ISAC MIMO-OFDM joint waveform optimization method based on dynamic range constraint

Abstract

The invention relates to an ISAC MIMO-OFDM joint waveform optimization method based on dynamic range constraint. Firstly, an ideal radar waveform design model is established, waveform design is realized based on a rapid iteration method, the waveform design is taken as an output condition, a communication evaluation index system is further introduced, performance adjustment between an ideal radar function and a communication function is realized in combination with a weight factor, and meanwhile, a communication system needs to work in a linear working area. In order to prevent signal distortion, a dynamic range constraint condition is introduced, and a joint optimization model is further established. In consideration of a multi-constraint problem, an alternating direction multiplier algorithm is introduced to carry out alternating solution, and the local optimal solution and the operation efficiency are balanced. Simulation test data analysis shows that the method can realize adjustment of communication and sensing performance under system power constraint through a weight factor and a dynamic range factor, and has extremely high engineering application value in a fast response scene based on target feedback.

Description

Technical Field

[0001] This invention discloses a joint waveform optimization method for ISAC MIMO-OFDM based on dynamic range constraints, belonging to the field of microwave sensing technology. Background Technology

[0002] Due to the constraints of limited spectrum resources, hardware redundancy, and application-driven limitations in traditional communication and radar systems, previously separate communication and radar functions have been integrated to form a communication-radar hybrid. As an emerging technology, this hybrid aims to achieve the synergistic integration of wireless communication and radar detection functions by sharing hardware and spectrum resources. This technology can significantly improve the operational efficiency of sensing systems, reduce hardware costs, and is applicable to scenarios such as 5G / 6G, intelligent transportation, and the low-altitude economy.

[0003] Waveform design is crucial for the integration of communication and radar. OFDM waveform integration, primarily for communication, has attracted widespread attention due to its more mature technology, high spectrum utilization, and ability to combat inter-symbol interference. Liu et al. explored the coexistence and spectrum sharing between downlink multi-user multiple-input multiple-output (MIMO) communication systems and multiple-input multiple-output radar systems, focusing on how to design transmission beams to maximize radar detection probability while meeting the performance requirements of the downlink communication system. Furthermore, they conducted communication and radar target detection in a dual-function MIMO-ISAC system with multiple antenna transmitters and downlink cellular users. Through shared waveform design, a single platform can implement a dual-function radar-communication (DFRC) system using the same transmit waveform within the same spectrum range. Wu proposed a space-time coding scheme for DFRC transmit beamforming, embedding communication information to avoid interference. Information symbols are embedded into the waveform using constellation mapping and phase rotation methods. Liu proposed dividing the antenna into two groups, deploying radar and communication antennas separately. By optimizing the communication beam, its beam shape is matched with the radar beam shape, while simultaneously meeting the performance requirements of downlink users. He also proposed a joint waveform design method to support sharing between the radar and communication systems, enabling the radar signal to avoid interference from the downlink communication channel. Jiang proposed a linear superposition method that cleverly synthesizes radar and communication waveforms in the far-field region of a uniform linear array, thereby achieving efficient waveform design. He also proposed an alternative projection method to reduce the peak-to-average power ratio (PAPR) of the transmitted waveform.

[0004] However, the waveform designs mentioned above all fall under the category of static waveform design, failing to consider the practical constraints of systems engineering and the collaborative design of communication-radar system performance indicators. This invention designs a distributed solution framework based on the Alternating Direction Multiplier Method (ADMM), which effectively reduces the computational complexity of complex system models and ensures convergence through variable splitting and closed-loop iterative updates. To improve sensing performance, an improved fuzzy function sidelobe suppression criterion is proposed, combined with the finite-memory quasi-Newton method (L-BFGS) to generate high-resolution radar reference waveforms. This provides theoretical support and a technical implementation path for waveform optimization in integrated sensing systems, and has significant reference value for the design of next-generation intelligent wireless networks. Summary of the Invention

[0005] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a joint waveform optimization method for ISAC MIMO-OFDM based on dynamic range constraints, which solves the problem of dynamically adjustable balance design of communication and radar sensing performance under constant mode constraints of communication systems.

[0006] The technical solution of this invention is: a joint waveform optimization method for ISAC MIMO-OFDM based on dynamic range constraints, comprising: Design and construct an ideal radar waveform under Nyquist sampling conditions; Based on the ideal radar waveform obtained from the design, an ideal condition integrated sensing waveform optimization model considering the integrated performance of communication and sensing is established. Based on the waveform optimization model, a nonlinear inductive integrated waveform design model considering constant modulus constraints is established; An oversampling factor is introduced into the nonlinear waveform design model to obtain a nonlinear inductive integrated waveform design model under oversampling conditions; For the oversampled nonlinear inductive integrated waveform design model, a waveform optimization solution method based on the alternating direction multiplier method is adopted to obtain an inductive integrated spatial radiation waveform that takes into account communication, sensing and system power constraints.

[0007] The design constructs an ideal radar waveform under Nyquist sampling conditions, including: From space Directional transmission signal Represented as:

[0008] in For a complete time-domain signal transmission vector, For time-domain vectors considering only the precoding and symbol matrices, It is a spatial guiding vector. To introduce the selection matrix for the cyclic prefix, Indicates a length of N s +N cp The identity matrix, N s N is the number of subcarriers. cp The length of the cyclic prefix; definition for

[0009] but The waveform of directional radiation is used It means, that is

[0010] Further detection angle The fuzzy function is , where J is the delay matrix and D is the normalized Doppler frequency matrix; With the optimization objective of minimizing the integral of the ambiguity function within the finite time delay and Doppler frequency range, the optimization function is obtained as follows:

[0011] In the formula, This represents the set of probe angles of interest. This represents the set of time delay pairs and Doppler frequency pairs of interest; The similarity between the actual radiation pattern and the ideal radiation pattern is expressed as: , in For ideal direction charts; Taking into account both the ambiguity function and the beam pattern, the optimization problem for designing the ideal radar waveform can be expressed as:

[0012] in Represents the set of all angles covered by the beam; The above optimization problem is solved using a finite-memory quasi-Newton method; the optimal solution obtained by the finite-memory quasi-Newton method is denoted as the ideal radar waveform. After energy normalization, it can be expressed as: .

[0013] The process of establishing an ideal condition integrated sensing waveform optimization model based on the ideal radar waveform obtained from the design, considering the integrated performance of communication and sensing, includes: With the objectives of minimizing the MUI and the similarity between the design waveform and the ideal radar waveform, the following objective function is established:

[0014] in, Let H be the symbol matrix for all users, H be the frequency domain channel state matrix corresponding to all subcarriers, and x be the precoding symbol matrix for all users. It is an ideal radar waveform, where F is the Fourier transform matrix. For length unit array, ρ represents the number of antennas, and ρ is the weighting factor. Assume there is a total Each antenna has an independent radio frequency channel; compared to the first... One antenna, define the selection matrix. :

[0015] The constraint on the total energy of all symbols in the time domain is transformed into a constraint on the total energy of the effective symbols:

[0016] in Indicates corresponding to One antenna and Total available energy for a system with OFDM symbols.

[0017] Based on the ideal waveform optimization model, a nonlinear inductive integrated waveform design model considering the constant modulus constraint of the system is established, including: With the total energy fixed, the power of each symbol is placed within a dynamic range, as follows:

[0018] in This represents the energy of each OFDM symbol. This is a dynamic range constraint parameter, which corresponds to the upper limit of fluctuation in the linear power region of a single channel; Establish a nonlinear inductive integrated waveform design model considering the constant modulus constraint of the system:

[0019] The process of introducing an oversampling factor into the nonlinear inductive integrated waveform design model to obtain the nonlinear inductive integrated waveform design model under oversampling conditions includes: Define the selection matrix for oversampling :

[0020] Where κ is the oversampling rate; This leads to the nonlinear inductive integrated waveform design model under oversampling conditions:

[0021] Based on the aforementioned nonlinear inductive integrated waveform design model, combined with The model can be further described as follows:

[0022] in, ; The nonlinear sensing integrated waveform design model under the obtained oversampling conditions employs a waveform optimization solution method based on the alternating direction multiplier method to obtain a spatial radiation waveform that takes into account communication, sensing, and system power constraints, including: Introduce two auxiliary vectors and The nonlinear synesthetic waveform design model under oversampling conditions is transformed into:

[0023] The corresponding augmented Lagrangian function is:

[0024] in and It is a Lagrange multiplier, and η is the penalty factor; The augmented Lagrangian function is solved using the alternating direction multiplier method, specifically as follows: First step, update : When solving When the problem is minimized, it can be expressed by the formula:

[0025] in, It is a selection matrix whose elements are either 0 or 1; selection matrix Select the option corresponding to the first... The elements of each antenna are combined into a new vector; Including the One antenna 1 symbol, of which Indicates the first One symbol; because the constraint covers all and Then the minimization problem is equivalently transformed into

[0026] Further decomposed into The individual problem, among which The subproblem can be further written as

[0027] The solution to the above subproblem is expressed as:

[0028] The value of is updated by minimizing the problem as follows:

[0029] Because the constraints cover all Therefore, the problem is broken down into The issue of size;

[0030] The optimal solution to the above problem is

[0031] In calculating all All Then, the vector The elements are arranged into a vector; because The vectors are contained in a regular form. The vector obtained by rearranging all the elements of is . ; Step 2, update : Through the obtained To solve and update, that is:

[0032] Based on the augmented Lagrangian function, the problem can be specifically represented as:

[0033] The above problem is an unconstrained quadratic optimization problem; assuming the derivative is 0, simplifying the equation, and solving the equation yields the optimal solution as follows:

[0034] in,

[0035]

[0036] but Least squares solution:

[0037] Step 3, Update :

[0038] ; The ADMM iteration process described above, repeated multiple times, yields the solution 's' of the ISAC waveform design model under oversampling conditions. Combined with the cyclic prefix selection matrix, the inductively coupled transmitted waveform radiating into space can then be obtained. .

[0039] The advantages of this invention compared to the prior art are as follows: This invention studies the transmit waveform optimization design problem of an ISAC MIMO-OFDM system under dynamic range constraints to improve the system's detection capability under constant mode and dynamic range constraints, while also achieving certain communication symbol transmission performance. First, by combining multi-user interference (MUI) minimization with radar waveform similarity constraints, a non-convex quadratic constraint optimization model that balances communication service quality and sensing performance is constructed. Furthermore, a sub-antenna dynamic range constraint is introduced to control the peak-to-average power ratio (PAPR), resulting in more stable transmit waveform performance. To address the complexity of model solving, a distributed optimization framework based on the Alternating Directional Multiplier Method (ADMM) is proposed. Through variable splitting and closed-loop iterative updates, the computational complexity is effectively reduced while ensuring convergence. The proposed method achieves a balance between communication and sensing performance under dynamic range constraints, exhibiting excellent performance in radar beam pattern matching and ambiguity function sidelobe suppression. In addition, experiments verify the improvement of hardware compatibility brought by dynamic range constraints, as well as the flexible adjustment capability of different trade-off factors and dynamic range constraint parameters on system performance. This allows for adjustments based on different integrated sensing application scenarios, demonstrating high engineering application value. Attached Figure Description

[0040] Figure 1 This shows the beam pattern under different weighting factors.

[0041] Figure 2 This is the waveform ambiguity function under different weighting factors.

[0042] Figure 3 The waveform ISLR under different weighting factors.

[0043] Figure 4 The waveform symbol error rate is denoted by different weighting factors.

[0044] Figure 5 This represents the total waveform communication rate under different weighting factors.

[0045] Figure 6 The beam pattern is shown under different dynamic range constraint factors.

[0046] Figure 7 This refers to the waveform fuzziness function under different dynamic range constraint factors.

[0047] Figure 8The ISLR waveforms under different dynamic range constraint factors.

[0048] Figure 9 The waveform symbol error rate under different dynamic range constraint factors.

[0049] Figure 10 The total waveform communication rate under different dynamic range constraint factors. Detailed Implementation

[0050] The implementation and effects of the present invention will be described in further detail below.

[0051] This invention proposes a joint waveform optimization method for ISAC MIMO-OFDM based on dynamic range constraints. The method first establishes an optimization problem for an ideal radar waveform and solves it using a fast optimization method. Second, it establishes a joint optimization problem under the constraints of a comprehensive sensing index system, and introduces dynamic range constraints, further extending it to optimization problems under oversampling conditions. Finally, for multivariate coupled and constrained mathematical optimization problems, the ADMM method is used for variable decoupling and alternating solution to obtain a sensing-integrated waveform that meets the set indicators.

[0052] The main implementation steps of this invention are as follows: A joint waveform optimization method for ISAC MIMO-OFDM based on dynamic range constraints includes: Design and construct an ideal radar waveform under Nyquist sampling conditions; Based on the ideal radar waveform obtained from the design, an ideal condition integrated sensing waveform optimization model considering the integrated performance of communication and sensing is established. Based on the waveform optimization model, a nonlinear inductive integrated waveform design model considering constant modulus constraints is established; An oversampling factor is introduced into the nonlinear waveform design model to obtain a nonlinear inductive integrated waveform design model under oversampling conditions; For the oversampled nonlinear inductive integrated waveform design model, a waveform optimization solution method based on the alternating direction multiplier method is adopted to obtain an inductive integrated spatial radiation waveform that takes into account communication, sensing and system power constraints.

[0053] The design constructs an ideal radar waveform under Nyquist sampling conditions, including: From space Directional transmission signal Represented as:

[0054] in For a complete time-domain signal transmission vector, For time-domain vectors considering only the precoding and symbol matrices, It is a spatial guiding vector. To introduce the selection matrix for the cyclic prefix, Indicates a length of N s +N cp The identity matrix, N s N is the number of subcarriers. cp The length of the cyclic prefix; definition for

[0055] but The waveform of directional radiation is used It means, that is

[0056] Further detection angle The fuzzy function is , where J is the delay matrix and D is the normalized Doppler frequency matrix; With the optimization objective of minimizing the integral of the ambiguity function within the finite time delay and Doppler frequency range, the optimization function is obtained as follows:

[0057] In the formula, This represents the set of probe angles of interest. This represents the set of time delay pairs and Doppler frequency pairs of interest; The similarity between the actual radiation pattern and the ideal radiation pattern is expressed as: , in For ideal direction charts; Taking into account both the ambiguity function and the beam pattern, the optimization problem for designing the ideal radar waveform can be expressed as:

[0058] in Represents the set of all angles covered by the beam; The above optimization problem is solved using a finite-memory quasi-Newton method; the optimal solution obtained by the finite-memory quasi-Newton method is denoted as the ideal radar waveform. After energy normalization, it can be expressed as: .

[0059] The process of establishing an ideal condition integrated sensing waveform optimization model based on the ideal radar waveform obtained from the design, considering the integrated performance of communication and sensing, includes: With the objectives of minimizing the MUI and the similarity between the design waveform and the ideal radar waveform, the following objective function is established:

[0060] in, Let H be the symbol matrix for all users, H be the frequency domain channel state matrix corresponding to all subcarriers, and x be the precoding symbol matrix for all users. It is an ideal radar waveform, where F is the Fourier transform matrix. For length unit array, ρ represents the number of antennas, and ρ is the weighting factor. Assume there is a total Each antenna has an independent radio frequency channel; compared to the first... One antenna, define the selection matrix. :

[0061] The constraint on the total energy of all symbols in the time domain is transformed into a constraint on the total energy of the effective symbols:

[0062] in Indicates corresponding to One antenna and Total available energy for a system with OFDM symbols.

[0063] Based on the ideal waveform optimization model, a nonlinear inductive integrated waveform design model considering the constant modulus constraint of the system is established, including: With the total energy fixed, the power of each symbol is placed within a dynamic range, as follows:

[0064] in This represents the energy of each OFDM symbol. This is a dynamic range constraint parameter, which corresponds to the upper limit of fluctuation in the linear power region of a single channel; Establish a nonlinear inductive integrated waveform design model considering the constant modulus constraint of the system:

[0065] The process of introducing an oversampling factor into the nonlinear inductive integrated waveform design model to obtain the nonlinear inductive integrated waveform design model under oversampling conditions includes: Define the selection matrix for oversampling :

[0066] Where κ is the oversampling rate; This leads to the nonlinear inductive integrated waveform design model under oversampling conditions:

[0067] Based on the aforementioned nonlinear inductive integrated waveform design model, combined with The model can be further described as follows:

[0068] in, ; The nonlinear sensing integrated waveform design model under the obtained oversampling conditions employs a waveform optimization solution method based on the alternating direction multiplier method to obtain a spatial radiation waveform that takes into account communication, sensing, and system power constraints, including: Introduce two auxiliary vectors and The nonlinear synesthetic waveform design model under oversampling conditions is transformed into:

[0069] The corresponding augmented Lagrangian function is:

[0070] in and It is a Lagrange multiplier, and η is the penalty factor; The augmented Lagrangian function is solved using the alternating direction multiplier method, specifically as follows: First step, update : When solving When the problem is minimized, it can be expressed by the formula:

[0071] in, It is a selection matrix whose elements are either 0 or 1; selection matrix Select the option corresponding to the first... The elements of each antenna are combined into a new vector; Including the One antenna 1 symbol, of which Indicates the first One symbol; because the constraint covers all and Then the minimization problem is equivalently transformed into

[0072] Further decomposed into The individual problem, among which The subproblem can be further written as

[0073] The solution to the above subproblem is expressed as:

[0074] The value of is updated by minimizing the problem as follows:

[0075] Because the constraints cover all Therefore, the problem is broken down into The issue of size;

[0076] The optimal solution to the above problem is

[0077] In calculating all All Then, the vector The elements are arranged into a vector; because The vectors are contained in a regular form. The vector obtained by rearranging all the elements of is . ; Step 2, update : Through the obtained To solve and update, that is:

[0078] Based on the augmented Lagrangian function, the problem can be specifically represented as:

[0079] The above problem is an unconstrained quadratic optimization problem; assuming the derivative is 0, simplifying the equation, and solving the equation yields the optimal solution as follows:

[0080] in,

[0081]

[0082] but Least squares solution:

[0083] Step 3, Update :

[0084] ; The ADMM iteration process described above, repeated multiple times, yields the solution 's' of the ISAC waveform design model under oversampling conditions. Combined with the cyclic prefix selection matrix, the inductively coupled transmitted waveform radiating into space can then be obtained. .

[0085] The effects of the present invention will be further illustrated below through simulation experiments.

[0086] The ground-based communication sensing system was simulated using the simulation parameters shown in Table 1. The communication channel was assumed to be a Rayleigh fading channel, with its principal component direction aligned with the radar target direction. The channel consisted of four non-zero channel taps, and the system operating frequency was 28 GHz. The basic MIMO-OFDM waveform applied 40 effective OFDM symbols in the time domain. The subcarrier spacing was set to 300 kHz. We assumed the required detection range for the scenario was 400 m, and the maximum speed of the target to be detected was 120 km / h. To obtain satisfactory sidelobes in pulse compression, the distance corresponding to the CP length should be greater than 400 m. Based on calculations, the CP length was set to 32. When designing the ideal radar waveform, we minimized the ambiguity function of the target delay Doppler region. From the sampling frequency perspective, the Doppler frequency introduced by the maximum speed of 120 km / h is negligible. Therefore, for the range-Doppler region of interest, we set the normalized Doppler frequency to 0. Based on the sampling interval and a detection range of 400m, the Nyquist rate sampling delay range is [-32, 32], and the oversampling delay range is [-64, 64], within the delayed Doppler region of interest. The constellation selected for the communication users is Quadrature Phase Shift Keying (QPSK).

[0087] Table 1 Target and Radar Parameters

[0088] Figure 1 This paper presents beammaps of the integrated communication and sensing waveforms designed under oversampling conditions and with different ρ values, including ideal beammaps, ideal radar waveform beammaps, and design results for different ρ values. When ρ=0, the waveform beammap is closest to the ideal beammap. As ρ increases, the difference between the beammap and the ideal beammap gradually widens. Figure 2 The data shows that the smaller ρ is, the lower the sidelobe level of the fuzzy function; as ρ increases, the performance of the fuzzy function deteriorates. Figure 3 It is clearly shown that the integrated sidelobe ratio (ISLR) increases with increasing ρ. When ρ increases from 0 to 1, the design is more geared towards communication performance, and the radar performance gradually decreases. Figure 4 and Figure 5 The symbol error rate (SER) and average total rate are shown to change with ρ. When ρ approaches 1, SER decreases and the average total rate increases, indicating that communication performance is given higher priority; while when ρ approaches 0, radar performance becomes dominant.

[0089] Adjusting the dynamic range constraint parameter m (i.e., the upper limit of power fluctuation of a single antenna) is crucial to the robustness of the system. Figures 6 to 8 It is clearly shown that more relaxed dynamic range constraints can significantly improve radar performance. Additionally, as... Figure 9 and Figure 10 As shown, the decrease in communication performance is not significant when the dynamic range constraint is tightened, and the algorithm can still maintain stable communication performance even under a certain degree of dynamic range constraint.

Claims

1. A joint waveform optimization method for ISAC MIMO-OFDM based on dynamic range constraints, characterized in that, include: Design and construct an ideal radar waveform under Nyquist sampling conditions; Based on the ideal radar waveform obtained from the design, an ideal condition integrated sensing waveform optimization model considering the integrated performance of communication and sensing is established. Based on the waveform optimization model, a nonlinear inductive integrated waveform design model considering constant modulus constraints is established; An oversampling factor is introduced into the nonlinear waveform design model to obtain a nonlinear inductive integrated waveform design model under oversampling conditions; For the oversampled nonlinear inductive integrated waveform design model, a waveform optimization solution method based on the alternating direction multiplier method is adopted to obtain an inductive integrated spatial radiation waveform that takes into account communication, sensing and system power constraints.

2. The ISAC MIMO-OFDM joint waveform optimization method based on dynamic range constraints according to claim 1, characterized in that, The design constructs an ideal radar waveform under Nyquist sampling conditions, including: From space Directional transmission signal Represented as: in For a complete time-domain signal transmission vector, For time-domain vectors considering only the precoding and symbol matrices, It is a spatial guiding vector. To introduce the selection matrix for the cyclic prefix, Indicates a length of N s +N cp The identity matrix, N s N is the number of subcarriers. cp The length of the cyclic prefix; definition for but The waveform of directional radiation is used Indicates, that is Further detection angle The fuzzy function is , where J is the delay matrix and D is the normalized Doppler frequency matrix; With the optimization objective of minimizing the integral of the ambiguity function within the finite time delay and Doppler frequency range, the optimization function is obtained as follows: In the formula, This represents the set of probe angles of interest. This represents the set of time delay pairs and Doppler frequency pairs of interest; The similarity between the actual radiation pattern and the ideal radiation pattern is expressed as: ， in For ideal direction charts; Taking into account both the ambiguity function and the beam pattern, the optimization problem for designing the ideal radar waveform can be expressed as: in Represents the set of all angles covered by the beam; The above optimization problem is solved using a finite-memory quasi-Newton method; the optimal solution obtained by the finite-memory quasi-Newton method is denoted as the ideal radar waveform. After energy normalization, it can be expressed as: 。 3. The ISAC MIMO-OFDM joint waveform optimization method based on dynamic range constraints according to claim 2, characterized in that, The process of establishing an ideal condition integrated sensing waveform optimization model based on the ideal radar waveform obtained from the design, considering the integrated performance of communication and sensing, includes: With the objectives of minimizing the MUI and the similarity between the design waveform and the ideal radar waveform, the following objective function is established: in, Let H be the symbol matrix for all users, H be the frequency domain channel state matrix corresponding to all subcarriers, and x be the precoding symbol matrix for all users. It is an ideal radar waveform, where F is the Fourier transform matrix. For length unit array, ρ represents the number of antennas, and ρ is the weighting factor. Assume there is a total Each antenna has an independent radio frequency channel; compared to the first... One antenna, define the selection matrix. : The constraint on the total energy of all symbols in the time domain is transformed into a constraint on the total energy of the effective symbols: in Indicates corresponding to One antenna and Total available energy for a system with OFDM symbols.

4. The ISAC MIMO-OFDM joint waveform optimization method based on dynamic range constraints according to claim 3, characterized in that, Based on the ideal waveform optimization model, a nonlinear inductive integrated waveform design model considering the constant modulus constraint of the system is established, including: With the total energy fixed, the power of each symbol is placed within a dynamic range, as follows: in This represents the energy of each OFDM symbol. This is a dynamic range constraint parameter, which corresponds to the upper limit of fluctuation in the linear power region of a single channel; Establish a nonlinear inductive integrated waveform design model considering the constant modulus constraint of the system: 。 5. The ISAC MIMO-OFDM joint waveform optimization method based on dynamic range constraints according to claim 4, characterized in that, The process of introducing an oversampling factor into the nonlinear inductive integrated waveform design model to obtain the nonlinear inductive integrated waveform design model under oversampling conditions includes: Define the selection matrix for oversampling : Where κ is the oversampling rate; This leads to the nonlinear inductive integrated waveform design model under oversampling conditions: Based on the aforementioned nonlinear inductive integrated waveform design model, combined with The model can be further described as follows: in, .

6. The ISAC MIMO-OFDM joint waveform optimization method based on dynamic range constraints according to claim 5, characterized in that, The nonlinear sensing integrated waveform design model under the obtained oversampling conditions employs a waveform optimization solution method based on the alternating direction multiplier method to obtain a spatial radiation waveform that takes into account communication, sensing, and system power constraints, including: Introduce two auxiliary vectors and ， The nonlinear inductive integrated waveform design model under oversampling conditions is transformed into: The corresponding augmented Lagrangian function is: in and It is a Lagrange multiplier, and η is the penalty factor; The augmented Lagrangian function is solved using the alternating direction multiplier method, specifically as follows: First step, update : When solving When the problem is minimized, it can be expressed by the formula: in, It is a selection matrix whose elements are either 0 or 1; selection matrix Select the option corresponding to the first... The elements of each antenna are combined into a new vector; Including the One antenna 1 symbol, of which Indicates the first One symbol; because the constraint covers all and Then the minimization problem is equivalently transformed into Further decomposed into The individual problem, among which The subproblem can be further written as The solution to the above subproblem is expressed as: The value of is updated by minimizing the problem as follows: Because the constraints cover all Therefore, the problem is broken down into The issue of size; The optimal solution to the above problem is In calculating all All Then, the vector The elements are arranged into a vector; because The vectors are contained in a regular form. The vector obtained by rearranging all the elements of is . ; Step 2, update : Through the obtained To solve and update, that is: Based on the augmented Lagrangian function, the problem can be specifically represented as: The above problem is an unconstrained quadratic optimization problem; assuming the derivative is 0, simplifying the equation, and solving the equation yields the optimal solution as follows: in, but The least squares solution: Step 3, Update : ； The ADMM iteration process described above, repeated multiple times, yields the solution 's' of the ISAC waveform design model under oversampling conditions. Combined with the cyclic prefix selection matrix, the inductively coupled transmitted waveform radiating into space can then be obtained. .

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