Transmitted waveform and reflected beam forming joint design method for intelligent reflector-assisted radar communication integrated system

Abstract

A DFRC (radar-communication integration) system solves the problem of spectrum congestion caused by increase of wireless application requirements, but in an actual environment, a wireless transmission environment is very complex, functions of radar and communication can be affected, and the DFRC system can be applied to a wireless communication system, so that the DFRC system can be applied to a wireless communication system, and the DFRC system can be applied to the wireless communication system. The invention researches a joint design method for forming a DFRC transmitted waveform and an RIS reflected wave beam by taking a Cramer-Rao bound for minimizing radar parameter estimation as a criterion and taking a signal to interference plus noise ratio demand of a communication user under a symbol-level precoding system as a constraint. According to the invention, the proposed joint design problem is described as a non-convex optimization problem with the non-convex objective function and the constant modulus constraint. According to the method, multiple variables are decoupled under an alternating optimization framework, and a transmitted waveform optimization sub-problem and an RIS reflection coefficient optimization sub-problem are solved respectively. According to the method, an auxiliary variable is introduced, a non-convex problem is converted into a convex problem on the basis of Scherr relaxation, and then effective solving is conducted through an existing convex optimization method; according to the method, a fractional part in a target function is processed through an auxiliary variable and Scherr complementary processing, the fractional part is converted into a differential convex function form, and then iterative solution is carried out through a successive convex approximation algorithm in combination with a penalty method.The numerical experiment result shows that the radar detection parameter estimation performance is improved while the communication performance is guaranteed through the algorithm.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar signal processing, and specifically relates to a joint design method for transmitting waveform and reflecting beam forming for an intelligent reflector-assisted radar communication integrated system. Background Art

[0002] With the rapid growth of demand for wireless applications, the RF spectrum has become increasingly crowded, and spectrum sharing is seen as a necessary way to solve this problem. In practice, a large number of radar and wireless communication systems have spectrum overlaps, and their RF front-end structures are similar, so they can share spectrum and hardware platforms. This situation has given rise to the development of dual-function radar-communication (DFRC) technology to fundamentally solve the mutual interference problem between the two. At present, according to the different work focuses, DFRC waveform design can be divided into three categories: radar-centric, communication-centric, and joint optimization of radar and communication functions. Among them, DFRC with joint optimization of radar and communication functions adopts a common waveform design method, does not impose special structural constraints on the waveform form, and can be directly and flexibly adjusted based on radar and communication performance requirements, becoming one of the hot topics today.

[0003] In the radar part, the Cramér-Rao bound (CRB) is an important performance indicator in the target parameter estimation scenario, which provides the lower bound of the unbiased parameter estimation variance as the basic performance limit. In the communication part, the disadvantage of the traditional linear block-level precoding system is that it cannot guarantee the communication performance of each user, but can only guarantee the average communication performance. In contrast, the symbol-level precoding (SLP) technology can use the known symbol information to convert destructive multi-user interference into constructive interference (Cl) to achieve more reliable multi-user communication. In addition, for radar functions, SLP can design the instantaneous transmission waveform at each snapshot to provide higher degrees of freedom.

[0004] In practical applications, the performance of the DFRC system depends largely on the wireless propagation environment. However, obstacles are inevitable in practical application scenarios, which affect the normal implementation of radar and communication functions. The rise of Reconfigurable Intelligent Surface (RIS) provides a new solution to this problem. RIS adjusts the phase shift of the incident signal by adjusting the reflective element, which can reconfigure the wireless propagation environment. In systems involving RIS, traditional algorithms based on semi-definite relaxation usually ignore the rank-one constraint it brings and reconstruct the rank-one solution through randomization steps, but this may not be feasible, so a better processing method needs to be developed.

[0005] Based on the above advantages, the present invention designs a RIS-assisted DFRC system based on SLP and customizes a special method to solve the joint design problem of transmit waveform and beamforming. Summary of the invention

[0006] The purpose of the present invention is to propose a joint design method for transmission waveform and reflection beamforming for an intelligent reflector-assisted radar communication integrated system. The present invention proposes an algorithm based on alternating optimization (AO), Shure relaxation, successive convex approximation (SCA) and penalty method for the joint design of transmission waveform and reflection beamforming of RIS-assisted DFRC system. The results show that the design of the present invention has better performance than the traditional RIS-assisted DFRC system. A joint design method for transmission waveform and reflection beamforming for RIS-assisted DFRC system, the specific steps are as follows:

[0007] Step 1: Construct the joint design optimization problem of the transmit waveform and reflection beamforming of the RIS-assisted DFRC system, where the radar optimization adopts the minimization CRB criterion, and the communication constraint adopts the user's minimum signal-to-interference-plus-noise ratio (SINR) requirement constraint under the CI precoding criterion.

[0008] Step 2: Use the AO framework to decouple the variables and obtain the emission signal optimization subproblem and the reflection vector optimization subproblem.

[0009] Step 3, by introducing auxiliary variables and Schur relaxation technology, the transmission signal optimization sub-problem is transformed into a convex optimization problem, and is effectively solved by existing convex optimization technology.

[0010] The RIS-assisted DFRC system of the present invention has the following advantages over the existing system:

[0011] (1) Compared with the traditional RIS-assisted DFRC system based on block-level precoding, the present invention is based on a symbol-level precoding system, which can improve the estimation accuracy of radar target detection parameters while ensuring the user's minimum SINR requirement.

[0012] (2) The present invention realizes the radar target detection task in complex environments by introducing RIS, and greatly improves the radar performance while ensuring the user communication quality. BRIEF DESCRIPTION OF THE DRAWINGS

[0013] In order to more clearly illustrate the embodiments of the present application or the existing technical solutions, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments recorded in the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0014] Figure 1 Comparison of CRB performance for target parameter estimation in different systems;

[0015] Figure 2 The influence of the number of RIS array elements in different systems on the target parameter estimation CRB;

[0016] Figure 3 The influence of the communication user SINR requirement on the target parameter estimation CRB in the algorithm of the present invention under different numbers of users;

[0017] Figure 4 The figure shows the influence of the number of RIS array elements in the algorithm of the present invention on the target parameter estimation CRB under different numbers of users. DETAILED DESCRIPTION

[0018] The present invention, namely a joint design method for transmit waveform and reflection beamforming for a RIS-assisted DFRC system, is further described below with reference to the accompanying drawings and examples.

[0019] The present invention is a method for the joint design of transmission waveform and reflection beamforming for RIS-assisted DFRC system. The present invention adopts the idea of ​​alternating optimization, uses SCA and penalty method, and proposes an effective algorithm for the joint design of transmission waveform and reflection beamforming for RIS-assisted DFRC system under the AO framework. The basic idea of ​​the algorithm is to alternately optimize the transmission signal matrix and the RIS reflection coefficient vector, where the transmission signal matrix subproblem can be transformed into a convex problem and effectively solved by introducing auxiliary variables and Schur relaxation technology, and the RIS reflection coefficient vector subproblem is iteratively solved using the SCA algorithm, and a solution close to rank one is constructed in combination with the penalty method. The specific implementation steps of this method are as follows:

[0020] Step 1: For the joint design problem of the transmission waveform and reflection beamforming of the RIS-assisted DFRC system, the criterion is to minimize the CRB of radar parameter estimation, and the SINR requirement of the communication user under the CI precoding criterion is constrained. Consider that the number of base station antennas in the RIS-assisted DFRC system is M, the number of RIS reflection array elements is N, and it is a uniform linear array. The base station serves U single-antenna users and detects point targets at the same time. Point targets are only located on non-line-of-sight paths, and communication users are located on both line-of-sight and non-line-of-sight paths. Let the channel vectors from the base station and RIS to the u-th user be and The Rayleigh fading channel from the base station to the RIS is expressed as:

[0021]

[0022] Where κ is the Rayleigh factor, and the line-of-sight channel from the base station to the RIS is The steering vectors are expressed as:

[0023]

[0024] where θ R With θ B are the departure angle and arrival angle from the base station to the RIS respectively. Let the reflection coefficient matrix of the RIS be Φ=diag(v), and the total channel from the base station to the user is expressed as:

[0025]

[0026] Define φ as half of the angle of region Cl, and the expected received symbol of the uth user at time l is s u，l Therefore, under the Cl precoding criterion, the distance from the actual received noise-free signal of the u-th user to its decision boundary is expressed as:

[0027]

[0028] Define the minimum SINR required for the uth user as T u , the received noise power is The distance from the CI boundary to the decision boundary under the minimum receiving SINR requirement is expressed as:

[0029]

[0030] The radar receiving noise power is defined as The estimated CRB at angle θ is expressed as:

[0031]

[0032] Where ɑ is the complex channel coefficient, B = G T Φ T a(θ)aT (θ)ΦG is the channel matrix that the echo signal passes through, Defined as the partial derivative of B with respect to θ. is the steering vector of RIS at angle θ, d RIs is the spacing between adjacent RIS array elements, and λ is the carrier wavelength.

[0033] The transmit signal matrix to be optimized is R x =XX H , RIS reflection coefficient vector Define the distance between the actual received noise-free signal of the uth user and its decision boundary under the CI precoding criterion as d u,l , the distance from the Cl boundary to the decision boundary under the minimum receiving SINR requirement is The total energy emitted is P T The overall optimization model is expressed as:

[0034]

[0035] Step 2: Use a method based on the AO framework to transform problem (8) into two relatively easy-to-solve sub-problems to alternately optimize the transmit signal matrix and the RIS reflection coefficient vector.

[0036] Given a RIS reflection coefficient vector v, in the objective function, the irrelevant constant terms are removed and the communication constraint is equivalently transformed into a compact form, expressed as in:

[0037]

[0038] Question (8) about X and R x The sub-problem can be expressed as:

[0039]

[0040] Given X and R x , in the objective function, CRB is represented by v, which is expressed as:

[0041]

[0042] Where R1 = diag(a H (θ))G * G T diag(a(θ)), R2=diag(a H (θ))G * R x * G Tdiag(a(θ)). Similarly, we use v to represent the communication constraint, which can be expressed as in:

[0043]

[0044] By removing the irrelevant constant term in the objective function, the subproblem about v in problem (8) can be expressed as:

[0045]

[0046] Step 3: By introducing auxiliary variables and Schur relaxation techniques, problem (10) is transformed into a convex problem that is easy to solve. Introduce auxiliary variables u, apply Schur complement theorem to process the objective function, and apply Schur complement theorem to the non-convex constraint R. x =XX H Performing the Shure relaxation, problem (10) is transformed into:

[0047]

[0048] where e l Represents the L×L unit matrix I L The problem is convex and can be solved efficiently using existing convex optimization algorithms.

[0049] Step 4: By introducing auxiliary variables, combining the SCA algorithm and the penalty method, problem (13) can be transformed into a convex problem for iterative solution. Define the following matrix to handle the constant modulus constraint:

[0050]

[0051] Problem (13) is equivalently converted to:

[0052]

[0053] Introducing two auxiliary variables u1 and u2, using Schur's complement theorem to transform them into convex semi-positive definite constraints, problem (16) is equivalently transformed into:

[0054]

[0055] The objective function of problem (17) can be transformed into the sum of a convex function and a concave function through algebraic transformation, which is defined as:

[0056]

[0057] in:

[0058]

[0059] Problem (17) regards the objective function as a differential convex function, that is, For problems with non-convex rank-one constraints, the SCA algorithm and penalty method are used to handle the problem and rank-one constraints. The basic process is as follows:

[0060] 1). In the iteration, use the first-order Taylor expansion to find the convex function A linear lower bound for The current number of iterations is r, which is defined as:

[0061]

[0062] 2). Replace with Ignoring the rank-one constraint, the subproblem under the SCA algorithm in the rth iteration is constructed as:

[0063]

[0064] This problem is a convex problem and can be solved using existing convex optimization algorithms.

[0065] 3). When the SCA cycle converges, we will get a general The suboptimal solution of Equivalent conversion to in for The maximum eigenvalue of is converted into a differentiable form and placed in the objective function as a penalty term. The suboptimal solution obtained by the iteration of the SCA algorithm is used as the initial value, and the rank-one constraint is approximated by an iterative method. In the p+1th iteration of the penalty method, the optimization problem is expressed as:

[0066]

[0067] in for The eigenvector corresponding to the largest eigenvalue of .

[0068] The specific process of solving problem (17) is as follows:

[0069] 1) Input parameters F i,I , convergence condition δ th

[0070] 2) Initialization settings: r = 0, v (0) ,

[0071] 3) Start iteration:

[0072] i. Calculate by formula (20)

[0073] ii. Update via (21)

[0074] iii. r=r+1

[0075] 4) If the objective function converges or reaches the upper limit of the number of iterations, stop the loop.

[0076] 5) Initialization setting: p = 1,

[0077] 6) Start iteration:

[0078] i. Update via (22)

[0079] ii. If Then ρ=2ρ, otherwise p=p+1,

[0080] 7) If the objective function converges or reaches the upper limit of the number of iterations, stop the loop

[0081] 8) Output And reconstructed into vector v

[0082] In summary, the complete solution process of the joint design method of the transmission waveform and reflection beamforming of the RIS-assisted DFRC system based on the AO framework is as follows:

[0083] 1) Input parameter B, γ i , P T

[0084] 2) Initialization settings: r = 1, v (0)

[0085] 3) Start iterating

[0086] i. Calculate X through (14) (r+1)

[0087] ii. Update by the method of claim 6

[0088] iii. r=r+1

[0089] 4) If the objective function converges or reaches the upper limit of the number of iterations, stop the loop.

[0090] Example

[0091] The combined design method of the transmission waveform and reflection beam forming for the RIS-assisted DFRC system of the present invention is further illustrated by Matlab simulation.

[0092] 1) Simulation system parameter settings

[0093] Unless otherwise specified, the number of transmitting antennas of the DFRC system in each simulation is M = 8, the number of RIS array elements is N = 32, the number of transmitted signal snapshots is L = 50, and the total transmission power is P T =30dBm, noise σ R =σ u =-80dBm, u = 1, ..., U, the SINR requirements of all users are equal, that is, Γ u =Γ, u = 1, ..., U. The base station, RIS and point target are located at coordinates (0m, 0m), (4m, 5m), (4m, 0m) respectively, and the four users are located at coordinates (40m, 0m), (40m, -10m), (50m, -10m) and (50m, 10m) respectively. Considering the distance-dependent path fading model K0=-30dB, d0=1m, the fading exponents on the base station-RIS, RIS-user, and base station-user links are 2.2, 2.2, and 3 respectively, and the Rice factor is set to 0.5.

[0094] 2) Compare system settings

[0095] In order to intuitively display the changes of CRB as an objective function in different scenarios, this embodiment uses the method (SLP) proposed in the present invention to draw a CRB change diagram, and compares it with several other typical RIS-assisted system design methods: RIS-assisted DFRC system (BLP) with block-level precoding for communication constraints; symbol-level DFRC system (max SNR) based on the maximum radar detection signal-to-noise ratio criterion for radar optimization criteria; symbol-level DFRC system (Random RIS) under random RIS reflection coefficients; and RIS-assisted radar-only system (Raday only). The unit of the image vertical coordinate CRB is dB.

[0096] 3) Simulation experiment and result analysis

[0097] The present invention carries out four example simulations in total, covering the CRB changes under different communication parameters, RIS array element numbers and different methods. Figure 1 For the case where the number of users U = 4, Figure 2 This is the case where the number of users U=2.

[0098] pass Figure 1, it can be seen that the algorithm proposed in the present invention is superior to other cases in performance under different SINR requirements. When the communication service quality requirement is low, the CRB value in the SLP case is about 0.02dB higher than the CRB value in the BLP and Radar only cases. This is because when the communication service quality requirement is low, the user's SINR requirement is easy to meet. At this time, the radar function dominates the dual-function waveform optimization, so it is very close to the radar only case, and due to the randomness of BLP, under low user SINR requirements, BLP is about 0.02dB lower than the CRB value in the SLP case, which can be ignored. However, under high SINR requirements, the radar performance of SLP shows superiority. It can be seen that with the further improvement of the communication service quality requirement, the CRB in the SLP case is lower than that in the BLP case. This is because the SLP design can provide a higher degree of freedom for the radar detection function compared to the BLP design.

[0099] pass Figure 2 ,It can be seen that under different RIS array element numbers, the SLP case is still better than the max SNR, Random RIS, and BLP cases. The performance gap between the SLP and Random RIS cases increases with the increase of N, which reflects the importance of reflection coefficient optimization.

[0100] pass Figure 3 It can be seen that the increase in the number of users and the SINR requirement Γ of the users both lead to a decrease in radar performance, reflecting the trade-off between radar and communication performance in the DFRC system. This illustrates the feasibility of the algorithm proposed in the present invention.

[0101] pass Figure 4 It can be seen that as the number of RIS elements N increases from 4 to 48, the CRB in the cases of U=1, 2, 3, and 4 decreases by 28.71dB, 28.42dB, 28.33dB, and 27.78dB, respectively. This shows that RIS greatly improves the radar function, and this improvement will slightly decrease as the number of users increases, that is, the performance gap between different users will also slightly increase as N increases.

[0102] In summary, the method described in the present invention has good comprehensive performance. Compared with other representative methods, the proposed method has almost the lowest CRB value, that is, the best parameter estimation performance. The present invention is applied in the DFRC system, and can simultaneously ensure the SINR requirements of communication users under the Cl precoding criterion and ensure the feasibility of the solution, and has high practical value.

Claims

1. A joint design method of transmission waveform and reflection beamforming for a dual-function radar-communication (DFRC) system assisted by a reconfigurable intelligent surface (RIS) is proposed. The method is characterized in that: based on alternating optimization (AO), successive convex approximation (SCA) algorithm and penalty method, a joint design method of DFRC transmission signal and RIS beamforming under the AO framework is proposed. This method is applicable to the communication user signal-to-interference-plus-noise ratio (Signal-to-Interference-plus-Noise) under the constructive interference (CI) precoding criterion with the Cramér-Rao Bound (CRB) of radar parameter estimation as the criterion. Ratio, SINR) requirement as the constraint of DFRC transmission signal and RIS reflection coefficient optimization method. Under the AO framework, the algorithm reconstructs the original problem into a fractional optimization problem with equality constraints and a fractional optimization problem with constant modulus constraints. And the sub-problems are effectively solved by using methods such as Shure relaxation, SCA algorithm, and penalty method. Specifically, it includes the following steps: S1. For the joint design problem of the transmit waveform and reflected beamforming of the RIS-assisted DFRC system, the criterion is to minimize the CRB of radar parameter estimation and the SINR requirement of the communication user under the CI precoding criterion is the constraint. The CRB of the radar parameter estimation at angle θ is defined as CRB(θ), and the transmit signal matrix to be optimized is R X =XX H , RIS reflection coefficient vector Under the CI precoding criterion, the distance between the actual received noise-free signal of the uth user and its decision boundary is d u，l , the distance from the CI boundary to the decision boundary under the minimum receiving SINR requirement is d u , the total emission energy is P T .The optimization model is: Where tr(·) represents the operation of taking the matrix trace, R X ≥0 means matrix R X is a positive semidefinite matrix; S2. A method based on the AO framework is used to transform problem (1) into two relatively easy-to-solve sub-problems to alternately optimize the transmit signal matrix and the RIS reflection coefficient vector; S3. The sub-problem about the transmit signal matrix can be expressed as: Among them, B is the total channel through which the radar echo signal passes, is its partial derivative with respect to the target direction. The present invention transforms subproblem (2) into an easy-to-solve convex problem by introducing auxiliary variables and Schur relaxation; S4. Converting CRB and communication constraints into a form about v, the subproblem of RIS reflection coefficient can be expressed as By introducing auxiliary variables and Schur's complement theorem to deal with the fraction in the objective function of problem (3), the problem can be transformed into an optimization problem with a rank-one constraint in which the objective function is a differential convex function. The present invention solves this complex non-convex optimization problem through the SCA algorithm and the penalty method.

2. The method according to step 1 of claim 1, characterized in that: Consider the RIS-assisted DFRC system with M base station antennas, N RIS reflector elements, and a uniform linear array. The base station serves U single-antenna users and detects point targets at the same time. Point targets are only located on non-line-of-sight paths, and communication users are located on both line-of-sight and non-line-of-sight paths. The channel vectors from the base station and RIS to the u-th user are respectively and Base station to RIS channel representation The reflection coefficient matrix of RIS is The total channel from the base station to the user is expressed as: Define φ as half of the CI area angle, and the expected received symbol of the uth user at the lth time is s u，l Therefore, under the CI precoding criterion, the distance from the actual received noise-free signal of the u-th user to its decision boundary is expressed as: Define the minimum SINR required for the uth user as Γ u , the received noise power is The distance from the CI boundary to the decision boundary under the minimum receiving SINR requirement is expressed as: The radar receiving noise power is defined as The estimated CRB at angle θ is expressed as: Where ɑ is the complex channel coefficient, B = G T Φ T a(θ)a T (θ)ΦG, is the partial derivative of B with respect to θ. is the steering vector of RIS at angle θ, d RIS is the spacing between adjacent RIS array elements, and λ is the carrier wavelength.

3. The method according to step 2 of claim 1, characterized in that: Given a RIS reflection coefficient vector v, in the objective function, the irrelevant constant terms are removed and the communication constraint is equivalently transformed into a compact form, expressed as in: Question (1) about X and R X The sub-problem can be expressed as problem (2). Given X and R X , in the objective function, CRB is represented by v, which is expressed as: Where R1 = diag(a H (θ))G * G T diag(a(θ)), R2=diag(a H (θ))G * R X * G T diag(a(θ)). Similarly, we use v to represent the communication constraint, which is expressed as in: By removing the irrelevant constant term in the objective function, the subproblem about v in problem (1) can be expressed as problem (3).

4. The method as claimed in step 3 of claim 1, characterized in that By introducing auxiliary variables and Schur complement, problem (2) is transformed into a convex problem that is easy to solve. By introducing auxiliary variables u, the objective function is processed using the Schur complement theorem, and the non-convex constraint R X =XX H By performing Shure relaxation, problem (2) is transformed into: where e l Represents the L×L unit matrix I L The problem is a convex problem and can be solved efficiently using existing convex optimization algorithms.

5. The method as claimed in step 4 of claim 1, characterized in that By introducing auxiliary variables and Schur complement, combined with SCA algorithm and penalty method, problem (3) can be transformed into a convex problem for iterative solution. The following matrix is ​​defined to handle constant modulus constraints: Problem (3) is equivalently converted to: Introducing two auxiliary variables u1 and u2, using Schur's complement theorem to transform them into convex semi-positive definite constraints, problem (13) is equivalently transformed into: The objective function of problem (14) can be transformed into the sum of a convex function and a concave function through algebraic transformation, which is defined as: in: Problem (14) is regarded as a problem with a non-convex rank-one constraint and an objective function as a difference convex function. The SCA algorithm and penalty method are used to deal with this problem and the rank-one constraint.

6. The method as claimed in claim 5, characterized in that The objective function in subproblem (14) can be expressed as a difference convex function Then use the SCA algorithm and penalty method to solve the problem. The basic process is as follows: 1). In the iteration, use the first-order Taylor expansion to find the convex function A linear lower bound for The current number of iterations is r, which is defined as: 2). Replace with Ignoring the rank-one constraint, the subproblem under the SCA algorithm in the rth iteration is constructed as: This problem is a convex problem and can be solved efficiently using existing convex optimization algorithms. 3). When the SCA cycle converges, we will get a general The suboptimal solution of Equivalent conversion to in for The maximum eigenvalue of is converted into a differentiable form and placed in the objective function as a penalty term. The suboptimal solution obtained by the iteration of the SCA algorithm is used as the initial value, and the rank-one constraint is approximated by an iterative method. In the p+1th iteration of the penalty method, the optimization problem is expressed as: in for The eigenvector corresponding to the largest eigenvalue of . The specific process of solving problem (14) is as follows: 1) Input parameters F i，l , convergence condition δ th 2) Initialization settings: r = 0, v (0) , 3) Start iteration: i. Calculate by formula (17) ii. Update via (18) iii. r=r+1 4) If the objective function converges or reaches the upper limit of the number of iterations, stop the loop. 5) Initialization setting: p = 1, 6) Start iteration: i. Update via (19) ii. If Then ρ=2ρ, otherwise p=p+1, 7) If the objective function converges or reaches the upper limit of the number of iterations, stop the loop 8) Output And reconstructed into vector v In summary, the complete solution process of the joint design method of the transmission waveform and reflection beamforming of the RIS-assisted DFRC system based on the AO framework is as follows: 1) Input parameter B, γ i , P T 2) Initialization settings: r = 1, v (0) 3) Start iterating i. Calculate X through (11) (r+1) ii. Update by the method of claim 6 iii. r=r+1 4) If the objective function converges or reaches the upper limit of the number of iterations, stop the loop.

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