Mixed field channel representation and acquisition method based on main lobe energy coverage criterion

Abstract

The invention discloses a mixed field channel representation and acquisition method based on a main lobe energy coverage criterion. The method comprises the following steps: S1, determining a Rayleigh distance and a minimum sampling distance based on communication system parameters; s2, determining an angle domain sampling interval, and generating angle sampling points according to 3dB power overlapping; s3, deriving and applying a 3dB distance recursion analytic expression, determining a self-adaptive distance sampling interval and iteratively generating distance sampling points at each angle according to a nonlinear attenuation rule of beam focusing, and ensuring that main lobe energy coverage overlapping of adjacent dictionary code words in a distance domain meets a preset threshold value; and S4, gathering the beam atoms corresponding to all angle and distance sampling points to obtain a final dictionary as mixed field channel representation. According to the invention, the design of the code words in the adjacent angle and distance domains meets the condition that the power overlap is not less than 3dB, so that the channel representation blind area is greatly reduced in the whole angle and distance domain range.

Description

Technical Field

[0001] This invention relates to channel estimation in the field of wireless communication, and in particular to a hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion. Background Technology

[0002] With the development of 6G technology, XL-MIMO and RIS technologies have become research hotspots in order to meet the requirements of extremely high data transmission rates and coverage. As antenna apertures increase significantly and operating frequencies evolve towards millimeter-wave and terahertz bands, the Rayleigh distance of antenna arrays extends considerably, causing communication multipath to often occur in the near-field, far-field, and mixed field regions of the base station or smart reflector.

[0003] In traditional communication systems, electromagnetic waves are typically assumed to be plane waves, and a Discrete Fourier Transform (DFT) dictionary is designed based on the far-field assumption to represent the channel for channel estimation and beam training. However, in the near-field and mixed-field regions, electromagnetic waves exhibit significant spherical wavefront characteristics, meaning the signal is not only angle-dependent but also distance- and height-dependent. In these cases, continuing to use a DFT dictionary containing only angle information leads to severe model mismatch, producing an "energy diffusion" effect that significantly reduces the accuracy of channel estimation and severely impairs spectral efficiency.

[0004] Existing technologies have proposed dictionary design schemes based on polar coordinates, such as polar coordinate dictionaries based on non-uniform sampling and polar coordinate dictionaries based on sparse optimization. These schemes usually perform grid division in both the angle domain and the distance domain at the same time, attempting to capture near-field features by increasing the sampling of the distance dimension. However, they make incorrect assumptions or simplifications regarding the physical coverage characteristics of dictionary codewords. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion. The adjacent angle and distance domain codewords are designed to satisfy a power overlap of not less than 3dB, thereby ensuring a significant reduction in channel representation blind spots across the entire angle and distance domain.

[0006] The objective of this invention is achieved through the following technical solution: a method for representing and acquiring a hybrid field channel based on the main lobe energy coverage criterion, comprising the following steps:

[0007] Step S1: Determine the Rayleigh distance and minimum sampling distance based on the communication system parameters;

[0008] Step S2: Determine the angle domain sampling interval and generate angle sampling points based on 3dB power overlap;

[0009] Step S3: Derive and apply the 3dB recursive analytical formula for the distance. At each angle, determine the adaptive distance sampling interval based on the nonlinear attenuation law of beam focusing, and iteratively generate distance sampling points. And ensure that the main lobe energy coverage overlap of adjacent dictionary codewords in the distance domain meets the preset threshold;

[0010] Step S4: Gather the beam atoms corresponding to all angle and distance sampling points to obtain the final dictionary, which serves as the hybrid field channel representation.

[0011] The beneficial effects of the present invention are: (1) The adjacent angle and distance domain codewords in the dictionary of the present invention are designed to meet the requirement that the power overlap is not less than 3dB, thereby ensuring a significant reduction in the channel representation blind zone throughout the entire angle and distance domain.

[0012] (2) The dictionary of the present invention achieves blind-zone-free coverage of near field, far field and mixed field.

[0013] (3) Compared with dictionaries based on low coherence design, the invented dictionary has a smaller codebook dimension while ensuring the accuracy of channel recovery. Attached Figure Description

[0014] Figure 1 This is a flowchart of the method of the present invention;

[0015] Figure 2 This diagram illustrates the accuracy of channel recovery using sparse Bayesian learning for different dictionaries with the same number of pilots. Detailed Implementation

[0016] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings, but the scope of protection of the present invention is not limited to the following description.

[0017] Considering the prevailing technical bias of prioritizing low-coherence dictionaries in existing technologies, cross-correlation is reduced by artificially increasing the codeword spacing. However, this strategy creates coverage blind spots in the near-field, mixed-field, and far-field physical spaces, leading to severe rasterization mismatch. This invention proposes for the first time: using 3dB main lobe overlap as a sampling criterion at the boundary between the range and angular domains; automatically adjusting the spacing between adjacent codewords through recursive non-uniform sampling to precisely satisfy 3dB overlap; and ensuring that the dictionary remains consistent with the real spherical wave propagation model, thereby reducing quantization errors at their source. Specifically:

[0018] like Figure 1 As shown, a hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion includes the following steps:

[0019] Step S1: Determine the Rayleigh distance and minimum sampling distance based on the communication system parameters;

[0020] This invention is not limited to XL-MIMO or RIS-assisted communication systems. The technical solution of this invention has a wide range of applications and can be efficiently applied in scenarios that meet the following two basic conditions:

[0021] (1) Basic condition one: The system operates in the radiation near-field region and the mixed field region. With the application of high frequency bands and the gradual trend of arrays towards ultra-large scale, the Rayleigh distance is greatly extended. This invention assumes that the user or scatterer is outside the reaction near-field.

[0022] (2) Basic condition two: The channel exhibits both spherical and planar wavefront characteristics simultaneously. On the one hand, this means that traditional representation methods based solely on planar wavefronts are ineffective, and the phase nonlinearity of the range dimension must be considered; on the other hand, this condition ensures that the channel has resolving power in the range domain, providing a physical basis for eliminating interference through range domain beamforming. By utilizing this energy focusing characteristic, this invention can accurately determine the effective range sampling points without relying on ultra-dense grids, making the representation of the hybrid field model more consistent with the laws of physical propagation.

[0023] Assuming the base station or RIS is equipped with A uniform linear array of antenna elements, with an element spacing of . The system operates at the carrier frequency. The corresponding wavelength is Due to the large antenna aperture, the user may be located in the near-field radiation region. In this case, the channel exhibits spherical wave characteristics, and its steering vector... Determined by both angle and distance, it can be represented as:

[0024]

[0025] in, Let be the sine of the angle. The distance from the user to the center of the array. For users to the first The distance between each antenna element.

[0026] It is worth emphasizing that this invention is applicable not only to single-hop direct links, but also to multi-hop propagation scenarios (such as RIS-assisted single-user single-base station communication systems). In multi-hop systems, although the reflection path may physically include near-field and far-field segments, rigorous analysis reveals that the overall channel received by the base station still exhibits a mixed-field form, and its mathematical structure can be described by a set of equivalent near-field models. For example, for a typical single-user single-base station RIS-assisted communication system, its equivalent steering vector can be written as:

[0027]

[0028] in, Describes the angle and distance of the near-field path. The angular contribution of the far-field path is described. This expression shows that the synthesis result of a multi-hop channel can still be transformed into a spherical wave form determined by both the "equivalent angle" and the "equivalent distance". Based on this equivalent model, this invention constructs a main lobe energy coverage criterion, enabling the proposed method to maintain high coverage and high-precision channel acquisition performance even in multi-hop hybrid field links.

[0029] Input system parameters: Number of antennas / Reflector elements Array element spacing ,wavelength Calculate Rayleigh distance :

[0030]

[0031] Set the starting boundary for near-field sampling In this example, we set... = That is, the boundary between the reaction near field and the radiation near field.

[0032] Step S2: Determine the angle domain sampling interval and generate angle sampling points based on 3dB power overlap;

[0033] The angle sampling point is determined using the half-power beamwidth characteristics of the ULA. The sampling interval in the angle domain of the ULA's 3dB beamwidth in the sinusoidal domain is... To cover the entire space [-1,1], the angle sampling points... The calculation formula is:

[0034]

[0035] in, This represents the total number of angle samples.

[0036] Step S3: Derive and apply the 3dB recursive analytical formula for the distance. At each angle, determine the adaptive distance sampling interval based on the nonlinear attenuation law of beam focusing, and iteratively generate distance sampling points. And ensure that the main lobe energy coverage overlap of adjacent dictionary codewords in the distance domain meets the preset threshold;

[0037] Utilizing the characteristics of ULA, sampling points are selected at each defined angle. In the distance domain [ ,+ Recursive sampling is performed on the beam. Since the near-field beam depth is non-linear and varies with angle... As the power changes, this invention uses Fresnel parameters to derive the distance sampling boundary for 3dB power.

[0038] S301. To avoid performing independent distance recursion calculations at each angle sampling point, this invention... Within the range of values, the sampling point closest to zero angle is selected and denoted as the anchor angle. ;

[0039] At the anchor point angle Below, a range sampling sequence is generated based on the 3 dB beam depth criterion:

[0040] The maximum distance for distance sampling is the distance to the far boundary:

[0041]

[0042] in, The Rayleigh distance parameter is related to the array.

[0043] For anchor point angle Distance from sampling point The generation logic is as follows:

[0044] A1. Initialization: Set the current distance from the upper bound. Distance index ;

[0045] A2. Recursive calculation: Calculate the next sampling center point based on the current distance upper bound. When the normalized power drops to 1 / 2, i.e. -3dB, the Fresnel parameter is approximately 1.32.

[0046] In distance domain recursive sampling, fixed angle sampling points Afterwards, the main lobe power of adjacent codewords is controlled by only one dimensionless variable (i.e., the Fresnel parameter), and the effective focusing constant under the defined angle is:

[0047]

[0048] in, Determined by parameters such as array aperture and wavelength, such that... It has the dimension of length, and thus is related to distance. By comparison, Fresnel parameters are introduced:

[0049]

[0050] According to the main lobe power attenuation law, when the distance offset causes the normalized power to decrease to That is, at -3dB, the corresponding Fresnel parameter reaches a fixed threshold:

[0051]

[0052] Will Substitution and The relationship yields:

[0053]

[0054] For ease of explanation, Approximately denoted as a constant The final recursive formula for the next distance sampling center point is:

[0055]

[0056] A3. Calculate the distance to the sampling point The corresponding outer 3dB power boundary serves as the starting lower bound for the next point. :

[0057]

[0058] A4. Loop check will Updated to The threshold is set based on the maximum beamwidth. ,like Less than the threshold of the far-field boundary ,but Repeat the above steps; otherwise, stop near-field sampling.

[0059] A5. Under different values ​​of q, through the above recursive process, a set of non-uniformly distributed distance sampling centers are obtained at the anchor point angle. The sampling is more dense at smaller distances to characterize the curvature changes of near-field spherical waves.

[0060] S303. For anchor point angles The generated arbitrary distance sampling center Map it to the remaining angle sampling points During the mapping process, the following equivalent focusing invariants remain unchanged:

[0061]

[0062] This yields the corresponding angle. The following distance sampling points:

[0063]

[0064] At the same time, ensure that Greater than ;

[0065] S304. After completing near-field range sampling, in order to uniformly represent the far-field plane wave propagation characteristics, at each angle sampling point... Add a far-field proxy distance point at the specified location, with its distance being a constant value much larger than the Rayleigh distance, for example:

[0066]

[0067] As the propagation distance approaches infinity, the spherical wave model naturally degenerates into a plane wave model. Therefore, the far-field proxy atom can be equivalently represented by the far-field channel component.

[0068] This is based on the principle that far-field plane waves can be approximated as spherical waves at their maximum distance.

[0069] Step S4: Gather the beam atoms corresponding to all angle and distance sampling points to obtain the final dictionary, which serves as the hybrid field channel representation.

[0070] S401. Construction of elements in a single atom vector:

[0071] For the arbitrary angle sampling pairs generated in steps S2-S3 , in a given column for containing A uniform forward array ULA with n elements and an element spacing of n is 1. The working wavelength is Under the condition of, determine the first The coordinates of the center of each array element are:

[0072]

[0073] Based on geometric relationships, calculate the sampling pair up to the th Distance between array elements:

[0074]

[0075] Furthermore, a guide vector determined by both angle and distance is constructed. Therefore, the first Each component satisfies:

[0076]

[0077] S402. Construction of Atomic Vectors:

[0078] For the guide vector Energy normalization yields the atomic vectors corresponding to the sampled pairs, i.e., the dictionary column vectors:

[0079]

[0080] S403. Angle sampling point Corresponding dictionary matrix sub-moments :

[0081] In step S3, for the angle sampling point For each q value, obtain Then, calculate the corresponding dictionary column vector according to step S402. Then, concatenate them column by column to obtain the corresponding dictionary matrix submatrices. ;

[0082] S404. Repeat step S403 for m = 1, 2, ..., M to form the final mixed-field 3dB power dictionary matrix. That is, the hybrid field channel representation:

[0083] .

[0084] The core of this invention lies in proposing a codebook construction method based on physical beamwidth and beam depth criteria, and designing a hybrid field 3dB energy dictionary matrix. The channel is represented using a method designed to ensure that the power overlap between atoms in the angular and distance domains is no less than 3 dB, and to minimize sampling blind spots.

[0085] This invention is primarily applied to next-generation ultra-large-scale multiple-input multiple-output (XL-MIMO) systems or intelligent reflector systems (RIS) scenarios, especially in the near-field region. The RIS is a plane composed of numerous passive reflectors. By programming and controlling the phase shift of each unit, controllable reflection of the incident signal is achieved, thereby intelligently reshaping the wireless channel environment and enhancing signal coverage and transmission quality. It is worth noting that this invention can be applied more broadly.

[0086] (1) High frequency communication: suitable for millimeter wave and terahertz communication because these frequency bands have short wavelengths, large array sizes, and significant near-field effects.

[0087] (2) Indoor coverage / blind spot elimination: In complex indoor environments or scenarios with obstacles, reliable connection is achieved by bypassing obstacles through RIS intelligent reflection.

[0088] (3) Positioning and sensing: The angles and distances in the dictionary can also be used for high-precision positioning and environmental sensing.

[0089] Furthermore, in one embodiment, the hybrid field 3dB power dictionary matrix constructed based on the main lobe energy coverage criterion... (Generated offline and stored on the base station or RIS controller side by steps S1–S4) is used for online channel acquisition and subsequent beam / reflection configuration. Specifically, during the pilot training phase, the terminal sends a pilot sequence, and the base station receives and obtains the observation vector. And use hybrid field channels to represent relationships. Transform the channel estimation problem into dictionary field coefficients. The problem of channel recovery can be addressed by using coefficient Bayesian learning, orthogonal matching pursuit, and other coefficient reconstruction methods to estimate dictionary domain coefficients, thereby recovering the channel and completing precoding design, beam selection, or RIS phase shift configuration accordingly.

[0090] Based on this, the application of the dictionary of this invention in the following typical scenarios can be briefly summarized as follows:

[0091] High-frequency communication (millimeter wave / terahertz, more pronounced in the near field):

[0092] Under large-scale arrays and short wavelength conditions, the probability of users being in the near-field of radiation increases significantly, and the channel exhibits spherical wave characteristics. Simple plane wave dictionaries are prone to model mismatch. When using the dictionary of this invention for pilot training and sparse reconstruction, it can simultaneously consider both the scientific angle and the distance dimension, thereby achieving higher-accuracy channel estimation with fewer codewords and reducing beam scanning / training overhead and computational complexity.

[0093] RIS-assisted obstacle avoidance:

[0094] In the RIS-assisted link, the cascaded channel can be equivalently represented as a hybrid field form "determined by both equivalent angle and equivalent distance". Based on this equivalent model, the dictionary of this invention is applied for channel acquisition. After obtaining the estimated channel, the optimal reflection direction / equivalent propagation component can be selected accordingly, and the RIS unit phase shift can be configured to enhance target area coverage, thereby achieving perception and tracking of environmental geometry or target motion state.

[0095] Localization and environmental perception (angle-distance parameters can be directly used for perception)

[0096] Since dictionary atoms correspond one-to-one with angles and distances, the angles and distances corresponding to the salient components of the dictionary domain coefficients can be used as the angle-distance estimation structure for the target, enabling users to locate with high precision. Under multi-frame / multi-time-slot observation, the changes of these salient components over time can also be tracked to achieve perception and tracking of environmental geometry or target motion state.

[0097] The dictionary proposed in this invention can significantly improve channel estimation performance using fewer codewords, while effectively eliminating the coverage blind spots in physical space inherent in traditional methods. Figure 2Taking a RIS-assisted single-base station-single-user communication system as an example, a hybrid-field channel is constructed in a simulation environment with an approximate multipath distribution. The figure compares the accuracy of channel recovery via sparse Bayesian learning using different dictionaries with the same number of pilots. "Proposed" indicates the hybrid-field physically consistent dictionary generated by this invention. It can be observed that compared to existing codebooks, the dictionary of this invention achieves at least a 10% improvement in estimation accuracy, demonstrating lower model mismatch in spherical wave modeling and near-field, far-field, and hybrid-field representations. Furthermore, Table 1 shows that, under the same angular-range resolution, the dictionary of this invention reduces the number of codewords by approximately 37% compared to traditional polar-domain codebooks. This significant reduction in the number of codewords not only lowers the computational complexity of channel estimation but also further shortens the processing latency during actual system operation.

[0098] Table 1 Number of dictionary codewords

[0099]

[0100] The dictionary exhibits excellent sparse representation capabilities. As shown in Table 2, the proposed dictionary construction method enables a highly sparse representation of the real channel within the dictionary domain. Extensive simulation experiments verify that, under known path parameters, orthogonal matching pursuit is used to reconstruct the channel. Whether in typical sparse scenarios with a small number of paths or complex scenarios with dense multipaths and incompletely sparse spatial structures, the proposed dictionary achieves significantly lower reconstruction errors than traditional plane wave or uniform angle sampling dictionaries.

[0101] Table 2. NMSE under different path count scenarios

[0102]

[0103] The foregoing description illustrates and describes a preferred embodiment of the present invention. However, as previously stated, it should be understood that the present invention is not limited to the forms disclosed herein and should not be construed as excluding other embodiments. It can be used in various other combinations, modifications, and environments, and can be altered within the scope of the inventive concept described herein through the foregoing teachings or techniques or knowledge in related fields. Any modifications and variations made by those skilled in the art that do not depart from the spirit and scope of the present invention should be within the protection scope of the appended claims.

Claims

1. A method for representing and acquiring hybrid field channels based on the main lobe energy coverage criterion, characterized in that: Includes the following steps: Step S1: Determine the Rayleigh distance and minimum sampling distance based on the communication system parameters; Step S2: Determine the angle domain sampling interval and generate angle sampling points based on 3dB power overlap; Step S3: Derive and apply the 3dB recursive analytical formula for the distance. At each angle, determine the adaptive distance sampling interval based on the nonlinear attenuation law of beam focusing, and iteratively generate distance sampling points. And ensure that the main lobe energy coverage overlap of adjacent dictionary codewords in the distance domain meets the preset threshold; Step S4: Gather the beam atoms corresponding to all angle and distance sampling points to obtain the final dictionary, which serves as the hybrid field channel representation.

2. The hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion according to claim 1, characterized in that: Step S1 includes: S101. Given the conditions that the communication system needs to satisfy: (1) The communication system operates in the near-field region and the mixed field region, while the user is outside the reaction near-field region; (2) The communication system's channel exhibits both spherical wavefront characteristics and planar wavefront characteristics simultaneously; S102. Given communication system parameters: Assume that in the communication system, the radiation source is equipped with... A ULA with antenna elements, wherein the ULA is a uniform linear array with an element spacing of . The wavelength at which the communication system operates is ; S103. Calculate Rayleigh distance : ; Set the minimum sampling distance for near-field sampling = This distance marks the initial boundary, which is the boundary between the reaction near field and the radiation near field.

3. The hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion according to claim 2, characterized in that: The radiation source includes a base station or a RIS, where RIS refers to a smart reflector.

4. The hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion according to claim 2, characterized in that: Step S2 includes determining the angle sampling point using the half-power beamwidth characteristics of the ULA, including: ULA's 3dB beamwidth in the sinusoidal domain and angular domain sampling interval ; To cover the entire space [-1,1], the angle sampling points... The calculation formula is: ； in, This represents the total number of angle samples.

5. The hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion according to claim 4, characterized in that: Step S3 includes: Utilizing the characteristics of ULA, sampling points are selected at each defined angle. In the distance domain [ ,+ Recursive sampling is performed on the first ( ), for the ( ) From one angle Distance from sampling point The generation logic is as follows: S301. To avoid performing independent distance recursion calculations at each angle sampling point, from Within the range of values, the sampling point closest to zero angle is selected and denoted as the anchor angle. ; At the anchor point angle Below, a range sampling sequence is generated based on the 3 dB beam depth criterion: The maximum distance for distance sampling is the distance to the far boundary: in, For array-related Rayleigh distance parameters; For anchor point angle Distance from sampling point The generation logic is as follows: A1. Initialization: Set the current distance from the upper bound. Distance index ; A2. Recursive calculation: Calculate the next sampling center point based on the current distance upper bound. When the normalized power drops to 1 / 2, i.e. -3dB, the Fresnel parameter is approximately 1.

32. In distance domain recursive sampling, fixed angle sampling points Subsequently, the main lobe power of adjacent codewords is controlled by only one dimensionless variable, namely the Fresnel parameter, and the effective focusing constant under the defined angle is: in, Determined by parameters such as array aperture and wavelength, such that... It has the dimension of length, and thus is related to distance. By comparison, Fresnel parameters are introduced: According to the main lobe power attenuation law, when the distance offset causes the normalized power to decrease to That is, at -3dB, the corresponding Fresnel parameter reaches a fixed threshold: Will Substitution and The relationship yields: Will Approximately denoted as a constant The final recursive formula for the next distance sampling center point is: A3. Calculate the distance to the sampling point The corresponding outer 3dB power boundary serves as the starting lower bound for the next point. : A4. Loop check will Updated to The threshold is set based on the maximum beamwidth. ,like Less than the threshold of the near-field boundary ,but Repeat the above steps; otherwise, stop near-field sampling. A5. Under different values ​​of q, through the above recursive process, a set of non-uniformly distributed distance sampling centers are obtained at the anchor point angle. }; S303. For anchor point angles The generated arbitrary distance sampling center Map it to the angle sampling point During the mapping process, the following equivalent focusing invariants remain unchanged: This yields the corresponding angle. The following distance sampling points: At the same time ensure Greater than ; S304. After completing near-field range sampling, in order to uniformly represent the far-field plane wave propagation characteristics, at each angle sampling point... Add a far-field proxy distance point at this location, with its distance being a constant value much larger than the Rayleigh distance: 。 6. The hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion according to claim 5, characterized in that: Step S4 includes: S401. Construction of elements in a single atom vector: For the arbitrary angle sampling pairs generated in steps S2-S3 , in a given column for containing A uniform forward array ULA with n elements and an element spacing of n is 1. The working wavelength is Under the condition of, determine the first The coordinates of the center of each array element are: Based on geometric relationships, calculate the sampling pair up to the th Distance between array elements: Furthermore, a guide vector determined by both angle and distance is constructed. Therefore, the first Each component satisfies: S402. Construction of Atomic Vectors: For the guide vector Energy normalization yields the atomic vectors corresponding to the sampled pairs, i.e., the dictionary column vectors: S403. Angle sampling point Corresponding dictionary matrix sub-moments : In step S3, for the angle sampling point For each q value, obtain Then, calculate the corresponding dictionary column vector according to step S402. Then, concatenate them column by column to obtain the corresponding dictionary matrix submatrices. ; S404. Repeat step S403 for m = 1, 2, ..., M to form the final mixed-field 3dB power dictionary matrix. That is, the hybrid field channel representation: 。 7. The hybrid field channel representation and acquisition method based on the main lobe energy coverage criterion according to claim 6, characterized in that: The generation of atomic vectors using the guiding vector formula ( ),include: Given a guide vector determined by both angle and distance , is represented as: exist Pick , Pick At that time, the corresponding guiding vector is calculated and used as the atomic vector. ( ).

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