A two-dimensional angle of arrival estimation method under fluid antenna energization

Abstract

The application discloses a two-dimensional angle of arrival estimation method under fluid antenna energization, which comprises the following steps: moving two orthogonally distributed fluid antennas along the x-axis and the z-axis at the same speed, respectively, moving a fixed distance each time, and receiving K narrowband pilot signals irrelevant to each other at each position; based on the received data after the fluid antenna moves multiple times, jointly using the autocovariance and cross-covariance information, respectively constructing the virtual aperture extended covariance vectors in the x-axis and z-axis directions; based on the two virtual aperture extended covariance vectors, using a two-step non-eigenvalue decomposition based subspace algorithm to coarsely estimate and then finely estimate the angle between the central transmission path of the narrowband pilot signal incidence and the x-axis positive direction and the angle between the central transmission path of the narrowband pilot signal incidence and the z-axis positive direction; and finally completing high-precision pairing of the two-dimensional angle of arrival; the method has the advantages that high-performance non-ambiguous estimation of the two-dimensional angle of arrival can be realized at a lower calculation cost and system overhead.

Description

Technical Field

[0001] This invention belongs to the field of array signal processing technology, and in particular relates to a two-dimensional angle of arrival estimation method enabled by a fluid antenna (FA). Background Technology

[0002] Two-dimensional angle-of-arrival (AOA) estimation under multipath transmission conditions plays a crucial role in both military and civilian applications. For example, in wireless communication systems, accurate AOA estimation is essential for acquiring channel state information (CSI) and performing effective downlink beamforming. Similarly, in radar and sonar detection systems, accurate AOA estimation is a fundamental basis for target detection and tracking, and a key component in achieving air and sea superiority.

[0003] To date, scholars both domestically and internationally have proposed a relatively rich set of two-dimensional angle-of-arrival (AOA) estimation methods within various frameworks, including the Feature Subspace Multiple Signal Classification (MUSIC) method, the Rotation Invariant Subspace (ESPRIT) method, and the Rank Reduction Criterion (RARE) method; sparse Bayesian learning and Approximate Message Passing (AMP) methods within the sparse signal reconstruction framework; and convolutional neural network (CNN) methods within the deep learning framework. However, it is necessary to point out that most of these methods are based on fixed-position antennas. Such antennas typically introduce three drawbacks / limitations: First, to achieve multi-target detection and estimation, a sufficient number of antenna elements and corresponding RF links are required to form a fixed antenna array system to ensure degrees of freedom, resulting in high costs for fixed antenna array systems. Second, the mutual coupling effect between antenna elements is unavoidable, and the phase errors generated by multiple RF links are difficult to effectively suppress and reliably calibrate, which severely degrades communication and parameter estimation performance. Third, the steering vector of a fixed antenna array system is static and corresponds to fixed degrees of freedom, which not only weakens reliable beamforming gain but also limits the realization of super-resolution two-dimensional AOA estimation and the ability to detect a large number of signal sources in practice. While massive MIMO antenna arrays could be used to address these issues, their high hardware costs are neither economical nor sustainable.

[0004] As a novel antenna design approach, fluid antennas have received significant research and attention in recent years due to their ability to overcome the shortcomings and limitations of fixed antenna array systems. In fluid antenna array systems, conductive fluids or liquid metals can be moved within a tubular container via software-controlled microfluidic devices or nanopumps, allowing for flexible and dynamic adjustment of the fluid antenna's position (also known as the port). This means that moving the fluid antenna can significantly increase spatial freedom. Furthermore, the ability to form a virtual array based on the movement / flow of a single fluid antenna provides a new solution for avoiding array coupling and reducing the impact of phase errors. However, current developments in fluid antenna-enabled technologies primarily focus on optimizing the fluid antenna's position to enhance channel estimation and beamforming design in wireless communication systems, while research on angle-of-arrival (AOA) estimation based on fluid antennas has been unreported. Therefore, researching a two-dimensional AOA estimation method based on fluid antennas, providing a novel technical solution for low-cost, high-performance, ambiguity-free estimation of the two-dimensional AOA, is of significant research importance. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to provide a two-dimensional angle of arrival estimation method under fluid antenna empowerment, which can achieve high-performance unambiguous estimation of two-dimensional angle of arrival with low computational cost and system overhead.

[0006] The technical solution adopted by this invention to solve the above-mentioned technical problems is: a two-dimensional angle of arrival estimation method under fluid antenna empowerment, characterized by including the following steps:

[0007] Step 1: Set the channel to a block fading channel with rich scattering and fast time-varying characteristics; and set that within a time block, the two fluid antennas move at the same speed along the positive x-axis and positive z-axis respectively, with each movement distance being d, and the total movement distance being Md, where d is greater than 0 and less than or equal to 1 / 2 of the carrier wavelength λ, and M is greater than 1; set that each fluid antenna receives K identical far-field and uncorrelated narrowband pilot signals with signal length N after each movement, where K is greater than or equal to 1;

[0008] Step 2: Represent the received data of the fluid antenna located on the x-axis after P movements as X. P The received data of the fluid antenna located on the z-axis after P movements is represented as Z. P Where K≤P≤M, X P It contains the angle between the center transmission path of each narrowband pilot signal incident and the positive x-axis, Z. P It contains the angle between the center transmission path of each narrowband pilot signal and the positive z-axis. The two angles corresponding to each narrowband pilot signal constitute the two-dimensional angle of arrival of this narrowband pilot signal.

[0009] Step 3: Calculate X P The relevant autocovariance vector r x,P Calculate Z P The relevant autocovariance vector r z,P Calculate X P With Z P The cross-covariance vector r in the first row xz,P Calculate Z P With X P The cross-covariance vector r in the first row zx,P Then use r x,P and r xz,P Construct the covariance vector r of the first virtual aperture extension. 1,P ; and utilize r z,P and r zx,P Construct the covariance vector r of the second virtual aperture extension. 2,P ;

[0010] Step 4: Based on r 1,P and r 2,P Using a two-step subspace algorithm based on non-eigenvalue decomposition, we first obtain a coarse estimate of the angle between the center transmission path of the K narrowband pilot signals and the positive x-axis. A rough estimate of the angle between the center transmission path of K narrowband pilot signals incident and the positive z-axis. Next, obtain a detailed estimate of the angle between the center transmission path of the K narrowband pilot signals and the positive x-axis. A detailed estimate of the angle between the center transmission path of K narrowband pilot signals incident and the positive z-axis.

[0011] Step 5: [Regarding...] and By performing one-to-one pairing, the included angles of each pairing constitute an estimate of the two-dimensional angle of arrival of a narrowband pilot signal.

[0012] In step 1, there are four ways the two fluid antennas can move: First, the fluid antenna on the x-axis moves from the origin along the positive x-axis to position Md, and the fluid antenna on the z-axis moves from position Md along the positive z-axis to the origin; Second, the fluid antenna on the z-axis moves from the origin along the positive z-axis to position Md, and the fluid antenna on the x-axis moves from position Md along the positive x-axis to the origin; Third, the fluid antenna on the x-axis moves from the origin along the positive x-axis to position Md, and the fluid antenna on the z-axis moves from the origin along the positive z-axis to position Md; Fourth, the fluid antenna on the x-axis moves from position Md along the positive x-axis to the origin, and the fluid antenna on the z-axis moves from position Md along the positive z-axis to the origin.

[0013] In step 2, X P ≈A α,P C0+A′ α,P C1+N x,P Z P ≈A β,P C0+A′ β,P C2+N z,P , where A α,P Let A represent the manifold matrix of the fluid antenna located on the x-axis. β,P This represents the manifold matrix of the fluid antenna located on the z-axis. This represents a complex matrix of dimension (P+1)×K.

[0014] j is represented by the imaginary part. This represents the random phase generated by the RF link corresponding to the fluid antenna located on the x-axis under non-ideal conditions. Let ψ represent the complex phase error generated by the RF link corresponding to the fluid antenna located on the x-axis, ψ represent the random phase generated by the RF link corresponding to the fluid antenna located on the z-axis under non-ideal conditions, and exp(jψ) represent the complex phase error generated by the RF link corresponding to the fluid antenna located on the z-axis. α k β represents the angle between the center transmission path of the k-th narrowband pilot signal incident on the positive x-axis. k Let α represent the angle between the center transmission path of the k-th narrowband pilot signal and the positive z-axis. k ,β k ξ represents the two-dimensional angle of arrival of the k-th narrowband pilot signal, with the superscript "T" indicating the transpose of the vector or matrix. k =exp((j2πdcosα) k ) / λ), A′ α,P For A α,P The derivative of A′ β,P For A β,P The derivative of N x,P N represents the Gaussian white noise matrix of the fluid antenna located on the x-axis after P movements. z,P C0 represents the Gaussian white noise matrix of the fluid antenna located on the z-axis after P movements, where C0 = [c 0,1 ,...,c 0,t ,...,c 0,N ], C1 = [c 1,1 ,...,c 1,t ,...,c1,N ], C2 = [c 2,1 ,...,c 2,t ,...,c 2,N ], t=1,…,N,c 0,t =[c 1,0,t ,...,c K,0,t ] T c 1,t =[c 1,1,t ,...,c K,1,t ] T c 2,t =[c 1,2,t ,…,c K,2,t ] T , s k,t L represents the t-th sampling point of the k-th narrowband pilot signal. k This represents the number of transmission paths for the k-th narrowband pilot signal, where l = 1, 2, ..., L k γ k,l,t s k,t Complex path gain propagating along the l-th transmission path, and All are s k,t The random variable generated along the l-th transmission path, and All follow a Gaussian distribution with a mean of 0.

[0015] In step 3 The superscript "H" indicates the conjugate transpose of a vector or matrix. X represents P The first line, Z represents P In the first row, p represents the power vector of the narrowband pilot signal receiver after transmission through the channel, and the k-th element of p is p k , The superscript "*" indicates the conjugate of a vector or matrix, i P+1 This represents a vector of dimension (P+1)×1, where the first element is 1 and all other elements are 0. This represents the noise variance of the Gaussian white noise generated by the fluid antenna located on the x-axis after each movement. This represents the noise variance of the Gaussian white noise generated by the fluid antenna located on the z-axis after each movement.

[0016] In step 3

[0017] in, Let J represent a complex vector of dimension (2P+1)×1. P This represents an anti-diagonal matrix of dimension (P+1)×(P+1), where the elements on the anti-diagonal are 1 and the elements at other positions are 0. Indicates by r xz,P The subvector formed by the 2nd to the (P+1)th elements, Indicates by The subvector consisting of the first P elements, Indicates r xz,P The first element, Indicates by r x,P The subvector formed by the 2nd to the (P+1)th elements, This represents the first virtual aperture extension steering matrix. This represents a complex matrix with dimensions (2P+1)×K. Indicates by r zx,P The subvector formed by the 2nd to the (P+1)th elements, Indicates by The subvector consisting of the first P elements, Indicates r zx,P The first element, Indicates by r z,P The subvector formed by the 2nd to the (P+1)th elements, This represents the second virtual aperture extension steering matrix.

[0018] The specific process of step 4 is as follows:

[0019] Step 4.1: Set K≤P<M, based on r 1,P Construct the first matrix R with dimensions (2P+2-K)×K. α,P , And based on r 2,P Construct a second matrix R with dimensions (2P+2-K)×K. β,P , Among them, U α,P V α,P U β,P and V β,P All are full-rank matrices with a Vandermonde structure. and All are full-rank matrices with a diagonal matrix structure. The function diag{·} represents diagonalization of a vector; then R α,PThe first orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. And for R β,P The second orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. Then through the spectral function A one-dimensional spectral search is performed within the entire spatial range [-90°, 90°] with a step size of 1°. All obtained spectral function values ​​are sorted in descending order. The angle values ​​corresponding to the first K spectral function values ​​are taken as coarse estimates of the angles between the center transmission paths of the K narrowband pilot signals and the positive x-axis, denoted as […]. Where α' is the angle variable, f(·) is the spectral function representation, and I 2P+2-K Let u represent an identity matrix of dimension (2P+2-K)×(2P+2-K). P (α')=[1,ξ',...,(ξ') 2P+1-K ] T ξ'=exp((j2πdcosα') / λ); Similarly, through the spectral function A one-dimensional spectral search is performed across the entire [-90°, 90°] spatial domain with a step size of 1°. All obtained spectral function values ​​are sorted in descending order. The angle values ​​corresponding to the first K spectral function values ​​are taken as coarse estimates of the angles between the center transmission paths of the K narrowband pilot signals and the positive z-axis, denoted as […]. Where β' is the angle variable,

[0020] Step 4.2: Set P = M, based on r 1,M Construct a third matrix R with dimensions (2M+2-K)×K. α,M , And based on r 2,M Construct a fourth matrix R with dimensions (2M+2-K)×K. β,M , Among them, U α,M V α,M U β,M and V β,M All are full-rank matrices with a Vandermonde structure; then, for R... α,M The third orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. And for R β,M The fourth orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. Then iterate k from 1 to K, and for each iteration, use the spectral function. Within the airspace A one-dimensional spectral search is performed with a step size of 0.1°. The angle value corresponding to the maximum value among all spectral function values ​​obtained in each traversal is used as a fine estimate of the angle between the center transmission path of the narrowband pilot signal and the positive x-axis. After the traversal, K fine estimates of the angle between the center transmission path of the narrowband pilot signal and the positive x-axis are obtained, denoted as follows: Among them, I 2M+2-K Let u represent an identity matrix of dimension (2M+2-K)×(2M+2-K), where Δα' ranges from [2°, 3°]. M (α')=[1,ξ',...,(ξ') 2M+1-K ] T Similarly, let k iterate from 1 to K, and for each iteration, use the spectral function. Within the airspace A one-dimensional spectral search is performed with a step size of 0.1°. The angle value corresponding to the maximum value among all spectral function values ​​obtained in each traversal is used as a fine estimate of the angle between the center transmission path of the narrowband pilot signal and the positive z-axis. After the traversal, K fine estimates of the angle between the center transmission path of the narrowband pilot signal and the positive z-axis are obtained, denoted as follows: Where Δβ' takes values ​​in the range [2°, 3°].

[0021] The specific process of step 5 is as follows:

[0022] Step 5.1: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] Substitution In the middle, we obtained And Substitution In the middle, we obtained

[0023] Step 5.2: Based on the least squares method, obtain and Their respective estimates are denoted as: and Among them, superscript This indicates a pseudo-inverse operation;

[0024] Step 5.3: Calculation The difference between exp(jψ) and The estimated value in, express The kth element, express The kth element;

[0025] Step 5.4: Calculate XP With Z P The cross-covariance matrix, and using Complex phase error compensation is performed to obtain matrix R. xz , Among them, Λ c0 =diag{p};

[0026] Step 5.5: For R xz Perform column-wise vectorization to obtain

[0027] The function vec(·) represents the column vectorization of the matrix. Represents the Kronecker product;

[0028] Step 5.6: [The sentence is incomplete and requires more context to be translated accurately.] Substitution In the middle, we obtained And Substitution In the middle, we obtained Then Finding linear operations The angle at which the maximum value is taken is set to... in, Then and The pairing constitutes an estimate of the two-dimensional angle of arrival for a narrowband pilot signal.

[0029] Compared with the prior art, the advantages of the present invention are as follows:

[0030] 1) By using a fluid antenna that moves on two orthogonal axes (x-axis and z-axis) to simultaneously acquire spatial information of the signal, a joint estimation of the two-dimensional angle of arrival is achieved. In the joint estimation process, a two-step subspace algorithm based on non-eigenvalue decomposition (coarse estimation and fine estimation) is used, which significantly improves the estimation accuracy of the two-dimensional angle of arrival.

[0031] 2) Both the x-axis and z-axis form a virtual array output by moving a single fluid antenna, which effectively avoids the generation of array mutual coupling. By constructing a covariance vector of virtual aperture extension and applying least squares phase error estimation technology, the impact of array phase error and fast time-varying channel multipath transmission is significantly reduced. It has good universality and high estimation accuracy.

[0032] 3) The two-step subspace algorithm based on non-eigenvalue decomposition and the linear operation pairing test are used to complete the efficient and unambiguous estimation of the two-dimensional angle of arrival, making full use of computing resources and reducing computing costs and system overhead.

[0033] 4) Two-dimensional angle of arrival estimation can be achieved with only two fluid antennas, reducing system overhead.

[0034] 5) The design of the moving distance of the fluid antenna satisfies the spatial sampling theorem to avoid the grating lobe problem, and allows the number of moves to be adjusted according to the actual scenario to balance performance and cost. Attached Figure Description

[0035] Figure 1 This is a flowchart illustrating the implementation process of the method of the present invention;

[0036] Figure 2 This is a schematic diagram of the receiving model for two-dimensional angle of arrival estimation under fluid antenna empowerment in the method of the present invention;

[0037] Figure 3 The figure shows the simulation results of the root mean square error of the two-dimensional angle of arrival as a function of signal-to-noise ratio (SNR), which was estimated using the rotation-invariant subspace method and the method of this invention, respectively.

[0038] Figure 4 The figure shows the simulation results of the root mean square error of the two-dimensional angle of arrival as a function of the length of the narrowband pilot signal, estimated using the rotationally invariant subspace method and the method of this invention, respectively. Detailed Implementation

[0039] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can implement it based on the description. The scope of protection of the present invention is not limited to the specific embodiments.

[0040] The present invention proposes a two-dimensional angle-of-arrival estimation method under fluid antenna empowerment, the flowchart of which is shown below. Figure 1 As shown, it includes the following steps:

[0041] Step 1: Set the channel as a block fading channel with rich scattering and fast time-varying characteristics, that is, in a channel with a length of The two-dimensional angle of arrival and channel information remain unchanged within the time block; and a length of [missing information] is set. Within a time block, the two fluid antennas move at the same speed (the specific speed is not limited, it only needs to be determined within a length of...). Within a time block, move at least M times, moving along the positive x-axis and positive z-axis respectively, such as... Figure 2 As shown, the distance moved each time is d, and the total distance moved is Md, where d is greater than 0 and less than or equal to 1 / 2 of the carrier wavelength λ. Generally, d is taken to be equal to 1 / 2 of λ, and M is greater than 1. Each fluid antenna is configured to receive K identical, uncorrelated, far-field narrowband pilot signals of length N after each movement. Figure 2 As shown, K is greater than or equal to 1.

[0042] Here, there are four ways the two fluid antennas can move. The first way is as follows: Figure 2 As shown, the fluid antenna on the x-axis moves from the origin along the positive x-axis to position Md, and the fluid antenna on the z-axis moves from position Md along the positive z-axis to the origin; the second method is: the fluid antenna on the z-axis moves from the origin along the positive z-axis to position Md, and the fluid antenna on the x-axis moves from position Md along the positive x-axis to the origin; the third method is: the fluid antenna on the x-axis moves from the origin along the positive x-axis to position Md, and the fluid antenna on the z-axis moves from the origin along the positive z-axis to position Md; the fourth method is: the fluid antenna on the x-axis moves from position Md along the positive x-axis to the origin, and the fluid antenna on the z-axis moves from position Md along the positive z-axis to the origin.

[0043] Step 2: Represent the received data of the fluid antenna located on the x-axis after P movements as X. P The received data of the fluid antenna located on the z-axis after P movements is represented as Z. P Where K≤P≤M, X P It contains the angle between the center transmission path of each narrowband pilot signal incident and the positive x-axis, Z. P It contains the angle between the center transmission path of each narrowband pilot signal incident and the positive z-axis. The two angles corresponding to each narrowband pilot signal constitute the two-dimensional angle of arrival of this narrowband pilot signal.

[0044] In this specific embodiment, X P ≈A α,P C0+A′ α,P C1+N x,P Z P ≈A β,P C0+A′ β,P C2+N z,P , where A α,P Let A represent the manifold matrix of the fluid antenna located on the x-axis. β,P This represents the manifold matrix of the fluid antenna located on the z-axis. This represents a complex matrix of dimension (P+1)×K.

[0045] exp(·) denotes an exponential function with the natural constant as its base, where the natural constant is 2.71…, and j represents the imaginary part. This represents the random phase generated by the RF link corresponding to the fluid antenna located on the x-axis under non-ideal conditions. Let ψ represent the complex phase error generated by the RF link corresponding to the fluid antenna located on the x-axis, ψ represent the random phase generated by the RF link corresponding to the fluid antenna located on the z-axis under non-ideal conditions, and exp(jψ) represent the complex phase error generated by the RF link corresponding to the fluid antenna located on the z-axis. and The superscript in α indicates the power, α k β represents the angle between the center transmission path of the k-th narrowband pilot signal incident on the positive x-axis. k Let α represent the angle between the center transmission path of the k-th narrowband pilot signal and the positive z-axis. k ,β k ξ represents the two-dimensional angle of arrival of the k-th narrowband pilot signal, with the superscript "T" indicating the transpose of the vector or matrix. k =exp((j2πdcosα) k ) / λ), A′ α,P For A α,P The derivative of A′ β,P For A β,P The derivative of

[0046] N x,P N represents the Gaussian white noise matrix of the fluid antenna located on the x-axis after P movements. z,P C0 represents the Gaussian white noise matrix of the fluid antenna located on the z-axis after P movements, where C0 = [c 0,1 ,…,c 0,t ,…,c 0,N ], C1 = [c 1,1 ,…,c 1,t ,…,c 1,N ],

[0047] C2 = [c 2,1 ,...,c 2,t ,...,c 2,N ], t=1,…,N,c 0,t =[c 1,0,t ,...,c K,0,t ] T c 1,t =[c 1,1,t ,...,c K,1,t ] T ,

[0048] c 2,t =[c 1,2,t ,...,c K,2,t ] T ,

[0049] s k,t L represents the t-th sampling point of the k-th narrowband pilot signal. k This represents the number of transmission paths for the k-th narrowband pilot signal, where l = 1, 2, ..., L k γ k,l,t s k,t Complex path gain propagating along the l-th transmission path, and All are s k,t The random variable generated along the l-th transmission path, and All follow a Gaussian distribution with a mean of 0.

[0050] Step 3: Calculate X P The relevant autocovariance vector r x,P Calculate Z P The relevant autocovariance vector r z,P Calculate X P With Z P The cross-covariance vector r in the first row xz,P Calculate Z P With X P The cross-covariance vector r in the first row zx,P Then use r x,P and r xz,P Construct the covariance vector r of the first virtual aperture extension. 1,P ; and utilize r z,P and r zx,P Construct the covariance vector r of the second virtual aperture extension. 2,P .

[0051] In this specific embodiment,

[0052] The superscript "H" indicates the conjugate transpose of a vector or matrix. X represents P The first line, Z represents P In the first row, p represents the power vector of the narrowband pilot signal receiver after transmission through the channel, and the k-th element of p is p k , The superscript "*" indicates the conjugate of a vector or matrix, i P+1 This represents a vector of dimension (P+1)×1, where the first element is 1 and all other elements are 0. This represents the noise variance of the Gaussian white noise generated by the fluid antenna located on the x-axis after each movement. This represents the noise variance of the Gaussian white noise generated by the fluid antenna located on the z-axis after each movement.

[0053] In this specific embodiment,

[0054] in, Let J represent a complex vector of dimension (2P+1)×1. P This represents an anti-diagonal matrix of dimension (P+1)×(P+1), where the elements on the anti-diagonal are 1 and the elements at other positions are 0. Indicates by r xz,P The subvector formed by the 2nd to the (P+1)th elements, Indicates by The subvector consisting of the first P elements, Indicates r xz,P The first element, Indicates by r x,P The subvector formed by the 2nd to the (P+1)th elements, This represents the first virtual aperture extension steering matrix. This represents a complex matrix with dimensions (2P+1)×K. Indicates by r zx,P The subvector formed by the 2nd to the (P+1)th elements, Indicates by The subvector consisting of the first P elements, Indicates r zx,P The first element, Indicates by r z,P The subvector formed by the 2nd to the (P+1)th elements, This represents the second virtual aperture extension steering matrix.

[0055] Step 4: Based on r 1,P and r 2,P Using a two-step subspace algorithm based on non-eigenvalue decomposition, we first obtain a coarse estimate of the angle between the center transmission path of the K narrowband pilot signals and the positive x-axis. A rough estimate of the angle between the center transmission path of K narrowband pilot signals incident and the positive z-axis. Next, obtain a detailed estimate of the angle between the center transmission path of the K narrowband pilot signals and the positive x-axis. A detailed estimate of the angle between the center transmission path of K narrowband pilot signals incident and the positive z-axis.

[0056] In this specific embodiment, step 4 is performed as follows:

[0057] Step 4.1: Set K≤P<M, based on r 1,P Construct the first matrix R with dimensions (2P+2-K)×K. α,P , And based on r 2,P Construct a second matrix R with dimensions (2P+2-K)×K. β,P , in, Indicates by r 1,P The subvector consisting of the first to the second (2P+2-K)th elements, Indicates by r 1,P The subvector consisting of the 2nd to the 2P+3-Kth elements, Indicates by r 1,P The subvector consisting of the Kth to the (2P+1)th elements, Indicates by r 2,P The subvector consisting of the first to the second (2P+2-K)th elements, Indicates by r 2,P The subvector consisting of the 2nd to the 2P+3-Kth elements, Indicates by r 2,P The subvector formed by the Kth to the (2P+1)th elements, U α,P V α,P U β,P and V β,P All are full-rank matrices with a Vandermonde structure. and All are full-rank matrices with a diagonal matrix structure. The function diag{·} represents diagonalization of a vector; then R α,P The first orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. And for R β,P The second orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. Then through the spectral function A one-dimensional spectral search is performed within the entire spatial range [-90°, 90°] with a step size of 1°. All obtained spectral function values ​​are sorted in descending order. The angle values ​​corresponding to the first K spectral function values ​​are taken as coarse estimates of the angles between the center transmission paths of the K narrowband pilot signals and the positive x-axis, denoted as […]. Where α' is the angle variable, f(·) is the spectral function representation, and I 2P+2-KLet u represent an identity matrix of dimension (2P+2-K)×(2P+2-K). P (α')=[1,ξ',...,(ξ') 2P+1-K ] T ξ'=exp((j2πdcosα') / λ); Similarly, through the spectral function A one-dimensional spectral search is performed across the entire [-90°, 90°] spatial domain with a step size of 1°. All obtained spectral function values ​​are sorted in descending order. The angle values ​​corresponding to the first K spectral function values ​​are taken as coarse estimates of the angles between the center transmission paths of the K narrowband pilot signals and the positive z-axis, denoted as […]. Where β' is the angle variable,

[0058] Step 4.2: Set P = M, based on r 1,M Construct a third matrix R with dimensions (2M+2-K)×K. α,M , And based on r 2,M Construct a fourth matrix R with dimensions (2M+2-K)×K. β,M , in, Indicates by r 1,M The subvector consisting of the first to the second (M+2-K)th elements, Indicates by r 1,M The subvector consisting of the 2nd to 2M+3-Kth elements, Indicates by r 1,M The subvector formed by the Kth to 2M+1th elements, Indicates by r 2,M The subvector consisting of the first to the second (M+2-K)th elements, Indicates by r 2,M The subvector consisting of the 2nd to 2M+3-Kth elements, Indicates by r 2,M The subvector formed by the Kth to 2M+1th elements, U α,M V α,M U β,M and V β,M All are full-rank matrices with a Vandermonde structure; then, for R... α,M The third orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. And for R β,M The fourth orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. Then iterate k from 1 to K, and for each iteration, use the spectral function. Within the airspace A one-dimensional spectral search is performed with a step size of 0.1°. The angle value corresponding to the maximum value among all spectral function values ​​obtained in each traversal is used as a fine estimate of the angle between the center transmission path of the narrowband pilot signal and the positive x-axis. After the traversal, K fine estimates of the angle between the center transmission path of the narrowband pilot signal and the positive x-axis are obtained, denoted as follows: Among them, I 2M+2-K Let u represent an identity matrix of dimension (2M+2-K)×(2M+2-K), where Δα' ranges from [2°, 3°]. M (α')=[1,ξ',...,(ξ') 2M+1-K ] T Similarly, let k iterate from 1 to K, and for each iteration, use the spectral function. Within the airspace A one-dimensional spectral search is performed with a step size of 0.1°. The angle value corresponding to the maximum value among all spectral function values ​​obtained in each traversal is used as a fine estimate of the angle between the center transmission path of the narrowband pilot signal and the positive z-axis. After the traversal, K fine estimates of the angle between the center transmission path of the narrowband pilot signal and the positive z-axis are obtained, denoted as follows: Where Δβ' takes values ​​in the range [2°, 3°].

[0059] Step 5: [Regarding...] and By performing one-to-one pairing, the included angles of each pairing constitute an estimate of the two-dimensional angle of arrival of a narrowband pilot signal.

[0060] In this specific embodiment, step 5 is performed as follows:

[0061] Step 5.1: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] Substitution In the middle, we obtained And Substitution In the middle, we obtained

[0062] Step 5.2: Based on the least squares method, obtain and Their respective estimates are denoted as: and Among them, superscript This indicates a pseudo-inverse operation.

[0063] Step 5.3: Calculation The difference between exp(jψ) and The estimated value

[0064] in, express The kth element, express The kth element.

[0065] Step 5.4: Calculate X P With Z P The cross-covariance matrix, and using Complex phase error compensation is performed to obtain matrix R. xz , Among them, Λ c0 =diag{p}.

[0066] Step 5.5: For R xz Perform column-wise vectorization to obtain

[0067] The function vec(·) represents the column vectorization of the matrix. This represents the Kronecker product.

[0068] Step 5.6: [The sentence is incomplete and requires more context to be translated accurately.] Substitution In the middle, we obtained And Substitution In the middle, we obtained Then Finding linear operations The angle at which the maximum value is taken is set to... in, Then and The pairing constitutes an estimate of the two-dimensional angle of arrival for a narrowband pilot signal.

[0069] The performance of the method of the present invention will be analyzed through simulation experiments below.

[0070] The simulation was performed using MATLAB software. The two-dimensional angles of arrival of the two zero-mean and uncorrelated narrowband pilot signals were {α1=79.6°, β1=98.8°} and {α2=105.3°, β2=127.2°}, respectively. Each narrowband pilot signal was transmitted along L... k = 100 transmission paths propagating, complex path gain γ k,l,t It follows a Gaussian distribution with a mean of 0 and a variance of 1. and Both are random variables with a mean of 0 and a variance of 1.5°. The fluid antennas located on the x-axis and z-axis move 10 times each, i.e., M = 10. The distance d moved each time is equal to 1 / 2 of the carrier wavelength λ. The complex phase error generated by the RF link corresponding to the fluid antenna on the x-axis is exp(jπ / 12), and the complex phase error generated by the RF link corresponding to the fluid antenna on the z-axis is exp(jπ / 6).

[0071] Simulation Experiment 1: This experiment demonstrates the effectiveness of the two-dimensional angle of arrival (Angle of Arrival) estimation method obtained using this invention under different signal-to-noise ratios (SNR), and measures it using the root mean square error (RMSE) index of the two-dimensional Angle of Arrival estimation. During the simulation, the length of the narrowband pilot signal N = 200, and the SNR changes from -9 dB to 6 dB. The simulation results are as follows: Figure 3 As shown. From Figure 3 As can be seen, the method of the present invention can achieve significantly better estimation performance than the rotationally invariant subspace (ESPRIT) method used for comparison throughout the entire SNR range, and is close to the theoretical lower bound of estimation (i.e., the Cramer-Rao lower bound), which fully verifies the superiority of the method of the present invention in resisting multipath transmission and phase error.

[0072] Simulation Experiment 2: This experiment demonstrates the effectiveness of the two-dimensional angle of arrival (Angle of Arrival) estimation method obtained using this invention under different narrowband pilot signal lengths, and measures it using the root mean square error (RMSE) index of the two-dimensional Angle of Arrival (Angle of Arrival) estimation. During the simulation, the SNR was fixed at -3 dB, and the narrowband pilot signal length N varied from 50 to 300. The simulation results are as follows: Figure 4 As shown. From Figure 4 As can be seen, the method of the present invention achieves significantly better estimation performance than the rotationally invariant subspace (ESPRIT) method used for comparison throughout the entire observation range of the narrowband pilot signal, further verifying the superiority of the present invention. Furthermore, it should be noted that the method of the present invention requires only two fluid antennas to complete the unambiguous estimation of the two-dimensional angle of arrival for multiple narrowband pilot signals, while the traditional method using the same number of fixed-position antennas cannot achieve effective estimation due to the inability to obtain an effective subspace, demonstrating the advantages of the present invention in terms of extended degrees of freedom and flexibility of use.

Claims

1. A two-dimensional angle-of-arrival estimation method under fluid antenna empowerment, characterized in that... Includes the following steps: Step 1: Set the channel to a block fading channel with rich scattering and fast time-varying characteristics; Within a time block, two fluid antennas move at the same speed along the positive x-axis and positive z-axis, respectively. Each movement is a distance d, and the total movement distance is Md, where d is greater than 0 and less than or equal to 1 / 2 of the carrier wavelength λ, and M is greater than 1. Each fluid antenna receives K identical, uncorrelated, far-field narrowband pilot signals with a signal length of N after each movement, where K is greater than or equal to 1. Step 2: Represent the received data of the fluid antenna located on the x-axis after P movements as X. P The received data of the fluid antenna located on the z-axis after P movements is represented as Z. P Where K≤P≤M, X P It contains the angle between the center transmission path of each narrowband pilot signal incident and the positive x-axis, Z. P It contains the angle between the center transmission path of each narrowband pilot signal and the positive z-axis. The two angles corresponding to each narrowband pilot signal constitute the two-dimensional angle of arrival of this narrowband pilot signal. Step 3: Calculate X P The relevant autocovariance vector r x,P Calculate Z P The relevant autocovariance vector r z,P Calculate X P With Z P The cross-covariance vector r in the first row xz,P Calculate Z P With X P The cross-covariance vector r in the first row zx,P Then use r x,P and r xz,P Construct the covariance vector r of the first virtual aperture extension. 1,P ; and utilize r z,P and r zx,P Construct the covariance vector r of the second virtual aperture extension. 2,P ; Step 4: Based on r 1,P and r 2,P Using a two-step subspace algorithm based on non-eigenvalue decomposition, we first obtain a coarse estimate of the angle between the center transmission path of the K narrowband pilot signals and the positive x-axis. A rough estimate of the angle between the center transmission path of K narrowband pilot signals incident and the positive z-axis. Next, obtain a detailed estimate of the angle between the center transmission path of the K narrowband pilot signals and the positive x-axis. A detailed estimate of the angle between the center transmission path of K narrowband pilot signals incident and the positive z-axis. Step 5: [Regarding...] and By performing one-to-one pairing, the included angles of each pairing constitute an estimate of the two-dimensional angle of arrival of a narrowband pilot signal.

2. The two-dimensional angle of arrival estimation method under fluid antenna empowerment according to claim 1, characterized in that... In step 1, there are four ways to move the two fluid antennas. The first way is that the fluid antenna on the x-axis moves from the origin along the positive x-axis to the Md position, and the fluid antenna on the z-axis moves from the Md position along the positive z-axis to the origin. The second way is that the fluid antenna on the z-axis moves from the origin along the positive z-axis to the Md position, and the fluid antenna on the x-axis moves from the Md position along the positive x-axis to the origin. The third method: The fluid antenna on the x-axis moves from the origin along the positive x-axis to position Md, and the fluid antenna on the z-axis moves from the origin along the positive z-axis to position Md. The fourth method: The fluid antenna on the x-axis moves from position Md along the positive x-axis to the origin, and the fluid antenna on the z-axis moves from position Md along the positive z-axis to the origin.

3. A two-dimensional angle-of-arrival estimation method under fluid antenna empowerment according to claim 1 or 2, characterized in that... In step 2, X P ≈A α,P C0+A′ α,P C1+N x,P Z P ≈A β,P C0+A′ β,P C2+N z,P , where A α,P Let A represent the manifold matrix of the fluid antenna located on the x-axis. β,P This represents the manifold matrix of the fluid antenna located on the z-axis. This represents a complex matrix of dimension (P+1)×K. j is represented by the imaginary part. This represents the random phase generated by the RF link corresponding to the fluid antenna located on the x-axis under non-ideal conditions. Let ψ represent the complex phase error generated by the RF link corresponding to the fluid antenna located on the x-axis, ψ represent the random phase generated by the RF link corresponding to the fluid antenna located on the z-axis under non-ideal conditions, and exp(jψ) represent the complex phase error generated by the RF link corresponding to the fluid antenna located on the z-axis. k = 1, 2, ..., K, α k β represents the angle between the center transmission path of the k-th narrowband pilot signal incident on the positive x-axis. k Let α represent the angle between the center transmission path of the k-th narrowband pilot signal and the positive z-axis. k ,β k ξ represents the two-dimensional angle of arrival of the k-th narrowband pilot signal, with the superscript "T" indicating the transpose of the vector or matrix. k =exp((j2πdcosα) k ) / λ), A′ α,P For A α,P The derivative of A′ β,P For A β,P The derivative, N x,P N represents the Gaussian white noise matrix of the fluid antenna located on the x-axis after P movements. z,P C0 represents the Gaussian white noise matrix of the fluid antenna located on the z-axis after P movements, where C0 = [c 0,1 ,...,c 0,t ,...,c 0,N ], C1 = [c 1,1 ,...,c 1,t ,...,c 1,N ], C2 = [c 2,1 ,...,c 2,t ,...,c 2,N ], t=1,…,N,c 0,t =[c 1,0,t ,…,c K,0,t ] T c 1,t =[c 1,1,t ,…,c K,1,t ] T c 2,t =[c 1,2,t ,…,c K,2,t ] T , s k,t L represents the t-th sampling point of the k-th narrowband pilot signal. k This represents the number of transmission paths for the k-th narrowband pilot signal, e = 1, 2, ..., L k γ k,e,t s k,t Complex path gain propagating along the l-th transmission path, and All are s k,t The random variable generated along the l-th transmission path, and All follow a Gaussian distribution with a mean of 0.

4. The two-dimensional angle of arrival estimation method under fluid antenna empowerment according to claim 3, characterized in that... In step 3 The superscript "H" indicates the conjugate transpose of a vector or matrix. X represents P The first line, Z represents P In the first row, p represents the power vector of the narrowband pilot signal receiver after transmission through the channel, and the k-th element of p is p k , The superscript "*" indicates the conjugate of a vector or matrix, i P+1 This represents a vector of dimension (P+1)×1, where the first element is 1 and all other elements are 0. This represents the noise variance of the Gaussian white noise generated by the fluid antenna located on the x-axis after each movement. This represents the noise variance of the Gaussian white noise generated by the fluid antenna located on the z-axis after each movement.

5. The two-dimensional angle of arrival estimation method under fluid antenna empowerment according to claim 4, characterized in that... In step 3 in, Let J represent a complex vector of dimension (2P+1)×1. P This represents an anti-diagonal matrix of dimension (P+1)×(P+1), where the elements on the anti-diagonal are 1 and the elements at other positions are 0. Indicated by r xz,P The subvector formed by the 2nd to the (P+1)th elements, Indicates by The subvector consisting of the first P elements, Indicates r xz,P The first element, Indicated by r x,P The subvector formed by the 2nd to the (P+1)th elements, This represents the first virtual aperture extension steering matrix. This represents a complex matrix with dimensions (2P+1)×K. Indicates by r zx,P The subvector formed by the 2nd to the (P+1)th elements, Indicates by The subvector consisting of the first P elements, Indicates r zx,P The first element, Indicated by r z,P The subvector formed by the 2nd to the (P+1)th elements, This represents the second virtual aperture extension steering matrix.

6. The two-dimensional angle of arrival estimation method under fluid antenna empowerment according to claim 5, characterized in that... The specific process of step 4 is as follows: Step 4.1: Set K≤P<M, based on r 1,P Construct the first matrix R with dimensions (2P+2-K)×K. α,P , And based on r 2,P Construct a second matrix R with dimensions (2P+2-K)×K. β,P , Among them, U α,P V α,P U β,P and V β,P All are full-rank matrices with a Vandermonde structure. and All are full-rank matrices with a diagonal matrix structure. The function diag{·} represents diagonalization of a vector; then R α,P The first orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. And for R β,P The second orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. Then through the spectral function A one-dimensional spectral search is performed within the entire spatial range [-90°, 90°] with a step size of 1°. All obtained spectral function values ​​are sorted in descending order. The angle values ​​corresponding to the first K spectral function values ​​are taken as coarse estimates of the angles between the center transmission paths of the K narrowband pilot signals and the positive x-axis, denoted as […]. Where α' is the angle variable, f(·) is the spectral function representation, and I 2P+2-K Let u represent an identity matrix of dimension (2P+2-K)×(2P+2-K). P (α')=[1,ξ',…,(ξ') 2P+1-K ] T ξ'=exp((j2πdcosα') / λ); Similarly, through the spectral function A one-dimensional spectral search is performed across the entire [-90°, 90°] spatial domain with a step size of 1°. All obtained spectral function values ​​are sorted in descending order. The angle values ​​corresponding to the first K spectral function values ​​are taken as coarse estimates of the angles between the center transmission paths of the K narrowband pilot signals and the positive z-axis, denoted as […]. Where β' is the angle variable, Step 4.2: Set P = M, based on r 1,M Construct a third matrix R with dimensions (2M+2-K)×K. α,M , And based on r 2,M Construct a fourth matrix R with dimensions (2M+2-K)×K. β,M , Among them, U α,M V α,M U β,M and V β,M All are full-rank matrices with a Vandermonde structure; then, for R... α,M The third orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. And for R β,M The fourth orthogonal matrix is ​​obtained by performing Gram-Schmidt orthogonalization on each column. Then iterate k from 1 to K, and for each iteration, use the spectral function. Within the airspace A one-dimensional spectral search is performed with a step size of 0.1°. The angle value corresponding to the maximum value among all spectral function values ​​obtained in each traversal is used as a fine estimate of the angle between the center transmission path of the narrowband pilot signal and the positive x-axis. After the traversal, K fine estimates of the angle between the center transmission path of the narrowband pilot signal and the positive x-axis are obtained, denoted as follows: Among them, I 2M+2-K Let u represent an identity matrix of dimension (2M+2-K)×(2M+2-K), where Δα' ranges from [2°, 3°]. M (α')=[1,ξ',...,(ξ') 2M+1-K ] T Similarly, let k iterate from 1 to K, and for each iteration, use the spectral function. Within the airspace A one-dimensional spectral search is performed with a step size of 0.1°. The angle value corresponding to the maximum value among all spectral function values ​​obtained in each traversal is used as a fine estimate of the angle between the center transmission path of the narrowband pilot signal and the positive z-axis. After the traversal, K fine estimates of the angle between the center transmission path of the narrowband pilot signal and the positive z-axis are obtained, denoted as follows: Where Δβ' takes values ​​in the range [2°, 3°].

7. The two-dimensional angle of arrival estimation method under fluid antenna empowerment according to claim 6, characterized in that... The specific process of step 5 is as follows: Step 5.1: [The text appears to be incomplete and contains several grammatical errors. A more accurate translation would require the full context.] Substitution In the middle, we obtained And Substitution In the middle, we obtained Step 5.2: Based on the least squares method, obtain and Their respective estimates are denoted as: and Among them, superscript This indicates a pseudo-inverse operation; Step 5.3: Calculation The difference between exp(jψ) and The estimated value in, express The kth element, express The kth element; Step 5.4: Calculate X P With Z P The cross-covariance matrix, and using Complex phase error compensation is performed to obtain matrix R. xz , Among them, Λ c0 =diag{p}; Step 5.5: For R xz Perform column-wise vectorization to obtain The function vec(·) represents the column vectorization of the matrix. Represents the Kronecker product; Step 5.6: [The sentence is incomplete and requires more context to be translated accurately.] Substitution In the middle, we obtained And Substitution In the middle, we obtained Then Finding linear operations The angle at which the maximum value is taken is set to... in, Then and The pairing constitutes an estimate of the two-dimensional angle of arrival for a narrowband pilot signal.

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