Here, the expected amplitude of each frequency component, represented by E[𝑎_𝑖], can be computed as follows [21]:

\(\begin{align}\mu_{i}=\sqrt{2 * S\left(\omega_{i}\right) * \Delta \omega}\end{align}\),       (5)

where ∆𝜔 is the frequency interval of the spectrum S(𝜔_𝑖). It is noted that the wave’s elevation formulated by (3) provides a more accurate model to address the motions of the buoy and faithfully capture the characteristics of the practical channel model in maritime communications. We now investigate the buoy angle in terms of the wave’s elevation in (3). It shows that the dynamic angle of the buoy can be determined by the geometric structure as depicted in Fig. 2. Upon observing in Fig. 2, it becomes evident that 'heaving' and 'tilting' constitute the motion. Because of this movement, the horizontal beam of the receiving beamforming remains stationary, while the vertical beam undergoes motion. Therefore, it is essential to elucidate this phenomenon. As the sea level obtained by (3) changes, the antenna direction with respect to the elevation of zero varies. Thereby, the buoy’s angle with Angle of Arrival (AoA) is 0 denoted as Ω₀ can be obtained by the elevations at the time (t-1) and (t+1). Subsequently, the angle of the buoy is expressed as:

\(\begin{align}\Omega_{0}=\tan ^{-1}\left(\frac{\gamma(\mathrm{x}, t+1)-\gamma(x, t-1)}{2}\right)\end{align}\),       (6)

Fig. 2. Modeling the angle of a buoy in a dynamic marine environment

Equation (5) involves the practical wave elevation at the specific point x of the fixed buoy. To establish the Angle of Arrival(AoA) of the LoS path of the Rician channel model in (1) as time varies, this mathematical modeling of the buoy's antenna direction is employed. This approach enables us to track changes in the angle over time, leading to a deeper comprehension of ocean wave behavior. Fig. 3 validates the feasibility of the mathematical model (6), where the actual buoy’s motion is obtained at the moderate ocean with a variable height of the wave and angular frequency. These Actual values of buoys are measured when the sea state was moderated by the Korean Ministry of Oceans and Fisheries, and the values obtained based on this data represent the actual angles of buoy movement. The actual values of the angle of the buoy movement indicate that the average angle is 0.4 degrees, and the variance is 8.8 degrees. Through mathematical modeling for a moderate sea state level, the average movement angle of the buoy is about 0.3 degrees, and the variance is about 8.9 degrees. Mathematical modeling of the moving angle of the buoy is used to construct a time-varying channel. It is noted that the formulated AoA model. This evaluation confirms the reasonableness of the formulated AoA model (6), namely, the relaxations employed in the derived outcome.

Fig. 3. Compare simulation angle data Ω with real angle data variation over time (Sea state level 4)

3. The proposed method

Fig. 4 depicts the equivalent channel gain received at the buoy considering the oceanic channel. This shows that it is not consistently advantageous to change the received beamforming constantly and demonstrates that location information is available. It is only when the gain falls notably below a set threshold that adjusting the beamforming becomes relevant. Continually tweaking beamforming from a communications standpoint adds undue intricacy and can lead to lags due to regular recalculations. Moreover, if the received signal quality remains above an acceptable threshold, there's minimal to no discernible benefit in making those adjustments. Only when the gain falls below our defined threshold is there a tangible advantage to refining the beamforming, as it can considerably restore and enhance the link quality. This adaptive strategy not only ensures robust communication but also optimizes system resources by focusing on adjustments only when truly necessary.

Fig. 4. Probability density function of equivalent channel gain (Sea state 4)

3.1 Initial beam searching

The foundation for this adjustment is to determine the initial beam for the highest beam gain based on the initial AoA (𝜃_n) of the received signal. Through these, we can formulate the beam sweeping at buoy as follows [22]:

\(\begin{align}\boldsymbol{n}^{*}=\arg \max _{\mathbf{n}}\left|\overline{\mathbf{F}}\left(\boldsymbol{\theta}_{\mathrm{AoA}}\right) \mathbf{H}_{\mathbf{R}}\right|^{2}\end{align}\),       (7)

By multiplying the channel matrix with the steering beamforming vector \(\begin{align}\overline{\mathbf{F}}\left(\theta_{\mathrm{AoA}}\right)\end{align}\). Then, we can evaluate the resulting gains. From these, received beamforming vector with the highest gain is selected, allowing us to identify the initial beam direction to ensure maximum signal strength and performance.

3.2 Refinement and tracking

In maritime channels like Fig. 2, it can be readily noted that the heaving and tilting of buoys on ocean waves cause a misalignment of beamforming between BS and buoy that wouldn’t have occurred in terrestrial channels. To enhance the link reliability, The buoy adjusts its receiving beam autonomously using the estimated angles. In our study, we delve into the functionalities and distinctions between the Kalman Filter (KF) and the Extended Kalman Filter (EKF), highlighting their pivotal roles in dynamic state estimation and prediction. The KF is particularly efficient in environments characterized by linear models and Gaussian noise, making it an excellent choice for straightforward tracking and navigation tasks. However, the Extended Kalman Filter comes into its own in more complex scenarios due to its capacity to handle nonlinear dynamics. This is achieved through a linearization process, where the EKF approximates the nonlinear behaviors of a system using linear models, thereby allowing for more accurate predictions in situations where traditional linear models fail. The applicability of these filters extends significantly into maritime communications, particularly in beamforming technologies where precise signal direction and management are crucial. In dynamic maritime environments, where the conditions are unpredictable and often nonlinear, the EKF's ability to adapt and predict beam directions is invaluable. By utilizing the Extended Kalman Filter, it can enhance the efficiency and accuracy of beamforming techniques. This improvement is critical for maintaining robust communication links in maritime settings, ensuring that signal strength and clarity are maximized even under challenging conditions. Therefore, in this paper, we implemented the Extended Kalman Filter to advance the application of beamforming in maritime communication, showcasing its effectiveness in managing the complexities associated with this advanced technology. we next show how this mismatch could be overcome via a phase rotation operation using an Kalman filter [23],[24]. Before use EKF, we must set the angle the buoy:

\(\begin{align}B_{t}=\left[\begin{array}{c}\Omega_{\mathrm{t}} \\ \tau_{\mathrm{t}}\end{array}\right]\end{align}\),       (8)

where Ω_t and τ_t is angle and velocity of buoy. We now use the elevation value, so the measurement model H can be expressed to:

\(\begin{align}H=\left[\frac{\partial}{\partial \Omega} \arctan \left(\frac{\gamma(\mathrm{x}, t)-\gamma(x, t-1)}{2}\right), 0\right]\end{align}\),       (9)

This is non-linear system. Therefore, we compute the Jacobian matrix for the behavioral and measurement models of the system. For the behavioral model, the Jacobian will be a unit matrix since the system depends primarily on changes in state (i.e., elevation). However, in real-world situations, we rarely know the behavioral model exactly. For the measurement model, we use the change in elevation with respect to Ω, this Jacobian will contain the partial derivative of elevation with respect to Ω.

\(\begin{align}H_{t}=H\left(\Omega_{t-1}\right)+\left.\frac{\Omega H}{d \Omega}\right|_{\Omega_{t-1}}\left(\Omega-\Omega_{t-1}\right)\end{align}\),       (10)

To use this parameter, it must go through a prediction process. We set the predicted state estimate as follow:

𝐵_{𝑡|𝑡−1} = 𝐴_𝑡−1 ∗ 𝐵_𝑡,       (11)

where 𝐵_{𝑡|𝑡−1} is the predicted state of next time. Then, the predicted estimate covariance is as follow:

P_{𝑡|𝑡−1} = 𝐴_𝑡𝑃_𝑡−1𝐴^𝐻_𝑡 + 𝑄_𝑡,       (12)

where P_{𝑡|𝑡−1} represents the uncertainty of our prediction. 𝑄_𝑡 is a process noise covariance matrix, accounting for the unpredictability in the state evolution. When a new measurement is available, the EKF updates its state estimate. The difference between the actual measurement and our predicted measurement (from the predicted state) is the residual:

𝑦_𝑡 = 𝐵_𝑡 − 𝐻_𝑡𝐵_{𝑡|𝑡−1}, (      13)

where 𝐻_𝑡 maps the state to the measurement space. To account for uncertainties in our prediction and the measurement, we compute the residual covariance:

𝑆_𝑡 = 𝐻_𝑡P_{𝑡|𝑡−1}𝐻^𝐻_𝑡,       (14)

The Kalman gain, 𝐾_𝑡 is calculated next. This determines how much we should adjust our state prediction based on the new measurement:

𝐾_𝑡 = 𝑃_{𝑡|𝑡−1}𝐻_𝑡𝑆⁻¹_𝑡,       (15)

The state estimate is then updated:

\(\begin{align}\hat{B}_{t}=B_{t \mid t-1}+K_{t} y_{t}\end{align}\),       (16)

Finally, we update our estimate's uncertainty:

\(\begin{align}\widehat{P_{t}}=\left(I-K_{t} H_{t}\right) P_{t \mid t-1}\end{align}\),       (17)

By iteratively performing prediction and update steps, the EKF offers a real-time estimation framework. For our system, it provides a way to estimate the angle and its angular velocity even in the presence of noisy measurements. We compare the angle estimated by EKF with the actual angle in Fig. 5. Through this methodology, the system can dynamically react to changes and continue to provide accurate estimates over time. We summarized the process in Algorithm 1. To proceed with Algorithm 1 as described below, constraints is as follows:

\(\begin{align}\hat{P}_{\text {error }}<\gamma(t)_{\text {var }}\end{align}\),       (18)

Fig. 5. Comparison of actual buoy movement angle Ω with Kalman data \(\begin{align}\widehat{\Omega}\end{align}\) over time according to Sea State

where \(\begin{align}\hat{P}_{\text {error }}\end{align}\) is a remaining uncertainty of the state vector resulting in remaining beamforming misalignment while γ(𝑡)_var is ocean wave variance resulting in beamforming misalignment without tracking of time varying AoA due to motion of buoy, respectively. If the variation or inconsistency in tracking is greater than the variability of the sea state, the application of the EKF for estimation purposes may not be optimal. The fundamental reason behind this is that when tracking noise or variability is excessively high, it can overshadow or complicate the data, thus diminishing the effectiveness and predictive accuracy of the EKF. Given these considerations, it might be against relying on the EKF for estimation in scenarios where the tracking variation significantly surpasses that of the sea state.

Algorithm 1: EKF-based beam refinement method

 

3.3 Link failure

In the event of a link failure or network outage, communication in specific areas could be disrupted. As a result, communication coverage would be compromised, leading to reduced efficiency. To address this challenge, it becomes imperative to establish contingency plans for link failures and network outages. These measures are essential for maintaining seamless communication coverage and optimizing operational effectiveness. Link failure is that if the performance of the beam we have set is lower than the predefined threshold known, it is referred to as a link failure. The link failure inequality is as follows:

\(\begin{align}\left|\mathbf{F}_{\left[: ; n^{*}\right]} \boldsymbol{\Phi}\left(\theta_{\mathrm{n}}+\widehat{\Omega}_{t}\right) \mathbf{H}_{\mathrm{R}}\right|^{2}<\varphi_{\mathrm{LF}}\end{align}\),       (19)

where 𝜑_LF is the beam performance gain threshold that we configure. Therefore, we calculate the probability that the received gain becomes less than some threshold to get the outage. As described above, the initiation of a comprehensive beam adjustment procedure entails the initialization of the received signal y, establishment of a fixed beam F, and definition of a time duration T. Subsequent actions encompass the following steps. First, the base station and the buoy location is given information, it is assumed that the angle of arrival (AOA) is known. Consequently, the initial beamforming vector for the beamforming process is set as the steering vector with the highest gain, as follows:

\(\begin{align}n^{*}=\underset{n \in S_{N}}{\arg \max } \overline{\mathbf{F}}\left(\theta_{\mathrm{n}}\right) \mathbf{H}_{\mathrm{R}}\end{align}\),       (20)

where 𝑛^∗ is constant of beamforming vector. Then, we refinement and tracking the beamforming to use estimate angle:

\(\begin{align}\widehat{\Omega}_{t}=\Omega_{\mathrm{t} \mid \mathrm{t}-1}+K_{t} y_{t}\end{align}\),       (21)

Algorithm 2: Beam tracking method

 

Furthermore, we adjust the beam considering the direction fine-tuning factor, denoted as \(\begin{align}\widehat{\Omega}_t\end{align}\), based on the previously known AoA. The final calculated receive gain at the buoy is as follows:

\(\begin{align}g_{k}=\overline{\mathbf{F}}_{\mathrm{k}\left[; n^{*}\right]} \boldsymbol{\Phi}\left(\theta_{\mathrm{n}}+\widehat{\Omega}_{t}\right) \mathbf{H}_{\mathrm{R}}\end{align}\),       (22)

In Eq. (22), \(\begin{align}\overline{\mathbf{F}}_{\mathrm{k}\left[; n^{*}\right]} \end{align}\) is the optimal beam where \(\begin{align}\overline{\mathbf{F}}\left(\theta_{\mathrm{AoA}\left[n^{*}\right]}\right)\end{align}\) is steering vector with the structure \(\begin{align}\overline{\mathbf{F}}\left(\theta_{\mathrm{AoA}\left[n^{*}\right]}\right)=\left[1, e^{\mathrm{j} \pi \sin \theta_{\mathrm{AoA} A_{n}}}, \ldots, e^{j(N-1) \pi \sin \theta_{\mathrm{AoA} n^{*}}}\right]^{T}\end{align}\) , n = 1, ..., N . And \(\begin{align}\boldsymbol{\Phi}\left(\theta_{\mathrm{n}}+\widehat{\Omega}_{t}\right)\end{align}\) is the phase rotation of the original vector, where \(\begin{align}\boldsymbol{\Phi}\left(\widehat{\Omega}_{k}\right)=\operatorname{diag}\left\{1, e^{j \theta_{\mathrm{n}}+\widehat{\Omega}_{t}}, \ldots, e^{j(N-1) \theta_{\mathrm{n}}+\widehat{\Omega}_{t}}\right\}\end{align}\) is a diagonal matrix given by \(\begin{align}\boldsymbol{\Phi}\left(\theta_{\mathrm{n}}+\widehat{\Omega}_{t}\right)\end{align}\) and \(\begin{align}\widehat{\Omega}_{t}\end{align}\) is the corresponding shifted phase that varies from −π/N to π/N. Finally, we proceed with the process of beam adjustment as follows: If the received beam gain is lower than the threshold for link failure according to the inequality expressed in the following equation:

\(\begin{align}\left|\overline{\mathbf{F}}_{\left[: n^{*}\right]} \boldsymbol{\Phi}\left(\theta_{\mathrm{n}}+\widehat{\Omega}_{t}\right) \mathbf{H}_{\mathrm{R}}\right|^{2}<\varphi_{\mathrm{LF}}\end{align}\),       (23)

We initiate the process of searching for and readjusting the beam. Conversely, if the received beam gain is greater than the threshold, we perform beam adjustment alone. Based on the above steps, the entire process is summarized in Algorithm 2. We verify the performance in simulation based on the received beam gain at the buoy.

4. Simulation

In this section, numerical results are provided for the outage probability according to beamforming algorithm and the sea state of the ocean. We consider a maritime communication system network illustrated in Fig. 1, in which BS and buoy are equipped with multi antennas. The Automatic Identification System (AIS) maritime system operates within the frequency band of 1.7GHz, and the reason for utilizing a K-factor of 3 is rooted in the conventional application of the two-ray channel model in maritime channel scenarios [7]. Then, in maritime communication scenarios, it is customary to consider a base station's transmission power of 46 dBm. This choice aligns with the typical consideration of such scenarios in the maritime channel modeling context. The system parameters are summarized in Table 1.

Table 1. Simulation parameter.

Fig. 6 evaluates the beamforming methods in the maritime communication scenario, where blue, red, and yellow lines are EKF-based beam tracking with beam-switching in Algorithm 2, without beam-switching in Algorithm 1, and fixed initial beamformer in (7), respectively. In Fig. 6(a), where the number of bouy’s antennas is four, the EKF algorithm is always superior to other method in the whole region of sea state level due to the tracking and refinement process encountering the bouy’s motion by wave. Utilizing the EKF algorithm, a minor loss in the received channel gain is observed as the sea state surpasses level 4. However, a distinct improvement is evident when compared to the fixed beamforming method. In Fig. 6 (b), where the sea state level set to 8, we compare the received channel gain as the beam width is narrower (i.e., the number of antennas is increased.) When employing a Kalman-based beam tracking approach, it was observed that the performance was significantly improved by over 4 times compared to using only the initial beam. This improvement was particularly pronounced when encountering substantial variations in wave conditions. Notably, for the number of antennas 5, the algorithmic utilization and non-utilization yielded performance disparities exceeding 10 times. This discrepancy is attributed to the fact that when the beamwidth is narrow, the beam may possess favorable gain characteristics when received, but due to its limited coverage, instances of failure to intercept the signal result in diminished gain. Nevertheless, with proficient tracking performance, even with a narrow beamwidth, satisfactory signal reception can be ensured, resulting in improved gain characteristics. Hence, the conclusion drawn from Fig. 6 underscores the indispensability and significance of accurate motion estimation and consequent beam adjustment to ensure resilience under challenging wave conditions for the desired link quality assurance. We have compiled the detailed performance results in Table 2 to provide a clear and comprehensive representation of the data.

Fig. 6. Average received channel gain over time intervals for various beamforming methods: (a) Across a range of Sea States from 2 to 7, (b). when the increasing number of antenna from 2 to 6 when operating at Sea State 8.

Table 2. Numerical results

Fig. 7 shows that the difference in performance between these strategies was minimal in scenarios with minimal wave disturbance (sea state 2). However, the Kalman-based algorithm was slightly ahead of the others, showing resilience even in stable conditions. The simulation result of Sea State 8 shows that the difference was more pronounced in more turbulent sea conditions. Relying solely on the single selected beam resulted in a sharp decrease in normalized gain, approaching 0.05, increasing the probability of outages. In contrast, the strategy based on Kalman information consistently delivered superior results. These results highlight the effectiveness of Kalman-based beam adjustment techniques for receive beamforming, ensuring robust maritime communications with the movement of buoy, especially in dynamic sea states.

Fig. 7. Compare outage probability performance with algorithms at Sea State 2,8

5. Conclusion

In this paper, we address the limitations of conventional maritime communication by modeling the dynamic maritime channel and designing adaptive beamforming strategies using Kalman-based buoy motion tracking. We first delved into the mathematical representation of the maritime channel, juxtaposing it with actual measured data. After analyzing the simulated channel, the received beam gain at ocean surface buoys was verified. We assessed the similarity from the perspective of actual data-based beam gain to justify the modeling. By leveraging the anticipated wave motions, we employed the EKF to project future angles of the buoy, devising beamforming strategies that adapt to dynamic marine conditions. We proposed a Kalman-based beamformer that tracks marine motion. Our methodology centered on dynamic maritime channel modeling and the application of a Kalman-oriented buoy movement tracking system for refined beamforming. This was all in pursuit of bolstering the quality of the communication link. The simulated scenarios, taking realistic wave actions into account, underscored the heightened link quality attained through our refined beamforming approach, even amid obstacles such as turbulent waves and fluctuating sea conditions. This enhancement is markedly pronounced in maritime contexts dominated by elevated sea conditions. In conclusion, the advancements showcased in our research augur well for a transformative shift in the realm of maritime communication applications in the foreseeable future.