RSMA assistance-based ISAC system beam forming precoder design method under imperfect channel

Abstract

The invention relates to a method for designing an ISAC system beam forming precoder based on RSMA assistance under an imperfect channel, and belongs to the technical field of communication perception integration. The method comprises the following steps: establishing an RSMA-assisted multi-user large-scale multiple-input single-output (MU-MISO)-based ISAC system model under ICSI, establishing an optimization problem model of beamforming precoder design and communication sensing resource allocation under the constraint of limited resources, channel uncertainty and satisfaction of communication rate and radar sensing estimation CRB, solving an optimization problem in a joint optimization model, and establishing a beamforming precoder design and communication sensing resource allocation optimization problem model under the constraint of limited resources, channel uncertainty and satisfaction of communication rate and radar sensing estimation CRB. The optimal system utility under the worst condition is obtained, the optimization problem in the joint optimization model is solved, and the optimal system utility under the general condition is obtained. The method has the advantages that the maximum improvement of the system utility under the general condition is realized, the target detection capability is remarkably enhanced while the fair rate of the user is improved, and a new solution is provided for the performance optimization and robustness transmission of the ISAC system.

Description

Technical Field

[0001] The present invention belongs to the technical field of integrated communication and sensing, and specifically relates to the design of a precoder for beamforming in an integrated communication and sensing system based on RSMA under imperfect channel state information. Background Art

[0002] With the continuous evolution of wireless communication technologies, 5G and future 6G networks are developing towards a more intelligent and integrated direction. Especially in complex communication environments, the deep integration of sensing and communication has become a key requirement. The integrated sensing and communication (ISAC) technology is proposed to address this challenge. It combines wireless communication and sensing functions. By sharing spectrum resources, it can not only perform efficient data transmission but also achieve accurate sensing and monitoring. This integration not only improves the utilization efficiency of spectrum resources but also reduces system latency and improves real-time performance, especially suitable for application scenarios such as autonomous driving, intelligent manufacturing, and smart cities that require both communication and sensing. The core advantage of ISAC is that the integration of communication and sensing enables the same spectrum resource to serve multiple functions, enhancing the overall performance of the network. For example, in autonomous driving applications, ISAC can ensure that vehicles maintain a stable communication connection while driving at high speed and can simultaneously sense changes in the surrounding environmental state in real time, ensuring driving safety and the accuracy of path planning. This collaborative working method significantly improves the system's adaptability to environmental changes and reduces delays caused by improper resource allocation.

[0003] In addition, Rate Splitting Multiple Access (RSMA), as a new type of multiple access method, can improve the spectrum utilization efficiency in complex wireless communication environments. RSMA decomposes the transmission tasks of users into multiple sub-streams and dynamically adjusts the transmission rate of each sub-stream, thus effectively coping with the spectrum management challenges under high-density users and complex channel conditions. Traditional multiple access technologies, such as Orthogonal Multiple Access (OMA) and Non-Orthogonal Multiple Access (NOMA), although certain progress has been made in multi-user access, still have deficiencies when facing problems such as channel fluctuations, interference management, and quality of service guarantee. RSMA enables the system to flexibly adjust resource allocation through rate splitting, more efficiently supporting simultaneous multi-user access. The advantage of RSMA lies in its ability to dynamically adjust the resource allocation strategy according to the channel state and the needs of different users, enabling multiple users to share spectrum resources while ensuring the overall performance and interference control of the system. Especially in scenarios such as autonomous driving and industrial Internet that require low latency and high reliability, RSMA can provide a more flexible resource scheduling scheme to support diverse quality of service requirements.

[0004] However, in actual engineering design, the Channel State Information at Transmitter (CSIT) of the transmitter is affected by various imperfect factors, such as estimation errors in Time Division Duplex (TDD) systems and quantization errors in Frequency Division Duplex (FDD) systems. These situations may lead to asynchronous problems or CSIT mismatch problems, which in turn affect the precoding matrix at the transmitter. Since it is actually difficult to obtain perfect CSI, the robust design problem arises - ensuring that the system can still maintain good performance under imperfect CSIT conditions. Generally, there are two types of channel error models, the statistical error model and the ellipsoidal bounded uncertainty model. The former captures statistical estimation errors, usually modeled as random variables, while the latter reflects quantization errors by defining upper bounds. In actual communication systems, the Base Station (BS) faces many challenges in obtaining CSIT, which stem from uncertainties such as channel estimation errors, feedback delays, and quantization errors. However, if the user receiver can achieve perfect CSIR through appropriate training, its estimation error can be comparable to additive noise.

[0005] In summary, combining ISAC with RSMA can provide an innovative solution for modern communication networks. ISAC improves the comprehensive capabilities and resource utilization rate of the system by integrating sensing and communication functions, while RSMA further optimizes the spectral efficiency and quality of service through flexible rate splitting and dynamic resource allocation. Although certain progress has been made in the theory and preliminary applications of these technologies, how to effectively combine ISAC and RSMA in complex and highly dynamic environments, especially under imperfect CSIT conditions, remains an urgent issue to be solved. Therefore, considering the limitations of channel estimation errors and the inability of traditional schemes to achieve significant performance gains, it is of great significance to study the beamforming design of RSMA-assisted ISAC systems under imperfect channel state (Imperfect Channel State Information, ICSI). Summary of the Invention

[0006] The present invention provides a method for designing a beamforming precoder for an RSMA-assisted ISAC system under imperfect channels to solve the problem of system performance optimization in scenarios with limited spectrum resources.

[0007] The technical solution adopted by the present invention includes the following steps:

[0008] Step 1: Construct an ISAC system model based on multi-user massive multiple-input single-output (MU-MISO) assisted by RSMA under ICSI, including a communication model, a channel model, and a radar sensing model.

[0009] Step 2: Under the constraints of limited resources, channel uncertainty, and meeting the communication rate and radar sensing estimation Cramer-Rao bound (CRB), establish an optimization problem model for beamforming precoder design and communication sensing resource allocation in the RSMA-assisted ISAC system model based on MU-MISO under ICSI to maximize the minimum user rate.

[0010] Step 3: Adopt an alternating optimization (AO) algorithm combined with a weighted minimum mean square error (WMMSE) framework to solve the optimization problem in the joint optimization model and obtain the best system utility in the worst case.

[0011] Step 4: Adopt an algorithm combining the S-procedure algorithm, the semidefinite relaxation (SDR) algorithm, the penalty function method (PFD), and the successive convex approximation (SCA) algorithm to solve the optimization problem in the joint optimization model and obtain the best system utility in the general case.

[0012] The specific system model constructed in Step 1 of the present invention is as follows:

[0013] Consider an ISAC system with a dual-function base station BS with multiple antennas. The BS serves multiple single-antenna communication users and a radar sensing target at the same time. The antenna array of the base station is a uniform linear array (ULA), and the antenna element spacing is half a wavelength. The radar and communication subsystems share the antenna deployment. During the operation of the BS, communication services are provided to all communication users through RSMA access. At the same time, the shared base station antenna transmits detection signals to the radar sensing target. The number of antennas in the base station is N. t , the user set is represented as

[0014] The communication model constructed in step 1 of the present invention is specifically:

[0015] For the lth time slot, use W k Represents the message sent by the base station to user k, applying 1-layer RSMA, and W k Decomposed into user k's public message part W c,k and private message part W p,k , that is, W k ={W c,k ,W p,k}, and then all users' public messages {W c,1 ,W c,2 ,...,W c,K}Commonly encoded as a common data stream s c (l); Private messages of all users {W p,1 ,W p,2 ,...,W p,K} are encoded as private data streams {s1(l),s2(l),...,s K (l)}, the final transmission message can be expressed as:

[0016] in represents a complex matrix of size (K+1)×1, and the beamforming precoder at the transmitting end is represented by P. The precoder corresponds to the transmitted data stream, where p c Indicates the precoder corresponding to the public data stream, p1, p2, ..., p K Denote the precoder corresponding to each user's private data stream, and make the following assumptions: 1) Each data stream is independent of each other, that is, the data stream satisfies s(l)s H (l)=I K+1 , where I K+1 represents the K+1-dimensional identity matrix, (·) H represents the conjugate transpose, 2) let the antenna power of the system be P t , the signal transmission power needs to be less than the antenna power where ||·|| represents the corresponding Euclidean norm;

[0017] Therefore, for the l-th time slot, the transmitted signal of the base station is expressed as:

[0018]

[0019] The received signal of user k is:

[0020]

[0021] where represents the channel gain, represents the communication system noise, represents a circularly symmetric complex Gaussian random distribution with a mean of 0 and a variance of At the receiver, user k first decodes the common data stream and treats the remaining private data streams and radar data as interference. Thus, the signal-to-interference-plus-noise ratio (SINR) for user k to decode the common data stream is:

[0022]

[0023] When user k decodes the private data stream, it first removes the already decoded common data stream through the successive interference cancellation (SIC) technique. Thus, the SINR for user k to decode the private data stream is:

[0024]

[0025] Using the Shannon formula, the common achievable rate R c,k and the private achievable rate R k of user k are:

[0026] R c,k = log2(1 + γ c,k ) (5)

[0027] R k = log2(1 + γ k ) (6)

[0028] Since all users need to successfully decode the common data stream, the actual common rate under the entire RSMA scheme is:

[0029] R c = min{R c,k} (7)

[0030] Denote the actual common rate of user k as C k . Then the common rate of each user needs to satisfy:

[0031]

[0032] The final achievable rate of user k is:

[0033] R rate,k = C k + R k (9).

[0034] The channel model constructed in step 1 of the present invention is specifically:

[0035] In an actual communication system, the BS faces many challenges in obtaining perfect channel state information at the transmitter, which stem from uncertainties such as channel estimation errors, feedback delays, and quantization errors. On the other hand, if the user receiver is properly trained, perfect channel state information can be obtained at the receiver, and its estimation error can be comparable to additive noise. Here, consider a channel model in which the channel state information CSIT at the transmitter is imperfect, while the channel state information CSIR at the receiver is considered perfect. It is assumed that users are uniformly distributed within 50m - 200m near the BS;

[0036] First, for user k, the channel gain of communication can be represented by h k as:

[0037] h k = L(d k ) 1 / 2 · g k (10)

[0038] where L(d k ) represents the influence of large-scale fading, manifested as path loss L(d k ) = 32.6 + 36.7log 10 (d k ), d k represents the distance from user k to the BS; represents the influence of small-scale fading, which follows a complex Gaussian distribution, g k ~ CN(0,1);

[0039] Considering the ellipsoidal bounded uncertain channel model, the true channel can be obtained from the estimated channel and the estimation error. The specific channel model can be expressed as:

[0040]

[0041] where represents the estimated channel, represents the estimation error of the channel, ∈ k represents the upper bound size of the channel estimation error.

[0042] The sensing model constructed in step 1 of the present invention is specifically:

[0043] Using the CRB as the radar sensing metric, the CRB provides a theoretical lower bound for the variances of unbiased estimators, including target parameters such as range, velocity, and angle, thus becoming a fundamental criterion for evaluating the accuracy and resolution of radar systems. By obtaining as small a CRB as possible, the ability of the system to accurately estimate target characteristics is enhanced, ensuring robust and reliable sensing performance. Assume that the number of receiving antennas at the BS is N r , if the target response matrix G(θ) is given, the target is sensed through the transmitted signal x(l), and the reflected echo signal y r (l) obtained at the BS receiving antenna can be expressed as:

[0044] y r (l) = G(θ)x(l) + n r (l) (12)

[0045] where n r (l) represents additive white Gaussian noise AWGN with a variance of The target response matrix can be expressed as:

[0046] G(θ) = αb(θ)a H (θ) = αA(θ) (13)

[0047] where α represents the complex reflection coefficient of the radar, which is related to the target scattering cross-section and round-trip path loss. a(θ) and b(θ) represent the transmit array steering vector and the receive array steering vector, which are determined by the antenna element spacing d, the transmission signal wavelength λ, and the angle θ. They can be expressed as:

[0048]

[0049] The CRB of the angle θ can be expressed by Equation (16):

[0050]

[0051]

[0052] where, represents the derivative of A(θ), represents the covariance matrix of the signal, (·) -1 represents the inverse operation, (·) T represents the transpose operation, represents taking the real part, and Tr(·) represents the trace operation.

[0053] The optimization problem model established in step 2 of the present invention can be expressed as:

[0054] To ensure fairness for users, the concept of max-min fairness is introduced. By maximizing the minimum user rate, the quality of service (QoS) of all users is guaranteed. Under the constraints of limited resources, uncertain channels, and the need to meet the communication rate and radar sensing estimation Cramer-Rao bound (CRB), an optimization problem model for beamforming precoder design based on the RSMA-ISAC system is established with the goal of maximizing the minimum user rate. It is:

[0055]

[0056] where \(c = [C_1,...,C k ,...,C K \) represents the common rate size of each user; the objective function represents the minimum rate among all users; constraint \(C_1\) is the common rate constraint required by RSMA; constraint \(C_2\) is the limit on the common rate size of each user, ensuring that the final common message can be successfully decoded by all users; constraint \(C_3\) is the constant modulus constraint of the antenna, restricting the dynamic range of the waveform amplitude, aiming to make the radar non-linear amplifier work at the maximum efficiency state (saturated or near-saturated state) to avoid non-linear distortion of the output waveform; constraint \(C_4\) is the CRB constraint of the radar, ensuring the performance of the radar subsystem. The smaller the value of \(\rho\), the more accurate the parameter estimation; constraint \(C_5\) represents the uncertainty set of the channel.

[0057] In step 3 of the present invention, the AO principle combined with the WMMSE framework is adopted to solve the max-min user fairness rate optimization problem and obtain the best system utility in the worst case. The steps are as follows:

[0058] 1) The minimum value function in problem affects the convexity of the overall problem. By introducing a variable \(R t \) to replace this function, which is used to represent the minimum rate. At this time, the transformed objective function will become an affine function, and at the same time, a new constraint will be introduced: the minimum rate \(R t \) should be less than the rate of each user to ensure that it is the minimum, that is:

[0059] R t =\(\min(C k + R k )\) (21)

[0060] 2) Through the WMMSE framework, the rate function is transformed into the MMSE form. A linear receiver \(g c,k \) and \(g k \) are introduced at user \(k\), corresponding to the common stream and the private stream respectively. During the reception of the private stream, the influence of the common stream is eliminated by SIC. Taking the common stream as an example, the mean square error (MSE) between its estimated symbol and the true symbol is:

[0061]

[0062] For the optimal MMSE receiver it can be obtained by solving as follows:

[0063]

[0064] Using the optimal MMSE receiver the optimal received MSE can be expressed by Equation (24):

[0065]

[0066] Therefore, the rate function can be expressed in the form of optimal MMSE reception:

[0067]

[0068] By introducing the weight variable w c,k , the MMSE is rewritten into the WMMSE framework, and the received WMSE can be expressed by Equation (26):

[0069]

[0070] And the optimal weight variable w c,k can be obtained by solving as follows:

[0071]

[0072] Therefore, the rate function can be converted into the representation form of the WMMSE framework:

[0073]

[0074] The derivation of the private rate is exactly the same as that of the public rate. Thus, the worst-case public achievable rate and private achievable rate can be respectively expressed through the WMMSE framework as:

[0075]

[0076] Then the original problem is constructed into another optimization problem under the WMMSE framework

[0077]

[0078] where

[0079] 3) Fix \(P\) and optimize \(g\). According to the Schur complement algorithm and its extended form, since the infinite number of uncertain channel sets introduced by the ICSIT condition can be converted into the form of Linear Matrix Inequalities (LMIs), for the processing of common data streams: Let where The MMSE problem under uncertainty is expressed as:

[0080]

[0081] By adding a new constraint and applying the Schur complement algorithm, this constraint is expressed in the form of LMIs. Therefore, the MMSE problem under uncertainty can be expressed as:

[0082]

[0083] Through the extended form of the Schur complement algorithm, let \(C = -[1\ 0]\), \(\epsilon=\epsilon\) k , since \(A\geq B\) is satisfied H \(DC+(B\) H \(DC)\) H , condition, so (33) can be converted into an SDP problem:

[0084]

[0085] where \(0\) represents the zero matrix, \(\Delta\) c,k is the newly introduced variable, and at the same time this variable needs to satisfy the condition \(\Delta\) c,k \(\geq0\);

[0086] Similarly, the private data stream is expressed as the MMSE problem under uncertainty:

[0087]

[0088] where \(\Delta\) k \(\geq0\), and These two problems can be quickly solved by the interior point method;

[0089] 4) Fix \(g\) and optimize \(P\). Fix the variable \(g\) in the problem and optimize the beamforming matrix \(P\). Since the expression of \(CRB(\theta)\) in the constraint \(C4\) of the problem is non-convex with respect to the variable \(P\), it needs to be relaxed; first, convert the constraint \(C4\) into equation (36):

[0090]

[0091] Apply the Schur complement algorithm to convert it into the form of LMIs:

[0092]

[0093] For the problem after substituting the constraint C5 in it with (34) and (35), the problem is updated to the problem Then it can be expressed as:

[0094]

[0095] where the constraint C5 is a newly added constraint, which is a relaxed form of the equation R x = PP H For the optimization problem in ξ c,k (g c,k , w c,k ) and ξ k (g k , w k ), first obtain the corresponding g c,k , w c,k , g k , w k by solving (34) and (35), and then substitute them into (26) to obtain the result of WMMSE;

[0096] In summary, first fix the receivers g and w, use the WMMSE framework combined with the Schur complement algorithm to calculate the beamforming matrix P; then fix P and solve (34) and (35) to calculate the optimal MMSE receivers g and w.

[0097] In step 4 of the present invention, an algorithm combining the S-procedure algorithm, the SDR algorithm, the PFD algorithm, the SCA algorithm, etc. is adopted to solve the optimization problem in the joint optimization model and obtain the optimal system utility under general circumstances. The process is as follows:

[0098] 1) Adopt the SDR algorithm and let where rank(P c ) = 1, rank(P k ) = 1, and the covariance matrix of the signal is expressed as Therefore, the common rate and the private rate are respectively:

[0099]

[0100] Define variable a k , b k , c k , d k represent the upper and lower bounds of SINR in the rate function, and the specific relationship is:

[0101]

[0102] Therefore, the lower bounds of the public rate and the private rate can be obtained:

[0103]

[0104] 2) Since (41) and (42) are the difference of two logarithmic functions, which leads to the non-convexity of the problem , solve this problem through the SCA algorithm. At the m-th iteration, use the first-order Taylor series as the surrogate function:

[0105]

[0106] where and represent the results of b k and d k at the (m - 1)-th iteration respectively;

[0107] 3) Convert the uncertainty problem into a deterministic problem through the S-procedure algorithm. First, rewrite the upper and lower bound relationship of SINR into the following form:

[0108]

[0109] Adopt the S-procedure algorithm, introduce slack variables u a,k ≥0, u b,k ≥0, u c,k ≥0, u d,k ≥0, and rewrite it into the LMI form:

[0110]

[0111] The optimization problem can be reconstructed as

[0112]

[0113] where

[0114] 4) The PFD algorithm is used to relax the rank-one structure and an improvement is made on this basis to construct a two-layer penalty iterative structure. The optimization problem under a fixed penalty factor is iterated internally, and the penalty factor size is updated externally to strengthen the penalty intensity. The specific content is as follows:

[0115] First, the rank-one constraint is equivalent to the equations Tr(P c ) = ||P c || and Tr(P k ) = ||P k ||. Therefore, the overall penalty function term is:

[0116]

[0117] where η is the penalty factor, reflecting the influence degree of the penalty term on the overall optimization function objective. When η is small, the penalty term in the final solution can better satisfy the imposed constraints, but the cost is that the objective function will be severely affected by the penalty. Therefore, during the optimization process, the penalty intensity is gradually increased. The update of the penalty factor at the nth iteration is as follows:

[0118] η n = λ·η n-1 (55)

[0119] Since the 2-norm in the penalty function term makes the whole expression non-convex, the first-order Taylor series is used to represent its lower bound:

[0120]

[0121] [[ID=3{5]]where represents the eigenvector corresponding to the largest eigenvalue in at the nth external iteration. Therefore, (54) is expressed as:

[0122] <{

[0123] After adding the penalty term, the optimization problem is expressed as:

[0124]

[0125] By combining the S-procedure algorithm, the SDR algorithm, the PFD algorithm, and the SCA algorithm, the original problem is transformed into an easily handled convex optimization problem. (58) can be efficiently solved by the interior-point method. Specifically, the result is directly obtained through the CVX tool, so as to obtain the optimal system utility of the RSMA-ISAC system in general cases.

[0126] Advantages of the present invention:

[0127] 1. The present invention considers the situation of limited spectrum resources in the 6G communication network, incorporates communication and sensing integration into the communication network, and taking into account user fairness and imperfect channel state information, proposes a robust beamforming optimization problem aimed at achieving the max-min fairness of users in the RSMA-assisted ISAC system and accurately estimating radar parameters by minimizing the CRB. The initial non-convex optimization problem is relaxed and transformed into a convex optimization problem through the proposed algorithm. Compared with traditional ISAC systems, the algorithm of the present invention achieves a performance improvement of more than 1.5 bps / Hz in terms of the system fair rate. Especially in the case of obvious channel fluctuations and large estimation errors, it can achieve a performance improvement of more than 1.5 bps / Hz, significantly improving the overall utility of the system. This innovation not only effectively solves the problem of system performance optimization in scenarios with limited spectrum resources, but also provides a new solution for the practical application of communication and sensing integration technology.

[0128] 2. In a complex urban environment, it is difficult for existing ISAC systems to obtain perfect channel state information. At the same time, under the condition of a large number of users, multi-user communication brings a large communication overhead, which often leads to performance limitations of the radar system. Therefore, the Rate Splitting Multiple Access (RSMA) technology has become an important technical means to improve communication and sensing performance. It enables the system to flexibly adjust resource allocation through rate splitting, more efficiently supports multi-user simultaneous access, brings additional degrees of freedom (DoFs) to the system, and has stronger robustness to channel fluctuations. It can dynamically adjust the resource allocation strategy according to the channel state and the needs of different users, enabling multiple users to share spectrum resources while ensuring the overall performance and interference control of the system. Traditional multiple access technologies can only achieve limited DoFs gain, but have poor robustness. To overcome the impact of "imperfect channels", this invention patent uses RSMA to assist the ISAC system, solving the performance bottleneck problem of the communication system under imperfect channel conditions.

[0129] 3. The algorithm proposed in the present invention for combining the WMMSE framework and the AO principle in the worst case is different from traditional algorithms. This algorithm utilizes the efficiency of WMMSE and also uses the AO principle to handle complex non-convex problems. Experimental results show that this algorithm exhibits significant advantages in terms of optimization accuracy, convergence speed, and stability. Specifically, compared with traditional ISAC, the algorithm proposed in the present invention has achieved a large improvement in the accuracy of target angle estimation, significantly enhancing the communication performance and sensing gain in the system and obtaining a better trade-off between communication and sensing performance.

[0130] 4. The algorithm proposed in this paper differs from traditional algorithms in that it creatively combines the S-procedure, SDR, PFD, and SCA algorithms. Experimental results demonstrate that this algorithm exhibits significant advantages in optimization accuracy, convergence speed, and stability, significantly enhancing the performance gain of beamforming design in the system. This improvement not only validates the algorithm's effectiveness but also provides a new technical approach for optimizing the robust performance of integrated communication and perception systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0131] Figure 1 It is the technical roadmap of the present invention;

[0132] Figure 2 It is the system model of the present invention;

[0133] Figure 3 This is a flow chart of Algorithm 1 proposed in the present invention that combines the WMMSE framework and the AO principle;

[0134] Figure 4 It is a flow chart of Algorithm 2 proposed by the present invention that combines S-procedure, SDR, PFD and SCA;

[0135] Figure 5 This is a schematic diagram of the convergence of Algorithm 1 proposed in the present invention for the worst case;

[0136] Figure 6 This is a schematic diagram of the convergence of Algorithm 2 proposed in the present invention for a general case;

[0137] Figure 7 This is a visualization graph of the impact of the signal-to-noise ratio on the maximum and minimum fair rates of users in a simulation experiment comparing the present invention with the traditional algorithm;

[0138] Figure 8 This is a visualization of the impact of the upper bound of the estimation error on the maximum and minimum fair rates of users in a simulation experiment comparing the present invention with the traditional algorithm;

[0139] Figure 9 It is a visual diagram of the trade-off between the radar perception performance of Algorithm 1 and the maximum and minimum rates of the communication system in the simulation experiment of the present invention compared with the traditional algorithm;

[0140] Figure 10 This is a visualization diagram of the trade-off between radar perception performance and communication system and rate of Algorithm 2 in the simulation experiment comparing the present invention with traditional algorithms. DETAILED DESCRIPTION

[0141] like Figure 1The technical roadmap of the present invention is shown below. In view of the problems existing in the prior art, the present invention proposes a beamforming precoder design method for the ISAC system assisted by RSMA under ICSI. Under the constraints of limited resources, channel uncertainty, meeting the communication rate and the radar sensing estimation CRB, an optimization problem model for beamforming precoder design and communication sensing resource allocation is established to maximize the minimum user rate. In the optimization process, the weighted minimum mean square error (WMMSE) framework and the alternating optimization (AO) principle are combined to obtain the best system utility in the worst case; and the S-procedure algorithm, the semi-definite relaxation algorithm (SDR), the penalty function method (PFD), and the sequential convex approximation algorithm (SCA) are creatively combined to obtain the best system utility in the general case.

[0142] The technical solution adopted by the present invention includes the following steps:

[0143] Step 1: Construct an ISAC system model, a communication model, a channel model, and a radar sensing model based on RSMA-assisted multi-user massive multiple-input single-output (MU-MISO) under ICSI as shown in Figure 2 :

[0144] (1) System model:

[0145] Consider an ISAC system with a dual-functional base station (BS) having multiple antennas. The BS serves multiple single-antenna communication users and a radar sensing target simultaneously. The antenna array of the base station is a uniform linear array (ULA), and the antenna element spacing is half-wavelength spacing. The radar and communication subsystems share the antenna deployment. During the operation of the BS, communication services are provided to all communication users through the RSMA access method, and at the same time, a detection signal is transmitted to the radar sensing target by sharing the antennas of the base station. The number of antennas of the base station is N t , and the user set is represented as

[0146] (2) Communication model:

[0147] For the l-th time slot, use W k to represent the message sent by the base station to user k. Applying 1-layer RSMA, decompose W k into the common message part W c,k and the private message part W p,k , that is, W k = {W c,k , W p,k}. Then decompose the common messages {W c,1 , W c,2 ,..., W c,K}Jointly encoded into the common data stream s c (l); The private messages {W p,1 , W p,2 ,..., W p,K} of all users are respectively encoded into the private data streams {s1(l), s2(l),..., s K (l)}, and the final transmitted message is expressed as where represents a complex matrix of size (K + 1)×1, and the beamforming precoder at the transmitter is denoted by P, The precoder corresponds to the transmitted data stream, where p c represents the precoder corresponding to the common data stream, and p1, p2,..., p K respectively represent the precoders corresponding to the private data streams of each user. The following assumptions are made: 1) The data streams are independent of each other, that is, the data stream satisfies s(l)s H (l) = I K+1 , where I K+1 represents the (K + 1)-dimensional identity matrix, and (·) H represents the conjugate transpose. 2) Let the antenna power of the system be P t , and the transmission power of the signal needs to be less than the antenna power where ||·|| represents the corresponding Euclidean norm;

[0148] Therefore, for the l-th time slot, the transmitted signal of the base station is expressed as:

[0149]

[0150] The received signal of user k is:

[0151]

[0152] where represents the channel gain, represents the communication system noise, represents a circularly symmetric complex Gaussian random distribution with a mean of 0 and a variance of . User k first decodes the common data stream at the receiver and treats the remaining private data streams and radar data as interference. Thus, the signal-to-interference plus noise ratio (SINR) for user k to decode the common data stream is:

[0153]

[0154] When user k decodes the private data stream, it first removes the decoded public data stream through the Successive Interference Cancellation (SIC) technique. Thus, the SINR for user k to decode the private data stream is:

[0155]

[0156] The common achievable rate R of user k is obtained using the Shannon formula c,k and the private achievable rate R k are:

[0157] R c,k = log2(1 + γ c,k ) (5)

[0158] R k = log2(1 + γ k ) (6)

[0159] Since all users need to successfully decode the public data stream, the actual common rate under the entire RSMA system is:

[0160] R c = min{R c,k} (7)

[0161] Denote the actual common rate of user k by C k . Then the common rate of each user needs to satisfy:

[0162]

[0163] The final achievable rate of user k is:

[0164] R rate,k = C k + R k (9)

[0165] (3) Channel model:

[0166] In an actual communication system, the BS faces many challenges in obtaining perfect channel state information at the transmitter, which stem from uncertainties such as channel estimation error, feedback delay, and quantization error; on the other hand, if the user receivers are properly trained, perfect channel state information can be obtained at the receivers, and its estimation error can be comparable to additive noise; therefore, consider a channel model in which the CSIT is imperfect while the CSIR is considered perfect, assuming that users are uniformly distributed within 50m - 200m near the BS;

[0167] First, for user k, the channel gain of communication can be represented by hk Denote:

[0168] h k = L(d k ) 1 / 2 ·g k (10)

[0169] where L(d k ) represents the impact of large-scale fading, manifested as path loss L(d k ) = 32.6 + 36.7log 10 (d k ), d k denotes the distance from user k to BS; represents the impact of small-scale fading, following a complex Gaussian distribution, g k ~CN(0,1);

[0170] Considering the ellipsoidal bounded uncertainty channel model, the true channel is obtained from the estimated channel and the estimation error. The specific channel model can be expressed as:

[0171]

[0172] where represents the estimated channel, represents the estimation error of the channel, ∈ k represents the upper bound of the channel estimation error;

[0173] (4) Radar sensing model:

[0174] Using the CRB as the radar sensing metric, the CRB provides a theoretical lower bound for the variance of unbiased estimators, including target parameters such as distance, velocity, and angle, thus becoming the basic standard for evaluating the accuracy and resolution of radar systems. By obtaining as small a CRB as possible, the ability of the system to accurately estimate target characteristics is enhanced, ensuring robust and reliable sensing performance. Assume the number of receiving antennas at the BS is N r , if the target response matrix G(θ) is given, the target is sensed through the transmitted signal x(l), and the reflected echo signal y r (l) obtained at the BS receiving antenna is expressed as:

[0175] y r (l) = G(θ)x(l) + n r (l) (12)

[0176] where n r (l) represents additive white Gaussian noise (AWGN), with a variance of The target response matrix can be expressed as:

[0177] G(θ) = αb(θ)a H (θ) = αA(θ) (13)

[0178] Where α represents the complex reflection coefficient of the radar, which is related to the target scattering cross-section and the round-trip path loss. a(θ) and b(θ) represent the transmit array steering vector and the receive array steering vector, which are determined by the antenna element spacing d, the transmission signal wavelength λ, and the angle θ. They can be expressed as follows:

[0179]

[0180] And the CRB of the angle θ can be expressed by Equation (16):

[0181]

[0182] Where, represents the derivative of A(θ), represents the covariance matrix of the signal, (·) -1 represents the inverse operation, (·) T represents the transpose operation, represents taking the real part, Tr(·) represents the trace operation;

[0183] Step 2: Under the constraints of limited resources, channel uncertainty, and meeting the communication rate and the CRB of radar sensing estimation, establish an optimization problem model for beamforming precoder design and communication sensing resource allocation in the ICSI-based RSMA-assisted Multi-User Multiple-Input Single-Output (MU-MISO) ISAC system model to maximize the minimum user rate:

[0184] To ensure fairness among users, the concept of max-min fairness is introduced. By maximizing the minimum user rate, the quality of service (QoS) of all users is guaranteed. Under the constraints of limited resources, channel uncertainty, and the need to meet the communication rate and the CRB of radar sensing estimation, an optimization problem model for beamforming precoder design and communication sensing resource allocation in the RSMA-ISAC system is established with the goal of maximizing the minimum user rate as follows:

[0185]

[0186] Where c = [C1,..., C k ,..., C Krepresents the common rate size of each user; the objective function represents the minimum rate among all users; constraint C1 is the common rate constraint required by RSMA; constraint C2 is the limit on the common rate size for each user, ensuring that the final common message can be successfully decoded by all users; constraint C3 is the constant modulus constraint of the antenna, which limits the dynamic range of the waveform amplitude, aiming to make the radar non-linear amplifier work at the maximum efficiency state (saturated or near-saturated state) and avoid non-linear distortion of the output waveform; constraint C4 is the CRB constraint of the radar, which ensures the performance of the radar subsystem. The smaller the value of ρ, the more accurate the parameter estimation; constraint C5 represents the uncertainty set of the channel;

[0187] Step 3: Adopt the alternating optimization (AO) algorithm combined with the weighted minimum mean square error (WMMSE) framework to solve the optimization problem in the joint optimization model and obtain the best system utility in the worst case:

[0188] The WMMSE algorithm is an optimization algorithm widely used in communication systems proposed in 2011, usually used to optimize power allocation, beamforming or resource allocation problems. The core idea of the WMMSE algorithm is to indirectly optimize the objective of the system by minimizing the weighted mean square error. Its advantages lie in strong applicability, excellent performance and high flexibility.

[0189] The present invention combines the Schur complement algorithm with the weighted minimum mean square error framework for the first time. By alternately optimizing the base station transmit beamforming vector and the optimal MMSE receiver, the optimization problem in the joint optimization model is solved, and the best system utility in the worst case is obtained. The flow chart of the alternating optimization algorithm is shown in Figure 3, and the specific steps are as follows:

[0190] 1) Problem The minimum value function in the problem will affect the convexity of the overall problem. By introducing the variable R t to replace this function, which is used to represent the minimum rate. At this time, the transformed objective function will become an affine function, and at the same time, a new constraint - the minimum rate R t should be less than the rate of each user to ensure that it is the minimum, that is:

[0191] R t = min(C k + R k ) (21)

[0192] 2) Convert the rate function into the MMSE form through the WMMSE framework, and introduce the linear receiver g c,k and g kThey respectively correspond to the public stream and the private stream. During the reception of the private stream, the influence of the public stream is eliminated by SIC. Taking the public stream as an example, the mean square error (MSE) between its estimated symbol and the true symbol is as follows:

[0193]

[0194] For the optimal MMSE receiver It can be obtained by solving as follows:

[0195]

[0196] Using the optimal MMSE receiver the optimal received MSE can be expressed by Equation (24):

[0197]

[0198] Therefore, the rate function can be expressed in the form of optimal MMSE reception:

[0199]

[0200] By introducing the weight variable w c,k , the MMSE is rewritten into the WMMSE framework, and the received WMSE can be expressed by Equation (26):

[0201]

[0202] And the optimal weight variable w c,k can be obtained by solving as follows:

[0203]

[0204] Therefore, the rate function can be converted into the representation form of the WMMSE framework:

[0205]

[0206] The derivation of the private rate is exactly the same as that of the public rate. Thus, the public achievable rate and the private achievable rate in the worst case can be respectively expressed by the WMMSE framework as:

[0207]

[0208] Then the original problem is constructed into another optimization problem under the WMMSE framework

[0209]

[0210] where

[0211] 3) Fix P and optimize g. According to the Schur complement algorithm and its extended forms, since the infinitely many uncertain channel sets introduced by the ICSIT condition can be transformed into the form of Linear Matrix Inequalities (LMIs), taking the processing of common data streams as an example, let where The MMSE problem under uncertainty can be expressed as:

[0212]

[0213] By adding the constraint and applying the Schur complement algorithm, this constraint can be expressed in the form of LMIs. Therefore, the MMSE problem under uncertainty can be expressed as:

[0214]

[0215] Through the extended form of the Schur complement algorithm, let C = -[1 0], ∈ = ∈ k , since A ≥ B is satisfied H DC+(B H DC) H , condition, so (33) can be transformed into an SDP problem:

[0216]

[0217] where 0 represents the zero matrix, and △ c,k is the newly introduced variable, and at the same time this variable needs to satisfy the condition △ c,k ≥ 0.

[0218] Similarly, the private data stream can also be expressed as the MMSE problem under uncertainty:

[0219]

[0220] where △ k ≥ 0, and These two problems can be solved quickly by the interior point method;

[0221] 4) Fix g and optimize P. Fix the variable g in the problem and optimize the beamforming matrix P. Since the problem The expression of CRB(θ) in constraint C4 is non-convex with respect to the variable P, so it needs to be relaxed. First, constraint C4 is transformed into Equation (36):

[0222]

[0223] Apply the Schur complement algorithm to convert it into LMIs form:

[0224]

[0225] The problem After substituting (34) and (35) into the constraint C5 in the problem Update to question but It can be expressed as:

[0226]

[0227] Constraint C5 is a new constraint, which is the equation R x =PP H The relaxed form of the optimization problem Middle c,k (g c,k ,w c,k ) and ξ k (g k ,w k ), first solve (34), (35) to get the corresponding g c,k ,w c,k ,g k ,w k , and then substitute it into (26) to obtain the result of WMMSE.

[0228] In summary, first, fix the receiver g, w, and use the WMMSE framework combined with the Schur complement algorithm to calculate the beamforming matrix P; then fix P and solve (34) and (35) to calculate the optimal MMSE receiving g, w.

[0229] Step 4: Use the combined algorithm of S-procure algorithm, SDR algorithm, PFD algorithm and SCA algorithm to solve the optimization problem in the joint optimization model and obtain the best system utility under general conditions. Figure 4 The specific process is as follows:

[0230] The SDR algorithm is an effective method for transforming non-convex optimization problems into convex optimization problems. It is commonly used in fields such as signal processing, communication systems, and control, especially when dealing with matrix variables and matrix inequality constraints. Its core idea is to transform the matrix constraints (such as positive definiteness constraints) in the original problem into semi-definite constraints, thereby turning the original problem into a semi-definite programming (SDP) problem. The SDP problem is convex and can be efficiently solved by standard convex optimization algorithms (such as the interior point method). In specific operations, SDR "relaxes" certain non-convex constraints into semi-definite constraints by introducing slack variables, thus transforming the originally difficult-to-solve non-convex problem into a convex problem that can be handled. The advantage of SDR is that it can provide a guarantee of the global optimal solution by transforming non-convex problems into convex optimization problems, and existing efficient convex optimization algorithms can quickly solve semi-definite programming problems, especially suitable for dealing with large-scale complex optimization problems.

[0231] The S-procedure is an important mathematical tool for dealing with the implication relationship of quadratic inequality constraints and transforming them into computable linear matrix inequalities (LMIs), thereby simplifying the solution of optimization and control problems. This method has wide applications in fields such as robust control, convex optimization, signal processing, and machine learning, especially crucial in problems such as Lyapunov stability analysis, robust performance optimization, and semi-definite programming (SDP). For example, in robust control, the S-procedure can be used to handle system uncertainties and ensure that stability conditions are still met under perturbations or parameter variations; in machine learning, it can be used for optimization constraints in kernel methods or support vector machines (SVMs). Since LMIs can be solved by efficient algorithms such as the interior point method, the S-procedure greatly improves the computational feasibility of complex constraint optimization problems and becomes one of the important tools in modern control and optimization theory.

[0232] The SCA algorithm is an iterative optimization method that transforms non-convex problems into convex problems and gradually approaches the optimal solution by locally convexifying the non-convex objective function or constraints in each iteration. The basic idea of the SCA algorithm is to approximately process the objective function or constraints at the current iteration point, usually using Taylor expansion or other local linearization means to transform it into a convex function. Then, in each iteration, solve this local convex optimization problem and use the current optimal solution as the starting point for the next iteration. The SCA algorithm is particularly suitable for optimization problems that are non-convex globally but have convex properties locally. The advantages of the SCA algorithm are that it can gradually optimize non-convex problems through local convexification methods, is suitable for dealing with problems with strong local convexity, and can flexibly handle different types of constraints and objective functions. In addition, the SCA algorithm can better handle high-dimensional complex problems and continuously improve the quality of the solution during the iteration process.

[0233] The penalty function method is a classic constrained optimization algorithm. Its core idea is to transform the constrained optimization problem into a series of unconstrained optimization sub-problems by constructing penalty terms. This method introduces a penalty term for the constraint violation amount in the objective function and, in conjunction with a gradually increasing penalty coefficient, forces the iterative solution to converge to the feasible region. According to the form of the penalty term, it can be divided into the external penalty function method (such as the quadratic penalty function), the exact penalty function method (such as the L1 penalty function), and the barrier function method, etc. The advantages of the penalty function method are simple implementation and wide applicability. It can handle equality constraints, inequality constraints, and complex non-linear constraint problems, and is widely used in engineering optimization, machine learning parameter training, optimal control and other fields. However, this method also has some limitations: as the penalty coefficient increases, the condition number of the optimization problem may deteriorate, resulting in numerical calculation difficulties; the convergence speed is greatly affected by the penalty coefficient growth strategy; for non-convex problems, it may fall into a local optimum. To improve these deficiencies, the penalty function method is often combined with algorithms such as the augmented Lagrangian method and sequential quadratic programming (SQP) to balance convergence and computational efficiency. In addition, in the modern optimization framework, the introduction of adaptive penalty strategies and hybrid algorithms further enhances the robustness and practicality of this method.

[0234] The present invention creatively combines the S-procedure algorithm, the SDR algorithm, and the SCA algorithm, and improves on the basis of the penalty function algorithm to construct a two-layer penalty iteration structure. By solving the optimization problem in the joint optimization model, the best system utility in general cases can be obtained. The process is as follows:

[0235] 1) Using the SDR algorithm, let where rank(P c ) = 1, rank(P k ) = 1, and the covariance matrix of the signal is expressed as Therefore, the common rate and the private rate are respectively:

[0236]

[0237] Define variables a k , b k , c k , d k to represent the upper and lower bounds of the SINR in the rate function. The specific relationship is:

[0238] Therefore, the lower bounds of the common rate and the private rate can be obtained:

[0239]

[0240] 2) Since (41) and (42) are the difference of two logarithmic functions, it leads to the problem The non-convexity of the problem is solved by the SCA algorithm. Specifically, at the m-th iteration, we use the first-order Taylor series as the surrogate function:

[0241]

[0242] where and represent the results of b k and d k at the (m - 1)-th iteration, respectively.

[0243] 3) The uncertainty problem is converted into a deterministic problem through the S-procedure algorithm. First, the upper and lower bound relationships of the SINR are rewritten in the following form:

[0244]

[0245] By using the S-procedure algorithm, slack variables u a,k ≥0, u b,k ≥0, u c,k ≥0, u d,k ≥0 are introduced and rewritten in the LMI form:

[0246]

[0247] The optimization problem can be reconstructed as

[0248]

[0249] where

[0250] 4) The PFD algorithm is used to relax the rank-one structure and an improved two-layer penalty iteration structure is constructed on this basis. The optimization problem under a fixed penalty factor is iterated internally, and the penalty factor size is updated externally to strengthen the penalty. The specific content is as follows:

[0251] First, the rank-one constraint is equivalent to the equalities Tr(P c ) = ||P c || and Tr(P k ) = ||P k ||. Therefore, the overall penalty function term is:

[0252]

[0253] where η is the penalty factor, reflecting the influence degree of the penalty term on the overall optimization function objective. When η is small, the penalty term in the final solution can better satisfy the imposed constraints, at the cost that the objective function will be severely affected by the penalty. Therefore, the intensity of the penalty is gradually increased during the optimization process. The update of the penalty factor at the n-th iteration is as follows:

[0254] η n = λ·η n-1 (55)

[0255] Since the 2-norm in the penalty function term makes the whole expression non-convex, the first-order Taylor series is used to represent its lower bound:

[0256]

[0257] where represents the eigenvector corresponding to the largest eigenvalue in at the n-th outer iteration. Therefore, (54) can be expressed as:

[0258]

[0259] After adding the penalty term, the optimization problem can be expressed as:

[0260]

[0261] In summary, by combining algorithms such as the S-procedure algorithm, the SDR algorithm, the PFD algorithm, and the SCA algorithm, the original problem is transformed into an easily handled convex optimization problem, and (58) can be efficiently solved by the interior-point method. Specifically, the result can be directly obtained through the CVX tool, so that the optimal system utility of the RSMA-ISAC system in general cases can be obtained.

[0262] The technical effects of the present invention are further described in detail below in combination with simulation experiments: The communication and sensing integrated system based on rate-splitting multiple access proposed by the present invention, namely RSMA-ISAC, is compared with the existing schemes.

[0263] 1. Simulation parameter settings

[0264] In the simulation experiment, the simulation system includes a dual-functional base station, which has a uniform linear array with the antenna spacing being half a wavelength, the number of transmitting antennas N t = 6, the number of receiving antennas N r = 8; the number of users K = 4; the number of radar targets M = 1, and the radar target directions are 0° respectively; the large-scale fading path loss is modeled as 32.6 + 36.7lg 10(d), where d represents the distance between the base station and the user, with the unit of m; the small-scale fading follows a log-normal distribution with a mean of 0 and a standard deviation of 1; the antenna power P t = 20 dB; the upper bound of the channel estimation error ∈ = 0.1; the noise variance The radar signal-to-interference-plus-noise ratio SNR radar = -20 dB; the frame length of the signal L = 1024; the users and targets are randomly distributed within the range of 50 m - 200 m around the base station; the initial value of the penalty factor is η = 10 5 , the iterative control variable of the penalty factor λ = 0.6, and the maximum number of iterations is 100 times for all cases.

[0265] 2. Simulation Content and Result Analysis

[0266] Figure 5 This is the convergence diagram of Algorithm 1 for the worst-case scenario proposed in the present invention, where the CRB accuracy ρ = 5·10 -3 , and the user estimation error ∈ k = 0.15. Obviously, it can be observed from the figure that the algorithm proposed in the present invention has a fast convergence speed and few iteration times. The proposed AO-based algorithm ensures the convergence of the objective value. For RSMA-ISAC and SDMA-ISAC, convergence is usually achieved in less than fifty iterations. This shows that Algorithm 1 has a strong ability to handle non-convex problems and the convergence speed has been greatly improved.

[0267] Figure 6 This is the convergence diagram of Algorithm 2 for the general case proposed in the present invention. We set the weight parameter η = 1·10 4 , to achieve the best balance between the penalty function and the main objective function. In addition, we set the update speed of the penalty term to λ = 0.2, which is used to update η in formula (55), so that the value of ψ rank gradually increases during the iteration process, thereby enhancing the influence of the penalty term. For the RSMA-ISAC strategy and the traditional SDMA-ISAC strategy proposed in this paper, both can converge extremely fast. Specifically, the penalty term approaches zero at the 5th iteration and can achieve full convergence in about 10 iterations. The results show that this algorithm obtains a sub-optimal solution and effectively maintains the first-order constraint. This shows that Algorithm 2 has good convergence performance.

[0268] Figure 7 This is the viewable diagram of the influence of the signal-to-noise ratio on the maximum and minimum fairness rates of users in the simulation experiment comparing the present invention with the traditional algorithm. From Figure 7It can be seen that under higher signal-to-noise ratio conditions, the result of Algorithm 2 is better than that of Algorithm 1 because Algorithm 2 gives more general results, while Algorithm 1 considers the achievable rate in the worst case. When the transmission power is between 10 dB and 30 dB, the RSMA-assisted ISAC method always shows better performance than the SDMA-assisted ISAC. In the worst case, the rate gap of Algorithm 1 expands from 0.83 bps / Hz at 10 dB to 1.23 bps / Hz at 30 dB; while in general cases, the rate gap of Algorithm 2 expands from 0.83 bps / Hz at 10 dB to 1.79 bps / Hz at 30 dB, highlighting that the RSMA-ISAC method can obtain greater gains under high signal-to-noise ratio conditions.

[0269] Figure 8 This is a viewable graph of the impact of the upper bound of the estimation error on the maximum and minimum fair rates of users in the simulation experiment comparing the present invention with traditional algorithms. From Figure 8 It can be seen that as the upper bound of the estimation error increases, the RSMA-assisted ISAC method shows more excellent robust performance than the SDMA-assisted ISAC system. Specifically, in the worst case, using Algorithm 1, when the estimation error increases from 0.1 to 0.5, the performance of the SDMA-assisted ISAC drops by about 56%, while the RSMA-assisted ISAC only drops by 32.4%; while in general cases, using Algorithm 2, the performance of the SDMA-assisted ISAC drops by about 47%, while the RSMA-assisted ISAC only drops by 28.8%. This comparative analysis clearly proves that under imperfect CSIT conditions, the RSMA-assisted ISAC framework has stronger robustness.

[0270] Figure 9 This is a viewable graph of the trade-off between the radar sensing performance of Algorithm 1 and the maximum and minimum rates of the communication system in the worst case in the simulation experiment comparing the present invention with traditional algorithms. From Figure 9 We observe that the RSMA-assisted ISAC is significantly better than the SDMA-assisted ISAC, which is confirmed by the higher rate at the left inflection point. When the ρ value drops below a certain threshold - approximately 6·10 -3 - further improving the sensing accuracy will lead to a sharp decline in communication performance. Specifically, Figure 9 shows that when the estimation error ∈ k = 0.1, as ρ drops from 6·10 -3 to 2·10 -3 , the performance of the RSMA-assisted ISAC drops by about 29%, while the performance of the SDMA-assisted ISAC drops significantly by about 54%. And when the estimation error ∈ kWhen ∈ = 0.5, the corresponding performance metrics also decreased by 31% and 56% respectively. This result indicates that under the condition of actual imperfect channel state information, RSMA-assisted ISAC achieves a better communication and sensing trade-off.

[0271] Figure 10 This is a trade-off visualization diagram of the radar sensing performance of Algorithm 2 and the maximum and minimum rates of the communication system in the general case in the simulation experiment comparing the present invention with traditional algorithms. From Figure 10 we can obtain a conclusion similar to Figure 9 . Specifically, Figure 10 shows that when the estimation error ∈ k = 0.1, as ρ decreases from 5·10 -3 to 2·10 -3 , the performance of RSMA-ISAC decreases by approximately 27%, while the performance of SDMA-ISAC decreases significantly by approximately 58%. And when the estimation error ∈ k = 0.5, the corresponding performance metrics decrease by 36% and 50% respectively.

[0272] In summary, Figure 5 and Figure 6 effectively prove the effectiveness of the algorithm proposed in the present invention. The proposed algorithm can converge to the optimal value within very few iterations. Figure 7 Effectively prove the superiority of the proposed RSMA-assisted ISAC system in terms of user fairness. Under the same resource conditions, the RSMA-assisted ISAC system can effectively allocate communication resources and sensing resources, and at the same time can ensure fairness for each user as much as possible. Figure 8 Verify that the method proposed in the present invention has better robustness compared with other methods. Figure 9 and Figure 10 Verify that the proposed RSMA-assisted ISAC system has a better communication and sensing performance trade-off for the ISAC system. In short, the beamforming precoder design method of the RSMA-assisted ISAC system under imperfect channels proposed in the present invention can achieve better user fairness than traditional methods, realize the effective allocation of resources, solve the problems of spectrum resource shortage and robust transmission, and achieve a significant improvement in the performance of the ISAC system.

Claims

1. A beamforming precoder design method for an RSMA-assisted ISAC system under imperfect channels, characterized in that It includes the following steps: Step 1: Construct an ISAC system model based on multi-user massive multiple-input single-output (MU-MISO) assisted by RSMA under ICSI, including a communication model, a channel model, and a radar sensing model; Step 2: Under the constraints of limited resources, channel uncertainty, and meeting the communication rate and radar sensing estimation Cramer-Rao bound (CRB), establish an optimization problem model for beamforming precoder design and communication sensing resource allocation in the ISAC system model based on MU-MISO assisted by RSMA under ICSI to maximize the minimum user rate; Step 3: Adopt an alternating optimization (AO) algorithm combined with the weighted minimum mean square error (WMMSE) framework to solve the optimization problem in the joint optimization model and obtain the best system utility in the worst case; Step 4: Adopt an algorithm combining the S-procedure algorithm, the semidefinite relaxation (SDR) algorithm, the penalty function method (PFD), and the successive convex approximation (SCA) algorithm to solve the optimization problem in the joint optimization model and obtain the best system utility in the general case.

2. The beamforming precoder design method for the RSMA-assisted ISAC system under imperfect channels according to claim 1, characterized in that, The specific system model constructed in Step 1 is as follows: Consider an ISAC system with a dual-functional base station BS equipped with multiple antennas. The BS serves multiple single-antenna communication users and a radar sensing target simultaneously. The antenna array of the base station is a uniform linear array (ULA), and the spacing between antenna elements is half-wavelength. The radar and communication subsystems share the antenna deployment. During the operation of the BS, communication services are provided to all communication users through the RSMA access method, and at the same time, a detection signal is transmitted to the radar sensing target by sharing the antennas of the base station. The number of antennas of the base station is N t , the user set is denoted as 3. The beamforming precoder design method for the RSMA-assisted ISAC system under imperfect channels according to claim 1, characterized in that, The specific communication model constructed in Step 1 is as follows: For the l-th time slot, let W k denote the message sent from the base station to user k. Applying 1-layer RSMA, W k is decomposed into the common message part W c,k and the private message part W p,k , i.e., W k ={W c,k ,W p,k}. Then, the common messages of all users {W c,1 ,W c,2 ,...,W c,K} are jointly encoded into the common data stream s c (l); the private messages of all users {W p,1 ,W p,2 ,...,W p,K} are separately encoded into the private data streams {s1(l), s2(l),..., s K (l)}. The final transmitted message can be expressed as: Among them represents a complex matrix of size (K + 1)×1. The beamforming precoder at the transmitter is denoted by P The precoder corresponds to the transmitted data stream. Among them, p c represents the precoder corresponding to the common data stream, and p1, p2,..., p K respectively represent the precoders corresponding to the private data streams of each user. The following assumptions are made: 1) The data streams are independent of each other, that is, the data streams satisfy s(l)s H (l) = I K+1 , where I K+1 represents the (K + 1)-dimensional identity matrix, and (·) H represents the conjugate transpose. 2) Let the antenna power of the system be P t , and the transmission power of the signal needs to be less than the antenna power where ||·|| represents the corresponding Euclidean norm; Therefore, for the l-th time slot, the transmitted signal of the base station is expressed as: The received signal of user k is: where represents the channel gain, represents the communication system noise, represents a circularly symmetric complex Gaussian random distribution with a mean of 0 and a variance of At the receiver, user k first decodes the common data stream, and treats the remaining private data streams and radar data as interference. Thus, the signal-to-interference-plus-noise ratio (SINR) for user k to decode the common data stream is as follows: When user k decodes the private data stream, it first removes the already decoded common data stream through the successive interference cancellation (SIC) technique. Thus, the signal-to-interference-plus-noise ratio (SINR) for user k to decode the private data stream is: The public achievable rate \(R\) of user \(k\) is obtained using the Shannon formula c,k and the private achievable rate \(R\) k as follows: R c,k = log2(1 + γ c,k ) (5) R k = log2(1 + γ k ) (6) Since all users need to successfully decode the common data stream, the actual common rate under the entire RSMA regime is: R c = min{R c,k} (7) Using C k to represent the actual public rate of user k, then the public rate of each user needs to satisfy: The final achievable rate of user k is: R rate,k = C k + R k (9).

4. The beamforming precoder design method for the RSMA-assisted ISAC system under imperfect channels according to claim 1, wherein The specific channel model constructed in Step 1 is as follows: In an actual communication system, the base station (BS) faces many challenges in obtaining perfect channel state information (CSI) at the transmitter. These challenges stem from uncertainties such as channel estimation error, feedback delay, and quantization error. On the other hand, if the user receivers are properly trained, perfect CSI can be obtained at the receivers, and its estimation error can be comparable to additive noise. Here, consider a channel model in which the CSI at the transmitter (CSIT) is imperfect, while the CSI at the receiver (CSIR) is regarded as perfect. Assume that users are uniformly distributed within 50m - 200m near the BS; First, for user k, the channel gain of the communication can be represented by h k as follows: h k = L(d k ) 1 / 2 · g k (10) where \(L(d\) k ) represents the impact of large-scale fading, manifested as path loss \(L(d\) k ) = 32.6 + 36.7 log 10 (d k ), where \(d\) k represents the distance from user \(k\) to the BS; represents the impact of small-scale fading, which follows a complex Gaussian distribution, \(g\) k ~ CN(0, 1); Considering the ellipsoidal bounded uncertain channel model, the true channel can be obtained from the estimated channel and the estimation error. The specific channel model can be expressed as: wherein represents the estimated channel represents the estimation error of the channel, ∈ k represents the upper bound magnitude of the channel estimation error 5. The beamforming precoder design method for the RSMA-assisted ISAC system under imperfect channels according to claim 1, characterized in that, The specific sensing model constructed in Step 1 is as follows: Using the CRB as a radar sensing metric, the CRB provides a theoretical lower bound for the variances of unbiased estimators of target parameters such as range, velocity, and angle, thus becoming a fundamental criterion for evaluating the accuracy and resolution of radar systems. By obtaining as small a CRB as possible, the ability of the system to accurately estimate target characteristics is enhanced, ensuring robust and reliable sensing performance. Assume that the number of receiving antennas at the BS is N r , if the target response matrix G(θ) is given, the target is sensed through the transmitted signal x(l), and the reflected echo signal y r (l) can be expressed as: y r y(l) = G(θ)x(l) + n r y(l) (12) where n r (l) represents additive white Gaussian noise (AWGN) with a variance of The target response matrix can be expressed as: G(θ) = αb(θ)a H (θ) = αA(θ) (13) where α represents the complex reflection coefficient of the radar, which is related to the target scattering cross-section and the round-trip path loss. a(θ) and b(θ) represent the transmit array steering vector and the receive array steering vector, which are determined by the antenna element spacing d, the transmission signal wavelength λ, and the angle θ. They can be expressed as: And the CRB of the angle θ can be expressed by Equation (16): Among them, represents the derivative of A(θ), represents the covariance matrix of the signal, (·) -1 represents the inverse operation, (·) T represents the transpose operation, represents obtaining the real part, and Tr(·) represents the trace operation.

6. The beamforming precoder design method for the RSMA-assisted ISAC system under ICSI according to claim 1, wherein The optimization problem model established in Step 2 can be formulated as: To ensure fairness for users, the concept of max-min fairness is introduced. By maximizing the minimum user rate, the quality of service (QoS) of all users is guaranteed. Under the constraints of limited resources, uncertain channels, and the need to meet the communication rate and radar sensing estimation Cramer-Rao bound (CRB), an optimization problem model for communication-aware resource allocation is established with the goal of maximizing the minimum user rate, and a beamforming precoder design scheme based on the RSMA-ISAC system is developed. It is as follows: where \(c = [C_1,\ldots,C k ,\ldots,C K \) represents the common rate magnitude of each user; the objective function represents the minimum rate among all users; constraint \(C1\) is the common rate constraint required by RSMA; constraint \(C2\) is the limit on the common rate magnitude for each user, ensuring that the final common message can be successfully decoded by all users; constraint \(C3\) is the constant modulus constraint of the antenna, which restricts the dynamic range of the waveform amplitude, aiming to make the radar non-linear amplifier work at the maximum efficiency state (saturated or near-saturated state) to avoid non-linear distortion of the output waveform; constraint \(C4\) is the CRB constraint of the radar, ensuring the performance of the radar subsystem. The smaller the value of \(\rho\), the more accurate the parameter estimation; constraint \(C5\) represents the uncertainty set of the channel.

7. The beamforming precoder design method for the RSMA-assisted ISAC system under imperfect channels according to claim 1, characterized in that Step 3 adopts the AO principle combined with the WMMSE framework to solve the max-min user fairness rate optimization problem and obtain the best system utility in the worst case. The steps are as follows: 1) Problem The minimum value function in it will affect the convexity of the overall problem. By introducing a variable R t to replace this function, which is used to represent the minimum rate. At this time, the transformed objective function will become an affine function, and at the same time, a new constraint will be introduced: the minimum rate R t should be less than the rate of each user to ensure that it is the minimum, that is: R t = min(C k + R k ) (21) 2) Convert the rate function into the MMSE form through the WMMSE framework, and introduce the linear receiver \(g\) at user \(k\). c,k and \(g\) k respectively correspond to the common stream and the private stream. During the reception of the private stream, the influence of the common stream is eliminated by SIC. Taking the common stream as an example, the mean square error (MSE) between its estimated symbol and the true symbol is: of the optimal MMSE receiver can be obtained by solving to get: Using the optimal MMSE receiver The optimal received MSE can be expressed by Equation (24): Therefore, the rate function can be expressed in the form of optimal MMSE reception: By introducing a weight variable w c,k , the MMSE is rewritten into the WMMSE framework, and the received WMSE can be expressed by Equation (26): And the optimal weight variable w c,k can be obtained by solving as follows: Therefore, the rate function can be transformed into the representation form of the WMMSE framework: The derivation of the private rate is exactly the same as that of the public rate. Thus, the public achievable rate and the private achievable rate in the worst case can be respectively expressed through the WMMSE framework as: Therefore, the original problem is formulated as another optimization problem under the WMMSE framework. Among them 3) Fix \(P\), optimize \(g\). According to the Schur complement algorithm and its extended forms, since the infinite number of uncertain channel sets introduced by the ICSIT condition can be converted into the form of Linear Matrix Inequalities (LMIs), processing of the common data stream: Let where The MMSE problem under uncertainty is expressed as: By adding new constraints and applying the Schur complement algorithm, this constraint is expressed in the form of LMIs. Therefore, the MMSE problem under uncertainty can be expressed as: By a generalized form of the Schur complement algorithm, let C = -[10], ∈ = ∈ k , since A ≥ B is satisfied H DC+(B H DC) H , the condition ||D|| ≤ ∈ holds, so (33) can be transformed into an SDP problem: where 0 represents the zero matrix, and △ c,k is a newly introduced variable, and this variable needs to satisfy the condition △ c,k ≥ 0; Similarly, the private data stream is expressed as an MMSE problem under uncertainty: where △ k ≥ 0, and these two problems can be quickly solved by the interior point method; 4) Fix g, optimize P, and fix the problem Fix the variable g in the problem, optimize the beamforming matrix P, because the problem The expression of CRB(θ) in constraint C4 is non-convex with respect to the variable P, so it needs to be relaxed; first, transform constraint C4 into equation (36): Applying the Schur complement algorithm, it is transformed into the form of LMIs: Bring the constraint C5 in the problem into (34) and (35), then the problem is updated to the problem Then it can be expressed as: Among them, the constraint C5 is a newly added constraint, which is the relaxation form of the equation R x = PP H . For the optimization problem with ξ c,k (g c,k , w c,k ) and ξ k (g k , w k ), first solve (34) and (35) to obtain the corresponding g c,k , w c,k , g k , w k , and then substitute them into (26) to get the result of WMMSE; In summary, first fix the receivers g and w, and use the WMMSE framework combined with the Schur complement algorithm to calculate the beamforming matrix P; then fix P and solve (34) and (35) to calculate the optimal MMSE reception g and w.

8. The beamforming precoder design method for the RSMA-assisted ISAC system under ICSI according to claim 1, characterized in that Step 4 adopts an algorithm combined with the S-procedure algorithm, SDR algorithm, PFD algorithm, SCA algorithm, etc. to solve the optimization problem in the joint optimization model and obtain the optimal system utility in general cases. The process is as follows: 1) Using the SDR algorithm, let where rank(P c ) = 1, rank(P k ) = 1, the covariance matrix of the signal is expressed as Therefore, the common rate and the private rate are respectively: Define variable a k , b k , c k , d k represent the upper and lower bounds of SINR in the rate function, and the specific relationship is as follows: Therefore, the lower bounds of the public rate and the private rate can be obtained: 2) Since (41) and (42) are the difference between two logarithmic functions, it leads to the problem of non-convexity. To solve this problem using the SCA algorithm, at the m-th iteration, the first-order Taylor series is used as the surrogate function: Among them and respectively represent the results of b k and d k in the (m - 1)-th iteration; 3) Convert the uncertainty problem into a deterministic problem through the S-procedure algorithm. First, rewrite the upper and lower bound relationships of SINR into the following form: Using the S-procedure algorithm, introduce the slack variable u a,k ≥0, u b,k ≥0, u c,k ≥0, u d,k ≥0, and rewrite it in the form of LMIs: Optimization problem can be reformulated as Among them 4) Adopt the PFD algorithm to relax the rank-one structure and improve it on this basis to construct a two-layer penalty iterative structure. The internal iteration optimizes the problem under a fixed penalty factor, and the external iteration updates the size of the penalty factor to strengthen the penalty intensity. The specific content is as follows: First, the rank-one constraint is equivalent to the equations Tr(P c ) = ||P c || and Tr(P k ) = ||P k ||. Therefore, the overall penalty function term is: where η is the penalty factor, which reflects the influence degree of the penalty term on the overall optimization function objective. When η is small, the penalty term in the final solution can better satisfy the imposed constraints, but the cost is that the objective function will be severely affected by the penalty. Therefore, gradually increase the penalty intensity during the optimization process. The update of the penalty factor at the nth iteration is as follows: η n = λ·η n-1 (55) Since the 2-norm in the penalty function term makes the whole expression non-convex, the first-order Taylor series is used to represent its lower bound: Among them represents the eigenvector corresponding to the largest eigenvalue in at the $n$-th outer iteration. Therefore, (54) can be expressed as: After adding a penalty term, the optimization problem is expressed as: Combine the S-procedure algorithm, SDR algorithm, PFD algorithm, and SCA algorithm to transform the original problem into an easily handled convex optimization problem. (58) can be efficiently solved by the interior point method. Specifically, directly obtain the result through the CVX tool, so as to obtain the optimal system utility of the RSMA-ISAC system in general cases.

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