A method for profile optimization design in a positioning system based on RIS

Abstract

The application discloses a profile optimization design method in a positioning system based on RIS, and the method comprises the following steps: constructing a system model, determining system component elements, i.e., an access point (AP), RIS, UE and their position relationship, establishing a signal model and a measurement model, performing Fisher information analysis, calculating a correlation matrix and parameters, and deducing a position error bound expression. Furthermore, an optimal RIS profile design strategy is proposed, an optimization problem is constructed, and the optimal covariance matrix and RIS profile design are derived by using matrix analysis and deduction. Meanwhile, considering the hardware limitation, a constrained RIS profile design is proposed, a constrained optimization problem is constructed, and the RIS profile satisfying the actual hardware constraint is obtained by using the alternating minimization technology. The application improves the positioning accuracy of user equipment (UE).

Description

Technical Field

[0001] The present invention relates to a wireless positioning method, in particular to an optimization design method for a reconfigurable intelligent surface (RIS) profile in an orthogonal frequency division multiplexing (OFDM) system. Background Art

[0002] In the continuous evolution of modern wireless communication technology, reconfigurable smart surfaces (RIS) have gradually emerged as a highly promising emerging technology, showing unique advantages in meeting the growing demand for high-precision positioning in future communication systems.

[0003] The demand for high-precision positioning in next-generation wireless communication systems (such as 6G) is rapidly increasing. While millimeter-wave and terahertz communication technologies have the potential to significantly expand bandwidth, the increased carrier frequency results in more severe signal path attenuation, significantly limiting their coverage range. Against this backdrop, RIS has attracted widespread attention due to its unique operating mechanism. Composed of a large number of programmable components, it can flexibly control electromagnetic wave propagation by precisely manipulating parameters such as component phase. This opens up new avenues for improving system performance and presents broad application prospects in key areas such as wireless communication networks, the Internet of Things, and autonomous driving.

[0004] RIS offers significant advantages in positioning applications. When the line-of-sight (LoS) path in traditional positioning systems is blocked by various obstacles, RIS effectively introduces new multipath components, making positioning possible in scenarios that would otherwise be extremely difficult or even impossible. Even in the presence of a LoS path, RIS can leverage the information carried by the newly generated virtual LoS path to significantly improve positioning accuracy. Its fundamental principle is to add additional reference points to the positioning process, significantly enhancing positioning accuracy and reliability. Due to these outstanding characteristics, RIS has attracted considerable attention in the field of post-5G positioning technology and has become the focus of numerous research studies. For example, Keykhosravi et al. studied 3D downlink positioning and synchronization in a single-input, single-output (SISO) system using a planar RIS; Dardari et al. explored methods for LoS / NLoS positioning using large RIS in near-field conditions; Elzanaty et al. conducted an in-depth analysis of the position and orientation error bounds in RIS-assisted positioning; Abu-Shaban et al. proposed an innovative architecture for near-field positioning using a RIS as a lens; and Alexandropoulos et al. studied positioning using multiple RIS equipped with single-receive radio frequency (RF) chains. These research results fully demonstrate the huge potential and broad application prospects of RIS in the field of positioning technology.

[0005] However, despite the many research results, the current RIS profile design method still faces many challenges. On the one hand, the existing design strategy has obvious shortcomings in considering actual constraint conditions (such as hardware limitations, total power constraints, etc.). For example, when Song et al. studied RIS-aided non-line-of-sight wireless sensing, although the beamforming patterns of the AP and the RIS were jointly optimized to reduce the Cramer-Rao Bound (CRB) of the UE position-related parameters, the actual phase adjustment range and accuracy of the RIS elements and other factors were not fully considered, which may lead to the difficulty of accurately implementing the designed RIS profile in actual hardware, thereby affecting the positioning performance. Although the self-sensing RIS architecture proposed by Shao et al. is innovative, the total power constraint is not fully considered in the design process, which may lead to high system energy consumption and is not conducive to long-term stable operation in practical applications. On the other hand, the performance of RIS profile design strategies varies in different scenarios. Rahal et al. designed the RIS profile using near-field wavefront curvature, which can improve the positioning performance in specific scenarios, but when the scene parameters (such as distance, signal strength, etc.) change, the performance of this method may fluctuate greatly, and there is a lack of a general and efficient design method that can be widely used in various complex scenarios. This makes it difficult to quickly adjust the RIS profile according to different scene requirements in practical applications to balance various performance indicators, and it is difficult to fully meet the strict requirements of future communication systems for high-precision positioning.

[0006] In summary, in the RIS-aided positioning system, it is of great significance to deeply study the profile optimization design method, fully consider various actual constraint conditions, and improve the positioning accuracy, which is the core problem that the present invention aims to solve. SUMMARY

[0007] The present invention aims to provide a profile optimization design method in a RIS-based positioning system that improves the positioning accuracy of user equipment (UE).

[0008] The method of the present invention mainly includes: constructing a signal model based on RIS. Construct a positioning receiving model. Design the optimal profile design for RIS. Consider the RIS profile optimization design under the constraint condition, and apply a specific algorithm to realize high-precision user position estimation.

[0009] Technical solution: The profile optimization design method in the RIS-based positioning system according to the present invention comprises the following steps:

[0010] Step (1), reconfigurable intelligent surface system model construction: determine system composition, establish positioning system including AP, RIS, UE, and clarify position relationship; set signal transmission parameters; determine signal expression; derive received signal expression; measure model construction, determine discrete signal sampling method; decompose propagation delay; simplify discrete signal expression; obtain frequency domain received measurement signal expression;

[0011] Step (2), perform Fisher information analysis and PEB analysis related calculation; design optimal profile design for RIS: take PEB as an index, construct RIS profile optimization problem under total power constraint, and derive optimal covariance matrix and RIS profile design;

[0012] Step (3), consider RIS hardware limitation, define RIS profile constraint, construct optimization problem with constraint, update variables W and RIS profile matrix respectively by using alternating minimization technology, and iteratively optimize to obtain RIS profile meeting constraint and having good performance.

[0013] Further, the method of step (1) comprises:

[0014] The reconfigurable intelligent surface system model construction comprises the following specific steps:

[0015] Step (11), determine system composition, consider a typical RIS enabled non-line-of-sight (NLoS) positioning system, which is composed of a single antenna access point (AP), an RIS with L uniform linear reconfigurable elements, and a single antenna user equipment (UE) located in the non-line-of-sight coverage area of the AP, at this time, the line-of-sight (LoS) path between the AP and the UE is blocked by various obstacles, and the RIS is used to establish a virtual LoS path from the AP to the UE to achieve positioning, using a 2D Cartesian coordinate system to represent the position, defining as the AP position vector as the RIS reference point position vector, wherein l=0,1,…,L-1 is the lth element position vector, as the UE position vector, p UE which can be expressed in the Cartesian coordinate system as [p x ,p y ] T , which can be expressed in the spherical coordinate system as [r,θ] T r is the distance from the origin to the UE, and θ is the angle of the signal relative to the positive x-axis;

[0016] Step (12), setting signal transmission parameters: considering the RIS-enabled downlink communication scenario with blocked LoS path, assuming that the AP transmits a series of orthogonal frequency division multiplexing (OFDM) symbols with bandwidth B, B is divided into M subcarriers, i.e. B = MΔf, each subcarrier bandwidth is Δf, the OFDM symbol duration T = 1 / Δf, and the cyclic prefix (CP) duration is defined as T cp , the total symbol duration T s = T + T cp ;

[0017] Step (13), determining the signal expression: according to the above setting, the transmitted continuous-time OFDM signal is represented as

[0018]

[0019] where x[n,m] is the data sample transmitted on the mth carrier of the nth OFDM symbol, N is the number of OFDM symbols, rect T (t) is a rectangular pulse, equal to 1 when t ∈ [0, T] and 0 otherwise,

[0020] Step (14), deriving the received signal expression: based on the transmission signal model and system structure, the passband received signal through the AP-RIS-UE link is represented as

[0021]

[0022] where α is the concatenated channel coefficient of AP-RIS and RIS-UE path, f c is the carrier frequency, is the delay vector on different elements of RIS between AP and UE, and α = G G T is the AP antenna gain, P T is the total transmit power, λ c is the wavelength, φ0 is the phase offset, is the UE-side antenna gain,

[0023] Measurement model construction specific steps:

[0024] Step (17), determining the discrete signal sampling method: under the assumption of UE and AP synchronization, the discrete signal y[k,n] corresponding to the kth sample within the nth symbol is obtained by sampling the continuous-time signal y(t) at t = nT s + T cp + kT / M, and the expression is

[0025]

[0026] Step (18), decompose the propagation delay: the total propagation delay through the lth RIS element [τ(t)] l It can be decomposed into [τ(t)] l =[τ br ] l +[τ r (t)] l +[τ ru ] l ,in is the propagation delay between the AP and the lth RIS element, is the propagation delay between the lth RIS element and the UE, [τ r (t)] t is the delay of the lth RIS element at time t, and assuming that the RIS configuration is adjusted between OFDM symbols [τ r,n ] l =[τ r (nT s +T cp )] l feasible, and [τ r,n ] l ∈[0,1 / f c ),

[0027] Step (19), simplify the discrete signal expression: To make the expression more compact, some approximate simplifications y[k,n] are adopted to be

[0028]

[0029] in τ r =τ br +τ ru , assuming x[n,m]=1, arrange y[k,n] into a matrix Among them F DFT is the discrete Fourier transform DFT matrix, Y r =[y0,y1,…,y N-1 ], y n =[y[0,n],…,y[M-1,n]] T ,Γ r =[γ r,0 ,…,γ r,N-1 ],

[0030] Step (20) obtains the frequency domain received measurement signal expression: using the discrete Fourier transform property Y r Transformed to the frequency domain, considering the existence of noise, the frequency domain received measurement signal is finally expressed as,

[0031]

[0032] where Z is the frequency-domain observation noise, whose columns are drawn from a circularly symmetric complex Gaussian distribution with zero mean and covariance matrix

[0033] Further, the step (2) method:

[0034] The specific steps of Fisher information matrix design:

[0035] Step (21), measurement matrix transformation: first transform the measurement matrix in step (20) into a concise vector form, that is,

[0036]

[0037] where h r = vec (a (φ ru )b T (τ r )) and the covariance matrix of vec (Z T ) is

[0038] Step (22), calculate the Fisher information matrix (FIM) of h r,aug : let u = [(a (IMr) h)", (a (IM8) j)" and Calculate the FIM of h r,aug according to the given formula, that is,

[0039]

[0040] By substituting the relevant expressions, we get

[0041]

[0042] where

[0043] Step (23), define and calculate the FIM about θ: further define the channel parameter set to be estimated as Calculate the FIM about θ by the formula

[0044]

[0045] , which involves the calculation of , where h r = vec (a (φ ru )b T (τ r )). Here, each​ denotes h r the derivative of the corresponding element in h θ,i with respect to the parameter vector θ. Specifically, H

[0046] This term denotes the derivative of the element in h r with respect to τ r . is a coefficient related to signal propagation, a T (φ ru ) is a vector related to the angle φ ru , is the i-th element of the derivative vector of b(τ r ) with respect to τ r .

[0047] Here is the derivative of the element in h r with respect to φ ru . is the derivative vector of a(φ ru ) with respect to φ ru , [b(τ r )] i is the i-th element of the vector b(τ r ).

[0048] This expression denotes the derivative of the element in h r with respect to the real part of α, where a T (φ ru ) and [b(τ r )] i are as defined above.

[0049] This is the derivative of the element in h r with respect to the imaginary part of α, j is the imaginary unit, a T (φ ru ) and [b(τ r )] i have the same meaning as above.

[0050] Substituting into the above expression gives:

[0051]

[0052] Further expanding this expression, we get:

[0053]

[0054] where the calculation of each term f ij is as follows:

[0055] Here is the square of the modulus of the coefficient , is the square of the norm of the vector, denotes the trace of the matrix .

[0056] Similar to f 11 , is the combination of the correlation coefficient and the norm of the vector, is the trace of another matrix.

[0057] Also involves the calculation of the norm of the vector and the trace of the matrix.

[0058] Contains the conjugate transpose product of the vector and the trace of the matrix.

[0059] Here there is the conjugate of the coefficient and the conjugate transpose product of the vector with the trace of the matrix.

[0060] Also an expression composed of the coefficient, the norm of the vector, and the trace of the matrix.

[0061] Finally, a relatively complex but explicit FIM expression about θ is obtained, and it is pointed out that by choosing the center of the RIS as the reference point, the expression can be simplified (such as making ), which is of great significance in subsequent optimization problem solving and system performance analysis, which can help us more effectively study and design RIS profiles and improve the positioning performance of the system and other related indicators.

[0062] PEB expression design specific steps:

[0063] Step (24), define the relevant parameter vector: first define the parameter vector related to the UE position as where p UE = [p x , p y ] T represents the position of the UE in the Cartesian coordinate system. This step lays the foundation for subsequent calculation of the Jacobian matrix and derivation of the PEB expression, and clearly defines the key parameters related to the UE position and their representation in the vector,

[0064] Step (25), calculate the Jacobian matrix: calculate the Jacobian matrix T about θ and , its expression is

[0065]

[0066] The Jacobian matrix T describes the relationship between the changes of θ and It plays a key role in the conversion of the FIM about θ to the FIM about

[0067] Step (26), obtain the FIM about : By combining the FIM about θ obtained in the previous step and the Jacobian matrix T, the FIM expression about is obtained according to the formula

[0068] Step (27), derive the PEB expression: using the FIM about obtained in the previous step, the PEB expression is derived according to the relationship between PEB and FIM. The PEB can be expressed as Further, by calculating C p , the specific PEB expression is obtained. When calculating C p , first express and T as block matrix forms, that is, and Then, using the block matrix inversion formula , each term in it is calculated to obtain Therefore, At the same time, can be calculated Through these calculations, the closed-form expression of C p is finally obtained as

[0069]

[0070] where Substitute C p into to obtain the complete PEB expression, which provides a target function for subsequent optimization of the RIS profile,

[0071] Specific steps for optimal RIS profile design:

[0072] Step (28), construct the RIS profile optimization problem: take the position error bound (PEB) as an index to evaluate the positioning performance, and construct the RIS profile optimization problem under the total power constraint, that is,

[0073]

[0074] s.t.tr{R r}≤P T ,R​​r ≥ 0

[0075] where P T denotes the total transmit power, PEB is expressed in terms of the covariance matrix R r associated with the RIS profile, i.e.,

[0076]

[0077] further expressed as

[0078]

[0079] where

[0080] Step (29), optimization of R r related terms and reformulation of the optimization problem: First, R r related terms are simplified, i.e., tr{R φ}, and can be simplified using the structure of R r , and then the optimal form of R r is determined as R r = UΛU H , where, is a Hermitian matrix (i.e., Λ = Λ H ), and U represents a normalized subspace spanned by a * (φ ru ) and , specifically, where For ease of derivation, Λ is further expressed as tr{R φ}, and are further simplified as follows: tr{R φ} = λ 11 ‖a(φ ru )‖ 2 , Next, the optimization problem is reformulated, and after substituting the simplified results, we can obtain:

[0081]

[0082] Since the inequality always holds, we have Therefore, we have tr{C p} ≥ φ(λ 11 , λ 22 ), where and if and only if 12 = 0,

[0083] Step (210), reformulate and solve the optimization problem: Based on the previous derivation, reformulate the RIS profile optimization problem as an optimization problem about some newly defined parameters, i.e.

[0084]

[0085] s.t. λ 11 + λ 22 ≤ P T , λ 11 > 0, λ 22 > 0

[0086] where is the expression related to the previous derivation. Define auxiliary parameters and Solving this optimization problem gives:

[0087]

[0088] Step (211), determine the optimal covariance matrix and RIS profile design: According to the previous results, express the optimal covariance matrix R r as

[0089]

[0090] where Then discuss the optimal strategy to design the RIS profile Γ r by reasonably allocating the number of a * (φ ru ) and to balance the PEB related to the position coordinates, determine such that the RIS profile and satisfies

[0091] Further, the step (3) method:

[0092] The design specific steps of constraining the RIS profile problem are constructed as follows:

[0093] Step (31), define the RIS profile constraint: Considering the RIS hardware limit, for the RIS configuration [γ r,n ] l of the l-th element during the n-th OFDM symbol, define the actual RIS profile matrix as Γ r = [γ r,0 , γ r,1 ,…, γ r,N-1], whose effective constraint set is {[γ r,n ] l : | | γ r,n | l = 1, ∠[γ r,n ] l ∈ [0, 2π)}.

[0094] Step (32), constructing optimization problem: Let Γ u denote the unconstrained RIS profile set that can achieve the desired covariance R r , where W is an arbitrary semi-unitary matrix satisfying WW H = I L , under this constraint, the RIS profile design problem is constructed as

[0095]

[0096] Due to the non-convexity of the constraint and WW H = I L , this problem is difficult to solve directly, so we use the alternating minimization technique to deal with this problem in the next step,

[0097] The specific steps of the update scheme of W are as follows:

[0098] Step (33), optimization problem simplification: Fix Γ r , the optimization problem in the original step (32) is simplified as

[0099]

[0100] s.t. WW H = I L

[0101]

[0102] Combined with WW H = I L , expand :

[0103] Ignoring the constant term , the cost function is rewritten as

[0104]

[0105] s.t. WW H = I L

[0106] Step (34), singular value decomposition solution: Singular value decomposition is performed on , that is, where is a left unitary matrix, is a diagonal matrix containing non-zero real eigenvalues, is a right semi-unitary matrix and (assuming N > L), substituting it into yields:

[0107]

[0108] By analyzing the inequality relationship when the equality holds, and the optimal solution is satisfying thus maximizing the objective function,

[0109] Γ r The update scheme steps are as follows:

[0110] Step (35), substitute W to simplify the objective function: Substitute the estimated value of W into the original objective function to obtain the optimization problem about Γ r

[0111]

[0112] Since each element of RIS can be independently designed, the problem is further decomposed into

[0113]

[0114] s.t. | [γ r,n ] l | = 1

[0115] where

[0116] Step (36), solve considering the unit modulus constraint: Considering the unit modulus constraint, assuming then the optimization problem is rewritten as

[0117]

[0118] Using the trigonometric function relationship the closed-form solution is

[0119] Step (37), obtain the RIS profile design: Using the phase estimate and considering the unit modulus constraint, the final RIS profile design is obtained By alternately using the above optimization problem about Γ r ​and W are updated in an iterative manner to optimize them in a way of iterative minimization until a specific termination criterion is satisfied.

[0120] Compared with the prior art, the present application has the beneficial effects that:

[0121] The method focuses on a reconfigurable intelligent surface (RIS)-assisted orthogonal frequency division multiplexing (OFDM) positioning system, aiming at the case that the line-of-sight path between an access point (AP) and a user equipment (UE) is blocked, by constructing a system model containing a single-antenna AP, a multi-element RIS and a single-antenna UE, the positional relationship of each component is determined, the AP transmits an OFDM signal and the expression of the received signal is derived, the frequency domain measurement signal is sampled and simplified; further, Fisher information analysis and position error bound (PEB) analysis are carried out, on this basis, the optimal RIS profile design strategy is proposed under the total power constraint with PEB as the index, the optimal covariance matrix and configuration mode are obtained, considering the hardware limitation of the RIS, the alternating minimization technique is used to construct and solve the constrained RIS profile design problem, and the RIS profile that meets the constraints and has good performance is obtained through iterative optimization, which effectively improves the positioning performance, the proposed constrained method performs excellently in various scenarios and approaches the performance of the optimal scheme, which is of great significance for the fusion of sensing and communication functions in future 6G applications.

[0122] Through the above process, the RIS signal model and the positioning reception model are constructed, and in the system construction and analysis, the execution of each step should ensure that the communication characteristics and positional relationship of the system are accurately reflected, providing a reliable foundation for subsequent profile design and optimization. In the profile design and optimization, the construction and solution of the optimization problem should ensure that the RIS profile design that optimizes the positioning performance under actual constraint conditions can be obtained, thereby providing strong support for positioning services in wireless communication systems. BRIEF DESCRIPTION OF DRAWINGS

[0123] Figure 1 Typical RIS-assisted SISO system downlink transmission scenario diagram, this diagram shows the architecture of a typical RIS-assisted single-input single-output (SISO) system in a downlink transmission scenario, where the BS is the base station responsible for transmitting wireless signals;

[0124] Figure 2 Achievable error bound performance under different RIS profile design strategies varies with time-sharing factor λ1, this diagram presents the Cramer-Rao bound (CRB) for τ r and φ ru and the corresponding position error bound (PEB) performance varies with the time-sharing factor λ1, where (a) τ rCramer-Rao bound (CRB) performance diagram; (b)φ ru Cramer-Rao bound (CRB), (c) position error bound (PEB) performance;

[0125] Figure 3 The performance of the reachable error bound varies with the distance r under different RIS profile design strategies. This figure depicts the performance of the reachable error bound as the distance increases from to τ under different RIS profile design strategies. r CRB, φ ru The changing trend of the performance indicators of reachable error bounds such as CRB and PEB, where (a)τ r Cramer-Rao Bound (CRB) performance, (b)φ ru Cramer-Rao Bound (CRB) performance, (c) Position Error Bound (PEB) performance;

[0126] Figure 4 The performance of the achievable error bound under different RIS profile design strategies varies with the number of RIS elements L. This figure shows that under different RIS profile design strategies, when the number of RIS elements L increases from to, τ r 、φ ru and p UE The change of the error bound performance, where (a)τ r Cramer-Rao Bound (CRB) performance, (b)φ ru Cramer-Rao Bound (CRB) performance, (c) Position Error Bound (PEB) performance;

[0127] Figure 5 Different position uncertainty levels σ p The performance of the reachable PEB varies with distance, which shows the uncertainty level σ at different locations. p Variation of the reachable PEB performance with distance for the proposed optimal and constrained RIS profile design strategies;

[0128] Figure 6 Variation of achievable PEB performance with distance for different OFDM symbol numbers N: This figure depicts the variation of achievable PEB performance with distance for the proposed optimal and constrained RIS profile design algorithms for different OFDM symbol numbers N (N=200, N=500 and N=1000). DETAILED DESCRIPTION

[0129] In the continuous development process of modern wireless communication technology, the demand for high-precision positioning technology is increasingly urgent. As an emerging technology, reconfigurable intelligent surface (RIS) has shown great potential in improving positioning accuracy. Based on the RIS-aided orthogonal frequency division multiplexing (OFDM) positioning system, the present invention proposes a systematic and innovative RIS profile optimization design scheme to address the challenges faced by existing RIS profile design methods in practical applications, such as insufficient consideration of hardware limitations and power constraints, limited adaptability in different scenarios, and other issues.

[0130] The present invention first constructs a system model, accurately describes the relationship between each component in the system and the signal transmission characteristics, and lays a solid foundation for subsequent analysis. Then, Fisher information analysis and position error bound (PEB) analysis are conducted in depth to reveal the internal laws of the system through rigorous mathematical derivation, thereby clarifying the optimization objective. Based on this, the optimal RIS profile design strategy is elaborated, fully considering the total power constraint, and the optimal covariance matrix and RIS profile configuration method with practical guiding significance are obtained. At the same time, aiming at the hardware limitations of RIS, the alternating minimization technique is innovatively used to solve the constrained RIS profile design problem, and the RIS profile that meets the actual constraint conditions and has excellent performance is obtained through iterative optimization. In the entire implementation process, each step is closely connected and interdependent, forming a complete and efficient RIS profile optimization design system, which aims to provide a practical solution for high-precision positioning in future wireless communication systems and promote the development and progress of related technologies in practical applications.

[0131] The present invention will be described in detail below in conjunction with the drawings and specific embodiments.

[0132] Embodiment

[0133] Step 1. Establishment of reconfigurable intelligent surface (RIS) system model, signal model and measurement model

[0134] Step (11), determine the system composition. For example Figure 1 , consider a typical RIS-enabled non-line-of-sight (NLoS) positioning system, which consists of a single-antenna access point (AP), an RIS with L uniform linear reconfigurable elements, and a single-antenna user equipment (UE) located within the non-line-of-sight coverage area of the AP. At this time, the line-of-sight (LoS) path between the AP and the UE is blocked by various obstacles, and the RIS is used to establish a virtual LoS path from the AP to the UE to achieve positioning. The position is represented using a 2D Cartesian coordinate system, and is defined as the AP position vector is defined as the RIS reference point position vector (where l = 0, 1, …, L-1 is the lth element position vector), p is the UE position vector, UE which can be expressed in Cartesian coordinates as [p x , y ] T and in spherical coordinates as [r, θ] T , where r is the distance from the origin to the UE and θ is the angle of the signal with respect to the positive x-axis.

[0135] Step (12), setting signal transmission parameters: considering the RIS-enabled downlink communication scenario with blocked LoS path, it is assumed that the AP transmits a series of orthogonal frequency division multiplexing (OFDM) symbols with bandwidth B, B is divided into M subcarriers, i.e. B = MΔf, each subcarrier bandwidth is Δf, the OFDM symbol duration T = 1 / Δf, and the cyclic prefix (CP) duration is defined as T cp , and the total symbol duration T s = T + T cp .

[0136] Step (13), determining the signal expression: according to the above setting, the transmitted continuous-time OFDM signal is expressed as

[0137]

[0138] where x[n,m] is the data sample transmitted on the mth carrier of the nth OFDM symbol, N is the number of OFDM symbols, rect T (t) is a rectangular pulse (equal to 1 when t ∈ [0, T] and 0 otherwise) Step (14), deriving the received signal expression: based on the transmission signal model and system structure, the passband received signal through the AP-RIS-UE link is expressed as

[0139]

[0140] where α is the concatenated channel coefficient of the AP-RIS and RIS-UE paths, f c is the carrier frequency, is the delay vector on different elements of the RIS between the AP and the UE, and α = α br α ru , (G T is the AP antenna gain, P T is the total transmit power, λ c is the wavelength, and φ0 is the phase offset), (G R is the UE-side antenna gain).

[0141] Specific steps for constructing the measurement model:

[0142] Step (17), determine the discrete signal sampling method: Under the assumption that UE and AP are synchronized, the discrete signal y[k,n] corresponding to the kth sample in the nth symbol is obtained by sampling at t = nT s +T cp +kT / M is sampled by the continuous time signal y(t), and the expression is

[0143]

[0144] Step (18), decompose the propagation delay: the total propagation delay through the lth RIS element [τ(t)] l It can be decomposed into [τ(t)] l =[τ br ] l +[τ r (t)] l +[τ ru ] l ,in is the propagation delay between the AP and the lth RIS element, is the propagation delay between the lth RIS element and the UE, [τ r (t)] t is the delay of the lth RIS element at time t, and assuming that the RIS configuration is adjusted between OFDM symbols [τ r,n ] l =[τ r (nT s +T cp )] l feasible, and [τ r,n ] l ∈[0,1 / f c ).

[0145] Step (19), simplify the discrete signal expression: To make the expression more compact, some approximations (detailed in Appendix A) are used to simplify y[k,n] to

[0146]

[0147] in τ r =τ br +τ ru , assuming x[n,m]=1, arrange y[k,n] into a matrix Among them F DFT is the discrete Fourier transform (DFT) matrix, Y r =[y0,y1,…,y N-1 ], y n =[y[0,n],…,y[M-1,n]]T ,Γ r =[γ r,0 ,…,γ r,N-1 ]。

[0148] Step (20), get the frequency domain received measurement signal expression: use the discrete Fourier transform property Transform Y r to the frequency domain, considering the existence of noise, the frequency domain received measurement signal is finally expressed as

[0149]

[0150] Where Z is the frequency domain observation noise, whose columns are drawn from a circularly symmetric complex Gaussian distribution with zero mean and covariance matrix .

[0151] Step 2. Strategy for optimal design of RIS profile based on the assumption that UE location information is known

[0152] Specific steps of Fisher information matrix design:

[0153] Step (21), measurement matrix conversion: first convert the measurement matrix in step (20) into a concise vector form, that is,

[0154]

[0155] Where h r =vec(a(φ ru )b T (τ r )), and determine the covariance matrix of vec(Z T ) as

[0156] Step (22), calculate the Fisher information matrix (FIM) of h r,aug : let u = [(a(IMr)h)", (a(IM8)h)" and Calculate the FIM of h r,aug according to the given formula, that is,

[0157]

[0158] By substituting the relevant expressions, we get

[0159]

[0160] Where

[0161] Step (23), define and calculate the FIM about θ: further define the channel parameter set to be estimated as By formula

[0162]

[0163] Computing the FIM about θ involves Calculation of h r =vec(a(φ ru )b T (τ r )). Here, each Indicates h r The derivative of the corresponding element in with respect to the parameter vector θ. Specifically, H θ,i The rows are calculated as follows:

[0164] This term indicates h r The elements in r The derivative of . is a coefficient related to signal propagation, a T (φ ru ) is the angle φ ru The associated vector, is b(τ r ) About τ r The i-th element of the derivative vector.

[0165] Here is h r The elements in ru The derivative of . is a(φ ru ) About φ ru The derivative vector of [b(τ r )] i is b(τ r ) the i-th element of the vector.

[0166] This formula represents h r The derivative of the real part of the elements in with respect to α, where a T (φ ru ) and [b(τ r )] i As defined above.

[0167] This is h r The derivative of the elements in with respect to the imaginary part of α

[0168] number, j is the imaginary unit, a T (φ ru ) and [b(τ r )]i The meaning remains unchanged.

[0169] Will Substituting into the above formula we can get:

[0170]

[0171] Expanding this expression further, we get:

[0172]

[0173] Among them, each f ij The calculation of is as follows:

[0174] here is the coefficient The square of the modulus, yes The squared norm of a vector, Representation matrix traces.

[0175] With f 11 similar, is a combination of the correlation coefficient and the vector norm, is the trace of another matrix.

[0176] It also involves the calculation of vector norms and matrix traces.

[0177] Contains the conjugate transpose product of the vector and the matrix trace.

[0178] Here we have the conjugate of the coefficients and the conjugate transpose product of the vectors with the matrix trace.

[0179] It is also an expression consisting of coefficients, vector norms, and matrix traces.

[0180] Finally, a more complex but clear FIM expression for θ is obtained, and it is pointed out that the expression can be simplified by choosing the center of RIS as the reference point (e.g., using ), this simplification is of great significance in the subsequent optimization problem solving and system performance analysis. It can help us study and design RIS profiles more effectively and improve the system's positioning performance and other related indicators.

[0181] Specific steps for PEB expression design:

[0182] Step (24), define the relevant parameter vector: First, define the parameter vector related to the UE position as where p UE = [p x , p y ] T represents the position of the UE in the Cartesian coordinate system. This step lays the foundation for the subsequent calculation of the Jacobian matrix and derivation of the PEB expression, clarifying the key parameters related to the UE position and their representation in the vector.

[0183] Step (25), calculate the Jacobian matrix: Calculate the Jacobian matrix T with respect to θ and , whose expression is

[0184]

[0185] The Jacobian matrix T describes the relationship between θ and , and plays a key role in the conversion of the FIM with respect to θ to the FIM with respect to .

[0186] Step (26), get the FIM with respect to : By combining the previously obtained FIM expression with respect to θ and the Jacobian matrix T, the FIM expression with respect to is obtained according to the formula This step is to prepare for the subsequent calculation of PEB, because PEB is closely related to , and through this conversion, the problem can be focused on the parameter vector which is directly related to the UE position.

[0187] Step (27), derive the PEB expression: Using the previously obtained FIM with respect to , according to the relationship between PEB and FIM, the PEB expression can be derived, which can be expressed as Further, by calculating C p , the specific PEB expression is obtained. When calculating C p , first express and T as block matrix form, i.e. and Then use the block matrix inversion formula to calculate each term in it, respectively, to get Therefore, we have At the same time, we can calculate Through these calculations, the closed-form expression of C p is finally obtained as

[0188]

[0189] where C p is substituted into to obtain the complete PEB expression, which provides the objective function for subsequent optimization of the RIS profile.

[0190] The specific steps for optimal RIS profile design are as follows:

[0191] Step (28), constructing the RIS profile optimization problem: evaluate the positioning performance with the position error bound (PEB) as the indicator, and construct the RIS profile optimization problem under the total power constraint, that is,

[0192]

[0193] s.t.tr{R r}≤P T ,R r ≥0

[0194] where P T represents the total transmit power. Through a series of derivations, the PEB is expressed in the form related to the covariance matrix R r of the RIS profile, that is,

[0195]

[0196] Further expressed as

[0197]

[0198] where Step (29), optimizing R r related terms and reconstructing the optimization problem: first, simplify the R r related terms, that is, tr{R φ}、 and can be simplified by using the structure of R r . Then determine the optimal form of R r as R r = UΛU H . Here, is a Hermitian matrix (i.e., Λ = Λ H ). U represents a normalized subspace spanned by a * (φ ru ) and , specifically where For ease of derivation, further express Λ as Further simplify tr{R φ}、 and tr{R φ} = λ 11 ‖a(φ ru )‖2 2 , Then the optimization problem is reformulated by substituting the simplification results into the following:

[0199]

[0200] Since the inequality always holds, we have Therefore, we have tr{C p} ≥ φ(λ 11 , λ 22 ), where φ and the equality holds if and only if λ 12 = 0.

[0201] Step (210), reformulate and solve the optimization problem: Based on the previous derivation, the RIS profile optimization problem is reformulated as an optimization problem with respect to some newly defined parameters, i.e.,

[0202]

[0203] s.t. λ 11 + λ 22 ≤ P T , λ 11 > 0, λ 22 > 0

[0204] where is the expression related to the previous derivation. Define the auxiliary parameters and Solving this optimization problem yields:

[0205]

[0206] Step (211), determine the optimal covariance matrix and RIS profile design: According to the results obtained above, the optimal covariance matrix R r is re-expressed as

[0207]

[0208] where Then the optimal strategy for designing the RIS profile Γ r is discussed, which balances the PEBs associated with the position coordinates by reasonably allocating the number of times of a * (φ ru ) and to determine RIS profile and satisfy

[0209] Step 3. Considering the practical limitations of RIS hardware, a method of constraining RIS profile design is proposed to constrain the RIS profile problem. The specific steps of constructing the design are as follows:

[0210] Step (31), define the RIS profile constraint: considering the hardware limitations of RIS, for the RIS configuration [γ r,n ] l of the lth element during the nth OFDM symbol, the actual RIS profile matrix is defined as Γ r = [γ r,0 , γ r,1 , …, γ r,N-1 ], whose effective constraint set is {[γ r,n ] l : | |γ r,n ] l | = 1, ∠[γ r,n ] l ∈ [0, 2π)}. Step (32), construct the optimization problem: let Γ u represent the unconstrained RIS profile set that can achieve the desired covariance R r , where W is an arbitrary semi-unitary matrix that satisfies WW H = I L . Under this constraint, the RIS profile design problem is constructed as

[0211]

[0212] Due to the non-convexity of the constraint ( and WW H = I L ), this problem is difficult to solve directly, so we use the alternating minimization technique to solve this problem in the following steps.

[0213] The specific steps of updating W are as follows:

[0214] Step (33), simplify the optimization problem: fix Γ r , and the original optimization problem is simplified to

[0215]

[0216] s.t. WW H = I L

[0217]

[0218] Combining WW H = I L , we have Expand:

[0219]

[0220] Ignoring constant terms The cost function is rewritten as

[0221]

[0222] s.t. WW H = I L

[0223] Step (34), singular value decomposition solution: for singular value decomposition, that is where is the left unitary matrix, is a diagonal matrix containing non-zero real eigenvalues, is a right semi-unitary matrix and (hypothesis N>L). Substitute it into :

[0224]

[0225] By analyzing the inequality relationship

[0226] When , take the equality, get the optimal solution satisfies , thus maximizing the objective function. Γ r The update scheme steps:

[0227] Step (35), substitute W to simplify the objective function: Substitute the estimated value of W into the original objective function to get the optimization problem about Γ r

[0228]

[0229] Since each element of RIS can be designed independently, the problem is further decomposed into

[0230]

[0231] where Step (36), consider the unit modulus constraint solution: considering the unit modulus constraint, assuming The optimization problem is rewritten as

[0232] ​

[0233] Using trigonometric relations The closed-form solution is obtained as

[0234] Step (37), obtain the RIS profile design: using the phase estimates and considering the unit norm constraint, obtain the final RIS profile design By alternating the update schemes for Γ r and W, they are optimized in an iterative minimization fashion until a specific termination criterion is satisfied.

[0235] 1. Initialization phase: samples are drawn from a complex Gaussian distribution with mean 0 and variance 1 to initialize the elements of Γ r .

[0236] 2. Iterative loop phase: the following operations are repeated until a termination condition is satisfied (e.g., a maximum number of iterations I max is reached or the amount of change in the RIS profile between two iterations, ΔΓ r , is less than a set threshold ∈).

[0237] Step a: decomposition operation applies singular value decomposition (SVD) technique to decompose Γ into where is the left semi-matrix, is a diagonal matrix containing non-zero real eigenvalues, is the right semi-matrix (satisfying assuming N > L).

[0238] Step b: update W, according to the decomposition result of step a, calculate to complete the update of W.

[0239] Step c: update Γ calculate

[0240] Step d: update RIS phase profile, for l = 1 to L and n = 0 to N - 1, calculate to realize the update of RIS phase profile.

[0241] Step e: construct Γ r using the updated phase profile to construct the RIS profile matrix 3. Termination and output phase: when the termination condition is satisfied, output the final constrained RIS profile

Claims

1. A profile optimization design method for a positioning system based on RIS, characterized in that: The steps include: Step (1), reconfigurable smart surface system model construction: determine the system composition, establish a positioning system including AP, RIS, and UE, and clarify the position relationship; set signal transmission parameters; determine the signal expression; derive the received signal expression; construct a measurement model and determine the discrete signal sampling method; Decompose the propagation delay; simplify the discrete signal expression; obtain the frequency domain received measurement signal expression; Step (2), perform Fisher information analysis and PEB analysis related calculations; Design the optimal profile design for RIS: Using PEB as the indicator, construct the RIS profile optimization problem under the total power constraint, and derive the optimal covariance matrix and RIS profile design; Step (3), considering the RIS profile optimization design under the constraint conditions: considering the RIS hardware limitations, defining the RIS profile constraints, constructing the constrained optimization problem, using the alternating minimization technique, updating the variable W and the RIS profile matrix respectively, and iteratively optimizing to obtain a RIS profile that satisfies the constraints and has good performance. The method of step (3) is as follows: Specific steps for constraining the RIS profile problem construction and design: Step (31), define the RIS profile constraints: Considering the RIS hardware limitations, for the RIS configuration of the lth element during the nth OFDM symbol [γ r,n ] l , define the actual RIS profile matrix as Γ r =[γ r,0 ,γ r,1 ,…,γ r,N-1 ], its effective constraint set for Step (32), construct the optimization problem: let Γ u Indicates that the expected covariance R can be achieved r The set of unconstrained RIS profiles, where W is for WW H =I L Under this constraint, the RIS profile design problem is constructed as Due to the non-convexity of the constraints and WW H =I L , this problem is difficult to solve directly, so we use the alternating minimization technique to deal with this problem in the next step. The specific steps of W's update plan are: Step (33), the optimization problem is simplified: fix Γ r , the original optimization problem is simplified to Combined with WW H =I L ,right Expand: Ignore constant terms The cost function can be rewritten as Step (34), singular value decomposition solution: Perform singular value decomposition, that is in is a left unitary matrix, is a diagonal matrix containing nonzero real-valued eigenvalues, is a right semiunitary matrix and (Assuming N>L), Substitution have to: By analyzing the inequality relationship when When taking the equal sign, the optimal solution is satisfy To maximize the objective function, Γ r The update solution steps are as follows: Step (35), substitute W to simplify the objective function: Substitute the estimated value of W into the original objective function to obtain r Optimization problem Since each element of RIS can be designed independently, the problem can be further decomposed into in Step (36), consider the unit module constraint solution: Consider the unit module constraint, assuming The optimization problem can be rewritten as Using trigonometric relationships The closed form solution is Step (37), get the RIS profile design: using the phase estimate And considering the unit mode constraint, the final RIS profile design is obtained By alternating the above r and W, optimizing them in an iterative minimization manner until a specific termination criterion is met.

2. The cross-section optimization design method in a positioning system based on RIS according to claim 1, characterized in that: In the step (1): Specific steps for building a reconfigurable intelligent surface system model: Step (11), determine the system composition. Consider a typical RIS-enabled non-line-of-sight (NLoS) positioning system, which consists of a single-antenna access point (AP), a RIS with L uniform linear reconfigurable elements, and a single-antenna user equipment (UE) located in the non-line-of-sight coverage area of ​​the AP. At this time, the line-of-sight (LoS) path between the AP and the UE is blocked by various obstacles. The RIS is used to establish a virtual LoS path from the AP to the UE to achieve positioning. The position is represented by a 2D Cartesian coordinate system, and the following definitions are defined: is the AP position vector is the RIS reference point position vector, where is the position vector of the lth element, is the UE position vector, p UE In the Cartesian coordinate system, it can be expressed as [p x ,p y ] T , which can be expressed in spherical coordinates as [r,θ] T ,r is the distance from the origin to the UE, θ is the angle of the human-radiated signal relative to the positive x-axis; Step (12): Set the signal transmission parameters: Consider the downlink communication scenario where the RIS is enabled with the LoS path blocked. Assume that the AP transmits a series of orthogonal frequency division multiplexing (OFDM) symbols with a bandwidth of B. B is divided into M subcarriers, i.e., B = MΔf, each subcarrier has a bandwidth of Δf, and the OFDM symbol duration is T = 1 / Δf. The cyclic prefix (CP) duration is defined as T. cp , total symbol duration T s =T+T cp ; Step (13), determine the signal expression: According to the above settings, the transmitted continuous-time OFDM signal is expressed as Where x[n,m] is the data sample transmitted on the m-th carrier of the n-th OFDM symbol, N is the number of OFDM symbols, and rect T (t) is a rectangular pulse, which is equal to 1 when t∈[0,T] and equal to 0 otherwise. Step (14), derive the received signal expression: Based on the transmission signal model and system structure, the passband received signal through the AP-RIS-UE link is expressed as where α is the concatenated channel coefficient of the AP-RIS and RIS-UE paths, and f c is the carrier frequency, is the delay vector between AP and UE on different elements of RIS, and G T is the AP antenna gain, P T is the total transmit power, λ c is the wavelength, φ0 is the phase shift, is the UE side antenna gain, Specific steps for constructing the measurement model: Step (17), determine the discrete signal sampling method: Under the assumption that UE and AP are synchronized, the discrete signal y[k,n] corresponding to the kth sample in the nth symbol is obtained by sampling at t = nT s +T cp +kT / M is sampled by the continuous time signal y(t), and the expression is Step (18), decompose the propagation delay: the total propagation delay through the lth RIS element [τ(t)] l It can be decomposed into [τ(t)] l =[τ br ] l +[τ r (t)] l +[τ ru ] l ,in is the propagation delay between the AP and the lth RIS element, is the propagation delay between the lth RIS element and the UE, [τ r (t)] t is the delay of the lth RIS element at time t, and assuming that the RIS configuration is adjusted between OFDM symbols [τ r,n ] l =[τ r (nT s +T cp )] l feasible, and [τ r,n ] l ∈[0,1 / f c ), Step (19), simplify the discrete signal expression: To make the expression more compact, some approximate simplifications y[k,n] are adopted to be in Assume x[n,m]=1, arrange y[k,n] into a matrix Among them F DFT is the discrete Fourier transform DFT matrix, Step (20) obtains the frequency domain received measurement signal expression: using the discrete Fourier transform property Y r Transformed to the frequency domain, considering the existence of noise, the frequency domain received measurement signal is finally expressed as, where Z is the frequency domain observation noise, whose columns are transformed from zero mean and covariance matrix into is drawn from a circularly symmetric complex Gaussian distribution.

3. The cross-section optimization design method for a positioning system based on RIS according to claim 2, characterized in that: The method of step (2): Specific steps for designing Fisher information matrix: Step (21), measurement matrix conversion: First, the measurement matrix in step (20) is converted into a concise vector form, that is, where h r =vec(a(φ ru )b T (τ r )), and determine vec(Z T )’s covariance matrix is Step (22), calculate h r,aug Fisher Information Matrix (FIM): Let u = [(a(IMr)h)", (a(IM8)h)" and Calculate h according to the given formula r,aug FIM, that is By substituting the relevant expressions, we get in Step (23), define and calculate the FIM about θ: further define the channel parameter set to be estimated as By formula Computing the FIM about θ involves By substituting the relevant derivative expressions, we finally get a more complex but clear FIM expression about θ, and point out that the expression can be simplified by choosing the center of RIS as the reference point (such as using Specific steps for PEB expression design: Step (24), define the relevant parameter vector: First, define the parameter vector related to the UE position as where p UE =[p x ,p y ] T Indicates the position of the UE in the Cartesian coordinate system. This step lays the foundation for the subsequent calculation of the Jacobian matrix and the derivation of the PEB expression, and clarifies the key parameters related to the UE position and their representation in the vector. Step (25), calculate the Jacobian matrix: calculate the Jacobian matrix about θ and The Jacobian matrix T of The Jacobian matrix T describes the The changing relationship between them is to convert the FIM about θ into the FIM about plays a key role in the FIM process, Step (26), get the FIM: By combining the FIM expression about θ obtained previously And Jacobian matrix T, according to the formula Get about The FIM expression, Step (27), derive the PEB expression: using the previously obtained According to the relationship between PEB and FIM, the PEB expression is derived. It is known that PEB can be expressed as Further calculation of C p To get the specific PEB expression, calculate C p When and T are expressed as block matrices, i.e. and Then use the block matrix inversion formula Calculate each of them separately and get as well as Through these calculations, we finally get C p The closed form expression for in C p Substitution The complete PEB expression is obtained, which provides the objective function for the subsequent optimization of RIS profile. Specific steps for optimal RIS profile design: Step (28), construct the RIS profile optimization problem: use the position error bound (PEB) as an indicator to evaluate the positioning performance, and construct the RIS profile optimization problem under the total power constraint, that is, Among them, P T Represents the total transmit power, and through a series of derivations, PEB is expressed as the covariance matrix R with the RIS profile r The relevant form, i.e., Further expressed as in Step (29), optimize R r Related items and reconstruction optimization problems: First, R r Related terms are simplified, that is, tr{R φ }、 and Available R r Simplify the structure and then determine R r The optimal form is R r =UΛU H ,here, is a Hermitian matrix (i.e. Λ=Λ H ), U represents a * (φ ru )and Zhang Cheng's normalized subspace is specifically in For the convenience of derivation, we further express Λ as Further simplify tr{R φ }、 and As follows: tr{R φ }=λ 11 ‖a(φ ru )‖ 2 、 Then reconstruct the optimization problem and substitute the simplified result into it to get: Due to the inequality Always holds true, so Therefore, there is tr{C p }≥φ(λ 11 ,λ 22 ),in And if and only if λ 12 = 0, the equation holds true. Step (210), reformulate and solve the optimization problem: Based on the previous derivation, the RIS profile optimization problem is reformulated as an optimization problem with some newly defined parameters, namely in It is an expression related to the previous derivation, defining auxiliary parameters and Solving the optimization problem yields: Step (211), determine the optimal covariance matrix and RIS profile design: According to the results obtained above, the optimal covariance matrix R r Re-expressed as in Then we discuss the design of RIS profile Γ r The optimal strategy is to reasonably allocate a * (φ ru )and The number of times to balance the PEB related to the position coordinates, determine Make RIS profile And satisfy

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