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Abstract
Due to the unique features, orbital angular momentum (OAM) beams have been widely explored for different applications. Accurate determination of the topological charge (TC) of these beams is crucial for their optimal utilization. In this paper, we propose a method that combines adaptive image processing techniques with a simple, parameter-free attention module (SimAM) based convolutional neural network to accurately identify the TC of high-order superimposed OAM beams. Experimental results demonstrate that under the combined influence of non-extreme light intensity and turbulence, it can achieve >95% identification accuracy of TCs ranging from ±1 to ±40. Moreover, even under partial-pattern-missing conditions, our method maintains an accuracy rate of over 80%. Compared with traditional attention mechanisms, SimAM does not require additional network design, significantly reducing the computational costs. Our approach showcases remarkable efficiency, robustness, and cost-effectiveness, making it adaptable to challenging factors such as non-uniform lighting and partially occluded light paths. This research provides a new direction for recognizing OAM modes with valuable implications for the future of communication systems.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, optics research has shown increasing interest in light fields with unique polarization states, amplitudes, or phases, such as cylindrical beams and vortex beams [1]. Vortex beams, in particular, have gained significant attention due to their distinct circular intensity distribution and spiral wavefront structure. These beams possess both orbital angular momentum (OAM) and spin angular momentum. The wavefront of a vortex beam carrying OAM can be described by the phase factor exp(ilφ), where l represents the topological charge (TC), and ϕ represents the azimuth angle [2]. In such a beam, each photon carries an OAM of lħ, with ħ representing the reduced Planck constant. The value of l corresponds to the number of 2π phase shifts as the vortex beam rotates along the beam axis while propagating a wavelength. During transmission, vortex beams exhibit a phase singularity at their center, resulting in zero light intensity in that region [3].
OAM beams have demonstrated significant potential in a wide range of applications, covering both classical and quantum information domains [4]. These applications include optical communication [5–7], optical imaging [8–10], optical tweezers [11,12], quantum information processing [13,14], optical manipulation [15], microscopy [16,17], object detection [18–21], and numerous other fields, as depicted in Fig. 1. The TC of OAM light plays a crucial role not only in characterizing its specific state and influencing its behavior in various applications. Developing robust methods for accurately identifying the TC is essential for harnessing the capabilities of OAM beams and advancing the related fields. In optical communication, for example, the mutual orthogonality of the OAM beams with different TCs allows multiple information multiplexing on the same spatial path through stacked beams, effectively increasing the capacity of the communication system. Accurate determination of the TC is vital for ensuring reliable data transmission and decoding.
Fig. 1. Applications and development of orbital angular momentum (OAM) beams. The distinctive light field distribution of OAM beams enables a broad spectrum of applications, with the number of topological charges (TCs) playing a crucial role.
In recent years, the propagation of light beams through the atmosphere, an inherently unstable and turbulent medium, has gained significant attention within the field of optics [22]. This interest arises from the impact of atmospheric conditions on light beams, including molecular absorption, interactions with atmospheric gases, aerosol scattering, and turbulence. Turbulence, in particular, disrupts the orthogonality of the OAM state, causing variations in beam amplitudes [23]. Accurately measuring the TC associated with OAM beams under these conditions poses a fundamental challenge. A variety of methods have been developed to measure the TC, which can be broadly categorized into physical methods based on interferometry [24–26], diffraction [27–29], hologram techniques [30], etc. Specifically, the use of traditional optical methods requires precise calibration of the optical components and parameter adjustments, which are both time consuming and setup challenging. Holography offers an alternative approach, but requires specialized equipment and algorithms. It is difficult to maintain high quality of bifurcation, spiral, or petal stripe patterns generated by interference, diffraction phenomena, etc., when detecting higher TC orders. Additionally, the model employed in the aforementioned method cannot accurately account for the effects of atmospheric turbulence.
Nowadays, the integration of machine learning (ML) has brought a revolution across various research domains, encompassing fields such as image and speech recognition, natural language processing, and autonomous systems [31]. These techniques hold the potential to significantly enhance the accuracy and efficiency of data analysis and decision-making in a wide range of applications [32]. In the 21st century, deep learning (DL), built on multi-layer networks, has emerged as a prominent focal point [33]. This evolution in DL has paved the way for the application of convolutional neural networks (CNNs) in tasks such as image recognition.
CNNs are a type of DL model specifically designed for processing data with a grid-like structure, which have been widely applied in computer vision tasks and have achieved significant breakthroughs in various fields. CNNs could extract local features by performing sliding window convolution on the input data, leveraging the benefits of local connections and weight sharing. This mechanism reduces the number of parameters in the network and captures the translational in-variance, thereby enhancing the model generalization capability.
However, training CNNs with a large number of layers using random gradient descent can meet challenges such as gradient vanishing or exploding. These issues hinder the effective transmission of information within the network, ultimately degrading its performance [34]. To address this problem, residual modules were introduced as a solution to redefine the design and training approach of deep networks [35]. These modules utilizing shortcut connections, which allow the network to skip certain layers during forward propagation, enabling the direct flow of information. This architecture effectively resolves the gradient vanishing problem, facilitates the training of very deep networks, and has led to significant performance improvements in tasks such as image recognition and feature extraction [36].
The development of attention mechanism has greatly promoted the development of natural language processing and other fields, including computer vision [37]. It enables neural networks to focus on different regions of input data at the same time [38], allowing parallel processing and capturing complex dependencies in sequential or structured data tasks. Attention mechanism has made the latest achievements in tasks such as machine translation, language understanding and image analysis [39]. Existing attention-based network architectures often require large numbers of parameters and additional sub-networks to generate attention weights [40]. In contrast, a simple, parameter-free attention module (SimAM) offers a lightweight solution without extra parameters [41]. SimAM introduces an optimized energy function to minimize structural adjustments and generates 3D attention weights for feature maps. It enables the network to effectively focus on the main parts of the input without distinguishing between channel or spatial features.
The development of ML has led to new advancements in the identification of OAM superposition TCs, especially for the field of optical communication system. Generally, information can be encoded onto each petal of the superimposed OAM beams. At the receiving end, an appropriate decoding method can be implemented to extract and interpret the encoded information associated with each petal. This process may involve using optical elements, such as diffraction gratings, or optical signal processing algorithms. In the period between 2014 and 2016, the Krenn’s team conducted OAM beam propagation experiments at a wavelength of 532 nm in urban and oceanic environments. They successfully demonstrated the applicability of ML techniques in classifying OAM single as well as superimposed beam TCs, achieving classification accuracy for TCs up to ±15 [42,43]. In 2019, Sun’s team focused on the relationship between the level of atmospheric turbulence interference on OAM beams and the TCs. They manually extracted the features and proposed a pattern recognition method for single-state OAM of TC up to 10 based on support vector machine [44]. In 2017, Zhang’s team conducted a comparative study of traditional ML methods, deep learning networks, and CNN as classifiers for OAM TCs [23]. CNN demonstrated remarkable performance with recognition rates approaching 100%, which provided valuable insights for selecting DL models for OAM TCs classification. In the same year, Doster’s group utilized superimposed Bessel-Gaussian beams for OAM encoding, employing a deep-layered AlexNet-like network, which consisting of approximately 20 million training weights distributed across five convolutional layers [45]. Even in the presence of strong turbulent channels, recognition rates exceeding 99% were achieved. Subsequently, many other studies have also redesigned CNN model structures to achieve improvements in high-order and high-accuracy recognition of superimposed beam TCs with low training costs [46–50]. These works are mostly based on CNNs, and the recognition of superimposed OAM TCs does not exceed over ±20.
In this study, we experimentally generate superimposed OAM beams and capture images of their distinct petal-like features under different turbulence and light intensity conditions. By utilizing image processing technique and parameter-free SimAM based CNN, we successfully identify TCs in the range of ±1 to ±40 using a charge-coupled device (CCD) camera at an accuracy of >95% with significantly reduced computational complexity. Our approach has excellent robustness and maintains high performance even under varying turbulence levels, illumination conditions, and partial missing situations. This work proves the advantage of attention mechanism in TC recognition, and ensures accurate identification in a variety of challenging situations.
2. Theoretical support
2.1 Superposed Laguerre Gaussian beam
Researchers have extensively explored various approaches to generate vortex beams, utilizing different beam types, including Laguerre Gaussian (LG) beams, Hermite Gaussian beams, Bessel-Gaussian beams, and their variants [51]. In our study, we adopt the LG beam, a solution to the Helmholtz equation within the cylindrical coordinate system under the paraxial approximation (r, φ, z). The amplitude of the LG beam at the source plane is mathematically expressed as follows [52]:
The above equation can be simplified as:
The intensity distribution of a single-mode LG beam is primarily characterized by the radius and width of its ring-like pattern. As the beam’s state increases, the radius expands while the pattern width diminishes. However, when two LG beams, each carrying an opposing TC, are superimposed, the resulting light field intensity distribution takes on a distinct petal-shaped configuration. When the radial index p = 0, the intensity profile of the resulting superimposed light field can be mathematically expressed as:
Figure 2 illustrates the schematic of the OAM superimposed beam generation process. Figure 2(a) showcases the distinctive petal-shaped intensity pattern that arises from the superimposition of two LG beams. Correspondingly, Fig. 2(b) provides detailed phase information, complemented by computer-generated phase hologram masks corresponding to the previously mentioned beams. We also demonstrate the superposition results of simulated light field intensity distribution of superimposed LG beams with varying TCs. Simulation results of the superimposed LG beams with various TCs reveal that the number of petal-shaped patterns corresponds to twice the number of TCs, as depicted in Fig. 2(c).
Fig. 2. OAM superimposed beam generation diagram. (a) Intensity diagram for the superimposition of beams; (b) phase information and corresponding computer-generated phase masks; (c) simulated light field intensity distribution of superimposed LG beams with varying TCs (l=±1, ± 2, ± 3, ± 4, ± 5, ± 10, ± 20, ± 30, ± 40, ± 60, ± 80, ± 100).
Upon closer analysis of the petal-shaped pattern formed by the superimposed LG beam, it is evident that the pattern dimensions increase in correlation with the rise in TC over a standardized distance of beam propagation. In many real-world scenarios, the receiver field of view is often restricted, potentially impacting the collection of higher-order OAM beams. Additionally, atmospheric turbulence and varying lighting conditions introduce further complexities that can affect the functionality of OAM beams. Hence, when deploying higher-order beams in practical situations, meticulous consideration of these practical constraints and environmental factors is crucial to ensure their effectiveness and reliability.
2.2 Turbulence
In practical scenarios, atmospheric turbulence is present in airspace, characterized by a complex internal structure that is irregular, diffusive, and rotational. The turbulence induces geometric distortion and ambiguity, leading to fluctuations in the refractive index of the light beam before it reaches the acquisition equipment. This phenomenon significantly impacts the quality of images and photographs in various applications, thereby hindering subsequent image processing and perception.
In our experiment, we utilize a spectrum reverse method that follows Kolmogorov spectral statistics to generate pseudo-random phase masks [53,54]. Such masks utilize the discrete Fourier transform to obtain the phase distribution based on the power spectral density function to simulate turbulence characterized by Fried coherent length r0. Figure 3(a) gives out an example of phase mask for beam generation (TC=±10) and Fig. 3(b1)-(b3) shows turbulence phase masks with coherent lengths at different scales. Figure 3(c) displays the experimental collection with part of stray light and no turbulence corresponding to (a) and Fig. 3(d1) and 3(d3) show the diagrams of the OAM superimposed beam after accurate filtering under turbulence corresponding to (b1)-(b3), respectively.
Fig. 3. Phase masks of (a) OAM superimposed beam of TC = ±10; (b1)∼(b3) turbulence with coherent lengths r0 = ∞, r0 = 1e-4 and r0 = 1e-5. (c), (d1)∼(d3) show the collection corresponding to (a), (b1)∼(b3), respectively. The image shown in (c) is captured in the scene without using an accurate filter, but (d1) to (d3) is accurate filtered. w/o: without; w/: with; Turb.: turbulence.
3. Experimental setup and data collection
3.1 Experimental setup
Figure 4 shows the schematic diagram of the generation, propagation, and acquisition of the superimposed OAM beam with TCs of ± l. The system mainly consists of a laser, a variable optical attenuator (VOA), a collimator, a half-wave plate (HWP), lenses, pinholes, spatial light modulators (SLMs), a CCD camera, and a computer. At the transmission, a Gaussian beam is emitted by the laser within the 1550-nm band. This beam is then directed towards a VOA to add different attenuation. After passing through the VOA, the attenuated beam is transmitted through an optical fiber and then coupled into free space by a collimator.
Fig. 4. Experimental setup. VOA: variable optical attenuator; Col: collimator; HWP: half-wave plate; SLM: spatial light modulator; CCD: charge-coupled device. In the first part of the experiment, the setup used is represented by the orange dotted box and in the second part, SLM2 and CCD are added.
The beam is then directed towards the center of the SLMs via the HWP and lens. Depending on the experimental setup, the entire optical path can be classified into two categories: with turbulence and without turbulence. As depicted by the dotted box in Fig. 4, the first part of our optical path involves SLM1 and CCD to capture superimposed OAM beams in a turbulence-free environment. In this configuration, SLM1 is equipped with designed phase masks that transform the Gaussian beam into the desired superimposed beam. In the second part, we have introduced additional components including SLM2, a convex lens, and a pinhole to simulate the optical path with turbulence based on previous work. Specifically, the phase mask for simulating turbulence is loaded onto SLM2.
The working mechanism of SLM is based on grating diffraction and the CCD captures the first-order diffracted light from the beam in our experiments. Although placing a pinhole in the optical path can filter out some stray light, it becomes impractical to constantly adjust the position and size of the pinhole when dealing with beams of different TCs. Therefore, for the above two experimental parts, we set pinholes with fixed (for the first part) and adaptive (for the second part) size and location to filter out unwanted diffraction orders, respectively.
3.2 Data collection
Figure 5 shows the intensity images of the OAM beams with different TCs ranging from ±1 to ±40. The subplots within the figure showcase the intensity patterns of the OAM beams under different conditions, such as without turbulence, with turbulence, and with fixed or adaptive adjustment of the pinhole based on the TC value. We want to fully consider the influence of stray light in order to optimize the model performance when there is no turbulence. When turbulence is present, we choose to employ adaptive pinholes, which can help ensuring optimal performance and reliable operation in the presence of turbulence while also considering the influence of light intensity. Such system design aligns well with subsequent applications, such as efficient space optical communication.
Fig. 5. Examples of experimental light intensity distributions of OAM superimposed beams for various TCs from ±1, ± 2, ± 3, up to ±40 under different conditions.
Based on a −8 dBm base light intensity of the laser, we adjust the attenuation using the VOA in 0.5 dB increments. Figure 6 displays some examples of the superimposed beams with TC=±16 for a range of attenuation (Att. 1) from 0.0 dB to 12.0 dB of the first part of the experimental setup. In the absence of turbulence, fluctuations in relative light intensity have a significant effect on the light intensity distribution within the captured image. Higher light intensities tend to amplify secondary diffraction spots and stray spots caused by environmental factors. At the same time, we showcase the superimposed beam captured images under the effects of two types of turbulence with different coherent lengths and variable attenuation (Att. 2) from 0.0 dB to 5.0 dB for the second part of the experiment. The captured intensity distribution becomes more complex in the presence of turbulence. Furthermore, a smaller coherent length in the turbulence leads to a greater distortion. This makes identification more difficult, especially for higher-order TCs.
Fig. 6. Examples of experimental light intensity distributions of OAM superimposed beams for various TCs = ±16 under different conditions. Att. 1: attenuation of the first part of the experimental setup; Att. 2: attenuation of the second part of the experimental setup.
To mitigate the impact of stray light spots and external interference on the recognition process, we employ the adaptive thresholding method provided by OpenCV [55] and utilize the “cv2.adaptiveThreshold” function to process the images. In particular, we incorporate Gaussian weighting, which averages the values within a 15-pixel neighborhood around each target pixel. The threshold value is then calculated by subtracting 3 from the weighted average. Figure 7 shows a series of experimental images captured during the process, alongside the corresponding images obtained after applying the processing method. As demonstrated, the method effectively mitigates the impact of stray light points to some extent while preserving the distinctive features of the petals.
Fig. 7. Examples of the experimentally captured results and diagrams using adaptive thresholding method.
4. Network framework with SimAM and residual modules
For the aforementioned processed images, we divide them into three types based on their attributes: different attenuation without turbulence (total number of 21600 images of 40 TCs), different attenuation with a turbulence coherent length of 1e-4 (total number of 22000 images of 40 TCs), and different attenuation with a turbulence coherent length of 1e-5 (total number of 22000 images of 40 TCs). In this way, each type corresponds to a specific scenario. Each dataset is further divided into a training, validation, and test set in a ratio of 6:2:2. Before training, data augmentation techniques are applied to increase the effectiveness and diversity of the training data, such as random cropping, random rotation, and image size scaling. Figure 8 illustrates the framework of our designed CNN based on the attention mechanism. The diagram depicts the structure of some network components and provides example heatmap diagrams of different layers involved in feature extraction. The overall network framework can be broadly described as comprising two main parts: a feature extraction component based on convolution and a classifier component based on fully connected (FC) layers. The number of epoch, batch size, and learning rate is set as 100, 32, and 0.0075, respectively.
Fig. 8. Framework diagram of the designed network, combined with residual module and SimAM. Conv.: convolutional layer; MaxPool2d: max-pooling layer; FC: fully connected layer; ReLU: rectified linear unit; BN: batch normalization. SimAM and ResidualBlock are expanded in detail and the feature extraction visualization displays the features concerned by each layer of the feature extractor in the form of heat map.
4.1 Feature extraction
The feature extractor consists of three convolutional layers, two SimAMs, a Residual module, maximum pooling layers, and activation functions. These layers use filters to convolve over the input, extracting spatial information and learning important patterns. The initial convolutional layer utilizes a 3 × 3 kernel to generate 16 feature channels and applies a rectified linear unit (ReLU) activation function for non-linear transformation. Afterwards, a 2 × 2 max-pooling operation is performed to reduce the dimensionality of the feature map. The subsequent convolutional and maximum pooling layers follow the same structure.
The attention modules could project the data onto multiple attention heads through a series of linear transformations. The attention scores, which indicate the significance or relevance of different elements in the input, are calculated using distinct sets of weights. Previously proposed modules primarily focus on either channel or spatial attention, and their output results are obtained through series or parallel operations. In contrast, SimAM introduces an innovative energy function that directly computes 3D attention weights based on the feature map. This approach removes the need for explicit separation between channel and spatial attention. It can also eliminate the requirement for additional network layer structure design and parameter settings. As a result, it provides a more computationally efficient solution without compromising performance.
When TC is large, the petal features contain more detailed information and pixels. Neural networks with fewer layers may struggle to converge effectively on this complex information. On the other hand, increasing the number of layers can lead to the problems of gradient vanishing or explosion. To address these challenges, we have inserted an additional residual module to increase the model's robustness. In the forward propagation process, residual connections are established by adding the input to the output of the second convolutional layer. The residual connections facilitate the flow of gradients through the network, which allows the network with a higher number of layers to capture more intricate patterns and improve convergence on the petal features with large TC states.
Figure 8 additionally provides the heatmap visualization of the image features extracted by each network layer. The heatmap visualization shows the regions of the input image where the attention of the network is higher. At first, in shallower layers, the network attention may be spread across the entire image, indicating that the network has a broader understanding of the overall features. As the network layers get deeper, the attention becomes more localized, focusing on specific regions of interest. This shrinking and refining of attention suggests that the network is trained to recognize and extract more discerning features from the input image.
4.2 Classifier
The classification component consists of two FC layers, which take the extracted features as input and perform classification based on these features. The ReLU activation function is applied to introduce nonlinearity. To mitigate overfitting, a dropout layer with a 50% dropout rate is inserted between fully connected layers. This regularization technique enhances the model ability to generalize.
4.3 Evaluation metrics for test sets
In this study, we employ four evaluation metrics to assess the performance on the test set: accuracy, precision, recall, and F1 score. These metrics are used to analyze the accuracy and effectiveness of the categorization process. Accuracy is calculated as the ratio of the number of correctly classified instances to the total number of instances, indicating the overall correctness of the predictions. Precision is calculated as the ratio of true positive predictions to the sum of true positive and false positive predictions, which reflecting the model’s ability to accurately identify positive instances. Recall is calculated as the ratio of true positive predictions to the sum of true positive predictions and false negative predictions, which quantifies the proportion of correctly predicted actual correct instances. F1 Score provides an overall assessment of the model’s performance by computing the harmonic mean of precision and recall.
In addition, we compute the confusion matrix for each test to show the performance of our classification model by comparing the predicted labels with the actual labels. Each unit represents the number of instances belonging to a particular class. The positions of the pattern points near the diagonal indicate a good classification performance, suggesting that the model can accurately classify the samples.
5. Identification result and discussion
In our experimental setup, we have observed that the higher-order OAM superimposed beams we captured exhibit relatively high quality. However, the adaptive thresholding method is highly sensitive to variations in lighting conditions. These changes in lighting can negatively impact the integrity of the petal shape pattern, which is crucial for recognition. Traditionally, petal recognition methods rely on petal counts, which can present challenges for human observers when patterns are missing or overlapping. Figure 9 visually demonstrates how turbulence and large attenuation can affect the shape and integrity of the processed petals at a TC value of ±40.
Fig. 9. Petal-shaped patterns missing after image processing using adaptive thresholding method when TC=±40.
Figure 10(a) presents a view of the recognition accuracy of all 40 TCs in the form of a line graph under different attenuation without any turbulence. When the attenuation increases more than 10 dB, more information is lost in the images. It can be predicted that decreasing the attenuation will reduce the clarity and visibility of the petal shape pattern, thus reducing the recognition rate of the network. In real-world situations, various factors such as distance, atmospheric conditions, and signal limitations can result in optical attenuation. Even in situations where the integrity of the petal shape is compromised, the SimAM based CNN enables relevant regions of the input image and extract features with >90% accuracy for under most of intensities. The confusion matrix presented in Fig. 11(c) corresponds to the scenario where the attenuation is 12 dB. In this particular case, the petal image is most severely missing. Consequently, we observe a relatively higher number of wrong classifications, especially in the high-order TC range.
Fig. 10. Accuracy curves of OAM TC identification (a) under different attenuation without turbulence, (b) under different attenuation with turbulence.
Fig. 11. Confusion matrices for 40 classes of TC classification without turbulence using (a) no processing images under any attenuation; (b) processed images under any attenuation; (c) processed images with an attenuation of 12 dB. (d) Confusion matrices for 40 classes of TC classification using processed images under any attenuation with turbulence coherent length of (e) 1e-4 and (f) 1e-5.
When introducing turbulence into the optical path, we incorporate additional equipment. Consequently, the assessment of light intensity cannot be evaluated based on the standard protocol without turbulence. In this study, we consider the first 10 attenuation steps of 0.5 dB. For both types of turbulence, our model consistently achieves a recognition accuracy of over 92%, as Fig. 10(b) shows. We also observe a slight decline in accuracy as the light intensity weakens. Nonetheless, the recognition accuracy remains relatively high, indicating the robustness of our model combined with adaptive thresholding method in handling variations in both turbulence and light intensity.
Furthermore, we categorize all the attenuated images with the same TC into one class. Then we compare the performance of our designed networks aiming at the TC classification of 40 types based on whether the images are processed and whether they are in a turbulent environment. The detailed performance comparison is presented in Table 1. In the absence of turbulence, the accuracy of the model on the unprocessed dataset is observed to be significantly lower, almost reaching the standard for random classification. This suggests that the network faces difficulties in recognizing TCs without any preprocessing. With the introduction of an adaptive thresholding approach, the accuracy improved to 96.60%. Furthermore, our model demonstrates a relatively high accuracy in distinguishing TC under different levels of turbulence.
Table 1. Test results across diverse datasets
By comparing the performances listed in Table 1 and analyzing the confusion matrices depicted in Fig. 11, we can draw the conclusion that the designed network is effective and robust in handling various conditions. These conditions include different light intensities and varying degrees of turbulence. When the field distortion caused by turbulence is significant, the accuracy will slightly decrease, but still remain above 95%.
6. Conclusion
Through experimental demonstrations, we successfully generate and collect high-quality OAM high-order superimposed beams while considering the impact of light intensity and turbulence on recognition performance. The experimental results show that the combination with our network model with adaptive threshold method can achieve over 95% recognition accuracy in most cases for TCs from ±1 to ±40. Even in the absence or partial obstruction of patterns, there can still be an accuracy rate of 80%. The robustness and effectiveness of the parameter-free attention module SimAM in extracting and grasping global features have played a significant role in enhancing the overall performance of our model in the challenging task of flower TC recognition. Compared with traditional CNN and other attention mechanisms, SimAM offers resource savings by eliminating the need for network structure and parameter design. These research findings hold significant application potential for the future of the communication field.
Funding
Shaanxi Province Innovation Talent Promotion Program-Science and Technology Innovation Team (2023-CX-TD-03); Natural Science Foundation of Shanxi Province (2023-JC-YB-587).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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