Spectrum reconstruction method and device based on tensorized unfolding network

Abstract

The application relates to the technical field of signal reconstruction, in particular to a frequency spectrum reconstruction method and device based on a tensorization unfolding network, which comprises the following steps: acquiring a multi-observation compressed signal; performing Hankel-lift transformation on the multi-observation compressed signal to obtain a third-order Hankel tensor; inputting the third-order Hankel tensor into a tensorization unfolding reconstruction network, which is constructed by deep unfolding of a tensor optimization iterative algorithm; and outputting a tensor sparse representation representing a multi-dimensional frequency spectrum structure in response to multi-layer tensor update processing of the tensorization unfolding reconstruction network, wherein in each tensor update layer, a learnable parameter-driven tensor factor and core tensor update operation is performed to obtain an updated Hankel tensor. By embedding the multi-observation compressed signal into a Hankel tensor space, joint modeling and high-precision recovery of cross-channel, cross-time and cross-subband frequency spectrum structures are realized, and problems such as strong dependence on artificial parameters, low convergence efficiency and poor adaptability to complex scenes of traditional tensor reconstruction algorithms are solved.

Description

Technical Field

[0001] This application relates to the field of signal reconstruction technology, and in particular to a spectrum reconstruction method and apparatus based on tensor expansion networks. Background Technology

[0002] With the rapid development of communications, radar, and electronic countermeasures towards ultra-wideband, high-frequency, and multi-mode technologies, the external electromagnetic environment exhibits significant characteristics such as a wide spectral span, high signal density, strong time-varying signals, and low signal-to-noise ratio. Signals in the frequency domain often exhibit quasi-sparse or even weakly sparse structures. In the ultra-wide frequency range of 10MHz–40GHz, traditional receiver systems based on Nyquist sampling theory require ultra-high-speed analog-to-digital converters, resulting in high hardware costs, high power consumption, and complex implementation, making it difficult to meet the comprehensive requirements of real-time performance, flexibility, and scalability in practical systems.

[0003] Compressed sensing theory, leveraging the sparsity of signals in the transform domain, overcomes the Nyquist sampling rate limitation. By constructing low-dimensional measurement models, it can acquire key signal information at low sampling rates, becoming a core technology for spectrum reconstruction and signal recovery. It is widely used in ultra-wideband spectrum monitoring, passive signal reception, and other scenarios. However, as engineering applications evolve towards multi-channel, array observation, and collaborative reception, multi-dimensional spectrum scenarios involving multiple observations and spanning time series are becoming increasingly common. Traditional algorithms that reconstruct signals using vector or matrix forms are no longer adequate for these scenarios and face numerous technical challenges.

[0004] Traditional spectrum reconstruction algorithms are mostly designed for single-channel observation signals, processing observation data in vector or matrix form. They fail to effectively capture the joint sparse structure between multiple observation channels, the consistency of spectral evolution over time, and the correlation between local spectral regions. In scenarios involving multi-channel observations, dense, weakly sparse, and overlapping spectra, severe model mismatch easily occurs, leading to a significant decrease in reconstruction accuracy. To adapt to multi-observation scenarios, some techniques attempt to use traditional tensor optimization iterative algorithms for spectrum reconstruction. By constructing multiple observation signals into tensor form, they utilize the multidimensional structural characteristics of tensors to achieve joint modeling. However, these pure tensor iterative algorithms have significant limitations.

[0005] On the one hand, traditional tensor optimization iterative algorithms require converting multiple observed signals into tensor form and then relying on manually set initialization parameters (such as spectral initialization and random initialization) to generate initial values ​​for tensor factors and core tensors. The initialization process is easily affected by human priors, resulting in poor stability and difficulty in adapting to complex and variable electromagnetic environments. On the other hand, the iterative process of these algorithms uses static parameters such as fixed step size and fixed precondition matrix, requiring manual adjustment to control the iteration stopping conditions. This results in low iteration efficiency, slow convergence speed, and difficulty in obtaining optimal recovery results in complex scenarios such as weak sparsity, dense spectrum, and continuous frequency (off-grid), leading to significant performance degradation.

[0006] Furthermore, existing spectrum reconstruction schemes integrating tensor theory and deep learning often lack in-depth design of tensor optimization iterative algorithms, failing to map the iterative process into a learnable network structure. This results in models that, while possessing some data-driven expressive capabilities, lack clear physical interpretability, making it difficult to balance reconstruction accuracy, computational efficiency, and algorithmic transparency. Simultaneously, these schemes lack learnable update mechanisms, failing to optimize key parameters in the tensor update process through end-to-end training. This makes them ill-suited to adapting to complex changes in the spectrum structure under multi-observation scenarios, leading to insufficient robustness in reconstruction under non-ideal conditions such as low signal-to-noise ratio and multi-channel interference.

[0007] To address the reconstruction needs of multi-channel, multi-observation, and multi-dimensional spectral scenarios, some technologies have attempted to construct tensor quantization reconstruction networks. However, most existing tensor quantization networks do not incorporate deep unfolding techniques, resulting in a disconnect between the network structure and the physical process of tensor optimization, poor interpretability, and a lack of dedicated tensor construction and update mechanisms designed for the characteristics of multi-observation compressed signals. Consequently, they cannot fully utilize the multi-dimensional structural priors of multi-observation signals, and the reconstruction accuracy and efficiency are insufficient to meet the needs of engineering applications. Summary of the Invention

[0008] This application provides a spectrum reconstruction method and apparatus based on tensor expansion networks, which can solve at least one of the technical problems in the background art to a certain extent.

[0009] To achieve the above objectives, this application adopts the following technical solution:

[0010] Firstly, a spectrum reconstruction method based on tensorized expansion networks is provided, the method comprising: Acquire multi-observation compressed signals; Perform a Hankel-lift transform on the multi-observation compressed signal to obtain a third-order Hankel tensor; The third-order Hankel tensor is input into the tensor quantization expansion and reconstruction network, which is constructed by deep expansion of the tensor optimization iterative algorithm and contains at least one tensor update layer. In response to the multi-layer tensor update process of the reconstructed network after tensor expansion, a tensor sparse representation representing the multidimensional spectral structure is output. In each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor.

[0011] Secondly, a spectrum reconstruction device based on tensor expansion network is provided, comprising: The acquisition module is used to acquire multi-observation compressed signals; The transformation module is used to perform a Hankel-lift transformation on the multi-observation compressed signal to obtain a third-order Hankel tensor; The input module is used to input the third-order Hankel tensor into the tensor quantization expansion and reconstruction network, which is constructed by deep expansion of the tensor optimization iterative algorithm and contains at least one tensor update layer. The output module is used to respond to the multi-layer tensor update processing of the reconstructed network after tensor expansion and output a tensor sparse representation of the multi-dimensional spectral structure. In each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor.

[0012] Thirdly, embodiments of this application provide an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the spectrum reconstruction method based on tensor-quantized unfolded networks as described in any one of the first aspects above.

[0013] Fourthly, embodiments of this application provide a computer-readable storage medium storing a computer program that, when executed by a processor, implements the spectrum reconstruction method based on tensor-quantized unfolded networks as described in any one of the first aspects above.

[0014] Fifthly, embodiments of this application provide a computer program product that, when run on an electronic device, causes the electronic device to execute the spectrum reconstruction method based on tensor-quantized expanded networks as described in any of the first aspects above.

[0015] It is understood that the beneficial effects of the second to fifth aspects mentioned above can be found in the relevant descriptions in the first aspect mentioned above, and will not be repeated here.

[0016] In this embodiment, a multi-observation compressed signal is first acquired, then a Hankel-lift transform is performed on the compressed signal to obtain a third-order Hankel tensor. This third-order Hankel tensor is then input into a tensor quantization expansion and reconstruction network. This network is constructed through deep expansion of a tensor optimization iterative algorithm and contains at least one tensor update layer. Responding to the multi-layer tensor update processing by the tensor quantization expansion and reconstruction network, it outputs a tensor sparse representation of the multi-dimensional spectral structure. In each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor. By embedding the multi-observation compressed signal into the Hankel tensor space, combined with low-rank Tucker decomposition and a deep expansion optimization structure, joint modeling and high-precision recovery of cross-channel, cross-time, and cross-subband spectral structures are achieved. This also solves the problems of traditional tensor reconstruction algorithms, such as strong dependence on manual parameters, low convergence efficiency, and poor adaptability to complex scenarios.

[0017] The above description is only an overview of the technical solution of this application. In order to better understand the technical means of this application and to implement it in accordance with the contents of the specification, and to make the above and other objects, features and advantages of this application more obvious and understandable, the following are specific embodiments of this application. Attached Figure Description

[0018] Various other advantages and benefits will become apparent to those skilled in the art upon reading the detailed description of the preferred embodiments below. The accompanying drawings are for illustrative purposes only and are not intended to limit the scope of this application. Furthermore, the same reference numerals denote the same parts throughout the drawings. In the drawings: Figure 1 This is a flowchart illustrating the spectrum reconstruction method based on tensorized expanded networks provided in the embodiments of this application; Figure 2 This is a schematic diagram of the spectrum reconstruction device based on tensor expansion network provided in the embodiments of this application; Figure 3 This is a schematic diagram of the structure of the electronic device provided in the embodiments of this application. Detailed Implementation

[0019] The embodiments of the technical solutions of this application will now be described in detail with reference to the accompanying drawings. The following embodiments are only used to more clearly illustrate the technical solutions of this application, and are therefore merely examples and should not be used to limit the scope of protection of this application. When the following description relates to the accompanying drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. Various changes, modifications, and equivalents of the methods, apparatus, and / or systems described herein will become apparent upon understanding this disclosure. For example, the order of operations described herein is merely illustrative and is not limited to those orders set forth herein, but can be changed as will become apparent upon understanding this disclosure, except for operations that must be performed in a specific order. Furthermore, for clarity and conciseness, descriptions of features known in the art may be omitted.

[0020] The embodiments described in the following examples of this disclosure are not representative of all embodiments consistent with this disclosure. Rather, they are merely examples of apparatuses and methods consistent with some aspects of this disclosure as detailed in the appended claims.

[0021] It should be noted that existing technologies for spectrum reconstruction in multi-observation, multi-dimensional structured spectrum scenarios suffer from problems such as inability to effectively capture multi-dimensional structure correlations, poor tensor initialization stability, static fixation of iterative parameters, insufficient reconstruction accuracy and efficiency, and poor interpretability and scenario adaptability. These issues make it difficult to meet the needs of complex engineering scenarios such as multi-channel observation, dense spectrum, and weakly sparse spectrum. This application proposes a spectrum reconstruction scheme based on tensor expansion networks. Through reasonable tensor construction, deep expansion network structure, and learnable update mechanism, it achieves high-precision, high-efficiency, and robust spectrum reconstruction in multi-observation scenarios to meet the practical needs of engineering applications such as ultra-wideband spectrum monitoring and radar signal processing.

[0022] This embodiment discloses a spectrum reconstruction method based on tensor expansion networks, which is applied to ultra-wideband electromagnetic signal processing, multi-channel spectrum sensing, radar signal recovery, and electromagnetic spectrum monitoring. The core of this method targets multi-observation, multi-channel, and multi-dimensional spectrum scenarios. By embedding multi-observation compressed signals into the Hankel tensor space, combined with low-rank Tucker decomposition and deep expansion optimization, it achieves joint modeling and high-precision recovery of cross-channel, cross-time, and cross-subband spectrum structures. Simultaneously, it addresses the problems of traditional tensor reconstruction algorithms, such as strong dependence on manual parameters, low convergence efficiency, and poor adaptability to complex scenarios. It balances model physical interpretability with data-driven expressive capabilities, improving the robustness and engineering applicability of reconstruction in complex scenarios such as weak sparsity, dense spectrum, and continuous frequency (off-grid).

[0023] See Figure 1This is a flowchart illustrating the spectrum reconstruction method based on tensor-quantized expanded networks provided in this application. The specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0024] like Figure 1 As shown, the spectrum reconstruction method based on tensor-quantized expanded networks provided in this embodiment includes the following steps: Step 101: Obtain the multi-observation compressed signal; Among them, multi-observation compressed signal refers to a set of low-sampling-rate observation signals acquired by a multi-channel compressed sampling module. The sampling rate is lower than the Nyquist sampling rate. It includes ultra-wideband electromagnetic spectrum observation data across time series and observation channels. It can be represented in matrix form, where s is the number of observations (i.e., the number of channels) and n is the sampling length of a single observation channel, which is used as the basic input for subsequent tensor construction.

[0025] Specifically, the multi-observation compressed signal is acquired through a multi-channel compressed sampling module. This sampling module includes multiple parallel-deployed sampling units, signal conditioning circuits, analog-to-digital conversion units, and sampling control modules. Each sampling unit corresponds to an observation channel, enabling synchronous low-rate sampling of ultra-wideband electromagnetic signals (10MHz–40GHz). The sampling rate is set to 1 / 4 to 1 / 10 of the Nyquist sampling rate, which reduces hardware power consumption and cost while preserving the core spectral characteristics of the signal.

[0026] The acquired multi-observation compressed signals are stored in matrix form, denoted as . In this matrix, s represents the number of observations (i.e., the number of sampling channels), ranging from 2 to 32, which can be flexibly adjusted according to actual engineering needs (such as array observation and coordinated reception); n represents the sampling length of a single observation channel, ranging from 1024 to 8192, ensuring that complete spectral evolution characteristics can be captured. In the multi-observation compressed signal matrix, each row corresponds to the sampling sequence of one observation channel, and each column corresponds to the sampling values ​​of different observation channels at the same sampling time, which can completely reflect the signal distribution characteristics across channels and across time.

[0027] It should be noted that during the acquisition process, the sampling parameters of each channel can be synchronously calibrated through the sampling control module to avoid sampling deviations caused by hardware mismatch between channels, and to ensure the consistency and reliability of the multi-observation compressed signal. At the same time, the acquired multi-observation compressed signal can be preprocessed to reduce the interference of non-ideal factors on the subsequent reconstruction process.

[0028] Step 102: Perform a Hankel-lift transform on the multi-observation compressed signal to obtain a third-order Hankel tensor.

[0029] Among them, the Hankel-lift transform is an embedding transformation method that maps multi-observation signals in matrix form to third-order Hankel tensors. This transformation can preserve the structural correlation of multi-observation signals in the time dimension and observation channel dimension, providing a carrier for joint modeling of multi-dimensional spectral structures.

[0030] Among them, the third-order Hankel tensor is a three-dimensional tensor obtained through the Hankel lifting transformation.

[0031] First, the Hankel lift transform module is invoked to read the multi-observation compressed signal matrix obtained in step 101. Subsequently, the dimension parameters n1 and n2 of the third-order Hankel tensor are determined according to the sampling length n, satisfying n1 + n2 - 1 = n. The dimension parameters can be flexibly set according to the spectral resolution requirements, with n1 = n2 (when n is odd) or n1 = n2 + 1 (when n is even) preferred to ensure the symmetry and rationality of the tensor structure. Finally, the multi-observation compressed signal matrix X is embedded into the third-order Hankel tensor through Hankel lifting transformation, denoted as […]. .

[0032] Specifically, for each observation channel (i.e., each row of the multi-observation compressed signal matrix X), a corresponding Hankel matrix is ​​constructed, where the element in the i-th row and j-th column of the Hankel matrix corresponds to the (i+j-1)-th sample value of the sampling sequence. Then, the Hankel matrices corresponding to all observation channels are stacked according to the channel dimension to form a third-order Hankel tensor. Through this transformation, the third-order Hankel tensor can simultaneously capture three core structural features: first, the consistency of spectral evolution in the time dimension (corresponding to the n1×n2 dimension Hankel matrix structure); second, the joint sparse structure among different observation channels (corresponding to the s-dimensional channel stacking); and third, the correlation and structural prior between local spectral regions (corresponding to the local element correlation characteristics of the Hankel matrix).

[0033] Compared with traditional matrix processing, the construction of third-order Hankel tensors can effectively preserve the multidimensional structural information of multi-observation signals, avoid the loss of structural information during matrix processing, and provide a carrier for subsequent low-rank Tucker decomposition and cross-dimensional joint modeling. It can significantly alleviate the performance degradation problem of traditional matrix-type multi-measurement vector (MMV) methods in spectrum-dense or continuous frequency scenarios.

[0034] Step 103: Input the third-order Hankel tensor into the tensor quantization expansion and reconstruction network. The tensor quantization expansion and reconstruction network is constructed by deep expansion of the tensor optimization iterative algorithm and contains at least one tensor update layer. The tensor expansion reconstruction network is constructed by deep expansion of the traditional tensor optimization iterative algorithm. It contains at least one tensor update layer. Each tensor update layer corresponds to one iteration operation of the traditional tensor optimization algorithm. The parameters can be optimized through end-to-end training to achieve adaptive updates of tensor factors and core tensors. Tensor optimization iterative algorithm is a class of iterative solutions to sparse signal reconstruction problems based on tensor structure characteristics. Its core is to approximate the target tensor that satisfies low-rank constraints by iteratively updating tensor factors and core tensors, thereby realizing the spectral reconstruction of multi-observation signals. Deep unrolling involves modularizing and hierarchically mapping each iterative operation of the tensor optimization algorithm, transforming it into a single computational unit (i.e., the tensor update layer) of the tensor unrolled network. This transforms the iterative solution process into the forward propagation process of the network, while preserving the physical meaning of the algorithm and supporting end-to-end parameter optimization. The tensor update layer is the basic computational unit of the tensor expansion reconstruction network. It corresponds to a complete iteration operation of the tensor optimization iterative algorithm. Its core function is to perform learnable parameter-driven tensor factor and core tensor update operations, and output the updated Hankel tensor to provide input for the next layer processing or the final reconstruction result output. The tensor factor is used to characterize the features of the third-order Hankel tensor in different dimensions. It includes three modulus factor matrices: frequency modulus factor matrix L, time modulus factor matrix R, and observation channel modulus factor matrix V. The Hankel tensor can be reconstructed by multiplying the three with the core tensor through the Tucker tensor. Among them, the core tensor, denoted as S, is the core component obtained by Tucker decomposition of the third-order Hankel tensor. It has low Tucker rank and is used to characterize the interaction relationship between the three tensor factors. The size of its rank determines the structural complexity and sparsity of the Hankel tensor. The learning-based initialization module is the initialization unit of the MSFA-Tensor network. It is a dedicated module for generating the initial values ​​of tensor factors and core tensors. It takes the third-order Hankel tensor and the observation mask operator as input and directly predicts the initial factors through a data-driven approach, replacing the traditional manually set spectral initialization or random initialization, thereby improving the stability and rationality of the initialization.

[0035] Among them, the MSFA-Tensor network is based on the MSFA-Net framework. It adopts a multi-scale learnable dictionary, ISTA deep expansion structure and frequency attention-enhanced soft threshold sparsity mechanism to achieve high-precision and interpretable spectrum estimation and signal reconstruction in single-channel or simple multi-observation scenarios. The MSFA-Tensor network is extended with tensor quantization to enhance its adaptability to multi-channel and multi-dimensional structural scenarios.

[0036] Optionally, before inputting the third-order Hankel tensor into the tensor expansion reconstruction network, the method further includes: taking the third-order Hankel tensor and the observation mask operator as input, generating initial values ​​for tensor factors and core tensors through a learning initialization module. The tensor factors include frequency modulus factor matrix, time modulus factor matrix, and observation channel modulus factor matrix.

[0037] In each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor.

[0038] Optionally, based on the learnable update operator of the current tensor update layer, tensor update calculation is performed with the initial value of the tensor factor or the tensor factor, core tensor, third-order Hankel tensor, and observation mask operator output from the previous layer as input; wherein, the learnable update operator contains learnable parameters for layer adaptation, and the learnable parameters include at least one of the following: learning step size, preconditioning scaling factor, shrinkage strength, and threshold parameter.

[0039] Optionally, the tensor residual between the current Hankel tensor and the original third-order Hankel tensor is calculated. The current Hankel tensor is obtained by multiplying the current tensor factor and the core tensor through the Tucker tensor. Based on the tensor residual and the learnable parameters of the current layer, gradient-type updates are performed on the frequency modulus factor matrix, time modulus factor matrix, observation channel modulus factor matrix and core tensor respectively to obtain the updated tensor factor and core tensor.

[0040] Optionally, a low-rank constraint is applied to the updated core tensor to preserve its low-rank characteristics. The low-rank constrained core tensor is then fused with the updated tensor factor through the Tucker tensor product to obtain the updated Hankel tensor.

[0041] The tensor expansion reconstruction network is constructed by deep expansion of the traditional tensor optimization iterative algorithm. The number of layers K in the network matches the number of iterations of the traditional tensor optimization algorithm. The value of K ranges from 5 to 30 and can be flexibly adjusted according to the requirements of reconstruction accuracy and computational efficiency. The network contains at least one tensor update layer. Each tensor update layer corresponds to one complete iteration operation of the traditional tensor optimization iterative algorithm. It retains the physical interpretability of the traditional algorithm and combines the advantages of end-to-end parameter optimization in deep learning. It can effectively overcome the limitations of traditional algorithms, such as strong dependence on manual parameters and low convergence efficiency.

[0042] Among them, under continuous frequency and structured sparsity conditions, the third-order Hankel tensor satisfies the low-Tucker rank property, and can be expressed as follows: , where L, R, and V are the frequency modulus factor matrix, time modulus factor matrix, and observation channel modulus factor matrix, respectively, S is the core tensor, and (.) is the Tucker tensor product.

[0043] The core optimization objective of tensor quantization expansion for network reconstruction is low-rank Hankel tensor completion, namely:

[0044]

[0045] in This represents the observation mask operator. Let r be the original third-order Hankel tensor, r be the Tucker rank of the core tensor, and ||·||F² be the square of the Frobenius norm. This optimization objective ensures that the reconstructed Hankel tensor fits the original observation signal while maintaining its low-rank sparsity.

[0046] Before inputting the third-order Hankel tensor into the tensor expansion and reconstruction network, a learning-based initialization process can be performed. The specific implementation is as follows: The learning-based initialization module is called, using the third-order Hankel tensor and observation mask operator obtained in step 102 as input. The learning-based initialization module uses a pre-defined lightweight neural network (such as a convolutional neural network or a fully connected network) to extract and map features from the input Hankel tensor and observation mask operator, directly predicting the initial values ​​of the generated tensor factors and core tensor. The tensor factors include the initial value L1 of the frequency modulus factor matrix, the initial value R1 of the time modulus factor matrix, and the initial value V1 of the observation channel modulus factor matrix; the initial value of the core tensor is S1. The neural network of the learning-based initialization module is trained on a large-scale labeled dataset. During training, the tensor factors and core tensor corresponding to the standard Hankel tensor are used as labels, and the mean squared error between the predicted value and the label is used as the loss function. The network parameters are iteratively optimized using the backpropagation algorithm to ensure that the initialization module can output initial factors that match the actual spectral structure. After initialization, the learning initialization module transmits the generated initial values ​​of tensor factors (L1, R1, V1) and the initial value of core tensor S1 to the first tensor update layer of the tensor expansion reconstruction network as the initial input for tensor update calculation.

[0047] In this network, each tensor update layer is configured with a learnable update operator and a parameter storage unit. The core performs tensor factor and core tensor update operations driven by learnable parameters to obtain the updated Hankel tensor. The specific process is as follows: First, the learnable update operator of the current tensor update layer is called. Obtain input data, which includes: if it is the first tensor update layer, the input data is the initial value of the tensor factor (L) generated by the learning initialization module. k R k V k ), initial value S of the core tensor kThe input data includes the third-order Hankel tensor and the observation mask operator obtained in step 102; if it is not the first tensor update layer, the input data is the updated tensor factor (L) output by the previous tensor update layer. k R k V k ), updated core tensor S k , as well as the third-order Hankel tensor and the observation mask operator.

[0048] Among them, the learnable update operator Learnable parameters θ including hierarchical adaptation k The learnable parameters are stored in the parameter storage unit. The learnable parameters include at least one of the following: learning step size, preconditioning scaling factor, shrinkage strength, and threshold parameter. The learnable parameters of different tensor update layers are independent of each other. Layered optimization is achieved through end-to-end training to adapt to the optimization needs of different update stages.

[0049] To ensure that the updated core tensor retains its low Tucker rank property and to avoid structural distortion and overfitting during reconstruction, the updated tensor factor (L) is obtained. k+1 R k+1 V k+1 ) and the updated core tensor S k+1 Afterwards, low-rank constraint processing and tensor fusion operations can be performed, as follows: First, the low-rank constraint processing subunit is invoked to update the core tensor S. k+1 To perform low-rank constraint processing, this embodiment preferentially adopts the kernel tensor norm thresholding method. By setting a threshold parameter, the kernel tensor S is subjected to low-rank constraints. k+1 The singular values ​​are thresholded, setting singular values ​​below the threshold to 0 and retaining those greater than or equal to the threshold. This preserves the low-rank characteristics of the core tensor, resulting in the low-rank constrained core tensor S. k+1 '.

[0050] Subsequently, the tensor fusion subunit is invoked, and the low-rank constrained core tensor S is fused using the Tucker tensor product. k+1 ', and the updated tensor factor (L k+1 R k+1 V k+1 The fusion is performed to obtain the updated Hankel tensor, i.e.

[0051] The updated Hankel tensor will be transmitted to the next tensor update layer as the output of the current tensor update layer, serving as the input for the tensor update calculation of the next layer. If the current tensor update layer is the last layer of the tensor expansion reconstruction network, the updated Hankel tensor will be transmitted to step 104 as the basis for tensor sparse representation.

[0052] It should be noted that the unfolded network achieves hierarchical adaptive optimization trajectory learning. Early layers are responsible for fast coarse recovery and error-driven search; middle layers are responsible for structural consistency and cross-channel correlation enhancement; and later layers are responsible for fine convergence, noise suppression, and spectral peak refinement. This results in faster error reduction and lower termination error under the same computational budget. Through this hierarchical differentiated optimization design, the tensor unfolded reconstruction network achieves faster error reduction and lower termination error under the same computational budget, significantly outperforming traditional fixed-parameter iterative tensor optimization algorithms.

[0053] Step 104: In response to the multi-layer tensor update process of the reconstructed network after tensor expansion, output a tensor sparse representation representing the multi-dimensional spectral structure.

[0054] Optionally, the tensor sparse representation is subjected to Tucker tensor inverse decomposition to obtain a set of multi-channel spectral sparse matrices, which contains the frequency-time spectral sparse representations for each observation channel.

[0055] Specifically, the tensor sparse representation can be decomposed into three tensor factors (frequency modulus factor matrix L, time modulus factor matrix R, and observation channel modulus factor matrix V) and a core tensor S using the Tucker tensor inverse decomposition algorithm. Finally, based on the decomposed tensor factors, a multi-channel spectral sparse matrix set is constructed. This set contains s spectral sparse matrices (s being the number of observations, i.e., the number of channels), each corresponding to one observation channel. The matrix dimension is frequency × time, clearly reflecting the distribution characteristics and sparsity of the spectral signal within that channel at different frequencies and time points, providing intuitive data support for subsequent spectrum monitoring and signal identification.

[0056] Optionally, the reconstruction error and spectral false negative rate of the tensor sparse representation are calculated. If the reconstruction error or spectral false negative rate exceeds a preset threshold, the number of layers of the tensor expansion reconstruction network, the update step size of the learnable parameters, or the output dimension of the learning initialization module are dynamically adjusted, and the tensor update process is re-executed.

[0057] This application embeds multi-observation compressed signals into a third-order Hankel tensor space through Hankel lifting transform, effectively preserving multi-channel and cross-time structural correlations. It maps traditional tensor optimization iterative algorithms into a learnable network structure using deep expansion techniques, balancing physical interpretability and data-driven expressive power. Learning-based initialization replaces traditional manual initialization, improving initialization stability. Hierarchical learnable updates and low-rank constraints achieve rapid convergence and high-precision reconstruction. Optional inverse decomposition and closed-loop feedback adjustment further enhance engineering applicability and robustness. It is suitable for complex application scenarios such as multi-channel observation, weak sparseness, dense spectrum, and continuous frequencies. Compared with traditional matrix-based reconstruction methods and pure tensor iterative methods, it significantly improves reconstruction accuracy, convergence efficiency, and scenario adaptability, and can be widely applied in practical engineering fields such as radar, communication, and spectrum monitoring. This method elevates observation data from the matrix domain to the Hankel-tensor space and combines a low-rank Tucker structure with a deep expansion optimization framework to achieve joint modeling of cross-channel correlations, local spectral consistency, and weakly sparse structures. Compared to single-channel or vectorized reconstruction methods, MSFA-Tensor offers significant advantages in the following application scenarios: it is suitable for multi-channel compressed sampling, array observation, and cooperative reception systems; it achieves higher recovery accuracy under both dense and weakly sparse spectral conditions; it exhibits more stable convergence behavior in scenarios with noise interference and structural mismatch; and it can share structural constraints across different dimensions, thereby improving overall spectral recovery consistency and structural representation capabilities. MSFA-Tensor significantly outperforms traditional matrix-based methods in cross-channel modeling capabilities, structural representation capabilities, and robustness, providing a more robust and reliable algorithmic foundation for the application of engineering systems in complex broadband electromagnetic environments.

[0058] In this embodiment, a multi-observation compressed signal is first acquired, then a Hankel-lift transform is performed on the compressed signal to obtain a third-order Hankel tensor. This third-order Hankel tensor is then input into a tensor quantization expansion and reconstruction network. This network is constructed through deep expansion of a tensor optimization iterative algorithm and contains at least one tensor update layer. Responding to the multi-layer tensor update processing by the tensor quantization expansion and reconstruction network, it outputs a tensor sparse representation of the multi-dimensional spectral structure. In each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor. By embedding the multi-observation compressed signal into the Hankel tensor space, combined with low-rank Tucker decomposition and a deep expansion optimization structure, joint modeling and high-precision recovery of cross-channel, cross-time, and cross-subband spectral structures are achieved. This also solves the problems of traditional tensor reconstruction algorithms, such as strong dependence on manual parameters, low convergence efficiency, and poor adaptability to complex scenarios.

[0059] Reference Figure 2The spectrum reconstruction device 200 based on tensor expansion network includes: Acquisition module 210 is used to acquire multi-observation compressed signals; Transformation module 220 is used to perform Hankel-lift transformation on the multi-observation compressed signal to obtain a third-order Hankel tensor; The input module 230 is used to input the third-order Hankel tensor into the tensor quantization expansion and reconstruction network, which is constructed by deep expansion of the tensor optimization iterative algorithm and contains at least one tensor update layer. Output module 240 is configured to output a tensor sparse representation of the multidimensional spectral structure in response to the multi-layer tensor update processing of the reconstructed network after tensor expansion, wherein, in each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor.

[0060] Optionally, the input module is further configured to: Using the third-order Hankel tensor and the observation mask operator as input, the initial values ​​of the tensor factors and the core tensor are generated through the learning initialization module. The tensor factors include the frequency modulus factor matrix, the time modulus factor matrix, and the observation channel modulus factor matrix.

[0061] Optionally, the output module is further configured to: Perform Tucker tensor inverse decomposition on the tensor sparse representation to obtain a multi-channel spectral sparse matrix set, which contains the frequency-time spectral sparse representation for each observation channel.

[0062] Optionally, the output module is further configured to: Based on the learnable update operator of the current tensor update layer, tensor update calculation is performed with the initial value of the tensor factor or the tensor factor and core tensor output from the previous layer, as well as the third-order Hankel tensor and observation mask operator as input; wherein, the learnable update operator includes learnable parameters for layer adaptation, and the learnable parameters include at least one of learning step size, preconditioning scaling factor, shrinkage strength and threshold parameter.

[0063] Optionally, the learnable update operator based on the current tensor update layer, taking the initial value of the tensor factor or the tensor factor output from the previous layer, the core tensor, the third-order Hankel tensor, and the observation mask operator as input, performs tensor update calculation, including: Calculate the tensor residual between the current Hankel tensor and the original third-order Hankel tensor, wherein the current Hankel tensor is obtained by multiplying the current tensor factor and the core tensor through the Tucker tensor; Based on the tensor residuals and the learnable parameters of the current layer, gradient-type updates are performed on the frequency modulus factor matrix, time modulus factor matrix, observation channel modulus factor matrix, and core tensor respectively to obtain the updated tensor factors and core tensor.

[0064] Optionally, the output module is also used for: The updated core tensor is subjected to low-rank constraints to preserve its low-rank characteristics. The updated Hankel tensor is obtained by fusing the low-rank constrained core tensor with the updated tensor factor through the Tucker tensor product.

[0065] Optionally, the output module is further configured to: Perform Tucker tensor inverse decomposition on the tensor sparse representation to obtain a multi-channel spectral sparse matrix set, which contains the frequency-time spectral sparse representation for each observation channel.

[0066] Optionally, the output module is further configured to: Calculate the reconstruction error and spectral false negative rate of the tensor sparse representation; If the reconstruction error or the false negative rate of the spectrum exceeds a preset threshold, the number of layers of the tensor expansion reconstruction network, the update step size of the learnable parameters, or the output dimension of the learning initialization module are dynamically adjusted, and the tensor update process is re-executed.

[0067] In this embodiment, a multi-observation compressed signal is first acquired, then a Hankel-lift transform is performed on the compressed signal to obtain a third-order Hankel tensor. This third-order Hankel tensor is then input into a tensor quantization expansion and reconstruction network. This network is constructed through deep expansion of a tensor optimization iterative algorithm and contains at least one tensor update layer. Responding to the multi-layer tensor update processing by the tensor quantization expansion and reconstruction network, it outputs a tensor sparse representation of the multi-dimensional spectral structure. In each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor. By embedding the multi-observation compressed signal into the Hankel tensor space, combined with low-rank Tucker decomposition and a deep expansion optimization structure, joint modeling and high-precision recovery of cross-channel, cross-time, and cross-subband spectral structures are achieved. This also solves the problems of traditional tensor reconstruction algorithms, such as strong dependence on manual parameters, low convergence efficiency, and poor adaptability to complex scenarios.

[0068] in addition, Figure 2 The spectrum reconstruction device based on tensor expansion network shown can be a software unit, a hardware unit, or a combination of software and hardware built into existing electronic devices. It can also be integrated into electronic devices as an independent component or exist as an independent electronic device.

[0069] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the above-described division of functional units and modules is merely an example. In practical applications, the above functions can be assigned to different functional units and modules as needed, that is, the internal structure of the device can be divided into different functional units or modules to complete all or part of the functions described above. The functional units and modules in the embodiments can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit. Furthermore, the specific names of the functional units and modules are only for easy differentiation and are not intended to limit the scope of protection of this application. The specific working process of the units and modules in the above system can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.

[0070] Figure 3 This is a schematic diagram of the structure of the electronic device provided in an embodiment of this application. For example... Figure 3 As shown, the electronic device 5 of this embodiment includes: at least one processor 50 ( Figure 3 (Only one is shown in the diagram) a processor, a memory 51, and a computer program 52 stored in the memory 51 and executable on the at least one processor 50, wherein the processor 50 executes the computer program 52 to implement the steps in any of the above embodiments of the spectrum reconstruction method based on tensor quantization unfolded networks.

[0071] The electronic device may be a desktop computer, laptop, handheld computer, or cloud server, etc. This electronic device may include, but is not limited to, a processor and memory. Those skilled in the art will understand that... Figure 3 This is merely an example of electronic device 5 and does not constitute a limitation on electronic device 5. It may include more or fewer components than shown in the figure, or combine certain components, or different components. For example, it may also include input / output devices, network access devices, etc.

[0072] The processor 50 may be a central processing unit, or it may be other general-purpose processors, digital signal processors, application-specific integrated circuits, off-the-shelf programmable gate arrays or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. The general-purpose processor may be a microprocessor or any conventional processor.

[0073] In some embodiments, the memory 51 may be an internal storage unit of the electronic device 5, such as a hard disk or memory of the electronic device 5. In other embodiments, the memory 51 may be an external storage device of the electronic device 5, such as a plug-in hard disk, smart memory card, secure digital card, flash memory card, etc., equipped on the electronic device 5. Further, the memory 51 may include both internal storage units and external storage devices of the electronic device 5. The memory 51 is used to store operating systems, applications, boot loaders, data, and other programs, such as the program code of the computer program. The memory 51 can also be used to temporarily store data that has been output or will be output.

[0074] This application also provides a computer-readable storage medium storing a computer program that, when executed by a processor, can implement the steps in the above-described method embodiments.

[0075] This application provides a computer program product that, when run on an electronic device, enables the electronic device to implement the steps described in the various method embodiments above.

[0076] If the integrated unit is implemented as a software functional unit and used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a computer-readable storage medium, and when executed by a processor, it can implement the steps of the various method embodiments described above. The computer program includes computer program code, which can be in the form of source code, object code, executable files, or certain intermediate forms. The computer-readable medium can include at least: any entity or device capable of carrying computer program code to a device / electronic device, a recording medium, a computer memory, a read-only memory, a random access memory, an electrical carrier signal, a telecommunication signal, and a software distribution medium. Examples include USB flash drives, portable hard drives, magnetic disks, or optical disks.

[0077] In the above embodiments, the descriptions of each embodiment have different focuses. For parts that are not described in detail or recorded in a certain embodiment, please refer to the relevant descriptions of other embodiments.

[0078] Those skilled in the art will recognize that the units and algorithm steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, or a combination of computer software and electronic hardware. Whether these functions are implemented in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of this application.

[0079] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains; the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the application; the terms “comprising” and “having”, and any variations thereof, in the specification, claims, and foregoing description of the drawings are intended to cover non-exclusive inclusion.

[0080] In this document, the term "embodiment" means that a particular feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of this application. The appearance of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a separate or alternative embodiment mutually exclusive with other embodiments. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described herein can be combined with other embodiments.

[0081] The above-described embodiments are only used to illustrate the technical solutions of this application, and are not intended to limit them. Although this application has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of this application, and should all be included within the protection scope of this application.

Claims

1. A spectrum reconstruction method based on tensor-quantized expanded networks, characterized in that, include: Acquire multi-observation compressed signals; Perform a Hankel-lift transform on the multi-observation compressed signal to obtain a third-order Hankel tensor; The third-order Hankel tensor is input into the tensor quantization expansion and reconstruction network, which is constructed by deep expansion of the tensor optimization iterative algorithm and contains at least one tensor update layer. In response to the multi-layer tensor update process of the reconstructed network after tensor expansion, a tensor sparse representation representing the multidimensional spectral structure is output. In each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor.

2. The method according to claim 1, characterized in that, Before inputting the third-order Hankel tensor into the tensor expansion and reconstruction network, the method further includes: Using the third-order Hankel tensor and the observation mask operator as input, the initial values ​​of the tensor factors and the core tensor are generated through the learning initialization module. The tensor factors include the frequency modulus factor matrix, the time modulus factor matrix, and the observation channel modulus factor matrix.

3. The method according to claim 2, characterized in that, In each of the tensor update layers, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor, including: Based on the learnable update operator of the current tensor update layer, tensor update calculation is performed with the initial value of the tensor factor or the tensor factor and core tensor output from the previous layer, as well as the third-order Hankel tensor and observation mask operator as input; wherein, the learnable update operator includes learnable parameters for layer adaptation, and the learnable parameters include at least one of learning step size, preconditioning scaling factor, shrinkage strength and threshold parameter.

4. The method according to claim 3, characterized in that, The learnable update operator based on the current tensor update layer, taking the initial value of the tensor factor or the tensor factor output from the previous layer, the core tensor, the third-order Hankel tensor, and the observation mask operator as input, performs tensor update calculations, including: Calculate the tensor residual between the current Hankel tensor and the original third-order Hankel tensor, wherein the current Hankel tensor is obtained by multiplying the current tensor factor and the core tensor through the Tucker tensor; Based on the tensor residuals and the learnable parameters of the current layer, gradient-type updates are performed on the frequency modulus factor matrix, time modulus factor matrix, observation channel modulus factor matrix, and core tensor respectively to obtain the updated tensor factors and core tensor.

5. The method according to claim 4, characterized in that, Also includes: The updated core tensor is subjected to low-rank constraints to preserve its low-rank characteristics. The updated Hankel tensor is obtained by fusing the low-rank constrained core tensor with the updated tensor factor through the Tucker tensor product.

6. The method according to claim 1, characterized in that, After the tensor sparse representation of the output characterizing the multidimensional spectral structure is described, the method further includes: performing Tucker tensor inverse decomposition on the tensor sparse representation to obtain a set of multi-channel spectral sparse matrices, wherein the set of multi-channel spectral sparse matrices contains the frequency-time spectral sparse representations for each observation channel.

7. The method according to claim 1, characterized in that, Following the tensor sparse representation of the output characterizing the multidimensional spectral structure, the following is also included: Calculate the reconstruction error and spectral false negative rate of the tensor sparse representation; If the reconstruction error or the false negative rate of the spectrum exceeds a preset threshold, the number of layers of the tensor expansion reconstruction network, the update step size of the learnable parameters, or the output dimension of the learning initialization module are dynamically adjusted, and the tensor update process is re-executed.

8. A spectrum reconstruction device based on tensor-quantized expansion network, characterized in that, include: The acquisition module is used to acquire multi-observation compressed signals; The transformation module is used to perform a Hankel-lift transformation on the multi-observation compressed signal to obtain a third-order Hankel tensor; The input module is used to input the third-order Hankel tensor into the tensor quantization expansion and reconstruction network, which is constructed by deep expansion of the tensor optimization iterative algorithm and contains at least one tensor update layer. The output module is used to respond to the multi-layer tensor update processing of the reconstructed network after tensor expansion and output a tensor sparse representation of the multi-dimensional spectral structure. In each tensor update layer, learnable parameter-driven tensor factor and core tensor update operations are performed to obtain the updated Hankel tensor.

9. The apparatus according to claim 8, characterized in that, The input module is also used for: Using the third-order Hankel tensor and the observation mask operator as input, the initial values ​​of the tensor factors and the core tensor are generated through the learning initialization module. The tensor factors include the frequency modulus factor matrix, the time modulus factor matrix, and the observation channel modulus factor matrix.

10. The apparatus according to claim 8, characterized in that, The output module is also used for: Perform Tucker tensor inverse decomposition on the tensor sparse representation to obtain a multi-channel spectral sparse matrix set, which contains the frequency-time spectral sparse representation for each observation channel.

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