An ISAC waveform design method based on Wasserstein distribution robustness and CVaR risk constraint

Abstract

This invention discloses an ISAC waveform design method based on Wasserstein split-bar and CVaR risk constraints. First, an empirical probability distribution is constructed based on historical channel observation samples. Using Wasserstein distance as a metric, an uncertainty set of the channel statistical distribution is constructed within the neighborhood of the empirical distribution. Then, by introducing conditional risk value as a risk metric, the tail distribution of the interference index is constrained to suppress low-probability but high-impact extreme interference events. Based on this, a split-bar waveform optimization model that simultaneously considers sensing performance and communication risk constraints is constructed. Through equivalent reconstruction of distribution uncertainty and risk constraints, the original stochastic optimization problem is transformed into a solvable deterministic convex optimization problem. The problem is decomposed and solved by introducing variable splitting and alternating direction multiplier methods, ultimately obtaining the ISAC transmit waveform that satisfies the total power and peak power constraints.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication and signal processing technology, specifically relating to a sensing-integrated waveform design method based on Wasserstein squaring bars and Conditional Value-at-Risk (CVaR) risk constraints. It is mainly applied to integrated sensing and communication (ISAC) systems in sixth-generation mobile communication (6G), and is suitable for application scenarios with incomplete channel statistics, unknown environmental distribution characteristics, and complex engineering constraints, for optimizing the design of base station transmission waveforms. Background Technology

[0002] With the deepening research into sixth-generation mobile communication (6G), Integrated Sensing and Communication (ISAC) is widely regarded as one of its key enabling technologies. ISAC systems, by sharing spectrum, hardware, and waveform resources, simultaneously achieve wireless communication and environmental sensing functions within the same system architecture, helping to improve spectrum utilization efficiency and hardware resource utilization. In ISAC system design, the optimized design of the transmit waveform is one of the core issues, as its performance directly affects the comprehensive trade-off between communication and sensing functions.

[0003] However, existing ISAC waveform design methods still face several challenges in practical applications. First, in real-world wireless environments, it is difficult for the transmitter to obtain accurate Channel State Information (CSI). Limited by factors such as channel estimation errors, feedback delays, quantization noise, and time-varying environmental characteristics, the actual CSI obtained often deviates from the true channel. When the actual channel statistical characteristics are inconsistent with the model assumed during the design phase, waveforms designed based on nominal channel information cannot guarantee stable communication performance.

[0004] To address CSI uncertainty, existing research typically employs robust optimization (RO) methods. A common approach is uncertainty set modeling based on worst-case criteria, which assumes the channel error lies within a predefined bounded set. While this method improves the system's adaptability to uncertainty to some extent, it fails to fully utilize the statistical distribution information inherent in historical observation data. It often requires design for extremely unfavorable scenarios, leading to overly conservative system performance and limited resource utilization efficiency in most practical applications.

[0005] Furthermore, existing waveform design methods primarily focus on optimizing average communication performance metrics, while paying insufficient attention to tail risks in communication performance fluctuations. In ISAC applications such as autonomous driving and the Industrial Internet, where high communication performance stability is required, occasional extreme interference under random channel conditions can adversely affect system operation. However, existing methods rarely incorporate risk metrics such as Conditional Value at Risk (CVaR) to characterize and constrain the tail distribution characteristics of communication performance, making it difficult to effectively control extreme interference during the design phase.

[0006] On the other hand, from an engineering implementation perspective, the transmitted waveform typically needs to meet hardware constraints such as the peak-to-average power ratio (PAPR) to prevent the RF power amplifier from entering the nonlinear operating range and causing signal distortion. However, the PAPR constraint has significant non-convex characteristics and is highly coupled with the semidefinite programming objective involved in the subbulb bar optimization, making it difficult to solve directly using traditional convex optimization methods. Related engineering constraints are often simplified or ignored in existing research.

[0007] In summary, existing technologies struggle to simultaneously address multiple factors within a unified framework, including channel statistical uncertainty, communication performance tail fluctuations, and practical hardware constraints. Therefore, it is necessary to propose an integrated sensing waveform design method that can characterize channel distribution uncertainty based on historical data, introduce risk constraints to control communication performance fluctuations, and ensure engineering feasibility. Summary of the Invention

[0008] This invention aims to address the problems in existing Integrated Sensing and Communication (ISAC) waveform design techniques, such as incomplete channel statistics, overly conservative robust design, insufficient control of communication performance fluctuations, and difficulty in balancing hardware nonlinear constraints. To this end, an ISAC waveform design method based on Wasserstein sub-barrel and CVaR risk constraints is proposed.

[0009] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0010] An ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints includes the following steps:

[0011] S1. Hybrid channel modeling and data-driven construction of channel distribution uncertainty sets;

[0012] S2. Optimization modeling of ISAC waveforms with sub-Bruker bars, incorporating CVaR risk constraints;

[0013] S3. Convex reconstruction and computability of optimization problems;

[0014] S4. Two-stage iterative solution based on the alternating direction multiplier method;

[0015] S5, Output the optimized waveform.

[0016] As a preferred technical solution of the present invention, step S1 is specifically as follows:

[0017] Step S1 is as follows:

[0018] S11. Constructing a signal transmission and hybrid channel model:

[0019] The signal transmission model of the downlink multiple-input multiple-output (MIMO) sensing integrated system can be represented as follows:

[0020] ;

[0021] in, For the first Individual users The received signal row vector in each time slot For base station to the downlink channel vectors for each user, This is a dual-function waveform matrix transmitted by the base station, where N represents the number of antennas. For communication frame length, For conjugate transpose, H denotes conjugate transpose. This is the large-scale fading vector. This is a small-scale fast fading vector. It is additive Gaussian noise. Represents the field of complex numbers. Indicates the noise variance. Represents the identity matrix;

[0022] Furthermore, a hybrid channel modeling strategy is adopted to decouple the physical channel into deterministic large-scale fading components and small-scale fast fading components that follow an unknown distribution.

[0023] S12. Construct a data-driven set of channel distribution uncertainties:

[0024] Based on historical channel observation samples, statistical modeling is performed on small-scale fast fading components to construct corresponding empirical probability distributions. Using Wasserstein distance as a metric, an uncertainty set of the channel statistical distribution is constructed within the neighborhood of these empirical probability distributions to characterize the uncertainty caused by channel distribution modeling errors.

[0025] To characterize distributional uncertainty, a fuzzy set of probability distributions based on Wasserstein distance is introduced, using empirical distributions as an example. Centered on, with radius as Constructing a second-order Wasserstein fuzzy set is as follows:

[0026] ;

[0027] in, For the first A fuzzy set of small-scale fading distributions for individual users. It is the first An empirical distribution was constructed from M sets of historical samples for each user. Let be the probability distribution of the channel random variable. The support set for small-scale fading vectors. Indicates that it is defined in The set of all probability distributions on, Let the radius of the fuzzy set be . It is the second-order Wasserstein distance.

[0028] As a preferred technical solution of the present invention: in step S11, the hybrid channel modeling strategy includes:

[0029] The physical channel in the integrated sensing system is represented as the product of a large-scale fading component and a small-scale fast fading component. The large-scale fading component is used to characterize the path loss and shadowing fading effect related to the user's location, while the small-scale fast fading component is used to characterize the random channel changes caused by multipath propagation. It is assumed that the small-scale fast fading component follows an unknown probability distribution.

[0030] As a preferred technical solution of the present invention: In step S12, constructing the corresponding empirical probability distribution based on historical channel observation samples includes:

[0031] For each user, corresponding channel observation samples are obtained in multiple time slots or multiple measurements. The channel observation samples are normalized according to the hybrid channel model to remove the large-scale fading component, thereby obtaining a small-scale fast fading sample set. An empirical probability distribution is constructed based on the small-scale fast fading sample set.

[0032] As a preferred technical solution of the present invention, step S2 is specifically as follows:

[0033] Using the dual-function waveform matrix transmitted by the base station as the optimization variable, a sensing error index for characterizing sensing performance and a communication interference index for characterizing communication performance are defined respectively. In view of the possible fluctuations in communication performance under random channel conditions, conditional value of risk is introduced as a risk measurement method to constrain the tail distribution of the communication interference index in the uncertainty set of channel distribution.

[0034] Given a risk confidence level Then the first The Value at Risk (CVaR) for communication interference conditions for an individual user can be defined as:

[0035] ;

[0036] in, Represents the loss function. It is the auxiliary threshold variable in the CVaR definition. Representing the real number field, corresponding to the optimized representation of VaR quantiles, X is the transmitted waveform matrix. Indicates the confidence level;

[0037] Under the conditions of satisfying the total transmit power constraint and the peak-to-average power ratio constraint, we construct a sub-Blule waveform optimization problem that simultaneously considers the constraints of sensing performance and communication risk.

[0038] Combining the sensing error index and the communication interference risk measurement, the Bruker waveform optimization problem is expressed as follows:

[0039] ;

[0040] in, For the communication-sensing tradeoff coefficient, To normalize the perception error, Represents the ideal perceived waveform. Denotes the Frobenius norm. For the first Normalized loss function for each user, For power, For the transmitted waveform matrix X, the () ) elements, N represents the number of antennas, L represents the communication frame length, and L is the peak power threshold. PAPR threshold parameter, This represents the communication-sensing tradeoff factor. Represents the Wasserstein fuzzy set. Let represent the row vector of the expected signal for the k-th user.

[0041] As a preferred embodiment of the present invention, the conditional risk value constraint on the tail distribution of communication interference indices within the channel distribution uncertainty set includes:

[0042] The communication interference index is selected as the risk measurement object. The tail risk ratio parameter is set and the corresponding auxiliary variable is introduced. The tail distribution of the communication interference index within the uncertainty set of the channel statistical distribution is constrained.

[0043] As a preferred technical solution of the present invention, step S3 is as follows:

[0044] Using the strong duality theory of Wasserstein biplastic bar optimization, the stochastic optimization problem containing distributional uncertainty is equivalently transformed into a deterministic convex optimization problem, and the nonlinear constraints are reconstructed into a semidefinite programming form using matrix analysis tools.

[0045] As a preferred technical solution of the present invention: transforming a stochastic optimization problem containing distribution uncertainty into a deterministic convex optimization problem includes:

[0046] Based on the strong duality of Wasserstein partial bar optimization, the worst-case expectation or risk measure on the set of distributed uncertainties is transformed into a deterministic constraint with dual variables. The deterministic constraint is then equivalently reconstructed through matrix inequalities, thus obtaining a convex optimization problem solvable by semidefinite programming.

[0047] As a preferred technical solution of the present invention, step S4 is specifically as follows:

[0048] To address the non-convexity introduced by the peak-to-average power ratio constraint, the variable splitting and alternating direction multiplier method is used to decompose and solve the non-convexity introduced by the peak-to-average power ratio constraint. By alternately updating the waveform variables, splitting variables, and dual variables, the ISAC transmission waveform that simultaneously satisfies the total transmit power constraint and the peak-to-average power ratio constraint is obtained.

[0049] As a preferred technical solution of the present invention, the nonconvexity introduced by the peak-to-average power ratio constraint is decomposed and solved using the variable component and alternating direction multiplier method, including:

[0050] By splitting variables, the peak-to-average power ratio constraint is decoupled from the convex optimization subproblem, and the original variables, split variables, and dual variables are updated alternately. The update of the split variables is a projection operation on the feasible set that simultaneously satisfies the total transmit power not exceeding a preset power threshold and the amplitude of a single sample not exceeding a preset peak threshold, thereby obtaining the ISAC transmit waveform that satisfies the total power constraint and the peak-to-average power ratio constraint.

[0051] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0052] This invention enables the sub-bar optimization design of the ISAC transmit waveform without the need for an accurate prior channel distribution model. It maintains high average communication performance while suppressing tail fluctuations in communication performance, and takes into account both sensing accuracy and engineering feasibility, thereby improving the performance stability and practicality of the integrated sensing system in complex and dynamic wireless environments. Attached Figure Description

[0053] Figure 1 This is a flowchart of the method of the present invention;

[0054] Figure 2 A diagram of a MIMO-ISAC system model;

[0055] Figure 3 This is a schematic diagram of a dual-function transmit signal matrix;

[0056] Figure 4 Simulation Experiment Result 1: Comparison of average reachability and rate under different CSI errors;

[0057] Figure 5 Simulation Experiment Result 2: Comparison of CVaR Interference Risk under Different CSI Errors. Detailed Implementation

[0058] The present invention will be further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that the following specific embodiments are for illustrative purposes only and are not intended to limit the scope of the invention.

[0059] This invention proposes an ISAC waveform design method based on Wasserstein subbulb bars and CVaR risk constraints, aiming to improve the communication reliability and perception robustness of ISAC systems under conditions of incomplete channel statistics and extreme interference events. The specific technical solution is as follows:

[0060] Step S1: Hybrid channel modeling and data-driven channel distribution uncertainty set construction.

[0061] This step is used to construct a hybrid channel model for the ISAC system and establish an uncertainty set of channel statistical distribution based on historical observation data, providing a search space for subsequent sub-blob waveform optimization.

[0062] First, establish the signal transmission model of the ISAC system, such as Figure 2 As shown. Consider the following multi-input multi-output (MIMO) sensor system:

[0063] The base station is configured with One antenna, used to simultaneously direct to Each single-antenna communication user transmits communication signals and performs environmental sensing of targets of interest. The base station's transmitting antenna adopts a uniform linear array (ULA) structure, and the communication and sensing functions share the same transmitting and receiving hardware resources.

[0064] Suppose the dual-function waveform matrix transmitted by the base station is:

[0065] (1);

[0066] in Indicates the number of base station antennas. Indicates the waveform length; let This represents the expected communication symbol matrix, whose elements have unit average power. Transmitted waveform matrix. Structural diagram as follows Figure 3 As shown, the base station transmits signals both to transmit communication symbols to various communication users and to form a detection beam pointing towards the sensing target.

[0067] To characterize the non-stationary characteristics of real wireless channels, a hybrid channel modeling strategy is adopted to decouple the downlink communication channel into large-scale fading components and small-scale fast fading components.

[0068] Specifically, let the first The downlink channel vector for each user is:

[0069] (2);

[0070] in, This represents the large-scale fading coefficient, used to characterize path loss and shadow fading effects; This represents the small-scale fast fading vector, used to characterize random channel changes caused by multipath propagation. Since user locations change slowly over short timescales, large-scale fading coefficients can be obtained through long-term average measurements and are considered known deterministic parameters; however, small-scale fast fading components are affected by environmental complexity, and their statistical distribution is difficult to accurately characterize using simple parametric models.

[0071] Based on this, historical channel observation data accumulated by base stations across multiple time slots or measurements is preprocessed to remove the influence of large-scale fading components, thereby obtaining a normalized sample set containing only small-scale fast fading characteristics. This processing improves the statistical stability of the samples, making them more suitable for subsequent distribution modeling.

[0072] Furthermore, based on a normalized set of small-scale fast fading samples, a corresponding empirical probability distribution is constructed. Using this empirical distribution as the center, the uncertainty set of the channel statistical distribution is constructed using the Wasserstein distance. This uncertainty set describes the possible offset range of the true channel distribution under finite historical sample conditions, thus providing a data-driven search domain for subsequent sub-Bruker waveform optimization without requiring a pre-defined specific distribution model.

[0073] Step S2: Introduce CVaR risk constraints for the sub-bar ISAC waveform optimization model.

[0074] Based on the channel distribution uncertainty set constructed in step S1, this step is used to establish an ISAC waveform optimization model that simultaneously considers the stability of sensing performance and communication performance.

[0075] Dual-function waveform matrix transmitted by base station As optimization variables, a perception error index is defined to characterize perception performance, and a communication interference index is defined to characterize communication performance. The perception error index describes the degree of matching of the transmitted waveform to the target perception direction, while the communication interference index describes the degree of deviation of the transmitted waveform from the expected communication symbol in a multi-user communication scenario.

[0076] To address potential performance fluctuations in communication links under random channel conditions, Conditional Value-at-Risk (CVaR) is introduced as a risk metric to constrain the tail distribution of communication interference indicators within the uncertainty set of channel distribution. This risk constraint can suppress low-probability but high-impact extreme interference scenarios, thereby improving the stability of communication performance in uncertain channel environments.

[0077] Based on this, a sub-Blu-rod ISAC waveform optimization model is constructed. Its goal is to comprehensively optimize the sensing error index and communication interference risk measurement within the channel distribution uncertainty set, under the premise of satisfying the total transmit power constraint and peak-to-average power ratio constraint, so as to achieve a trade-off between sensing performance and communication performance stability.

[0078] The optimization model is formally represented as a stochastic optimization problem with uncertainties in channel distribution. Communication interference risk is characterized by CVaR, and sensing performance is measured by corresponding error indicators, providing a foundation for subsequent problem reconstruction and solution.

[0079] The mathematical implementation of step S2:

[0080] The following section, using a specific mathematical model, provides a feasible mathematical implementation for "optimization modeling of the ISAC waveform of the sub-blob bar with CVaR risk constraints" in step S2. This embodiment is only used to illustrate the technical idea and implementation path of the present invention and does not constitute a limitation on the scope of protection of the present invention. Those skilled in the art can make equivalent substitutions or adjustments to the relevant mathematical forms, performance indicators, or weight settings without departing from the core idea of ​​the present invention, and all such substitutions or adjustments should fall within the scope of protection of the present invention.

[0081] 2.1 Mathematical definition of perceived performance index.

[0082] To characterize the performance of the transmitted waveform in the sensing dimension, a normalized sensing error metric is introduced:

[0083] (3);

[0084] in Denotes the Frobenius norm. This is a pre-constructed ideal reference waveform designed to meet the given sensing mission requirements and the same power constraints. This metric measures the deviation between the current transmitted waveform and the ideal sensing waveform.

[0085] 2.2 Mathematical definition of communication interference index;

[0086] In multi-user communication scenarios, a communication interference index is introduced to measure the impact of transmitted waveforms on communication performance.

[0087] No. The received signal of a single communication user can be represented as:

[0088] (4);

[0089] in, For the first Individual users The received signal row vector in each time slot For base station to the downlink channel vectors for each user, For the dual-function waveform matrix transmitted by the base station, For communication frame length, This is the conjugate transpose. This is the large-scale fading vector. This is a small-scale fast fading vector. It is additive Gaussian noise.

[0090] Definition of the first The normalized interference ratio (INR) for each user is:

[0091] (5);

[0092] By summarizing the multi-user communication interference, we can obtain the overall communication interference index:

[0093] (6);

[0094] 2.3 Normalization processing and empirical distribution construction of historical channel data;

[0095] In practical ISAC systems, the true statistical distribution of small-scale channel fast fading is usually unknown and difficult to accurately characterize using finite parameter models. Therefore, this invention utilizes historical channel observation data available during system operation to perform data-driven modeling of the uncertainty of small-scale fast fading.

[0096] Specifically, assuming that a total of [number] data points were obtained during the historical observation phase... users Group channel observation samples The corresponding large-scale fading coefficient This can be obtained through long-term averaging measurements. To eliminate the impact of large-scale fading, historical channel samples are normalized to obtain pure small-scale fast fading samples:

[0097] (7);

[0098] After normalization, the sample set It primarily reflects the multipath scattering characteristics of the channel, exhibits better statistical stationarity, and is suitable for subsequent sub-Bruker bar modeling.

[0099] Based on the above normalized samples, construct the first... Small-scale fast fading empirical distribution for individual users:

[0100] (8);

[0101] in Indicates in The Dirac measure at that location.

[0102] 2.4 Construction of Wasserstein fuzzy sets;

[0103] Under the condition of relying only on a limited historical sample, the empirical distribution There is an inevitable deviation between the distribution and the true channel distribution. To characterize this distribution uncertainty, a fuzzy set of probability distributions based on Wasserstein distance is introduced.

[0104] To characterize the uncertainty of the channel distribution for the k-th user, this embodiment uses an empirical distribution. Centered on, with radius as Construct a second-order Wasserstein fuzzy set.

[0105] (9);

[0106] in, For the first A fuzzy set of small-scale fading distributions for individual users. It is the first An empirical distribution was constructed from M sets of historical samples for each user. The probability distribution of the channel random variable. The support set for small-scale fading vectors. Indicates that it is defined in The set of all probability distributions on, Let the radius of the fuzzy set be . It is the second-order Wasserstein distance.

[0107] Under the assumption of a light-tailed distribution, based on the results of the Wasserstein measure set, the probability bound between the empirical distribution and the true distribution can be obtained, thus yielding the theoretical scale of the radius:

[0108] (10);

[0109] in, This represents the number of historical CSI samples for the k-th user, as... As the value increases, the empirical distribution becomes closer to the true distribution, and the radius of the fuzzy set decreases accordingly. The confidence parameter controls the probability level at which the fuzzy set contains the true distribution; An index determined by the distribution dimension. This is a constant scaling factor.

[0110] In practical communication systems, since theoretical constants are usually unknown and the CSI error statistical characteristics differ among different users, this embodiment further combines channel estimation error statistics to calibrate the robust radius. Specifically, it utilizes the CSI error sample of the k-th user:

[0111] (11);

[0112] Calculate its average error energy:

[0113] (12);

[0114] Finally, the robust radius of the k-th user is set as follows:

[0115] (13);

[0116] In numerical experiments, The coefficients can be selected through validation set analysis or sensitivity analysis to achieve a reasonable trade-off between system performance and robustness. The Wasserstein fuzzy set constructed in this way does not require prior assumptions about the specific distribution of the channel and can adaptively characterize statistical uncertainties in complex environments. This fuzzy set will serve as the distribution search space for communication interference risk assessment in subsequent risk modeling.

[0117] 2.5 Introduction of CVaR risk measurement;

[0118] To suppress tail fluctuations in communication interference metrics under random channel conditions, Conditional Value-at-Risk (CVaR) is introduced to measure the risk of communication interference. Given a risk confidence level... (For example, take) ), No. The communication interference CVaR of an individual user can be defined as:

[0119] (14);

[0120] in, Represents the loss function. It is the "auxiliary threshold variable" in the definition of CVaR, corresponding to the optimized representation of VaR (quantiles). (15).

[0121] 2.6. CVaR risk modeling using a multi-bar structure;

[0122] Given that the true distribution of the small-scale fast fading vector is unknown, we only assume that it belongs to the Wasserstein distribution uncertainty set constructed in step S1. Then the partial CVaR of communication interference can be further expressed as:

[0123] (16);

[0124] superscript Representing the expected value and risk measure with respect to the distribution The calculation. This form characterizes the worst-case scenario for communication interference risk across all possible channel distributions.

[0125] 2.7 Optimization of ISAC waveform with split bar;

[0126] Combining the sensing error index and the communication interference risk measure, the optimization problem of the ISAC waveform of the Bloom bar can be expressed as:

[0127] (17);

[0128] in, For the communication-sensing tradeoff coefficient, To normalize the perception error, For the first Normalized loss function for each user, For power, For the waveform matrix X, the ( ) elements, , which is the peak power threshold. This is the PAPR threshold parameter.

[0129] Step S3: Convex reconstruction and computability of the optimization problem (SDP / LMI transformation).

[0130] This step, based on the CVaR waveform optimization model (Problem P1) constructed in step S2, addresses the infinite-dimensional optimization difficulties caused by its distributional uncertainty (Wasserstein fuzzy set) and tail risk metric (CVaR). It utilizes the strong duality of Wasserstein CVaR optimization and the equivalent transformation of the risk metric to transform the "CVaR under the worst distribution" into a finite-dimensional deterministic constraint. Furthermore, for the quadratic supremum problem concerning the channel perturbation variable that arises after the transformation, it reconstructs the LMI using the quadratic matrix criterion method and Schur's complement lemma. Finally, Problem P1 is equivalently reconstructed into a convex optimization substructure containing the LMI constraint (denoted as Problem P2). Step S3 yields a convex reconstruction form that is easy to solve. Further considering the PAPR constraint and its implementation, step S4 solves the problem by variable splitting combined with the ADMM algorithm.

[0131] The mathematical implementation of step S3:

[0132] 3.1 Vectorization and Quadratic Form Representation of Physical Models;

[0133] To facilitate the handling of quadratic uncertainties using matrix analysis tools, the signal model is first vectorized. Let...

[0134] (18);

[0135] This represents the vectorized form of the transmitted waveform matrix. Using identities... The k-th user in the th... The received signal in each time slot can be represented as: (19);

[0136] in, It is a selection matrix. It is the first A unit vector with each element equal to 1. Therefore, , Corresponding to waveform matrix The List.

[0137] The normalized interference loss function for the kth user is:

[0138] (20);

[0139] in for The Okay. Rewriting it in a time-slot summation form and substituting it into the vectorized representation, we get: (twenty one);

[0140] Expand the above equation term by term (note the conjugate relationship):

[0141] (twenty two);

[0142] Further organized into information about Hermitian quadratic form function: (twenty three);

[0143] in,

[0144] (twenty four);

[0145] (25);

[0146] (26);

[0147] As can be seen from the structure, It is a Hermitian positive semi-definite matrix, therefore... about It is a convex quadratic function. This representation provides the foundation for the dual reconstruction and constraint transformation of the risk term of the CVaR of the sub-Bruker bar.

[0148] 3.2 Decompose "sup–CVaR" into a finite-dimensional form (introduce relaxation and dual variables);

[0149] From step S2, we can obtain the overall optimization problem (P1), for each user The communication risk item is defined as follows:

[0150] (27);

[0151] Using the variational form of CVaR (Rockafellar–Uryasev form), the above equation can be written as:

[0152] (28);

[0153] Using Sion's minimax theorem, we can interchange sup and inf to obtain:

[0154] (29);

[0155] Further definition:

[0156] (30);

[0157] The above formula can then be written as:

[0158] (31);

[0159] This form transforms the original Blue Bar CVaR risk term into a "threshold variable". The two-layer structure of "one-dimensional optimization + worst-case expectation on Wasserstein fuzzy sets" will be transformed in the next step by applying Wasserstein DRO strong duality theory.

[0160] For fixed and ,definition: (32);

[0161] in, The small-scale fading support set defined in step S2. This represents the deterministic optimization variable introduced by dual reconstruction, used to replace the original random channel vector. .

[0162] Under the support set and integrability assumptions given in step S2, and combined with The lower semicontinuity of the problem can be used to apply the strong duality conclusion of Wasserstein's DRO, which equivalently reconstructs the above infinite-dimensional distributed optimization problem into a finite-dimensional dual form:

[0163] (33);

[0164] in The first one obtained in step S1 Normalized small-scale fast-fading samples for each user (normalized samples). For the Lagrange dual variable corresponding to the Wasserstein distance constraint, it is used to characterize the intensity of the penalty introduced when the probability distribution deviates from the empirical distribution.

[0165] Therefore, the equivalent expression for the CVaR risk term of the k-th user is:

[0166] (34);

[0167] To further eliminate the internal supremum term corresponding to each sample point, for Introducing auxiliary variables Then the above equation can be equivalently rewritten as:

[0168] (35);

[0169] Further substitution The explicit form of the constraint can be obtained as follows:

[0170] (36);

[0171] The above results transform the original infinite-dimensional worst-case distribution expectation term into a finite number of "worst-case function upper bounds" constraints on sample points, laying the foundation for the next step of LMI reconstruction based on quadratic structure and S-procedure.

[0172] 3.3 LMI Restructuring Based on S-Procedure;

[0173] The sample-level constraints obtained in the previous step are (for a fixed user k and sample m):

[0174] (37);

[0175] The constraint still includes information about The internal maximization operation. To obtain a computable form, we will use the following... The quadratic structure is reconstructed into a linear matrix inequality (LMI) constraint.

[0176] For ease of derivation, we adopt the full-space support approximation (i.e., taking the full-space support approximation) consistent with many communication / signal processing papers. ); if the original support set The following reconstruction corresponds to... Relaxed to Thus, a conservative but computable sufficient condition is obtained.

[0177] Equivalent partitioning of maximum constraint: Let , (38);

[0178] but:

[0179] (39);

[0180] Therefore, the original sample-level constraints are equivalent to the following two conditions being met simultaneously:

[0181] (1) Trivial constraint: (corresponding to) Take 0)

[0182] (40);

[0183] when When, the maximum value of the left side of the above equation is 0 (in = (obtained from), therefore obtained .

[0184] (2) Quadratic constraints: (corresponding to) (Take the positive branch)

[0185] (41);

[0186] Equivalently, for any ,have:

[0187] (42);

[0188] Substitute the quadratic form, expand, and rearrange:

[0189] (43);

[0190] Substituting it into the above inequality and expanding the penalty term: − (44);

[0191] It is possible to obtain information about Hermitian quadratic inequality: (45);

[0192] To simplify the notation, the following definition is given:

[0193] (46);

[0194] (47);

[0195] (48);

[0196] The quadratic inequality can then be written as:

[0197] (49);

[0198] Matrix criterion for complex Hermitian quadratic forms: For complex quadratic forms,

[0199] (50);

[0200] (in ),condition: It holds true if and only if its augmented Hermitian matrix satisfies:

[0201] (51);

[0202] Therefore, equation (49) is equivalent to:

[0203] (52);

[0204] Multiplying both sides by -1 can be written in positive semidefinite form:

[0205] (53);

[0206] Using Schur to obtain information about Computable LMI:

[0207] It was observed that equation (53) still contains:

[0208] (54);

[0209] Its against This is a quadratic form. To obtain the linear matrix inequality with respect to the optimization variables, we introduce:

[0210] (55);

[0211] Then we have:

[0212] (56);

[0213] Simultaneously define:

[0214] (57);

[0215] (58);

[0216] Then equation (54) can be written as:

[0217] (59);

[0218] Further utilizing Schur complement (complementing the bottom-right extended unit block), the above equation is equivalent to deriving the following augmented LMI:

[0219] (60);

[0220] Combination and This transforms the sample-level supremum constraint from the previous step into a finite-dimensional positive semi-definite constraint form. This result provides a crucial foundation for the computable reconstruction of the subsequent global optimization problem.

[0221] 3.4. Obtain the solvable SDP form (Problem P2);

[0222] Based on the above equivalent transformations, we introduce a set of decision variables:

[0223] (61);

[0224] (P1) can be refactored into the following optimization problem (P2):

[0225] (62);

[0226] At this point, (P2) can be reconstructed into a convex substructure with SDP / LMI as the core and a non-convex structure constrained by PAPR; step S4 further uses variable splitting and ADMM for solving.

[0227] In order to handle the non-convex feasible region decomposition caused by PAPR constraints in subsequent step S4 using ADMM, the constraints in (P2) are usually split into two categories: ① Convex SDP subproblem part: by This consists of linear matrix inequalities; ② Hardware non-convex constraint part: and (Corresponding to PAPR constraints). Thus, in step S4, variable splitting can be used to... It is split into "SDP solvable variables" and "amplitude projectable variables" and updated alternately.

[0228] Step S4: Two-stage iterative solution based on the alternating direction multiplier method (ADMM).

[0229] This step addresses the contradiction in the convex reconstruction problem (P2) obtained in step S3, namely, that "semi-positive definite constraints can be solved convexly" and "peak-to-average power ratio (PAPR) constraints are difficult to directly incorporate into the same convex framework." It employs the variable splitting and alternating direction multiplier method (ADMM) to decompose the original problem into two easily tractable subproblems:

[0230] ① Global Robust Optimization Subproblem: Under the condition of fixed hardware constraint splitting variables, solve the semidefinite programming (SDP) subproblem containing split robust CVaR objective and LMI constraint, and update the waveform matrix and related robust dual parameters;

[0231] ② Hardware-constrained projection subproblem: Under the condition of fixed waveform variables, project and update the split variables to satisfy the total power constraint and PAPR (equivalent peak amplitude) constraint.

[0232] By alternately updating the original variable, split variable, and dual variable, the iteration converges to a realizable ISAC transmission waveform that satisfies the power and PAPR constraints.

[0233] The mathematical implementation of step S4:

[0234] 4.1. Using (P2) from step S3 as input, perform variable splitting;

[0235] The objective function obtained in step S3 can be simplified as follows: ,in To separate hardware constraints from the main optimization variables, a feasible set of waveforms that satisfy the emission energy and peak value constraints is defined:

[0236] (63);

[0237] Introducing auxiliary variables and make it consistent with the main variable. Through consistency constraints Coupling. The optimization problem (P2) obtained in step S3 can then be rewritten in the following equivalent form:

[0238] (64)

[0239] in, For set Indicator functions:

[0240] (65);

[0241] 4.2 Construct the augmented Lagrangian function;

[0242] Dual variables based on scaling The augmented Lagrangian function is constructed as follows:

[0243] (66);

[0244] in, >0 is the ADMM penalty parameter.

[0245] 4.3 ADMM Iterative Update (Two-stage: Robust Optimization + Hardware Projection);

[0246] In the In the next iteration, updates will be performed in the following three steps:

[0247] ① Global Robust Optimized Update (x-Update):

[0248] exist In the next iteration, the auxiliary variable is fixed. Dual variable with scaling Update waveform variables and the set of reconstructed variables of the blue bar :

[0249] (67);

[0250] in .

[0251] This SDP subproblem can be solved using the interior-point method or a general SDP solver.

[0252] ②Hardware constraint projection update ( -Update);

[0253] fixed and Update auxiliary variables :

[0254] (68);

[0255] make Therefore, the above problem is equivalent to solving:

[0256] (69);

[0257] That is, in the set The problem of finding the shortest distance on the surface. According to step 4.1 in S4... The problem can be defined as follows:

[0258] (70);

[0259] objective function The phase of each component acts only through relative phase, while the constraints act only on the magnitude (energy and peak amplitude). The optimal solution can be taken as... In phase form (when hour): (71)

[0260] when When the phase is chosen arbitrarily, taking any fixed value does not affect the objective function value. Therefore, the problem in the complex field can be reduced to a modulus optimization problem in the real field: (72);

[0261] This problem is a real-variable quadratic optimization problem with single-sphere equality constraints and box constraints. Although the original set is non-convex, its structure allows us to obtain a semi-closed-form update form.

[0262] Introducing Lagrange multipliers Equality constraints By modeling and combining the KKT conditions of the box constraint, the optimal modulus value can be obtained:

[0263] (73);

[0264] The truncation operator is defined as follows: , ride By satisfying the energy equation To determine. Because the left end is about For monotonically changing problems, the bisection method can be used for efficient solutions. After obtaining Then, the auxiliary variable is updated to:

[0265] (74);

[0266] 4.4 Update the scaling dual variable ;

[0267] According to the scaled form of ADMM, the dual variable is updated as follows:

[0268] (75);

[0269] 4.5 Stopping Criteria and Output Waveform;

[0270] When the original residual satisfies:

[0271] (76);

[0272] The iteration terminates when the maximum number of iterations is reached, and the final waveform solution is output. This is a preset threshold. In this embodiment, the output can be... As the transmit waveform that satisfies hardware constraints;

[0273] Step S5: Output the optimized waveform.

[0274] The final output waveform is as follows:

[0275] (77).

[0276] The ADMM algorithm framework proposed in this invention cleverly decouples the complex bibliometric optimization problem from non-convex hardware constraints. Through alternating iterations of global optimization and local projection, it achieves an efficient and stable solution. This design is the core technical support that ensures the entire method is practically feasible in engineering and computationally efficient.

[0277] The four steps described above are interconnected and work synergistically to transform the sub-Bruker optimization and risk control theory into a complete, feasible, and engineering-promising ISAC waveform design scheme. This scheme is suitable for integrated sensing applications with high requirements for communication performance stability and sensing accuracy. Its overall process is as follows: Figure 1 As shown.

[0278] To further illustrate the effectiveness of the method of the present invention in typical application scenarios, a numerical simulation analysis of the proposed waveform design scheme is performed in conjunction with an embodiment.

[0279] The simulation environment is set as follows: In a downlink MIMO-ISAC system, the base station is equipped with an 8-antenna uniform linear array, simultaneously serving 4 single-antenna users and detecting a point target. The transmitted waveform length can be set to... The system satisfies the total transmit power constraint and can be subject to a peak-to-average power ratio (PAPR) constraint, for example, its threshold can be set to 2. The receiver noise power can be set to 0.01. In the sub-Bruker modeling, an empirical distribution can be constructed using several historical channel samples, for example, the number of samples can be set to 50, and a Wasserstein fuzzy set can be constructed based on this, with its radius set, for example, 0.05. To characterize the risk of the system under uncertain channel conditions, the conditional value of risk (CVaR) is introduced as a risk measure, with its confidence level set, for example, 0.95. In the integrated sensing system, the tradeoff between communication performance and sensing performance can be adjusted by a tradeoff coefficient, for example, the tradeoff coefficient can be set to 0.5. The optimization problem can be solved using an algorithm based on the alternating direction multiplier method (ADMM), for example, the penalty parameter can be set to 1, the maximum number of iterations can be set to 200, when both the original residual and the dual residual are less than 1. The algorithm is considered to have converged at this point. In terms of channel modeling, it can be assumed that the channel follows a complex Gaussian-Rayleigh fading model, and an additive error model can be used to describe the imperfect channel state information.

[0280] Experimental results are as follows Figure 4 and Figure 5 As shown. Figure 4 This paper presents a comparison of average reachability and rate under different CSI errors. As the channel estimation error (NMSE) gradually increases from −30dB to 0dB, the system rate of each method generally decreases. This is because the increase in CSI error leads to a deviation between the transmitted beam and the real channel, thereby reducing the signal-to-interference-plus-noise ratio (SNR) of the received signal and affecting the system communication performance. The perfect CSI design, acting as the upper bound of performance, maintains a relatively stable rate, unaffected by changes in channel estimation error. The expectation optimization method based on nominal CSI exhibits high performance when the error is small, but its rate decreases significantly with increasing error, indicating that this method is sensitive to CSI error. The worst-case robust method based on the norm sphere uncertainty set can mitigate the performance degradation caused by error to some extent, but its performance is relatively low in the low-error region due to its conservative design. The sub-Brutal robust method based on the Wasserstein distance to construct fuzzy sets achieves a better balance between robustness and performance. In contrast, the sub-Brutal CVaR method proposed in this embodiment achieves a high average reachability throughout the entire error range, demonstrating better robustness and verifying the effectiveness of the proposed method under imperfect CSI conditions.

[0281] Figure 5This paper compares the interference risk performance of various methods under different channel estimation error levels, using Conditional Value at Risk (CVaR) to measure the tail interference risk of the system under uncertain channel conditions. The overall trend shows that as the channel estimation error (NMSE) gradually increases from −30dB to 0dB, the CVaR interference risk of each method gradually increases. This is because an increase in channel estimation error leads to a deviation between the transmitted beam and the actual channel, resulting in an increase in system interference level and further amplifying the tail risk under extreme channel conditions. Among these methods, the expectation optimization method based on nominal CSI, which does not consider channel uncertainty, exhibits the most significant increase in interference risk and displays a high tail risk when the error is large. The worst-case robust method based on norm sphere uncertainty sets can suppress the increase in interference risk to some extent, but its conservative design strategy still has certain limitations in overall risk control. The fuzzy set-based robust method based on Wasserstein distance achieves a better balance between robustness and system performance by characterizing the uncertainty of channel distribution, thus its overall interference risk is lower than that of traditional worst-case robust methods. In contrast, the sub-Brutal CVaR method proposed in this embodiment exhibits the lowest interference risk level across the entire NMSE range, indicating that this method can more effectively suppress interference risk under extreme channel conditions, thereby improving the robustness of the system in imperfect CSI environments.

[0282] In summary, addressing key issues in ISAC systems such as incomplete channel statistics, unknown environmental distribution characteristics, and hardware nonlinear constraints, this invention proposes a data-driven sub-Brønsted bar CVaR waveform design method. This method constructs a set of channel distribution uncertainties using Wasserstein distance, introduces conditional risk value to characterize communication performance fluctuations, and combines the alternating direction multiplier method to achieve feasible waveform solutions under total power and peak-to-average power ratio constraints, thus achieving a robust trade-off between communication and sensing performance.

[0283] The above embodiments are only used to illustrate the technical solutions and implementation methods of the present invention, and do not constitute a limitation on the scope of protection of the present invention. Any equivalent substitutions, parameter adjustments, or structural changes made by those skilled in the art to the embodiments without departing from the technical concept of the present invention should fall within the scope of protection of the present invention.

Claims

1. A method for designing ISAC waveforms based on Wasserstein split-bars and CVaR risk constraints, characterized in that, Includes the following steps: S1. Hybrid channel modeling and data-driven construction of channel distribution uncertainty sets; S2. Optimization modeling of ISAC waveforms with sub-Bruker bars, incorporating CVaR risk constraints; Step S2 is as follows: Using the dual-function waveform matrix transmitted by the base station as the optimization variable, a sensing error index for characterizing sensing performance and a communication interference index for characterizing communication performance are defined respectively. In view of the possible fluctuations in communication performance under random channel conditions, conditional value of risk is introduced as a risk measurement method to constrain the tail distribution of the communication interference index in the uncertainty set of channel distribution. Given a risk confidence level Then the first The Value at Risk (CVaR) for communication interference conditions per user is defined as: ； in, Represents the loss function. It is the auxiliary threshold variable in the CVaR definition. Representing the real number field, corresponding to the optimized representation of VaR quantiles, X is the transmitted waveform matrix. Indicates the confidence level; Under the conditions of satisfying the total transmit power constraint and the peak-to-average power ratio constraint, we construct a sub-Blule waveform optimization problem that simultaneously considers the constraints of sensing performance and communication risk. Combining the sensing error index and the communication interference risk measurement, the Bruker waveform optimization problem is expressed as follows: ； in, For the communication-sensing tradeoff coefficient, To normalize the perception error, Represents the ideal perceived waveform. Denotes the Frobenius norm. For the first Normalized loss function for each user, This is the large-scale fading vector. Let be the probability distribution of the channel random variable. This is a small-scale fast fading vector. This is the large-scale fading vector. For power, For the transmitted waveform matrix X, the () ) elements, N represents the number of antennas, L represents the communication frame length, and L is the peak power threshold. PAPR threshold parameter, Represents the Wasserstein fuzzy set. This represents the row vector of the expected signal for the k-th user; S3. Convex reconstruction and computability of optimization problems; S4. Two-stage iterative solution based on the alternating direction multiplier method; S5, Output the optimized waveform.

2. The ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints according to claim 1, characterized in that, Step S1 is as follows: S11. Constructing a signal transmission and hybrid channel model: A signal transmission model for a downlink multiple-input multiple-output (MIMO) integrated sensing system is established, represented as follows: ； in, For the first Individual users The received signal row vector in each time slot For base station to the downlink channel vectors for each user, This is a dual-function waveform matrix transmitted by the base station, where N represents the number of antennas. For communication frame length, For conjugate transpose, H denotes conjugate transpose. This is the large-scale fading vector. Let the radius of the fuzzy set be . This is a small-scale fast fading vector. It is additive Gaussian noise. Represents the field of complex numbers. Indicates the noise variance. Represents the identity matrix; Furthermore, a hybrid channel modeling strategy is adopted to decouple the physical channel into deterministic large-scale fading components and small-scale fast fading components that follow an unknown distribution. S12. Construct a data-driven set of channel distribution uncertainties: Based on historical channel observation samples, statistical modeling is performed on small-scale fast fading components to construct corresponding empirical probability distributions. Using Wasserstein distance as a metric, an uncertainty set of the channel statistical distribution is constructed within the neighborhood of these empirical probability distributions to characterize the uncertainty caused by channel distribution modeling errors. To characterize distributional uncertainty, a fuzzy set of probability distributions based on Wasserstein distance is introduced, using empirical distributions as an example. Centered on, with radius as Constructing a second-order Wasserstein fuzzy set is as follows: ； in, For the first A fuzzy set of small-scale fading distributions for individual users. It is the first An empirical distribution was constructed from M sets of historical samples for each user. Let be the probability distribution of the channel random variable. The support set for small-scale fading vectors. Indicates that it is defined in The set of all probability distributions on, Let the radius of the fuzzy set be . It is the second-order Wasserstein distance.

3. The ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints according to claim 2, characterized in that, In step S11, the hybrid channel modeling strategy includes: The physical channel in the integrated sensing system is represented as the product of a large-scale fading component and a small-scale fast fading component. The large-scale fading component is used to characterize the path loss and shadowing fading effect related to the user's location, while the small-scale fast fading component is used to characterize the random channel changes caused by multipath propagation. It is assumed that the small-scale fast fading component follows an unknown probability distribution.

4. The ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints according to claim 2, characterized in that, In step S12, based on historical channel observation samples, the corresponding empirical probability distribution is constructed, including: For each user, corresponding channel observation samples are obtained in multiple time slots or multiple measurements. The channel observation samples are normalized according to the hybrid channel model to remove the large-scale fading component, thereby obtaining a small-scale fast fading sample set. An empirical probability distribution is constructed based on the small-scale fast fading sample set.

5. The ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints according to claim 1, characterized in that, Conditional Value at Risk (VaR) imposes tail distribution constraints on communication interference metrics within the channel distribution uncertainty set, including: The communication interference index is selected as the risk measurement object. The tail risk ratio parameter is set and the corresponding auxiliary variable is introduced. The tail distribution of the communication interference index within the uncertainty set of the channel statistical distribution is constrained.

6. The ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints according to claim 1, characterized in that, Step S3 is as follows: Using the strong duality theory of Wasserstein biplastic bar optimization, the stochastic optimization problem containing distributional uncertainty is equivalently transformed into a deterministic convex optimization problem, and the nonlinear constraints are reconstructed into a semidefinite programming form using matrix analysis tools.

7. The ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints according to claim 6, characterized in that, Transforming stochastic optimization problems involving distributional uncertainty into deterministic convex optimization problems includes: Based on the strong duality of Wasserstein partial bar optimization, the worst-case expectation or risk measure on the set of distributed uncertainties is transformed into a deterministic constraint with dual variables. The deterministic constraint is then equivalently reconstructed through matrix inequalities, thus obtaining a convex optimization problem solvable by semidefinite programming.

8. The ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints according to claim 1, characterized in that, Step S4 is as follows: To address the non-convexity introduced by the peak-to-average power ratio constraint, the variable splitting and alternating direction multiplier method is used to decompose and solve the non-convexity introduced by the peak-to-average power ratio constraint. By alternately updating the waveform variables, splitting variables, and dual variables, the ISAC transmission waveform that simultaneously satisfies the total transmit power constraint and the peak-to-average power ratio constraint is obtained.

9. The ISAC waveform design method based on Wasserstein split bar and CVaR risk constraints according to claim 8, characterized in that, The nonconvexity introduced by the peak-to-average power ratio constraint is decomposed and solved using the variable component and alternating direction multiplier method, including: By splitting variables, the peak-to-average power ratio constraint is decoupled from the convex optimization subproblem, and the original variables, split variables, and dual variables are updated alternately. The update of the split variables is a projection operation on the feasible set that simultaneously satisfies the total transmit power not exceeding a preset power threshold and the amplitude of a single sample not exceeding a preset peak threshold, thereby obtaining the ISAC transmit waveform that satisfies the total power constraint and the peak-to-average power ratio constraint.

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