A method for underwater acoustic OTFS off-grid channel estimation based on adaptive non-uniform grid refinement

Abstract

The application discloses a kind of water acoustic OTFS off-grid channel estimation methods based on adaptive non-uniform grid refinement, first acquisition OTFS received signal and demodulation, according to pilot symbol region constructs pilot observation vector;Coarse integer time delay-doppler grid and its perception matrix are constructed and initial channel estimation is carried out, to determine effective coarse support set;According to the degree of dispersion of local doppler energy of candidate support and the degree of residual correlation, the path set to be refined is screened;Around the candidate support in the path set to be refined, construct local fractional doppler refinement atom and merge with coarse integer grid atom, form local non-uniform enhanced dictionary;Support auxiliary sparse bayesian re-estimation is carried out on enhanced dictionary.The present application only introduces fractional doppler refinement atom in the local area where off-grid mismatch may exist, while controlling the size of dictionary and computational complexity, effectively alleviates the energy leakage and off-grid mismatch problem caused by fractional doppler in water acoustic OTFS system, and improves the channel estimation accuracy.

Description

Technical Field

[0001] This invention belongs to the field of underwater acoustic communication technology, specifically relating to an off-grid channel estimation method for Orthogonal Time Frequency Space (OTFS) underwater acoustic communication systems. Background Technology

[0002] Underwater acoustic communication is a crucial technology for long-distance underwater information transmission, marine observation, resource development, and equipment coordination. However, due to factors such as low sound speed, limited bandwidth, complex boundary reflections, and relative motion between the transmitter and receiver, underwater acoustic channels typically exhibit severe multipath spread, significant Doppler effects, and strong time-varying characteristics, increasing the difficulty of channel estimation and equalization detection at the receiver. Orthogonal time-frequency spatial modulation (OTFS) is a two-dimensional modulation technique that carries and processes information in the time-delay-Doppler domain. It can map time-varying channels to a relatively stable time-delay-Doppler domain, enabling double-spread channels to exhibit a sparse or near-sparse structure. Therefore, OTFS provides an effective approach for high-dynamic channel processing in underwater acoustic mobile communication.

[0003] In underwater acoustic OTFS systems, accurately acquiring time-delay-Doppler domain channel state information is crucial for reliable detection. Existing channel estimation methods typically construct a sensing matrix based on a pre-defined integer time-delay-Doppler grid and recover the sparse channel using methods such as compressed sensing, approximate message passing, or sparse Bayesian learning. However, due to limitations in frame length, bandwidth, and Doppler resolution, the Doppler parameters of the actual path often do not strictly fall on integer grid points. Even after Doppler compensation or time-domain resampling, residual Doppler may still exist, causing the actual path to deviate from the pre-defined grid.

[0004] When fractional Doppler shift exists, the energy of a single physical path diffuses along the Doppler dimension to adjacent grid cells, leading to a decrease in the sparsity of the time-delay-Doppler domain channel. In this case, a model mismatch exists between the fixed-integer grid sensing matrix and the actual channel response, easily causing support misjudgments, energy leakage, and estimation error propagation, thereby reducing the performance of subsequent equalization detection.

[0005] To mitigate off-mesh mismatch, a denser global finer mesh can be constructed across the entire time-delay-Doppler domain. However, global uniform refinement significantly increases the dictionary size and introduces a large number of redundant atoms unrelated to the actual path, leading to increased computational complexity. Furthermore, an overly dense dictionary enhances inter-atomic correlations, making sparse recovery or Bayesian estimation more prone to slower convergence, support confusion, and noise amplification.

[0006] In recent years, researchers have proposed various non-uniform grid schemes to reduce dictionary size. For example, some literature proposes a non-uniform dynamic grid partitioning strategy based on exponential growth, allocating higher resolution in high Doppler frequency regions and maintaining lower sampling density in low Doppler frequency regions. Other literature proposes an off-grid sparse Bayesian inference method based on grid evolution, which evolves an initial uniform coarse grid into a non-uniform dense grid. However, the above methods still have the following shortcomings: the grid structure of the exponentially growing non-uniform grid is a pre-defined global pattern, and the refinement position cannot be dynamically adjusted according to the feedback of the channel estimation results; the grid evolution method adopts a "grid point migration" strategy, which carries the risk of losing correct integer support during the migration process, and the global evolution still involves updating a large number of grid points. In addition, the above methods do not fully utilize the supporting information provided in the coarse estimation stage to construct supporting priors in the re-estimation stage, nor do they provide an effective low-complexity solution to the problem of enhanced dictionary correlation after local refinement.

[0007] Therefore, how to improve the ability of the OTFS system to represent fractional Doppler off-grid paths while controlling the dictionary size, and to enhance the robustness and computational efficiency of channel estimation, is an important problem that needs to be solved in current OTFS channel estimation. Summary of the Invention

[0008] The purpose of this invention is to provide an underwater acoustic OTFS off-grid channel estimation method based on adaptive non-uniform grid refinement, so as to solve the problems of model mismatch, energy leakage and support misjudgment in existing fixed integer grid channel estimation methods under fractional Doppler conditions, while avoiding the dictionary size expansion and increased computational complexity caused by global uniform refinement.

[0009] The technical solution of this invention is as follows:

[0010] An underwater acoustic OTFS off-grid channel estimation method based on adaptive non-uniform grid refinement includes the following steps:

[0011] Step S1: Received signal processing and pilot observation construction: Obtain the OTFS received signal, and perform synchronization, Doppler compensation, down-conversion and OTFS demodulation on the OTFS received signal to obtain the time delay-Doppler domain received signal; extract local observation samples according to the region where the pilot symbols are located in the OTFS frame, and vectorize the local observation samples to obtain the pilot observation vector;

[0012] Step S2, Construction and initial estimation of coarse integer grid channel estimation model: Construct a coarse integer time delay-Doppler grid and its corresponding coarse integer grid sensing matrix, and perform initial channel estimation on the coarse integer grid sensing matrix to obtain the coarse integer grid channel estimation result;

[0013] Step S3, Effective coarse support selection: Determine the effective coarse support set based on the coarse integer grid channel estimation results;

[0014] Step S4, Path selection to be refined: Based on the local Doppler energy diffusion degree and residual correlation degree of the candidate supports in the effective coarse support set, the path selection to be refined is obtained;

[0015] Step S5, Local fractional Doppler mesh refinement: Construct local fractional Doppler refinement atoms around the candidate supports in the set of paths to be refined;

[0016] Step S6, Construction of Local Non-Uniform Enhancement Dictionary: The local fractional Doppler refinement atoms are merged with coarse integer grid atoms to form a local non-uniform enhancement dictionary;

[0017] Step S7, Support-Auxiliary Sparse Bayes Re-estimation and Channel Output: Perform support-aided sparse Bayes re-estimation on the locally non-uniform enhancement dictionary to obtain the channel estimation result on the enhancement grid; recover the time delay-Doppler off-grid channel estimation result based on the channel estimation result on the enhancement grid.

[0018] In a further preferred embodiment, in step S2, the coarse integer time delay-Doppler grid includes a coarse integer time delay grid and a coarse integer Doppler grid; the coarse integer time delay grid is... The coarse integer Doppler grid is ,in For the maximum delay tap, The largest integer Doppler tap; the coarse integer grid sensing matrix It is composed of pilot observation response atoms of each candidate grid point in the coarse integer time delay-Doppler grid arranged in columns.

[0019] In a further preferred embodiment, in step S2, a method is established based on the pilot observation vector. Coarse integer grid-aware matrix Coarse integer grid channel coefficient vector and observation noise Composition of coarse integer grid channel estimation model ,in The modeling error is caused by fractional Doppler offset grid mismatch; the initial channel estimation is performed using a unitary approximation message-passing sparse Bayesian learning method to obtain coarse integer grid channel estimation results. And based on the pilot observation vector Reconstructing signals with coarse integer grid The difference between them gives a rough estimate of the residual. .

[0020] In a further preferred embodiment, in step S3, the effective coarse support set is determined based on the energy distribution of the coarse integer grid channel estimation results; specifically, this includes: [the process of] calculating the coarse integer grid channel estimation results... Rearranged into a two-dimensional coarse estimation matrix Construct a coarse estimate of the energy distribution map , Indicates the delay index is Doppler index is coarse integer grid cell energy, Two-dimensional coarse estimation matrix Medium delay index is Doppler index is Elements; calculate the energy of each coarse integer grid atom. And sorted in descending order. Coarse integer grid channel estimation results Middle Channel coefficient estimates for each candidate grid point; effective coarse support number determined based on cumulative energy criterion. To obtain an effective coarse support set The cumulative energy criterion specifies conditions that must be met. The minimum number of supports is denoted as ,in The cumulative energy threshold, For coarse integer grid size, The number of candidate supports to meet the cumulative energy requirements, The number after sorting in descending order The energy of a coarse integer grid atom It is a very small positive number. , To support the lower limit, To support the upper limit of the number.

[0021] In a further preferred embodiment, in step S4, the set of paths to be refined is determined based on the local Doppler energy diffusion index and the residual correlation index; the local Doppler energy diffusion index... The energy level is determined by the ratio of the sum of the energies of the adjacent Doppler cells at the time delay tap where the candidate support is located to the energy of the principal Doppler cell. ,in For the candidate support Doppler neighborhood set, Support for candidates Coarse integer time delay-Doppler grid position, Let be the radius of the Doppler neighborhood. It is a very small positive number; the residual correlation index The response atoms of candidate supports and their adjacent Doppler units and the coarsely estimated residuals The normalized correlation between them is determined, i.e. ,in Position in coarse integer grid The corresponding response atom, This indicates the conjugate transpose.

[0022] In a further preferred embodiment, in step S4, the local Doppler energy diffusion index is... Residual related indicators Normalization is performed to obtain the normalized index. and And construct an equilibrium mesh mismatch score based on the geometric mean of the two. The set of paths to be refined is: ,in To refine the decision threshold.

[0023] In a further preferred embodiment, in step S5, for the set of paths to be refined... Candidate support Its normalization refinement priority is Its local refinement order is ,in To achieve the maximum refinement order, For floor operations; fractional Doppler refinement atoms are generated in the local Doppler neighborhood near the candidate support, and the set of local refinement Dopplers is: ,in The coordinates are the refined Doppler grid. Support for candidates Coarse integer Doppler index, For fractional Doppler fine-tuning, satisfying , For a set of integers, For the largest integer Doppler tap; construct the first... Local refinement delay near the support to be refined - Doppler mesh Refine all local meshes With coarse integer delay-Doppler grid Merging yields an enhanced delay-Doppler grid. .

[0024] In a further preferred embodiment, in step S6, for the enhanced delay-Doppler grid... Each grid point in Construct the response vector within its pilot observation region. Arrange all the enhanced grid response vectors column-wise to obtain the locally non-uniform enhanced sensing matrix. ,in To increase the number of atoms in the grid, Establish an enhanced dictionary observation model ,in For pilot observation vectors, To enhance the channel coefficient vector on the grid, To model observation noise and residual errors.

[0025] In a further preferred embodiment, in step S7, the set of paths to be refined is... The candidate supports and their associated local fractional Doppler refinement atoms together constitute the support auxiliary set. For the first An enhanced grid atom, defining support for auxiliary indicator variables. for: Initialize the sparse precision parameters according to the supporting auxiliary indicator variables. : ,and Let the sparse precision matrix of the enhanced lattice channel coefficients be... Noise accuracy is In Gaussian priors Below, the posterior covariance matrix of the enhanced trellis channel coefficients is: The posterior mean is .

[0026] In a further preferred embodiment, in step S7, the first... Posterior reliability of each enhanced grid atom , For the first The posterior mean of each enhanced grid atom. For the posterior covariance matrix The One diagonal element; select the effective support set on the enhanced mesh based on the posterior reliability. ,in The threshold for posterior reliability is used; low-dimensional posterior update is performed using the Woodbury matrix identity, and the posterior mean is rewritten as... The posterior covariance matrix is ​​rewritten as The matrix inversion dimension is derived from the enhanced dictionary dimension. Transformed into observation dimensions; after iterative convergence, the channel estimation result on the enhanced grid is: The time-delay-Doppler domain off-grid channel estimation results of the channel are recovered based on the time delay index, Doppler index, and fractional Doppler coordinates corresponding to the non-zero channel coefficients on the enhanced grid.

[0027] Beneficial effects

[0028] This invention provides an underwater acoustic OTFS off-grid channel estimation method based on adaptive non-uniform grid refinement, which has at least the following beneficial effects:

[0029] (1) This invention utilizes coarse integer grid estimation results to locate credible local regions where fractional Doppler offset mismatch may exist. Fractional Doppler refinement atoms are introduced only within these credible local regions, rather than performing global uniform refinement across the entire time-delay-Doppler domain. This reduces the number of redundant refinement atoms, lowers the size of the enhancement dictionary, and decreases computational overhead. Compared to existing technologies: Existing non-uniform grid methods (such as exponentially growing non-uniform grids) use a preset global grid structure that does not dynamically adjust with channel estimation results; existing grid evolution methods (such as GESBI) employ a "grid point migration" strategy, continuously adjusting the position of virtual grid points through external iteration. The refinement position and density of this invention are adaptively determined entirely by feedback information from the coarse estimation results, achieving a closed-loop process of on-demand refinement—sensing-location-refinement—and avoiding the problem that preset grid patterns cannot adapt to actual channel distribution.

[0030] (2) This invention combines path amplitude, Doppler neighborhood energy diffusion, residual correlation, and local posterior uncertainty to comprehensively judge the refinement requirements of different candidate paths, so that the limited fractional Doppler refinement atoms are concentrated in the region that is difficult to be fully represented by the coarse integer grid model. In this way, the ability to represent fractional Doppler off-grid paths can be improved, and the support misjudgment problem caused by energy leakage and model mismatch can be alleviated.

[0031] (3) This invention uses newly added refinement atoms as candidate prior supports (rather than fixed supports) and performs support-assisted sparse Bayesian re-estimation on a locally non-uniform enhancement dictionary. While utilizing local refinement information, it adaptively distinguishes between effective and redundant refinement atoms, reducing the impact of noise disturbances, path coupling, and coarse estimation errors on the final channel estimation result. Compared with existing technologies: Existing grid evolution methods adopt a "grid point migration" strategy, where grid points may deviate from the correct integer grid positions during iteration, resulting in the loss of integer support information. This invention adopts a "new rather than migrate" strategy, retaining all coarse integer grid points and only adding fractional Doppler refinement atoms in local regions. This avoids the risk of grid drift from an architectural perspective and enhances the robustness of the estimation results.

[0032] (4) Existing off-grid estimation methods under the sparse Bayesian learning framework usually adopt a uniform prior setting for all atoms. This invention dynamically constructs a support auxiliary set based on the coarse estimation result and sets an initial sparsity precision for the enhanced grid atoms differently. Atoms belonging to the support auxiliary set are given a smaller initial sparsity precision (higher prior activation tendency), while those not belonging to the set are given a larger initial sparsity precision (stronger initial sparsity constraint). This realizes a two-layer mechanism of prior guidance + iterative update, which can more accurately select effective support from the overcomplete dictionary.

[0033] (5) The present invention uses the Woodbury matrix identity to reduce the posterior update in the support-assisted sparse Bayes re-estimation, transforming the inversion of the dictionary dimension matrix into the inversion of the observation dimension matrix, which significantly reduces the complexity of matrix operations without changing the estimation accuracy, and improves the engineering feasibility of the method in underwater acoustic OTFS receivers.

[0034] In summary, this invention can alleviate the off-grid mismatch problem caused by residual Doppler and fractional Doppler in underwater acoustic OTFS systems, improve the channel estimation accuracy in the time-delay-Doppler domain, and improve the performance of subsequent equalization detection. It is suitable for underwater acoustic mobile communication scenarios with strong multipath, strong Doppler, and time-varying characteristics.

[0035] Additional aspects and advantages of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0036] The above and / or additional aspects and advantages of the present invention will become apparent and readily understood from the description of the embodiments taken in conjunction with the following drawings, in which:

[0037] Figure 1 This is a flowchart illustrating the overall steps of the underwater acoustic OTFS off-grid channel estimation method based on adaptive non-uniform grid refinement in this application.

[0038] Figure 2 This is a flowchart of the underwater acoustic OTFS received signal processing and pilot-aided channel estimation model construction in this application.

[0039] Figure 3 This is a schematic diagram of the path selection and local fractional Doppler grid refinement based on the coarse integer grid estimation results in this application.

[0040] Figure 4 This is a flowchart of the support-assisted sparse Bayes re-estimation process on the locally non-uniformly enhanced dictionary in this application.

[0041] Figure 5 This is a comparative schematic diagram of coarse integer mesh, globally uniform refined mesh, and locally non-uniform enhanced mesh in this application. Detailed Implementation

[0042] The embodiments of the present invention are described in detail below. These embodiments are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0043] As shown in Figure 1, this embodiment provides an underwater acoustic OTFS off-grid channel estimation method based on adaptive non-uniform grid refinement, including: received signal processing and pilot observation construction, coarse integer grid channel estimation, effective coarse support selection, path to be refined screening, local fractional Doppler grid refinement, local non-uniform enhancement dictionary construction, and support-assisted sparse Bayesian re-estimation.

[0044] Step S1: Receive signal processing and pilot observation construction.

[0045] like Figure 2 As shown, the receiving end acquires the OTFS received signal after propagation through the underwater acoustic channel, and performs synchronization, Doppler estimation and compensation, down-conversion, filtering and OTFS demodulation processing on the received signal in sequence to obtain the time delay-Doppler domain received signal.

[0046] In the specific implementation process, let the time-delay-Doppler domain received signal matrix obtained by the receiver after OTFS demodulation be: in, Indicates the number of time-delay grids. This represents the Doppler grid number. The signal matrix received from the time-delay-Doppler domain is determined based on the pilot symbol positions and pilot guard zone settings in the OTFS frame. The local observation region affected by the pilot symbols is extracted, and the received samples within the local observation region are vectorized according to a preset order to obtain the pilot observation vector: in, This indicates the number of observation samples that were extracted.

[0047] Step S2: Construction and initial estimation of the coarse integer grid channel estimation model. Based on the preset maximum delay tap and maximum integer Doppler tap, a coarse integer delay-Doppler grid is constructed, and a corresponding coarse integer grid sensing matrix is ​​generated on this grid. Based on the coarse integer grid sensing matrix, an initial channel estimation is performed using a unitary approximation message-passing sparse Bayesian learning method, yielding coarse integer grid channel coefficients, posterior statistics, and coarse estimation residuals. The coarse integer grid channel coefficients characterize the sparse distribution of the channel on the preset integer delay-Doppler grid, and the coarse estimation residuals characterize the observation components not yet fully explained by the coarse integer grid model.

[0048] Step S2 includes the following sub-steps:

[0049] Step S201: Construct a coarse integer time delay-Doppler grid based on the preset maximum time delay tap and the maximum integer Doppler tap.

[0050] In the specific implementation process, the maximum delay tap is set as follows: The largest integer Doppler tap is Construct a coarse integer time-delay grid: ;

[0051] Constructing a coarse integer Doppler grid: ;

[0052] This yields the coarse integer time delay-Doppler grid set: ;

[0053] The size of the coarse integer grid is: .

[0054] Step S202: For each candidate grid point in the coarse integer time delay-Doppler grid set, construct its response atom within the pilot observation region. Let the... The number of coarse integer grid points are: The corresponding pilot observation response atom is denoted as: .

[0055] Arrange all coarse integer time delay-Doppler grid response atoms column-wise to obtain the coarse integer grid sensing matrix:

[0056]

[0057] Step S203: Establish a pilot-assisted channel estimation model on the coarse integer time delay-Doppler grid:

[0058]

[0059] in, This represents the vector of channel coefficients to be estimated on a coarse integer delay-Doppler grid. Represents the noise vector. This represents the modeling error caused by fractional Doppler mesh mismatch.

[0060] It should be noted that in actual underwater acoustic OTFS channels, the first... The Doppler frequency shift of the propagation path can be expressed as:

[0061]

[0062] in, For integer Doppler taps, For fractional Doppler shift, and , For OTFS slot duration. At this time, the actual path no longer falls strictly on integer Doppler grid points, resulting in a mismatch between the coarse integer grid model and the actual path response.

[0063] Step S204: The unitary approximation message-passing sparse Bayesian learning method is used to perform an initial estimation of the coarse integer grid channel estimation model, and the coarse integer grid channel estimation result is obtained. .

[0064] Further, the coarse estimation residual is calculated based on the coarse integer grid channel estimation results:

[0065]

[0066] Among them, the coarse estimation residual This is used to characterize observational components that are not fully explained by the coarse integer grid model. When fractional Doppler shifts exist in the actual propagation path, the coarse estimation residuals typically contain local residual components caused by off-grid mismatch.

[0067] Step S3, Effective Coarse Support Selection. A two-dimensional delay-Doppler energy map is constructed based on the coarse integer grid channel coefficients. Specifically, the coarse integer grid channel coefficients are rearranged into a two-dimensional matrix according to their corresponding delay and Doppler indices, and the energy on each delay-Doppler grid cell is calculated. Then, based on the cumulative energy criterion, coarse grid atoms with significant energy contributions are selected from the two-dimensional delay-Doppler energy map to obtain the effective coarse support set. The effective coarse support set is used to represent candidate regions on the coarse integer grid that may contain actual propagation paths.

[0068] Step S3 includes the following sub-steps:

[0069] Step S301: The coarse integer grid channel estimation result is... Rearrange the corresponding time delay and Doppler indices to obtain the two-dimensional time delay-Doppler domain coarse estimation matrix: .

[0070] Step S302: Construct a coarsely estimated energy distribution map based on the two-dimensional time-delay-Doppler domain coarse estimation matrix:

[0071]

[0072] in, Indicates the delay index is Doppler index is coarse integer grid cell energy, Two-dimensional coarse estimation matrix Medium delay index is Doppler index is Element.

[0073] Step S303, calculate the first Energy of a coarse integer grid atom: , Coarse integer grid channel estimation results Middle Channel coefficient estimates for each candidate grid point;

[0074] All Sort in descending order to obtain the sorted index sequence: in, Indicates the sorted order of the first... A coarse grid atomic index.

[0075] Step S304: Determine the number of effective coarse supports based on the cumulative energy criterion. Let the cumulative energy threshold be... Then the minimum number of supports required to satisfy the cumulative energy condition is:

[0076]

[0077]

[0078] in, To avoid extremely small positive numbers with a denominator of zero, The number of candidate supports to meet the cumulative energy requirements.

[0079] The cumulative energy threshold The value range is from 0.85 to 0.95, and in this embodiment, it is preferably 0.90. This value range was determined through a large number of underwater acoustic OTFS channel simulation experiments: when Values ​​below 0.85 introduce more noise, increasing computational overhead in subsequent refinement stages; when... Values ​​higher than 0.95 may miss some weak paths, affecting the completeness of channel estimation. A value of 0.90 strikes a good balance between support retention and computational complexity.

[0080] To avoid selecting too many noisy supports under low signal-to-noise ratio conditions, or missing weak paths when strong principal paths dominate, a lower limit is further set for the number of supports. and the upper limit of the number of supports The final effective coarse support number is obtained:

[0081]

[0082] This yields the effective coarse support set: .

[0083] The effective coarse support set Used to represent candidate path regions with significant energy contributions on coarse integer grids.

[0084] Step S4: Path Refinement Screening. Candidate paths in the effective coarse support set are subjected to grid mismatch discrimination. Specifically, based on the path amplitude significance, Doppler neighborhood diffusion, residual correlation gain, and local posterior uncertainty of the candidate path, it is determined whether there is fractional Doppler grid mismatch near the candidate path. Here, path amplitude significance measures the energy contribution of the candidate path, Doppler neighborhood diffusion measures the energy leakage of adjacent Doppler cells around the candidate path, residual correlation gain measures whether there are still observational components unexplained by the coarse integer grid near the candidate path, and local posterior uncertainty measures the stability of the estimation results near the candidate path. Based on the above discrimination results, a set of paths to be refined is obtained.

[0085] like Figure 3 As shown, after obtaining the effective coarse support set Subsequently, the candidate supports in the effective coarse support set are subjected to mesh mismatch judgment to filter out the set of paths to be refined that may have fractional Doppler mesh mismatch. It should be noted that the effective coarse support set... Since the cumulative energy criterion has already been obtained from the coarse integer grid estimation results, this step no longer selects candidates solely based on path amplitude. Instead, it further combines local Doppler energy diffusion and coarse estimation residual information to determine whether candidate supports need to undergo local fractional Doppler refinement.

[0086] Step S4 includes the following sub-steps:

[0087] Step S401: Construct a coarse energy estimation map and define the Doppler neighborhood of candidate supports. Then, use the coarse integer grid channel estimation results... According to coarse integer time delay-Doppler grid Rearranged into a two-dimensional coarse estimation matrix And construct the corresponding coarse energy estimation map: .

[0088] For effective coarse support sets Candidate support Let its corresponding coarse integer time delay-Doppler grid position be... Tap at the same delay The Doppler neighborhood set of the candidate support is defined as follows:

[0089]

[0090] in, For coarse integer Doppler grid sets, Let be the radius of the Doppler neighborhood. The Doppler neighborhood set... Used to describe candidate support The surrounding Doppler units may be subject to energy diffusion.

[0091] Step S402: Calculate the local Doppler energy diffusion index of the candidate support. For the candidate support... Its local Doppler energy diffusion index is defined as:

[0092]

[0093] in, This indicates the energy of the principal Doppler cell where the candidate support is located. This represents the energy of adjacent Doppler cells at the same time delay tap. To avoid extremely small positive numbers with a denominator of zero, larger ones... This indicates that there is a more obvious Doppler neighborhood energy diffusion near the candidate support, which is more likely to correspond to the off-grid mismatch caused by fractional Doppler.

[0094] Step S403: Calculate the residual correlation index near the candidate support. To distinguish between local energy diffusion caused by fractional Doppler and weak neighborhood components caused by noise disturbance or path coupling, this application further utilizes the reconstructed residual estimated by coarse integer grid for discrimination. The coarse estimation residual is expressed as:

[0095]

[0096] in, For pilot observation vectors, For coarse integer grid-sensing matrix, This is the result of coarse integer grid channel estimation.

[0097] For candidate support Calculate the residual correlation indices within its main Doppler cell and adjacent Doppler cells:

[0098]

[0099] in, Represents the position in the coarse integer grid. The corresponding response atom, as described above Same meaning Indicates conjugate transpose. Larger This indicates that the atoms near the candidate support are still strongly correlated with the coarse estimation residual, meaning that the coarse integer grid model has not fully explained the observed components in this region.

[0100] Step S404: Construct an equilibrium mesh mismatch score. To comprehensively consider the correlation between local Doppler energy diffusion and coarse estimation residuals, firstly, in the effective coarse support set... Internally respectively to and Normalization is performed:

[0101]

[0102] Then construct candidate supports. Balanced mesh mismatch score: .

[0103] The scoring considers both the degree of energy diffusion in the Doppler neighborhood near the candidate support and the degree of residual correlation. Only when the candidate region simultaneously exhibits significant Doppler neighborhood diffusion and strong residual correlation will the score be considered valid. This will result in a larger size, thereby reducing unnecessary refinement caused by weak neighborhood components dominated by noise.

[0104] Step S405: Determine the set of paths to be refined based on the mesh mismatch score. Based on the balanced mesh mismatch score, the set of paths to be refined is defined as follows: .

[0105] in, To refine the decision threshold, the value range is 0.5 to 0.85, and in this embodiment, 0.65 is preferred. This value range was determined through numerous Monte Carlo simulations: when Values ​​below 0.5 introduce a large number of unnecessary refinement atoms, increasing computational overhead; when... Values ​​higher than 0.85 will miss many paths that need refinement, affecting the estimation accuracy. A value of 0.65 strikes a good balance between refined gain and computational complexity.

[0106] With effective coarse support set In comparison, the set of paths to be refined Only candidate regions with a higher probability of fractional Doppler mismatch with the grid are retained. Subsequent actions will only focus on... Candidate supports generate locally fractional Doppler refined atoms; if If the value is empty, local refinement is not performed, and the enhanced mesh degenerates into a coarse integer mesh.

[0107] Step S5, Local Fractional Doppler Mesh Refinement. A local fractional Doppler candidate mesh is constructed around the candidate paths in the set of paths to be refined. For each path to be refined, fractional Doppler refinement atoms are generated only in its corresponding local Doppler neighborhood, without global uniform refinement across the entire time-delay-Doppler domain. Further, the refinement density is adaptively determined based on the refinement scores of different candidate paths; when the Doppler neighborhood diffusion of a candidate path is significant, the residual correlation gain is high, or the local posterior uncertainty is strong, more fractional Doppler refinement points are generated in its vicinity; when a candidate path can be well represented by a coarse integer mesh, the number of refinement points is reduced. This results in a newly added refined mesh set.

[0108] like Figure 3 and Figure 5 As shown, after obtaining the set of paths to be refined Subsequently, this application generates fractional Doppler refinement atoms only around candidate supports in the set of paths to be refined, without performing global uniform refinement across the entire time-delay-Doppler domain. For Let its corresponding coarse integer time delay-Doppler grid position be... Since this application primarily addresses off-mesh mismatch caused by fractional Doppler, time-delay coordinates are maintained during local refinement. The settings remain unchanged, with only local refinement applied to the Doppler coordinates.

[0109] Step S5 includes the following sub-steps:

[0110] Step S501: Normalize the mesh mismatch score of the support to be refined. For Its normalization refinement priority is defined as follows:

[0111]

[0112] in, The equilibrium mesh mismatch score obtained in step S4, To avoid extremely small positive numbers with a denominator of zero. The larger the value, the more pronounced the fractional Doppler mesh mismatch is near the candidate support, thus requiring a higher local refinement density.

[0113] Step S502: Determine the local refinement order based on the normalization refinement priority. The local refinement order corresponding to each support to be refined is defined as:

[0114]

[0115] in, Indicates the maximum refinement order. This indicates the rounding up operation. The value ranges from 2 to 8, and in this embodiment, it is set to 4. The stronger the mismatch between the candidate support and the grid, the greater its corresponding... The larger the value, the more fractional Doppler refinement atoms are generated near the candidate support.

[0116] Step S503: Construct a local fractional Doppler candidate set. Let the fractional Doppler refinement interval be... And satisfy: In this embodiment, when At that time, set ,at this time The constraints are satisfied.

[0117] For the The support to be refined is defined as follows:

[0118]

[0119] in, This represents the refined Doppler grid coordinates, which can be non-integer values; This represents the coarse integer Doppler grid set. Because... The corresponding original integer Doppler atoms are already included in the coarse integer grid, so they will not be added again.

[0120] Step S504: Construct the first... Local refinement delay near the support to be refined - Doppler mesh: ;

[0121] Merge all locally refined meshes with the coarse integer time-delay-Doppler mesh to obtain the enhanced time-delay-Doppler mesh:

[0122]

[0123] in, For coarse integer time delay-Doppler grid sets, To increase the number of mesh atoms, the enhanced mesh undergoes fractional Doppler refinement only near the supports to be refined, rather than global refinement across the entire Doppler dimension. Although the fractional Doppler point spacing within each local region is... However, since the refinement region and the local refinement order are adaptively determined by the coarse estimation results, the overall enhanced mesh exhibits a non-uniform structure.

[0124] Step S6: Construction of the locally non-uniform enhancement dictionary. The newly added refined grid set is merged with the coarse integer grid set, and duplicate grid points are removed to obtain a locally non-uniform enhancement grid. Subsequently, for each grid point in the locally non-uniform enhancement grid, its response atom within the pilot observation region is constructed, and all response atoms are arranged column-wise to obtain the enhancement sensing matrix. Since the locally non-uniform enhancement grid only introduces fractional Doppler refinement atoms in local regions where there may be off-grid mismatch, it can improve the characterization ability of fractional Doppler off-grid paths while controlling the dictionary size.

[0125] Step S6 includes the following sub-steps:

[0126] Step S601, for the enhanced mesh set Each grid point in Construct its response vector within the pilot observation region: in, This indicates the number of pilot observation samples.

[0127] Step S602: Arrange all enhanced mesh response vectors column-wise to obtain the locally non-uniform enhanced sensing matrix:

[0128]

[0129] Step S603: Establish an enhanced dictionary observation model on the locally non-uniform enhanced sensing matrix:

[0130]

[0131] in, For pilot observation vectors, This represents the channel coefficient vector on the enhanced trellis. This represents observation noise and residual modeling error.

[0132] After the above processing, the dictionary is enhanced. It includes both coarse integer grid atoms and local fractional Doppler refinement atoms. Since fractional Doppler atoms are introduced only in local regions where grid mismatch may exist, the enhanced dictionary can improve the representation of fractional Doppler paths while controlling the dictionary size.

[0133] Step S7: Support Assisted Sparse Bayes Re-estimation and Channel Output. The newly added refined atoms are used as candidate prior supports, rather than fixed supports, to establish a support-assisted sparse Bayes re-estimation model on the enhanced sensing matrix. Specifically, newly added refined atoms generated by the local refinement process are labeled using candidate support indicator variables, and the channel coefficients, sparse accuracy parameters, candidate support parameters, and noise accuracy parameters are alternately updated during the sparse Bayes iteration process. For newly added refined atoms that can effectively explain the observed signal, their sparsity penalty is reduced and they are retained; for redundant refined atoms generated by noise perturbations or erroneous refinement, their sparsity penalty is increased and they are compressed. Further, the posterior mean and posterior covariance are updated in a low dimension using the Woodbury matrix identity, transforming the inversion of the dictionary-dimensional matrix into the inversion of the observation-dimensional matrix to reduce computational complexity. After iteration, the effective channel supports and channel coefficients on the enhanced grid are determined based on the final posterior mean, yielding the off-grid time delay-Doppler domain channel estimation results for the underwater acoustic OTFS system.

[0134] like Figure 4 As shown, while the locally non-uniform augmentation dictionary improves the representation of fractional Doppler paths, the refined atoms near the same coarse support often exhibit strong correlations. Directly retaining all refined atoms can easily introduce redundant supports. Therefore, this application performs support-assisted sparse Bayesian re-estimation on the augmentation dictionary to relearn the effective supports on the augmented grid.

[0135] Step S7 includes the following sub-steps:

[0136] Step S701: Construct a support auxiliary set. The coarse integer grid UAMP-SBL estimation results provide reliable coarse support information, and the local refinement process provides fractional Doppler candidate atoms that may exist near these coarse supports. This application uses the set of paths to be refined... The candidate supports and their associated local fractional Doppler refinement atoms together constitute the support auxiliary set. This definition ensures the conciseness and specificity of the support auxiliary set, avoiding the assignment of excessively strong a priori activation tendencies to too many unreliable atoms.

[0137] For the An enhanced mesh atom, defining support for auxiliary indicator variables:

[0138]

[0139] in, Indicates the first Each enhanced grid atom belongs to the support auxiliary set and has a high prior activation tendency; Indicates the first The enhanced mesh atoms do not belong to the support auxiliary set. It should be noted that the support auxiliary indicator variables are only used to introduce soft prior support information and are not fixed support constraints. All enhanced mesh atoms will still participate in the update process during subsequent Bayesian inference.

[0140] Step S702: Initialize the sparse precision parameters according to the supporting auxiliary indicator variable:

[0141]

[0142] Atoms belonging to the supporting auxiliary set are assigned a smaller initial sparsity precision, thus exhibiting a higher prior activation tendency; atoms not belonging to the supporting auxiliary set are assigned a larger initial sparsity precision, thus subjecting them to stronger initial sparsity constraints. The sparsity precision parameters of all atoms continue to be updated during iteration, therefore unreliable refinement atoms can be suppressed, while atoms outside the supporting auxiliary set are not completely excluded. In general, Pick to In this embodiment, we take This ratio setting ensures that the initial activation tendency of atoms within the support set is significantly higher than that of atoms outside the set, while at the same time preventing atoms outside the set from completely losing the opportunity to be activated.

[0143] Step S703, let the sparse precision matrix of the enhanced lattice channel coefficients be:

[0144]

[0145] Noise accuracy is In Gaussian priors: Based on the enhanced dictionary observation model, the posterior distribution of the enhanced trellis channel coefficients is as follows: The posterior covariance matrix is:

[0146]

[0147] The posterior mean is: Among them, the posterior mean Given the channel coefficient estimates on the augmented grid, the diagonal elements of the posterior covariance matrix are... Used to characterize the Posterior uncertainty of an enhanced grid atom.

[0148] Step S704: To suppress redundant support caused by related refined atoms, relying solely on the posterior mean amplitude for support determination may not be reliable enough. Therefore, this application further defines the... Posterior reliability of each enhanced grid atom:

[0149]

[0150] in, For the first The posterior mean of each enhanced grid atom. The larger the value, the greater the atom's posterior mean and the lower its posterior uncertainty, making it more likely to correspond to effective support.

[0151] The effective support set on the enhanced mesh is selected based on the posterior reliability.

[0152]

[0153] in, The posterior reliability decision threshold ranges from 0.3 to 0.7, and is preferably 0.5 in this embodiment. This range was determined statistically from a large number of Monte Carlo simulation results: when Values ​​below 0.3 will retain too much noise support; when... Values ​​higher than 0.7 will miss some valid paths. A value of 0.5 is the empirically optimal value that balances selection precision and recall. This approach retains effective refinement atoms while suppressing redundant candidate atoms.

[0154] It should be further explained that although this application introduces dense fractional Doppler candidate atoms in the local refinement stage, there may be strong correlations between refined atoms near the same coarse support. However, due to the inherent sparsity of the underwater acoustic channel (limited number of propagation paths) and the fact that the refinement operation is limited to the local neighborhood of a very small number of strong paths (rather than being expanded globally), the dictionary is enhanced by the differentiated prior guidance provided by the support auxiliary set and the posterior reliability screening mechanism in step S704. The effective condition number is always within a controllable range, and will not cause support confusion or convergence failure in the sparse Bayes inference process due to excessive local density.

[0155] Step S705: To reduce the computational complexity of the enhanced grid re-estimation process, this application employs the Woodbury matrix identity for low-dimensional posterior update. Directly calculating the posterior covariance matrix requires... Finding the inverse of a 3D matrix. Using the Woodbury matrix identity, the posterior mean can be rewritten as:

[0156]

[0157] The posterior covariance matrix can be rewritten as:

[0158]

[0159] Therefore, the matrix inversion dimension is transformed into an enhanced dictionary dimension. Transformed into observation dimensions .when In this context, the Woodbury low-dimensional posterior update can reduce the computational complexity of supporting sparse Bayesian re-estimation. In practical OTFS systems, the number of pilot observation samples is typically much smaller than the augmented dictionary dimension, thus this method has broad applicability.

[0160] Step S706: After iterative updates and convergence, the channel estimation results on the enhanced grid are obtained:

[0161]

[0162] Based on the time delay index, Doppler index, and fractional Doppler coordinates corresponding to the non-zero channel coefficients on the enhanced grid, the time delay-Doppler domain off-grid channel estimation results of the underwater acoustic OTFS system are recovered.

[0163] To illustrate the construction method of the locally non-uniform reinforced mesh in this application, Figure 5 A comparative schematic diagram of coarse integer mesh, globally uniform refined mesh, and locally non-uniform enhanced mesh is presented. The coarse integer mesh contains only preset integer time-delay-Doppler mesh points; the globally uniform refined mesh uniformly inserts fractional Doppler atoms throughout the entire time-delay-Doppler domain, which improves parameter characterization but significantly increases the dictionary size; the locally non-uniform enhanced mesh inserts fractional Doppler atoms only near the supports to be refined, thereby reducing the number of redundant atoms while improving the characterization ability of off-grid paths.

[0164] To verify the effectiveness of the proposed channel estimation method, an underwater acoustic OTFS communication system was constructed in a simulation experiment, and a time-delay-Doppler domain channel model with multipath spread and fractional Doppler offset was set up. Channel estimation was compared using the coarse integer grid UAMP-SBL method, the global uniform refinement method, and the adaptive non-uniform grid refinement method proposed in this application. During the experiment, normalized mean square error (MSE) and bit error rate (BER) were used as evaluation metrics. MSE was used to measure the deviation between the estimated channel and the actual channel, and BER was used to measure the impact of the channel estimation results on the subsequent equalization and detection performance.

[0165] Simulation results show that when fractional Doppler offset exists in the actual propagation path, the coarse integer grid method is prone to energy leakage and support mismatch, leading to increased channel estimation error. While the global uniform refinement method can improve the representation of off-grid paths, it significantly increases the size of the enhancement dictionary and computational complexity. In contrast, the method proposed in this application introduces fractional Doppler refinement atoms only in local regions where off-grid mismatch may exist, and filters effective supports through support-assisted sparse Bayesian re-estimation. Therefore, it can reduce channel estimation error while controlling the dictionary size.

[0166] Furthermore, the method was validated based on measured underwater acoustic OTFS data. The receiver first synchronized, performed Doppler compensation, down-converted, and demodulated the measured underwater acoustic signal using OTFS. Then, channel estimation was performed using both the coarse integer grid estimation method and the method described in this application, and the estimation results were used for subsequent equalization detection. Experimental results show that the method described in this application can improve the equalization detection effect under measured underwater acoustic channel conditions, reduce the uncoded bit error rate, and increase the clustering degree of constellation points after equalization. This indicates that the method described in this application can not only alleviate the fractional Doppler off-grid mismatch problem in simulated channels but is also applicable to actual underwater acoustic mobile communication scenarios with residual Doppler, multipath spread, and time-varying characteristics.

[0167] This application provides an underwater acoustic OTFS off-grid channel estimation method based on adaptive non-uniform grid refinement. It utilizes coarse integer grid estimation results to locate candidate regions where fractional Doppler off-grid mismatch may exist, and filters supports to be refined based on local Doppler energy diffusion and residual correlation. Subsequently, local fractional Doppler refinement atoms are constructed only near the supports to be refined, forming a local non-uniform enhancement dictionary. Furthermore, reliable coarse supports and their associated local fractional Doppler refinement atoms are used together as support-aided sparse Bayesian re-estimation on the local non-uniform enhancement dictionary, and the Woodbury matrix identity is used to reduce the posterior update complexity. Therefore, this method can alleviate the energy leakage and off-grid mismatch problems caused by fractional Doppler in underwater acoustic OTFS systems, improve channel estimation accuracy, and enhance subsequent equalization detection performance.

[0168] Although embodiments of the present invention have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of the present invention without departing from the principles and spirit of the present invention.

Claims

1. A method for estimating the off-grid channel in underwater acoustic OTFS based on adaptive non-uniform grid refinement, characterized in that: Includes the following steps: Step S1: Acquire the OTFS received signal, and perform synchronization, Doppler compensation, downconversion and OTFS demodulation on the OTFS received signal to obtain the time delay-Doppler domain received signal; extract local observation samples according to the region where the pilot symbols are located in the OTFS frame, and vectorize the local observation samples to obtain the pilot observation vector; Step S2: Construct a coarse integer time delay-Doppler grid and its corresponding coarse integer grid sensing matrix, and perform initial channel estimation on the coarse integer grid sensing matrix to obtain the coarse integer grid channel estimation result; Step S3: Determine the effective coarse support set based on the coarse integer grid channel estimation results; Step S4: Based on the local Doppler energy diffusion degree and residual correlation degree of the candidate supports in the effective coarse support set, a set of paths to be refined is obtained. Step S5: Construct local fractional Doppler refinement atoms around the candidate supports in the set of paths to be refined; Step S6: Merge the local fractional Doppler refinement atoms with the coarse integer grid atoms to form a local non-uniform enhancement dictionary; Step S7: Perform support-assisted sparse Bayes re-estimation on the locally non-uniform enhancement dictionary to obtain the channel estimation result on the enhancement grid; The delay-Doppler off-grid channel estimation results are recovered based on the channel estimation results on the enhanced grid.

2. The method according to claim 1, characterized in that, In step S2, the coarse integer time delay-Doppler grid includes a coarse integer time delay grid and a coarse integer Doppler grid; the coarse integer time delay grid is... The coarse integer Doppler grid is ,in For the maximum delay tap, The largest integer Doppler tap; the coarse integer grid sensing matrix It is composed of pilot observation response atoms of each candidate grid point in the coarse integer time delay-Doppler grid arranged in columns.

3. The method according to claim 1, characterized in that, In step S2, the pilot observation vector is established. Coarse integer grid-aware matrix Coarse integer grid channel coefficient vector and observation noise Composition of coarse integer grid channel estimation model ,in The modeling error is caused by fractional Doppler offset grid mismatch; the initial channel estimation is performed using a unitary approximation message-passing sparse Bayesian learning method to obtain coarse integer grid channel estimation results. And based on the pilot observation vector Reconstructing signals with coarse integer grid The difference between them gives a rough estimate of the residual. .

4. The method according to claim 1, characterized in that, In step S3, the effective coarse support set is determined based on the energy distribution of the coarse integer grid channel estimation results; specifically, it includes: [the process of] calculating the coarse integer grid channel estimation results... Rearranged into a two-dimensional coarse estimation matrix Construct a coarse estimate of the energy distribution map , Indicates the delay index is Doppler index is coarse integer grid cell energy, Two-dimensional coarse estimation matrix Medium delay index is Doppler index is Elements; calculate the energy of each coarse integer grid atom. And sorted in descending order. Coarse integer grid channel estimation results Middle Channel coefficient estimates for each candidate grid point; effective coarse support number determined based on cumulative energy criterion. To obtain an effective coarse support set The cumulative energy criterion specifies conditions that must be met. The minimum number of supports is denoted as ,in The cumulative energy threshold, For coarse integer grid size, The number of candidate supports to meet the cumulative energy requirements, The number after sorting in descending order The energy of a coarse integer grid atom It is a very small positive number. , To support the lower limit, To support the upper limit of the number.

5. The method according to claim 1, characterized in that, In step S4, the set of paths to be refined is determined based on the local Doppler energy diffusion index and the residual correlation index; the local Doppler energy diffusion index The energy level is determined by the ratio of the sum of the energies of the adjacent Doppler cells at the time delay tap where the candidate support is located to the energy of the principal Doppler cell. ,in For the candidate support Doppler neighborhood set, Support for candidates Coarse integer time delay-Doppler grid position, Let be the radius of the Doppler neighborhood. It is a very small positive number; the residual correlation index The response atoms of candidate supports and their adjacent Doppler units and the coarsely estimated residuals The normalized correlation between them is determined, i.e. ,in Position in coarse integer grid The corresponding response atom, This indicates the conjugate transpose.

6. The method according to claim 5, characterized in that, In step S4, the local Doppler energy diffusion index is... Residual related indicators Normalization is performed to obtain the normalized index. and And construct an equilibrium mesh mismatch score based on the geometric mean of the two. The set of paths to be refined is: ,in To refine the decision threshold.

7. The method according to claim 5, characterized in that, In step S5, for the set of paths to be refined Candidate support Its normalization refinement priority is Its local refinement order is ,in To achieve the maximum refinement order, For floor operations; fractional Doppler refinement atoms are generated in the local Doppler neighborhood near the candidate support, and the set of local refinement Dopplers is: ,in The coordinates are the refined Doppler grid. Support for candidates Coarse integer Doppler index, For fractional Doppler fine-tuning, satisfying , For a set of integers, For the largest integer Doppler tap; construct the first... Local refinement delay near the support to be refined - Doppler mesh Refine all local meshes With coarse integer delay-Doppler grid Merging yields an enhanced delay-Doppler grid. .

8. The method according to claim 7, characterized in that, In step S6, for the enhanced delay-Doppler grid Each grid point in Construct the response vector within its pilot observation region. Arrange all the enhanced grid response vectors column-wise to obtain the locally non-uniform enhanced sensing matrix. ,in To increase the number of atoms in the grid, Establish an enhanced dictionary observation model ,in For pilot observation vectors, To enhance the channel coefficient vector on the grid, To model observation noise and residual errors.

9. The method according to claim 1, characterized in that, In step S7, the set of paths to be refined is... The candidate supports and their associated local fractional Doppler refinement atoms together constitute the support auxiliary set. For the first An enhanced grid atom, defining support for auxiliary indicator variables. for: Initialize the sparse precision parameters according to the supporting auxiliary indicator variables. : ,and Let the sparse precision matrix of the enhanced lattice channel coefficients be... Noise accuracy is In Gaussian priors Below, the posterior covariance matrix of the enhanced trellis channel coefficients is: The posterior mean is .

10. The method according to claim 9, characterized in that, In step S7, the first... Posterior reliability of each enhanced grid atom , For the first The posterior mean of each enhanced grid atom. For the posterior covariance matrix The One diagonal element; select the effective support set on the enhanced mesh based on the posterior reliability. ,in The threshold for posterior reliability is used; low-dimensional posterior update is performed using the Woodbury matrix identity, and the posterior mean is rewritten as... The posterior covariance matrix is ​​rewritten as The matrix inversion dimension is derived from the enhanced dictionary dimension. Transformed into observation dimensions; after iterative convergence, the channel estimation result on the enhanced grid is: The time-delay-Doppler domain off-grid channel estimation results of the channel are recovered based on the time delay index, Doppler index, and fractional Doppler coordinates corresponding to the non-zero channel coefficients on the enhanced grid.

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