Mobile antenna assistance-based integrated perception communication and air computing system and joint optimization method

Abstract

The invention discloses an integrated perception communication and air computing system based on movable antenna assistance and a joint optimization method, and belongs to the technical field of wireless communication and networks. A traditional large-scale fixed antenna array is replaced by a receiving antenna (MA) of which the position can be dynamically adjusted, so that the hardware cost and the energy consumption are remarkably reduced. Compared with the prior art that the ISCCO system needs to configure a large-scale antenna array to ensure the sensing precision and the communication capacity, so that the equipment cost is sharply increased, and the energy consumption is proportionally increased along with the number of radio frequency chains, the method provided by the invention has the advantages that the characteristics of spatial diversity are enhanced by dynamically adjusting the position by virtue of the MA; compared with the prior art, the sensing-communication performance equivalent to that of a large-scale fixed array can be achieved with fewer antenna elements, the purchase and deployment cost of the antenna and a matched radio frequency chain is reduced, the transmitting power of a fixed transmitting antenna (FPA) can be dynamically adapted based on MA real-time position parameters, redundant energy consumption waste is avoided, and the dual optimization effect of low cost and low energy consumption is achieved.

Description

Technical Field

[0001] This invention belongs to the field of wireless communication and network technology, specifically an integrated sensing communication and over-the-air computing system and joint optimization method based on a movable antenna. Background Technology

[0002] With the rapid expansion of advanced IoT systems such as digital twins, augmented reality, and smart grids, the demand for high-precision sensing, high-speed computing, and efficient communication among large-scale IoT devices is becoming increasingly urgent. In traditional technologies, sensing, communication, and computing functions are often designed independently, leading to increased system overhead and significantly reduced overall efficiency. To address this issue, Integrated Sensing and Communication (ISAC) technology achieves collaborative data acquisition and transmission by sharing hardware and spectrum resources, while Air Computer (AC) technology utilizes analog waves superimposed on the physical layer to directly perform function calculations. The combination of ISAC and AC technologies to form the Integrated Sensing, Communication, and Computation Over-The-Air (ISCCO) framework has become an important direction supporting massive IoT applications. Currently, related research focuses on beamforming design within the ISCCO framework. For example, by balancing sensing and air computing performance through beam pattern tradeoffs, and by jointly designing transceiver beamforming to optimize system performance, some studies have also demonstrated that ISCCO outperforms the simple ISAC scheme in positioning estimation.

[0003] However, existing ISCCO systems still have significant bottlenecks. Specifically, to ensure the sensing accuracy and communication capacity of ISCCO systems, large-scale antenna arrays are typically required. This not only significantly increases equipment costs but also leads to a proportional increase in energy consumption with the number of RF chains. Meanwhile, although movable antennas (MA) technology has attracted attention due to its ability to dynamically adjust position and enhance spatial diversity with fewer antenna elements, and has been proven in the ISAC framework to reduce the Cramer-Rao Bound (CRB), improve sensing mutual information and communication capacity, and optimize the trade-off between sensing and communication performance, existing MA designs are mostly limited to downlink single-site systems and cannot directly adapt to the uplink multi-sensor joint sensing and computing scenarios required by ISCCO. Research in this area remains lacking. Therefore, there is an urgent need for a movable antenna optimization scheme suitable for ISCCO systems to improve spectral efficiency and reduce RF costs while adapting to the uplink multi-sensor joint sensing and computing scenarios required by ISCCO systems.

[0004] The above content is only used to help understand the technical solution of this application and does not represent an admission that the above content is prior art. Summary of the Invention

[0005] To address the problems existing in the background technology, the present invention provides an integrated sensing communication and over-the-air computing system based on a movable antenna and a joint optimization method, which can improve the integration of the mobile antenna system of the ISCCO system, thereby improving spectrum efficiency and reducing radio frequency costs.

[0006] To achieve the above objectives, this application proposes an integrated sensing communication and over-the-air computing system based on a movable antenna, comprising:

[0007] Base station and multiple ISAC sensors;

[0008] The base station includes at least one access point (AP), and the access point (AP) is configured with multiple movable receiving antennas (MA). The movable receiving antennas (MA) can dynamically adjust their positions within a preset spatial range.

[0009] Each of the ISAC sensors is equipped with a fixed transmitting antenna FPA and a movable receiving antenna MA. The fixed transmitting antenna FPA is used to transmit the data to be calculated to the access point AP. The movable receiving antenna MA of the ISAC sensor can dynamically adjust its position within a preset spatial range to receive the reflected signal of the sensing target.

[0010] The access point (AP) receives the superimposed signal of the data to be calculated transmitted by multiple ISAC sensors through the fixed transmitting antenna (FPA) via its movable receiving antenna (MA), and performs data aggregation calculation at the physical layer based on the AirComp principle.

[0011] The access point (AP) is also used to integrate the reflected signals received by the movable receiving antennas (MA) of each of the ISAC sensors to achieve joint sensing, and to construct an optimization model with the goal of minimizing the mean square error of aerial computation based on the Cramer-Rao boundary (CRB) of sensing accuracy, the transmit power of each of the ISAC sensors and the position constraints of the movable receiving antennas (MA).

[0012] The access point (AP) solves the optimization model using the block coordinate descent (BCD) algorithm, iteratively optimizing the transmit beamforming matrix of each of the ISAC sensors, its own receive beamforming matrix, the position of the movable receiving antenna (MA) of each of the ISAC sensors, and its own movable receiving antenna (MA) position, in order to adapt to the uplink multi-sensor joint sensing and computing scenario.

[0013] In a preferred embodiment of the present invention, both the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor are equipped with a position driving module and a position feedback module.

[0014] The position driving module is used to drive the antenna to translate or rotate within a preset spatial range according to the optimized position parameters output by the access point (AP).

[0015] The location feedback module is used to collect the actual location information of the antenna in real time and feed it back to the access point (AP) to correct the location constraint parameters in the optimization model.

[0016] In a preferred embodiment of the present invention, the Cramer-Rao bound constraint of the sensing accuracy in the optimization model constructed by the access point (AP) is specifically as follows:

[0017] Based on the reflected signals received by the movable receiving antenna MA of each of the ISAC sensors, the theoretical lower limit of the positioning error of the perceived target is calculated, and the lower limit does not exceed a preset threshold.

[0018] Specifically, the positional constraint of the movable receiving antenna MA is as follows: the distance between any two movable receiving antennas MA is not less than a preset safety distance, and the antenna position does not exceed the preset physical space boundary.

[0019] In a preferred embodiment of the present invention, the ISAC sensor is further configured with a signal preprocessing module;

[0020] The signal preprocessing module is used to perform noise reduction and modulation processing on the data to be transmitted by the fixed transmitting antenna FPA, and to adjust the signal transmission direction according to the transmit beamforming matrix issued by the access point AP.

[0021] The access point (AP) is also equipped with a calculation result verification module, which compares the data aggregation calculation result obtained based on the AirComp principle with a preset accuracy threshold. If the threshold is exceeded, the block coordinate descent (BCD) algorithm is triggered to iterate and optimize again.

[0022] Furthermore, this application also proposes an airborne computing method applied to an integrated sensing communication and airborne computing system based on a movable antenna, the airborne computing method comprising:

[0023] The multiple ISAC sensors of the integrated sensing communication and over-the-air computing system based on movable antennas transmit the data to be calculated to the access point (AP) of the base station through their respective fixed transmitting antennas (FPAs) and receive the reflected signals of the sensing targets through their respective movable receiving antennas (MAs).

[0024] The access point (AP) receives a superimposed signal formed by the superimposed data transmitted by multiple ISAC sensors through its configured movable receiving antenna (MA), and performs data aggregation calculation on the superimposed signal at the physical layer based on the AirComp principle.

[0025] The access point (AP) integrates the reflected signals received by the movable receiving antennas (MA) of each of the ISAC sensors to achieve joint sensing. Based on the Cramer-Rao boundary (CRB) of sensing accuracy, the transmit power of each ISAC sensor, and the positional constraints of the movable receiving antennas (MA), an optimization model is constructed with the goal of minimizing the mean square error of aerial computation (MSE).

[0026] The access point (AP) solves the optimization model using the block coordinate descent (BCD) algorithm, iteratively optimizing the transmit beamforming matrix of each of the ISAC sensors, its own receive beamforming matrix, the position of the movable receiving antenna (MA) of each of the ISAC sensors, and the position of its own movable receiving antenna (MA).

[0027] In a preferred embodiment of the present invention, before the plurality of ISAC sensors transmit the data to be calculated to the access point (AP), the over-the-air calculation method further includes:

[0028] Construct an integrated sensing communication and over-the-air computing system based on movable antenna assistance, which includes the sensing target, the access point (AP), and multiple ISAC sensors;

[0029] The access point (AP) is configured with multiple movable receiving antennas (MA), and each ISAC sensor is synchronously configured with a fixed transmitting antenna (FPA) and a movable receiving antenna (MA). The transmitting direction of the FPA is fixed to be towards the AP. The movement range of the movable receiving antennas (MA) of the access point AP and the movable receiving antennas (MA) of the ISAC sensor is limited to a preset physical space area.

[0030] The position vectors representing the movement range of the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor, which are limited to a preset physical space area, are expressed as follows:

[0031] );

[0032] Among them, the The position vectors of the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor are respectively represented as follows:

[0033] ;

[0034] .

[0035] In a preferred embodiment of the present invention, the steps of the plurality of ISAC sensors transmitting data to be calculated through a fixed transmitting antenna FPA and receiving the reflected signal of the sensed target through their respective configured movable receiving antennas MA include:

[0036] The ISAC sensor first acquires the locally collected data to be calculated and the reflected signal of the sensed target. Then, based on the current AP's MA position parameters and its own MA position parameters, it adjusts the signal transmission power of the FPA and transmits the data to be calculated to the AP in a directional manner through the FPA.

[0037] Each of the aforementioned ISAC sensors captures reflected signals through its own MA, filters out signal noise, and then transmits the signals to the AP to achieve multi-sensor joint sensing.

[0038] In a preferred embodiment of the present invention, the step of constructing an optimization model with the objective of minimizing the mean square error of aerial computation based on the Cramer-Rao boundary (CRB) of sensing accuracy, the transmit power of each of the ISAC sensors, and the positional constraints of the movable receiving antenna (MA) includes:

[0039] In time slot t, each sensor sends a data symbol vector to the access point (AP). , used for AirComp aggregation, where Let be the number of aggregation functions, with signs following an independent Gaussian distribution of zero mean and unit variance, satisfying . and hour ;

[0040] set up For the transmit beamforming matrix, the first The signals emitted by each sensor are transmitted to the AP via a channel. The channel matrix between the AP and the sensors is as follows:

[0041] ;

[0042] in This is the AP receive antenna response matrix. This is the response matrix of the sensor's transmitting antenna. This is the path gain matrix;

[0043] The aggregated signal received by the AP is:

[0044] ;

[0045] in To receive the beamforming matrix, It is Additive White Gaussian Noise (AWGN);

[0046] AirComp performance for receiving signals With launch symbols and Mean squared error (MSE) measurement:

[0047] ;

[0048] In the MA-ISCCO integrated sensing communication and airborne computing system based on a movable antenna, the data symbols transmitted by the sensors are simultaneously used to detect far-field point targets. The radar channel adopts a line-of-sight (LoS) propagation model, expressed as:

[0049] ;

[0050] in, and These are the transmit and receive steering vectors, respectively, and the position of the sensor's fixed transmit antenna. and the location of the movable receiving antenna Related; For complex channel coefficients, For the first The angle of the sensor to the target;

[0051] After matched filtering, the first The target signal received by each sensor is:

[0052] ;

[0053] in, For AWGN;

[0054] Target positioning accuracy is measured by CRB, and the expression is:

[0055] ;

[0056] in, for right The derivative, For the emission covariance matrix, Let be the number of coherent time slots. The trace term in the denominator can be simplified to... , These functions facilitate optimization and solution.

[0057] ;

[0058] ;

[0059] ;

[0060] in: , , ;

[0061] The target position estimates from each sensor are transmitted to the AP via AirComp, and the final positioning result is obtained after aggregation.

[0062] The optimization objective is to minimize the MSE of the AirComp by jointly designing the transmit and receive beamforming and MA position, while satisfying the constraints of CRB, transmit power and antenna movement area.

[0063] The issues related to optimization variables are described as follows:

[0064] ;

[0065] ;

[0066] ;

[0067] ;

[0068] ;

[0069] ;

[0070] ;

[0071] The above formula ensures that the estimation of each sensor meets the required accuracy; limits the transmit power of each sensor; restricts the received MA to a specific spatial region; and enforces a minimum spacing to prevent antenna coupling. Due to variable coupling and non-convex CRB constraints, this problem is NP-hard and difficult to solve directly.

[0072] In a preferred embodiment of the present invention, the step of solving the optimization model using the block coordinate descent BCD algorithm and iteratively optimizing the transmit beamforming matrix of each ISAC sensor, its own receive beamforming matrix, the position of the movable receiving antenna MA of each ISAC sensor, and its own movable receiving antenna MA position includes:

[0073] The first iteration is performed, fixing the MA position of each ISAC sensor and the MA position of the AP, and optimizing the transmit beamforming matrix of each ISAC sensor;

[0074] The second iteration is performed, fixing the transmit beamforming matrix and MA position, and optimizing the receive beamforming matrix of the AP;

[0075] The third iteration is performed, with the beamforming matrix fixed, to optimize the MA position of each of the ISAC sensors;

[0076] Perform the fourth iteration, fix the sensor MA position and beamforming matrix, and optimize the AP MA position; repeat the above four iterations until the difference between the mean square error (MSE) of the aerial computation between two adjacent iterations is less than the preset convergence threshold, stop the iteration and output the current optimization parameters.

[0077] In a preferred embodiment of the present invention, the step of solving the optimization model using the block coordinate descent BCD algorithm and iteratively optimizing the transmit beamforming matrix of each ISAC sensor, its own receive beamforming matrix, the position of the movable receiving antenna MA of each ISAC sensor, and the position of its own movable receiving antenna MA includes:

[0078] The non-convex optimization problem in step 102 is solved based on the BCD method. The optimization variables are divided into four blocks: transmit beamforming matrix, receive beamforming matrix, position of movable antenna (MA) at the sensor, and position of MA at the AP. The optimization is iteratively performed.

[0079] 1) Update the transmit beamforming matrix:

[0080] Optimize the transmit beamforming matrix while keeping other variables fixed. The subproblems are:

[0081] ;

[0082] Introducing auxiliary variables And using semidefinite relaxation (SDR), the subproblem is reconstructed as follows:

[0083] ;

[0084] ;

[0085] ;

[0086] ;

[0087] Transform it into a convex optimization problem using Schur complement:

[0088] ;

[0089] 2) Update the transmit beamforming matrix:

[0090] With other variables fixed, the optimization of the receiving beamforming matrix Z becomes an unconstrained convex problem:

[0091] ;

[0092] The optimal solution is MMSE beamforming.

[0093] ;

[0094] 3) Update the MA position at the sensor.

[0095] With other variables fixed, the MA position at the sensor is optimized by maximizing the denominator of the CRB. :

[0096] ;

[0097] in, Given a set of positional constraints, transform it into its equivalent form:

[0098] ;

[0099] The SCA method is used to linearize the nonconvex terms, and the update formula is as follows:

[0100] ;

[0101] in , , , The parameters are known.

[0102] 4) Update the MA position at AP.

[0103] Alternating optimization is used, optimizing the position of a single antenna each time. The subproblem is:

[0104] ;

[0105] in, , ;

[0106] Transform into:

[0107] ;

[0108] in, , ;

[0109] The surrogate function is constructed using the MM method, and the update formula is as follows:

[0110] ;

[0111] in, , , Given parameters, This is the projection operator.

[0112] The algorithm flow is as follows: initialize variables, iteratively update transmit beamforming, receive beamforming, sensor MA position, and AP-MA position until the MSE decreases below a threshold. The computational complexity mainly comes from each step, namely... , and .

[0113] The beneficial effects of this invention are as follows:

[0114] This application achieves a significant reduction in hardware cost and energy consumption by replacing traditional large-scale fixed antenna arrays with dynamically adjustable receiving antennas (MA). Compared to the existing technology where ISCCO systems require large-scale antenna arrays to ensure sensing accuracy and communication capacity, leading to a surge in equipment costs and a proportional increase in energy consumption with the number of RF chains, this application leverages the spatial diversity characteristics of MA's dynamic position adjustment. It achieves sensing-communication performance comparable to large-scale fixed arrays with fewer antenna components. This not only reduces the procurement and deployment costs of antennas and supporting RF chains but also dynamically adapts the transmit power of the fixed transmitting antenna (FPA) based on the real-time position parameters of the MA, avoiding redundant energy waste and achieving a dual optimization effect of low cost and low energy consumption.

[0115] Furthermore, this application achieves a comprehensive improvement in system performance through a collaborative optimization design of sensing, communication, and computing. Compared to existing technologies where the independent design of sensing, communication, and computing functions leads to resource waste and low efficiency, or where the combination lacks refined collaborative control, this application achieves the following synergy: First, it utilizes the AirComp principle to allow multi-sensor data to be computed to be naturally superimposed during transmission, with the receiving end directly performing aggregation computation at the physical layer, reducing latency and bandwidth waste; second, it uses multi-sensor MA synchronous reception of reflected signals, combined with Cramer-Rao boundary (CRB) dynamic adjustment of MA positions to reduce positioning errors and improve sensing accuracy; third, it incorporates sensing accuracy constraints into the computation optimization objective, ensuring a balance among the three, ultimately achieving a comprehensive performance improvement of more accurate computation, more reliable sensing, and more efficient communication, thus solving the efficiency bottleneck of independent functions in traditional technologies.

[0116] Furthermore, through targeted design involving hardware adaptation and algorithm optimization, the MA technology has been adapted to ISCCO uplink multi-sensor scenarios, significantly expanding its application scope. Compared to existing technologies where MA designs are mostly limited to ISCCO downlink single-station systems and cannot meet the joint sensing and computing needs of ISCCO uplink multi-sensor systems, this application configures FPA+MA for sensors in terms of hardware. The FPA ensures stable uplink data transmission, while the MA is responsible for receiving sensing signals. In terms of algorithms, an optimization model is constructed with the goal of minimizing the AirComp mean square error. Combined with multiple constraints, the block coordinate descent (BCD) algorithm is used to iteratively optimize beamforming and MA position, solving the problem of multi-sensor MA collaborative control. This successfully fills the gap in the application of MA technology in ISCCO uplink multi-sensor scenarios and meets the collaborative needs of advanced IoT systems such as digital twins and smart grids. Attached Figure Description

[0117] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with this application and, together with the description, serve to explain the principles of this application.

[0118] To more clearly illustrate the technical solutions in the embodiments of this application or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0119] Figure 1 This is a schematic diagram of a specific scenario of an integrated sensing communication and over-the-air computing system based on a movable antenna, as described in this application.

[0120] Figure 2 This is a flowchart illustrating an embodiment of an air computing method for an integrated sensing communication and air computing system based on a movable antenna assisted by this application.

[0121] Figure 3 This is a flowchart illustrating a second embodiment of an air computing method for an integrated sensing communication and air computing system based on a movable antenna assisted by this application.

[0122] Figure 4 This is a flowchart illustrating a third embodiment of an air computing method for an integrated sensing communication and air computing system based on a movable antenna assisted by this application.

[0123] Figure 5 This is a schematic diagram of a specific process for an air computing method for an integrated sensing communication and air computing system based on a movable antenna, according to this application.

[0124] Figure 6(a) is a data diagram illustrating the convergence and complexity of an aerial computation method according to this application;

[0125] Figure 6(b) is a data diagram illustrating another convergence and complexity of an aerial computation method according to this application;

[0126] Figure 7(a) is a performance comparison diagram of an aerial computing method according to this application;

[0127] Figure 7(b) is a schematic diagram showing another performance comparison of an aerial computing method according to this application.

[0128] Figure 7(c) is a schematic diagram showing another performance comparison of an aerial computing method according to this application.

[0129] The purpose, features, and advantages of this application will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0130] It should be understood that the specific embodiments described herein are merely illustrative of the technical solutions of this application and are not intended to limit this application.

[0131] To better understand the technical solution of this application, a detailed description will be provided below in conjunction with the accompanying drawings and specific implementation methods.

[0132] The executing entity in this embodiment can be a computing service device with multi-task collaborative processing capabilities, data aggregation and computing functions, and a hardware parameter optimization and control module. Examples include edge servers with edge computing capabilities, base station controllers with integrated signal processing units, and IoT gateway devices supporting distributed collaboration. Alternatively, it can be a dedicated embedded device equipped with high-precision antenna control algorithms and over-the-air computing logic, such as a customized integrated wireless communication and sensing controller or a multi-sensor collaborative scheduling terminal. The following description uses a resource allocation device (hereinafter referred to as the device) as an example to illustrate this embodiment and the subsequent embodiments.

[0133] This resource allocation equipment must simultaneously possess the capabilities of hardware status monitoring, data processing and calculation, and parameter optimization and distribution. On one hand, it can collect real-time hardware and signal information such as the location data of the movable receiving antenna (MA) of the access point (AP), the transmit power and data to be calculated by the ISAC sensor, and the reflected signals of the sensed target. On the other hand, it can run the block coordinate descent (BCD) optimization algorithm to build a model with the goal of minimizing the mean square error (MSE) of the aerial computation, and iteratively generate optimization instructions such as transmit / receive beamforming matrices and MA position adjustment parameters. At the same time, it can accurately distribute the optimization parameters to the AP and each ISAC sensor, driving hardware actions such as antenna position adjustment and transmit power adaptation, ensuring the stable progress of the collaborative process of sensing, communication, and computing, and adapting to the needs of uplink multi-sensor joint scenarios.

[0134] Based on this, the embodiments of this application provide an aerial computing method, referring to... Figure 1 , Figure 1 This is a schematic diagram of a specific scenario for the integrated sensing communication and over-the-air computing system based on a movable antenna assisted by this application.

[0135] like Figure 1 The embodiment of the present invention shown includes an integrated sensing communication and over-the-air computing system based on a movable antenna, comprising:

[0136] Base station and multiple ISAC sensors.

[0137] The base station includes at least one access point (AP), and the AP is configured with multiple movable receiving antennas (MA). The movable receiving antennas (MA) can dynamically adjust their positions within a preset spatial range.

[0138] Each ISAC sensor is equipped with a fixed transmitting antenna (FPA) and a movable receiving antenna (MA). The fixed transmitting antenna (FPA) is used to transmit the data to be calculated to the access point (AP). The movable receiving antenna (MA) of the ISAC sensor can dynamically adjust its position within a preset spatial range to receive the reflected signal of the target being sensed.

[0139] The access point (AP) receives the superimposed signal of the data to be calculated transmitted by multiple ISAC sensors through the fixed transmit antenna (FPA) via its movable receiving antenna (MA), and performs data aggregation calculation at the physical layer based on the AirComp principle.

[0140] The access point (AP) is also used to integrate the reflected signals received by the movable receiving antennas (MA) of each ISAC sensor to achieve joint sensing. Based on the Cramer-Rao boundary (CRB) of sensing accuracy, the transmit power of each ISAC sensor, and the positional constraints of the movable receiving antennas (MA), an optimization model is constructed with the goal of minimizing the mean square error (MSE) of aerial computation.

[0141] The access point (AP) solves the optimization model using the block coordinate descent (BCD) algorithm, iteratively optimizing the transmit beamforming matrix of each ISAC sensor, its own receive beamforming matrix, the position of the movable receiving antenna (MA) of each ISAC sensor, and its own movable receiving antenna (MA) position, in order to adapt to the uplink multi-sensor joint sensing and computing scenario.

[0142] Here, the integrated sensing, communication, and over-the-air computing system based on movable antennas consists of a base station and multiple integrated sensing and communication (ISAC) sensors. The base station includes at least one access point (AP), and each AP is equipped with multiple movable antennas (MAs). These movable antennas (MAs) can dynamically adjust their positions within a preset spatial range, the adjustment range of which is limited by the maximum range of motion d marked by the dashed box on the right side of the figure. max Meanwhile, adjacent antennas must satisfy the minimum spacing d. min The constraints are designed to ensure the signal reception performance of the antenna array.

[0143] Each ISAC sensor is equipped with a fixed transmit antenna (FPA) and a movable receive antenna (MA). The function of the fixed transmit antenna (FPA) is to transmit the data to be computed to the access point (AP). During transmission, the data will pass through multipath channels (H1 to H2 as shown in the upper right corner of the figure). M (Channel). The movable receiving antenna MA of the ISAC sensor can also dynamically adjust its position within a preset spatial range. The purpose of this position adjustment is to receive the signal reflected by the sensing target. This process is illustrated in the lower part of the figure below as a signal interaction between the sensing target and the sensor.

[0144] In the data computation phase, the Access Point (AP) receives superimposed signals of data to be computed from multiple ISAC sensors transmitted via a fixed transmit antenna (FPA) through its configured movable receiving antenna (MA). Subsequently, based on the principle of Airborne Computation (AirComp), the AP performs aggregation computation directly on this superimposed data at the physical layer. This computation method eliminates the steps of data demodulation and secondary forwarding, significantly improving data processing efficiency and reducing transmission latency.

[0145] Furthermore, in terms of joint sensing and optimized control, the access point (AP) integrates the reflected signals received by the movable receiving antennas (MAs) of each ISAC sensor to achieve joint sensing of the target. Simultaneously, the AP can comprehensively consider the Cramér-Rao Bound (CRB) of sensing accuracy, the transmit power constraints of each ISAC sensor, and the positional constraints of the movable receiving antennas (MAs) to construct an optimization model aimed at minimizing the mean square error (MSE) of aerial computation.

[0146] It should be noted that the Cramé-Rao boundary theoretically defines the lower limit of positioning error; the smaller the CRB, the higher the theoretical upper limit of sensing accuracy. The positional constraints of the movable receiving antenna (MA) include its spatial range of motion and antenna spacing.

[0147] To solve this optimization model, the access point (AP) employs a block coordinate descent (BCD) algorithm to iteratively optimize the transmit beamforming matrix of each ISAC sensor, its own receive beamforming matrix, and the positions of the movable receiving antennas (MA) of each ISAC sensor and its own movable receiving antennas (MA). This enables the integrated sensing, communication, and over-the-air computing system based on movable antenna assistance to be well-suited for uplink multi-sensor joint sensing and computing scenarios. While ensuring sensing accuracy, it improves the efficiency of computing and communication, effectively solving problems such as functional independence, high cost, and poor adaptability inherent in traditional technologies.

[0148] In one implementation, both the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor are equipped with a position driving module and a position feedback module.

[0149] The position driving module is used to drive the antenna to translate or rotate within a preset spatial range based on the optimized position parameters output by the access point (AP).

[0150] The location feedback module is used to collect the actual location information of the antenna in real time and feed it back to the access point (AP) to correct the location constraint parameters in the optimization model.

[0151] In this embodiment of the invention, both the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor are equipped with a position driving module and a position feedback module. The two work together to ensure the accuracy of antenna position adjustment and the feasibility of dynamic optimization.

[0152] from Figure 1 From the perspective of the scenario, whether it is the movable receiving antenna of the access point (AP) on the base station side or the movable receiving antenna of each ISAC sensor, it needs to be within the preset space d max d min Translation or rotation within the constrained range relies on the position drive module. This module can drive the antenna to make corresponding position changes based on the optimized position parameters output by the access point (AP). For example, when the optimization model calculates that the movable receiving antenna MA of a certain ISAC sensor needs to be adjusted to a position closer to the sensing target to enhance the reception of reflected signals, the position drive module will push the antenna to translate to that position.

[0153] The location feedback module can collect the actual location information of the antenna in real time and feed it back to the access point (AP). During actual antenna adjustments, factors such as mechanical errors and environmental interference may affect the actual location, causing a deviation from the theoretically optimized location. By feeding back the actual location information, the access point (AP) can promptly correct the location constraint parameters in the optimization model. For example, if the actual location of a movable receiving antenna (MA) deviates from the expected optimized location, the feedback information allows the access point (AP) to more accurately consider the actual location constraints of the antenna during subsequent optimization model iterations. This ensures that the optimization model is always based on the actual hardware state, thereby improving the accuracy and stability of the entire system's coordinated sensing, computing, and communication.

[0154] In one implementation, the Cramer-Rao bound constraint (CRB) for sensing accuracy in the optimization model constructed by the access point (AP) is specifically as follows:

[0155] Based on the reflected signals received by the movable receiving antenna MA of each ISAC sensor, the theoretical lower limit of the positioning error of the perceived target is calculated, and the lower limit does not exceed a preset threshold.

[0156] Specifically, the positional constraints of the movable receiving antenna MA are as follows: the distance between any two movable receiving antennas MA is not less than the preset safety distance, and the antenna positions do not exceed the preset physical space boundaries.

[0157] In this embodiment of the invention, the optimization model constructed by the access point (AP) clearly defines the constraints on sensing accuracy and antenna position, which together constitute a dual guarantee of system performance and hardware feasibility.

[0158] Specifically, on the one hand, the Cramer-Rao boundary (CRB) serves as the theoretical lower limit for measuring positioning error; the smaller the CRB value, the higher the theoretical upper limit of the positioning accuracy of the sensed target. During system operation, the movable receiving antennas (MA) of each ISAC sensor can continuously receive reflected signals from the sensed target. The access point (AP) calculates the theoretical lower limit of the positioning error of the sensed target (i.e., the CRB value) based on these reflected signals. The reflected signals can include characteristic information such as target distance and angle.

[0159] It should be noted that the optimization model strictly limits this theoretical lower bound within a preset threshold, ensuring that the system's sensing accuracy meets the requirements of practical applications through mathematical constraints. For example, in a smart grid scenario, if centimeter-level positioning of device status is required, the preset threshold will be set as the corresponding upper limit of error, and the CRB constraint will theoretically guarantee the feasibility of this accuracy target.

[0160] On the other hand, from the perspective of the positional constraints of the movable receiving antennas (MA), the distance between any two movable receiving antennas (MA) is not less than a preset safety distance, which can avoid electromagnetic coupling interference between antennas. Specifically, when the spacing between multiple antennas is too small, signal transmission will generate mutual interference, resulting in a decrease in the signal-to-noise ratio of the received signal. The preset safety distance can be set according to parameters such as the antenna operating frequency band and power to ensure the independence of signal transmission. In addition, the antenna position does not exceed the preset physical space boundary, which is directly related to the hardware deployment scenario. For example, in an industrial IoT environment, the antenna may be limited to moving within the equipment cabinet or a specific area to avoid hardware damage or system failure caused by the antenna displacement exceeding the physical space (such as colliding with other equipment).

[0161] Therefore, by incorporating the two types of constraints, CRB and MA, into the optimization model, the accuracy of joint sensing is ensured from a performance perspective, while the safety and feasibility of hardware operation are guaranteed from an engineering perspective. This ensures that when the access point (AP) iteratively optimizes through the block coordinate descent (BCD) algorithm, all variable adjustments are within a reasonable range that meets performance standards and is controllable by the hardware, ultimately achieving a balance between theoretical optimization and practical feasibility.

[0162] In one implementation, the ISAC sensor is also equipped with a signal preprocessing module.

[0163] The signal preprocessing module is used to perform noise reduction and modulation processing on the data to be transmitted by the fixed transmit antenna FPA, and to adjust the signal transmission direction according to the transmit beamforming matrix issued by the access point AP.

[0164] The access point (AP) is also equipped with a calculation result verification module, which compares the data aggregation calculation result obtained based on the AirComp principle with a preset accuracy threshold. If the threshold is exceeded, the block coordinate descent (BCD) algorithm is triggered to iterate and optimize again.

[0165] In this embodiment of the invention, the signal preprocessing module of the ISAC sensor and the calculation result verification module of the access point (AP) form a dual guarantee mechanism for data transmission and calculation accuracy, thereby improving the reliability of the system through refined processing and closed-loop verification.

[0166] Specifically, the signal preprocessing module of the ISAC sensor focuses on optimization processing before the data to be computed is transmitted. In the data transmission link, the original data to be computed may contain environmental noise or interference signals. Direct transmission will lead to a decrease in the quality of the superimposed signal received by the access point (AP), thus affecting the aggregation accuracy of AirComp. The signal preprocessing module filters out redundant noise through noise reduction algorithms (such as adaptive filtering) and modulates the data (such as using quadrature amplitude modulation, QAM) to convert the digital signal into an analog signal suitable for wireless transmission, improving the signal's anti-interference capability in the channel. More importantly, this module dynamically adjusts the signal transmission direction according to the transmit beamforming matrix issued by the access point (AP). Thus, the beamforming matrix is ​​essentially a set of weighted parameters. By weighting the signal phase and amplitude of the fixed transmit antenna (FPA), the transmitted signal can form a directional beam in space, accurately pointing to the movable receive antenna (MA) of the access point (AP), reducing signal radiation loss in non-target directions, and further improving transmission efficiency and signal quality.

[0167] Furthermore, the calculation result verification module of the access point (AP) can construct a closed-loop control mechanism for calculation accuracy. While physical layer aggregation calculation based on the AirComp principle can achieve efficient data fusion, the actual calculation results may deviate from the theoretical optimal value due to factors such as channel fading, noise interference, or antenna position deviation. The calculation result verification module compares the aggregation calculation results with a preset accuracy threshold, forming real-time monitoring of the calculation quality. When the result exceeds the threshold, the module triggers a block coordinate descent (BCD) algorithm for iterative optimization. Thus, the integrated sensing communication and over-the-air computing system based on movable antenna assistance can readjust parameters such as the transmit / receive beamforming matrix and the movable antenna position until the calculation result meets the accuracy requirements. This effectively compensates for the potential error accumulation problem in a single optimization, ensuring that the accuracy of over-the-air computing remains stable within the application requirements.

[0168] In this embodiment, the signal preprocessing module performs noise reduction and quality adjustment on the data to be calculated by preprocessing the data, thereby improving the data transmission quality. The calculation result verification module ensures the calculation accuracy from the terminal and triggers dynamic correction. The two work together to enable the integrated sensing communication and over-the-air computing system based on movable antenna assistance to maintain efficient and accurate sensing + computing + communication collaborative performance in complex wireless environments, further enhancing the engineering practicality of the solution.

[0169] Based on the aforementioned integrated sensing communication and over-the-air computing system assisted by a movable antenna, this application provides a first embodiment of an over-the-air computing method applied to this integrated sensing communication and over-the-air computing system assisted by a movable antenna, referring to... Figure 2 , Figure 2This is a flowchart illustrating the first embodiment of the aerial computing method of this application.

[0170] In this embodiment, the aerial calculation method includes steps S10 to S40:

[0171] In step S10, multiple ISAC sensors based on the integrated sensing communication and over-the-air computing system assisted by movable antennas transmit the data to be calculated to the access point AP of the base station through their respective configured fixed transmitting antennas FPA, and receive the reflected signals of the sensing targets through their respective configured movable receiving antennas MA.

[0172] In the embodiments of this application, each of the multiple ISAC sensors can simultaneously realize the output of computational data and the input of sensing information.

[0173] On the one hand, the ISAC sensor can transmit locally collected or generated data to the base station access point (AP) to provide raw data for subsequent over-the-air computing. On the other hand, it can capture the reflected signals of the target through its own sensing function, providing a signal source for joint sensing by the system, thus avoiding the waste of resources caused by the independent transmission and sensing acquisition in traditional technologies.

[0174] From a hardware collaboration perspective, this application's integrated sensing communication and over-the-air computing system, assisted by a movable antenna, achieves functional division through two types of antennas configured with ISAC sensors. The fixed transmit antenna (FPA) is responsible for the uplink transmission of the data to be computed. Its fixed characteristic is mainly reflected in the stability of the transmission direction and hardware position. The transmission direction is typically preset towards the access point (AP) to ensure the directionality of data transmission, reduce signal radiation loss in non-target directions, and improve transmission efficiency. The movable receive antenna (MA) is responsible for receiving reflected signals from the sensing target. Its mobility is a key design feature. The MA can dynamically adjust its position (e.g., translation, rotation) within a preset spatial range based on the target's location, environmental interference, and other factors to optimize the signal reception angle and gain, capturing high-quality reflected signals as much as possible (e.g., reducing multipath interference and improving the signal-to-noise ratio), laying the foundation for ensuring the accuracy of subsequent joint sensing.

[0175] From the perspective of signal interaction, the transmission and acquisition of the two types of signals in the integrated sensing communication and airborne computing system based on the movable antenna can be carried out simultaneously. When the ISAC sensor transmits the data to be computed to the AP via the wireless channel through the FPA, the MA can simultaneously receive the electromagnetic signals reflected by the sensing target (such as radar waves, radio frequency signals, etc.), without occupying hardware or spectrum resources in a time-sharing manner. This synchronous design further improves the system's time efficiency and spectrum utilization.

[0176] In addition, the transmission of data to be computed needs to be adapted to the requirements of subsequent air-to-air computing (AirComp). The data format and modulation method will be coordinated with the AP in advance (such as adopting a modulation scheme suitable for analog wave superposition) to ensure that the AP can directly perform physical layer aggregation calculation on the superimposed signals transmitted by multiple sensors. The acquisition of reflected signals needs to retain the key features of the perceived target (such as distance, angle, speed and other information).

[0177] Thus, by dividing the work between the two antennas and synchronizing the two tasks, the uplink transmission of the raw data required for aerial computing was completed, and the acquisition of the reflected signals required for joint sensing was also achieved.

[0178] In step S20, the access point AP receives the superimposed signal formed by the data to be calculated transmitted by multiple ISAC sensors through its configured movable receiving antenna MA, and performs data aggregation calculation on the superimposed signal at the physical layer based on the AirComp principle.

[0179] After the ISAC sensor transmits the data to be calculated to the access point (AP) of the base station and receives the reflected signal from the sensed target, and the access point (AP) receives the data to be calculated, it can receive the superimposed signal formed by the data to be calculated through its own configured movable antenna (MA), and perform data aggregation calculation based on the superimposed signal.

[0180] From the perspective of hardware adaptation logic for signal reception, the access point (AP) receives signals through its own configured movable receiving antenna (MA). On one hand, the AP's movable receiving antenna (MA) can dynamically adjust its orientation (e.g., translation to optimize the receiving angle, rotation to enhance signal gain) according to the distribution location of multiple sensors and the channel environment. Compared to a fixed antenna, it can more flexibly adapt to the differences in the uplink signal direction of multiple sensors, reducing signal attenuation caused by antenna pointing deviation. On the other hand, combined with the directional transmission characteristics of the fixed transmitting antenna (FPA) of the ISAC sensor, the AP's MA can optimize its position to focus on the signal transmission path of each FPA, maximizing the signal-to-noise ratio of the received signal, and laying a hardware foundation for ensuring the accuracy of subsequent aggregation calculations.

[0181] It should be noted that the AP does not receive independent signals from a single sensor, but rather superimposed signals naturally formed after multiple ISAC sensors transmit through the wireless channel. The superimposed signal is the core premise of the AirComp principle, eliminating the need for time-sharing transmission from each sensor or split-path reception by the AP, thus significantly improving the time efficiency of data transmission and processing.

[0182] Furthermore, based on the AirComp principle, data aggregation and calculation are performed directly at the physical layer, unlike the traditional process of receiving, demodulating, and then calculating. In contrast to traditional solutions where the AP needs to demodulate the wireless signals from each sensor into digital signals one by one before performing data aggregation (such as summation and averaging) through a processor, which not only incurs demodulation losses but also consumes significant computing resources and time, the AirComp principle utilizes the analog superposition characteristics of wireless signals to directly superimpose and calculate the analog signals corresponding to the data to be computed transmitted from multiple sensors at the physical layer.

[0183] For example, if each sensor needs to transmit data x1, x2, ..., x... n And by performing summation and aggregation, the superimposed signals received by the AP can directly correspond to x1 + x2 + ... + x n The analog equivalent value can be obtained without digital demodulation. The advantage of this physical layer computing mode lies in transmission-as-computation, which integrates data transmission and aggregation computing processes, significantly reducing computing latency, and is particularly suitable for the real-time requirements in IoT scenarios; in addition, it reduces signal conversion loss, avoids noise and errors introduced during demodulation, and improves the original accuracy of aggregation computing.

[0184] It should be noted that during the reception of superimposed signals, the AP can simultaneously sense information such as channel quality and signal strength. This information indirectly reflects the rationality of the current movable antenna position and beamforming parameters. Furthermore, the scheme of multi-signal superposition reception and physical layer computation fully utilizes the broadcast characteristics of the wireless channel and the superposition characteristics of analog signals. Without increasing spectrum resource usage, it achieves parallel processing of multi-sensor data, realizing the sensing, communication, computation, and coordination of an integrated sensing communication and over-the-air computing system based on movable antenna assistance.

[0185] Thus, by combining movable antenna-adaptive reception with AirComp physical layer computation, the problems of high latency and high loss in traditional data aggregation schemes are solved, and key computational support is provided for the efficient collaboration of the entire system.

[0186] In step S30, the access point AP integrates the reflected signals received by the movable receiving antennas MA of each ISAC sensor to achieve joint sensing. Based on the Cramer-Rao boundary (CRB) of sensing accuracy, the transmit power of each ISAC sensor, and the positional constraints of the movable receiving antennas MA, an optimization model is constructed with the goal of minimizing the mean square error (MSE) of aerial computation.

[0187] After the access point AP performs data aggregation calculation based on the superimposed signal formed by the data to be calculated and the principle of over-the-air computing, the access point AP can also integrate the target reflection signals collected by all ISAC sensors through the movable receiving antenna MA. These reflection signals come from the spatial positions of different ISAC sensors and carry multi-dimensional information of the target (such as distance and angular features at different angles).

[0188] It should be noted that the reflected signal of a single sensor is easily affected by occlusion and interference, resulting in limited sensing accuracy. However, by integrating multiple signals, multi-view complementarity can be achieved. For example, if a certain ISAC sensor fails to capture a clear reflected signal due to occlusion, the signals from sensors at other locations can make up for this deficiency, ultimately forming a joint perception of the target and ensuring the reliability and accuracy of the sensing results. This is also a key advantage of the system that distinguishes it from traditional single-station sensing schemes.

[0189] Based on the completion of joint sensing, the access point (AP) can further construct an optimization model with the goal of minimizing the mean square error (MSE) of over-the-air computation. The MSE can be used to reflect the accuracy of the previous physical layer aggregation calculation. The smaller the MSE, the smaller the deviation between the data aggregation result and the theoretical true value. It is a core indicator for measuring the system's computing performance. Therefore, using it as an optimization target can ensure the reliability of over-the-air computation and meet the requirements for data computation accuracy in IoT scenarios.

[0190] During the model construction process, the access point (AP) can focus on incorporating three key types of constraints.

[0191] The first category is the Cramer-Rao boundary (CRB) constraint for sensing accuracy. The CRB is the theoretical lower limit of positioning error, and its value is closely related to the quality of reflected signals and sensor positions. Incorporating the CRB into the constraint can prevent subsequent parameter optimization from sacrificing sensing performance for computational accuracy, ensuring that the positioning error of joint sensing is always controlled within an acceptable range for practical applications. The second category is the transmit power constraint of each ISAC sensor. Excessive transmit power of the sensor will lead to a surge in energy consumption and interference with other devices, while too low power will affect data transmission and reflected signal strength. Therefore, it is necessary to limit the power range through constraints to balance system performance, energy consumption, and anti-interference capability. The third category is the position constraint of the movable receiving antenna (MA), including the minimum safe distance between antennas and physical space boundaries. Too small a distance will cause electromagnetic coupling between antennas, leading to signal interference, while exceeding the physical boundaries may cause hardware collision damage. These constraints ensure the feasibility of the optimization process from an engineering implementation perspective, avoiding theoretically optimal but practically unachievable parameters.

[0192] Therefore, the above three types of constraints, together with the goal of minimizing the over-the-air computing MSE, constitute the optimization model. This model not only ensures the synergistic improvement of the system's computing and sensing performance, but also takes into account the safety and energy efficiency of hardware operation. At the same time, the construction of this model also incorporates the core data from the previous stages. The reflected signals that the joint sensing relies on are collected by the movable antenna of the ISAC sensor, and the optimization goal of the over-the-air computing MSE corresponds to the accuracy requirements of the previous physical layer aggregation calculation.

[0193] In step S40, the access point (AP) solves the optimization model using the block coordinate descent (BCD) algorithm, and iteratively optimizes the transmit beamforming matrix of each ISAC sensor, its own receive beamforming matrix, the position of the movable receiving antenna MA of each ISAC sensor, and the position of its own movable receiving antenna MA.

[0194] The access point (AP) employs the Block Coordinate Descent (BCD) algorithm to solve the optimization model. The optimization objective involves minimizing the mean square error (MSE) of the airborne computation, while constraints include the Cramer-Rao bound (CRB) for sensing accuracy, transmit power limitations, and antenna position restrictions. Furthermore, the parameters to be optimized (transmit beamforming matrix, receive beamforming matrix, and the positions of each movable antenna) are strongly coupled, making directly solving the entire non-convex optimization problem extremely difficult. The core idea of ​​the BCD algorithm is to decompose the multivariate optimization problem into multiple univariate subproblems, iteratively approximating the global optimum. This method effectively reduces the solution complexity while ensuring the stability of the optimization process.

[0195] Here, the iterative optimization process revolves around four core parameters, and the adjustment of each parameter serves to improve the overall system performance. Specifically, the first optimization is of the transmit beamforming matrix of each ISAC sensor. This transmit beamforming matrix, by weighting the signal amplitude and phase of the sensor's fixed transmit antenna FPA, can adjust the spatial radiation direction and intensity of the signal, making it more accurately pointed to the access point AP's movable receive antenna MA, reducing transmission loss and interference, thereby reducing the signal superposition error of over-the-air computing. The second optimization is of the access point AP's own receive beamforming matrix, which is used to weight the received superimposed signal, enhance the useful signal component, suppress noise and interference, and further improve the accuracy of physical layer aggregation computing. The third optimization is of the position of the movable receive antenna MA of each ISAC sensor. By optimizing the distribution of the antennas in the preset space, it can more efficiently capture the reflected signal of the sensed target, improve the CRB performance of joint sensing, and avoid electromagnetic coupling between antennas. Finally, the optimization of the position of the access point AP's own movable receive antenna MA allows it to form a better spatial match with the sensor's transmit beam, improving the reception quality of the superimposed signal and providing a better signal foundation for over-the-air computing.

[0196] It should be noted that when adjusting one type of parameter each time, the other parameters remain fixed. The optimal value of that type of parameter is obtained by solving the corresponding subproblems. After completing one round of optimization of the four types of parameters, the in-flight computation MSE and sensing CRB indicators can be re-evaluated. If the convergence condition is not met (e.g., the difference in MSE between two adjacent iterations is still greater than the preset threshold), the above optimization process is repeated. As the number of iterations increases, each parameter gradually converges towards the optimal configuration, ultimately enabling the system to minimize the mean square error of in-flight computation while satisfying all constraints, and simultaneously ensuring a balance between joint sensing accuracy, energy consumption control, and hardware security.

[0197] In this way, the previously constructed optimization model is transformed into specific hardware control instructions, enabling the system to dynamically adapt to changes in the wireless environment and the needs of multi-sensor collaboration. This solves the problem that fixed parameter configurations cannot take into account multi-objective optimization, and ultimately achieves global optimization of the performance of the entire joint computing sensing system.

[0198] In one implementation, the over-the-air computation method further includes the following steps before multiple ISAC sensors transmit the data to be computed to the access point (AP):

[0199] Establish an integrated sensing communication and over-the-air computing system based on movable antenna assistance, which includes sensing targets, access points (APs), and multiple ISAC sensors.

[0200] The AP is configured with multiple movable receiving antennas MA. Each ISAC sensor is synchronously configured with a fixed transmitting antenna FPA and a movable receiving antenna MA. The transmitting direction of the FPA is fixed to face the AP. The movement range of the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor is limited to a preset physical space area.

[0201] The position vectors representing the movement range of the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor, which are limited to a preset physical space area, are expressed as follows:

[0202] );

[0203] Among them, the The position vectors of the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor are respectively represented as follows:

[0204] ;

[0205] .

[0206] It should be noted that before executing the above-mentioned air computing method, the integrated sensing communication and air computing system based on movable antenna assistance can build a joint computing sensing system integrating sensing, communication and computing to sense the signal of the sensing target, and transmit the sensed signal to the access point (AP) after superimposing it. In the process of transmission, air computing is realized, thereby realizing the air computing method based on the integrated sensing communication and air computing system based on movable antenna assistance.

[0207] It should be noted that when building this integrated sensing, communication, and over-the-air computing system based on movable antennas, several core elements can be deployed, including sensing targets, access points (APs), and multiple ISAC sensors, which together constitute the integrated sensing, communication, and over-the-air computing system based on movable antennas. The AP is equipped with multiple movable receiving antennas (MAs), whose positions can be adjusted within a pre-defined physical space. Each ISAC sensor is equipped with both a fixed transmitting antenna (FPA) and a movable receiving antenna (MA). The transmission direction of the fixed transmitting antenna (FPA) is set to always face the AP to ensure the directionality and stability of data transmission. The movement range of the movable receiving antennas (MAs) of the AP and the ISAC sensors is limited to a pre-defined physical space. This design ensures the flexibility of antenna adjustment while avoiding hardware conflicts or signal interference caused by exceeding physical boundaries, providing a basic hardware layout support for the system's subsequent joint sensing and over-the-air computing functions.

[0208] For example, when setting up this system, a sensing target, an access point (AP), and multiple ISAC sensors can be deployed. The AP is equipped with multiple movable receiving antennas (MA), and each ISAC sensor is equipped with both a fixed transmitting antenna (FPA) and a movable receiving antenna (MA). The FPA's transmission direction is fixed towards the AP, while the movable receiving antennas (MA) of the AP and the ISAC sensors are limited to a preset physical space area.

[0209] The setup of movable receiving antennas (MAs) needs to be tailored to the specific scenario. First, based on the approximate location of the target and the deployment distribution of the access points (APs) and sensors, the initial activity area of ​​each MA can be defined to ensure that the antenna can cover the critical angles for signal reception during movement. Then, the initial position can be determined through testing and adjustment to guarantee stable signal reception. This allows the MA to flexibly adjust its position to optimize signal reception; for example, when the target moves or the signal is interfered with, the MA can enhance its ability to capture reflected signals by changing its position. On the other hand, limiting the movement range avoids hardware collisions or signal disruptions caused by excessive antenna movement. This facilitates joint sensing and data reception, improving the reception quality of reflected signals through position adjustment to provide accurate data for joint sensing, and better receiving signals transmitted from other devices, ensuring the overall coordinated operation of the system.

[0210] Based on the first embodiment of this application, a second embodiment of this application is proposed. In this second embodiment, content that is the same as or similar to that in the first embodiment described above can be referred to the above description and will not be repeated hereafter. Based on this, please refer to... Figure 3 , Figure 3 This is a flowchart illustrating the steps of a second embodiment of the over-the-air computing method. As a refinement of step 10 in the over-the-air computing method, it includes steps 110 to 120:

[0211] Step 110: The ISAC sensor first acquires the locally collected data to be calculated and the reflected signal of the sensed target. Then, based on the current MA position parameters of the AP and its own MA position parameters, it adjusts the signal transmission power of the FPA and transmits the data to be calculated to the AP in a directional manner through the FPA.

[0212] Step 120: Each ISAC sensor captures the reflected signal through its own MA, filters the signal noise, and then transmits it to the AP to achieve multi-sensor joint sensing.

[0213] In the second embodiment, the data transmission and signal acquisition process of the ISAC sensor is segmented. The ISAC sensor first acquires two key pieces of information: first, locally acquired data to be calculated, which may come from various monitoring devices or the sensor's own measurement results; second, the signal reflected back from the target, containing characteristic information related to the target. Then, the ISAC sensor can dynamically adjust the signal transmission power of the fixed transmitting antenna FPA by combining the position parameters of the movable receiving antenna MA of the current access point AP and its own movable receiving antenna MA. This ensures that the transmission power matches the current spatial location, avoiding energy waste or interference caused by excessive power, and preventing data transmission quality from being affected by insufficient power. After adjustment, the data to be calculated is directionally sent to the AP via the FPA, ensuring accurate data transmission.

[0214] Meanwhile, each ISAC sensor uses its own movable receiving antenna (MA) to capture the reflected signal of the target. Since the reflected signal may be mixed with various noises during propagation, affecting subsequent sensing accuracy, the sensor performs noise filtering on these signals, retaining only the valid information. The processed reflected signal is transmitted to the AP (Active Detector). After the signals from multiple sensors are aggregated, the AP can achieve joint sensing of the target based on this multi-source information, improving the accuracy and reliability of the sensing. This ensures both efficient transmission of the data to be processed and provides a high-quality signal foundation for joint sensing, allowing for more refined collaborative operation of the entire system.

[0215] In one implementation, the root access point (AP) receives a superimposed signal formed by the superimposed data transmitted from multiple ISAC sensors through its configured movable receiving antenna (MA), and performs data aggregation calculation on the superimposed signal at the physical layer based on the AirComp principle, including:

[0216] In time slot t, each sensor sends a data symbol vector to the access point (AP). , used for AirComp aggregation, where Let be the number of aggregation functions, with signs following an independent Gaussian distribution of zero mean and unit variance, satisfying . and hour ;

[0217] set up For the transmit beamforming matrix, the first The signals emitted by each sensor are transmitted to the AP via a channel. The channel matrix between the AP and the sensors is as follows:

[0218] ;

[0219] in This is the AP receive antenna response matrix. This is the response matrix of the sensor's transmitting antenna. This is the path gain matrix;

[0220] The aggregated signal received by the AP is:

[0221] ;

[0222] in To receive the beamforming matrix, It is Additive White Gaussian Noise (AWGN).

[0223] AirComp performance for receiving signals With launch symbols and Mean squared error (MSE) measurement:

[0224] ;

[0225] In the MA-ISCCO integrated sensing communication and airborne computing system based on a movable antenna, the data symbols transmitted by the sensors are simultaneously used to detect far-field point targets. The radar channel adopts a line-of-sight (LoS) propagation model, expressed as:

[0226] ;

[0227] in, and These are the transmit and receive steering vectors, respectively, and the position of the sensor's fixed transmit antenna. and the location of the movable receiving antenna Related; For complex channel coefficients, For the first The angle of the sensor to the target;

[0228] After matched filtering, the first The target signal received by each sensor is:

[0229] ;

[0230] in, For AWGN;

[0231] Target positioning accuracy is measured by CRB, and the expression is:

[0232] ;

[0233] in, for right The derivative, For the emission covariance matrix, Let be the number of coherent time slots. The trace term in the denominator can be simplified to... , These functions facilitate optimization and solution.

[0234] ;

[0235] ;

[0236] ;

[0237] in: , , ;

[0238] The target position estimates from each sensor are transmitted to the AP via AirComp, and the final positioning result is obtained after aggregation.

[0239] The optimization objective is to minimize the MSE of the AirComp by jointly designing the transmit and receive beamforming and MA position, while satisfying the constraints of CRB, transmit power and antenna movement area.

[0240] The issues related to optimization variables are described as follows:

[0241] ;

[0242] ;

[0243] ;

[0244] ;

[0245] ;

[0246] ;

[0247] ;

[0248] The above formula ensures that the estimation of each sensor meets the required accuracy; limits the transmit power of each sensor; restricts the received MA to a specific spatial region; and enforces a minimum spacing to prevent antenna coupling. Due to variable coupling and non-convex CRB constraints, this problem is NP-hard and difficult to solve directly.

[0249] Based on the first embodiment of this application, a third embodiment of this application is proposed. In this third embodiment, content that is the same as or similar to that of the first embodiment described above can be referred to the above description and will not be repeated hereafter. Based on this, please refer to... Figure 4 , Figure 4 This is a flowchart illustrating the steps of the third embodiment of the over-the-air computing method. As a refinement of step 40 in the over-the-air computing method, it includes steps 410 to 440:

[0250] Step 410: Perform the first iteration, fix the MA position of each ISAC sensor and the MA position of the AP, and optimize the transmit beamforming matrix of each ISAC sensor.

[0251] Step 420: Perform the second iteration, fix the transmit beamforming matrix and MA position, and optimize the AP's receive beamforming matrix.

[0252] Step 430: Perform the third iteration, fix the beamforming matrix, and optimize the MA position of each ISAC sensor.

[0253] Step 440: Perform the fourth iteration, fix the sensor MA position and beamforming matrix, and optimize the AP MA position; repeat the above four iterations until the difference between the mean square error (MSE) of the aerial calculation of two adjacent iterations is less than the preset convergence threshold, stop the iteration and output the current optimization parameters.

[0254] This process involves iterative optimization in stages to gradually improve the system parameters.

[0255] First, in the first iteration, the position of the movable receiving antenna MA of each ISAC sensor and AP remains unchanged, and the focus is on optimizing the transmit beamforming matrix of the sensor. The goal is to make the signal transmitted by the sensor more accurately fit the current antenna layout and reduce transmission loss.

[0256] The second iteration then fixes the optimized transmit beamforming matrix and MA position, and instead optimizes the AP's receive beamforming matrix, enabling the AP to extract useful signals more efficiently, suppress interference, and improve data aggregation accuracy.

[0257] In the third iteration, the beamforming matrix is ​​kept unchanged, and the MA position of each ISAC sensor is adjusted. By optimizing the spatial distribution of the antenna, the ability to capture reflected signals from the sensing target is enhanced, and the joint sensing effect is improved.

[0258] In the subsequent fourth iteration, the MA position of the fixed sensor and the beamforming matrix were fixed, and the MA position of the AP was optimized to further improve the AP's reception quality of signals superimposed from multiple sensors.

[0259] Therefore, the above four rounds of iteration will be repeated. After each cycle, the mean square error (MSE) is calculated in the air. When the difference between two adjacent iterations is less than the preset threshold, it indicates that the system performance has become stable. At this time, the iteration is stopped and the current optimization parameters are output to ensure that the system reaches a better state under the premise of meeting the constraints.

[0260] In one implementation, the steps of solving the optimization model using the block coordinate descent (BCD) algorithm and iteratively optimizing the transmit beamforming matrix of each ISAC sensor, its own receive beamforming matrix, the position of the movable receiving antenna MA of each ISAC sensor, and the position of its own movable receiving antenna MA include:

[0261] The non-convex optimization problem in step 102 is solved based on the BCD method. The optimization variables are divided into four blocks: transmit beamforming matrix, receive beamforming matrix, position of movable antenna (MA) at the sensor, and position of MA at the AP. The optimization is iteratively performed.

[0262] 1) Update the transmit beamforming matrix:

[0263] Optimize the transmit beamforming matrix while keeping other variables fixed. The subproblems are:

[0264] ;

[0265] Introducing auxiliary variables And using semidefinite relaxation (SDR), the subproblem is reconstructed as follows:

[0266]

[0267] ;

[0268] ;

[0269] ;

[0270] Transform it into a convex optimization problem using Schur complement:

[0271] ;

[0272] 2) Update the transmit beamforming matrix:

[0273] With other variables fixed, the optimization of the receiving beamforming matrix Z becomes an unconstrained convex problem:

[0274] ;

[0275] The optimal solution is MMSE beamforming.

[0276] ;

[0277] 3) Update the MA position at the sensor.

[0278] With other variables fixed, the MA position at the sensor is optimized by maximizing the denominator of the CRB. :

[0279] ;

[0280] in, Given a set of positional constraints, transform it into its equivalent form:

[0281] ;

[0282] The SCA method is used to linearize the nonconvex terms, and the update formula is as follows:

[0283] ;

[0284] in , , , The parameters are known.

[0285] 4) Update the MA position at AP.

[0286] Alternating optimization is used, optimizing the position of a single antenna each time. The subproblem is:

[0287] ;

[0288] in, , ;

[0289] Transform into:

[0290] ;

[0291] in, , ;

[0292] The surrogate function is constructed using the MM method, and the update formula is as follows:

[0293] ;

[0294] in, , , Given parameters, This is the projection operator.

[0295] The algorithm flow is as follows: initialize variables, iteratively update transmit beamforming, receive beamforming, sensor MA position, and AP-MA position until the MSE decreases below a threshold. The computational complexity mainly comes from each step, namely... , and .

[0296] The following describes an optimal embodiment of this application to illustrate the content protected by this application. It should be noted that this embodiment is only an example and does not limit the patent scope of this application. All equivalent structural transformations made using the contents of this application's specification and drawings under the technical concept of this application, or direct / indirect applications in other related technical fields, are included within the patent protection scope of this application.

[0297] Please refer to Figure 5 , Figure 5 This is a schematic diagram of a specific process for an air computing method proposed in this application, applied to an integrated sensing communication and air computing system assisted by a movable antenna. The air computing method includes the following exemplary steps:

[0298] Step S101: Construct a novel MA-ISCCO framework assisted by a movable antenna to achieve efficient joint sensing and computation:

[0299] The position vector of the receiving mobile antenna at the AP is represented as follows:

[0300] (1)

[0301] Among them, the The position vectors of the fixed-position transmitting antenna and the mobile receiving antenna at each sensor are respectively represented as follows:

[0302] (2)

[0303] (3)

[0304] Step S102: The optimization objective is to minimize the MSE of the AirComp, defined as the expected squared difference between the estimated received signal and the sum of the transmitted signals; while simultaneously satisfying constraints such as the sensing Cramer-Rao boundary (CRB), transmit power, and antenna position.

[0305] In time slot t, each sensor sends a data symbol vector to the access point (AP). , used for AirComp aggregation, where Let be the number of aggregation functions. The signs of these functions follow an independent Gaussian distribution with zero mean and unit variance, satisfying . and hour ;

[0306] set up For the transmit beamforming matrix, the first The signals emitted by each sensor are transmitted to the AP via a channel. The channel matrix between the AP and the sensors is as follows:

[0307] (4)

[0308] in This is the AP receive antenna response matrix. This is the response matrix of the sensor's transmitting antenna. This is the path gain matrix;

[0309] The aggregated signal received by the AP is:

[0310] (5)

[0311] in To receive the beamforming matrix, It is Additive White Gaussian Noise (AWGN).

[0312] AirComp performance for receiving signals With launch symbols and Mean squared error (MSE) measurement:

[0313] (6)

[0314] In the MA-ISCCO system, the data symbols transmitted by the sensor are also used to detect far-field point targets. The radar channel adopts a line-of-sight (LoS) propagation model, expressed as:

[0315] (7)

[0316] in: and These are the transmit and receive steering vectors, respectively, and the position of the sensor's fixed transmit antenna. and the location of the movable receiving antenna Related; For complex channel coefficients, For the first The angle of the sensor to the target;

[0317] After matched filtering, the first The target signal received by each sensor is:

[0318] (8)

[0319] in: For AWGN;

[0320] Target positioning accuracy is measured by CRB, and the expression is:

[0321] (9)

[0322] in: for right The derivative, For the emission covariance matrix, Let be the number of coherent time slots. The trace term in the denominator can be simplified to... , These functions facilitate optimization and solution.

[0323] (10a)

[0324] (10b)

[0325] (10c)

[0326] in: , , ;

[0327] The target position estimates from each sensor are transmitted to the AP via AirComp, and the final positioning result is obtained after aggregation.

[0328] The optimization objective is to minimize the MSE of the AirComp by jointly designing the transmit and receive beamforming and MA position, while satisfying the constraints of CRB, transmit power, and antenna movement area. The problem related to the optimization variables is described as follows:

[0329] ;

[0330] (11a)

[0331] (11b)

[0332] (11c)

[0333] (11d)

[0334] (11e)

[0335] (11f)

[0336] Specifically: (11a) ensures that the estimation of each sensor meets the required accuracy; (11b) limits the transmit power of each sensor; (11c) and (11e) restrict the received MA to a specific spatial region; (11d) and (11f) enforce a minimum spacing to prevent antenna coupling. Due to variable coupling and non-convex CRB constraints, this problem is NP-hard and difficult to solve directly.

[0337] Step S103: For the non-convex optimization problem in step 102, we developed a low-complexity algorithm based on BCD, which integrates Schur complement, MMSE beamforming SCA and MM methods;

[0338] The non-convex optimization problem in step 102 is solved based on the BCD method. The optimization variables are divided into four blocks: transmit beamforming matrix, receive beamforming matrix, position of movable antenna (MA) at the sensor, and position of MA at the AP. The optimization is performed iteratively.

[0339] 1) Update the transmit beamforming matrix:

[0340] Optimize the transmit beamforming matrix while keeping other variables fixed. The subproblems are:

[0341] (12)

[0342] Introducing auxiliary variables And using semidefinite relaxation (SDR), the subproblem is reconstructed as follows:

[0343]

[0344] (13a)

[0345] (13b)

[0346] (13c)

[0347] Transform it into a convex optimization problem using Schur complement:

[0348] (14)

[0349] 2) Update the transmit beamforming matrix:

[0350] With other variables fixed, the optimization of the receiving beamforming matrix Z becomes an unconstrained convex problem:

[0351] (15)

[0352] The optimal solution is MMSE beamforming.

[0353] (16)

[0354] 3) Update the MA position at the sensor:

[0355] With other variables fixed, the MA position at the sensor is optimized by maximizing the denominator of the CRB. :

[0356] (17)

[0357] in: Given a set of positional constraints, transform it into its equivalent form:

[0358] (18)

[0359] The SCA method is used to linearize the nonconvex terms, and the update formula is as follows:

[0360] (19)

[0361] in: , , , The parameters are known.

[0362] 4) Update the MA position at AP:

[0363] Alternating optimization is used, optimizing the position of a single antenna each time. The subproblem is:

[0364] (20)

[0365] in: , ;

[0366] Transform into:

[0367] (twenty one)

[0368] in: , ;

[0369] The surrogate function is constructed using the MM method, and the update formula is as follows:

[0370] (twenty two)

[0371] in, , , Given parameters, This is the projection operator.

[0372] The algorithm flow is as follows: initialize variables, iteratively update transmit beamforming, receive beamforming, sensor MA position, and AP-MA position until the MSE decreases below a threshold. The computational complexity mainly comes from each step, namely... , and .

[0373] The performance of the proposed algorithm was evaluated using Monte Carlo simulation. The system parameters are as follows:

[0374] Operating frequency 2GHz (wavelength) =015m, 4 ISAC sensors (each with 4 fixed transmit antennas and 4 movable receive antennas), AP with 8 movable receive antennas; the channel model contains 10 multipaths, and parameters such as path loss and radar channel coefficient are set according to the standard model; the comparison schemes include PSO, fixed antenna (FPA), movable antenna only for AP / sensor (AP-MA / Sensor-MA), and partitioned moving baselines.

[0375] Figures 6(a) and 6(b) show the convergence and complexity of the present invention, respectively. As can be seen from Figure 6(a), the proposed algorithm converges within 30 iterations, and the performance of AirCompMSE is comparable to that of PSO. As can be seen from Figure 6(b), the CPU time required for its convergence is about half that of PSO, and the computational complexity is lower.

[0376] Figures 7(a), 7(b), and 7(c) show the performance comparison between the proposed solution and the conventional solution. Figure 7(a) shows that the proposed solution has the lowest MSE at all transmit powers, with a more significant advantage in low-power scenarios, demonstrating the functional trade-off capability of the movable antenna in resource-constrained systems. Figure 7(b) shows that compared to the FPA, Sensor-MA, and AP-MA solutions, the MSE is reduced by 378%, 297%, and 128%, respectively, because the movable antenna efficiently utilizes spatial degrees of freedom to improve the channel through position optimization. Figure 7(c) shows that as the antenna movement area expands, the proposed solution consistently outperforms the partitioned movement solution, with a greater reduction in MSE.

[0377] Among them, the traditional schemes include: (1) POS: using the Particle Swarm Optimization (PSO) algorithm to update the position of MA; (2) FPA: both the sensor and the AP are equipped with FPA; (3) AP-MA: only the AP is equipped with MA; (4) Sensor-MA: only the sensor is equipped with MA; (5) Partitions: the motion area of ​​MAs is divided into partitions, and each MA is restricted to its designated area.

[0378] The integrated sensing communication and over-the-air computing system and joint optimization method based on movable antenna assistance proposed in this invention significantly outperforms the baseline scheme through spatial diversity enhancement using movable antenna.

[0379] Through the above description, the basic functions of the over-the-air computing method of the present invention, applied to an integrated sensing communication and over-the-air computing system based on a movable antenna, have been explained. This over-the-air computing method of the present invention, applied to an integrated sensing communication and over-the-air computing system based on a movable antenna, can achieve accurate solutions for power flow in distribution networks with hybrid interconnected inverters / grid-type inverters, and is of great significance for the accurate analysis of voltage limit exceeding problems in distribution networks with hybrid interconnected inverters / grid-type inverters.

[0380] This embodiment is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. An integrated sensing communication and over-the-air computing system based on a movable antenna, characterized in that, The integrated sensing communication and over-the-air computing system based on movable antenna assistance includes: Base station and multiple ISAC sensors; The base station includes at least one access point (AP), and the access point (AP) is configured with multiple movable receiving antennas (MA). The movable receiving antennas (MA) can dynamically adjust their positions within a preset spatial range. Each of the ISAC sensors is equipped with a fixed transmitting antenna FPA and a movable receiving antenna MA. The fixed transmitting antenna FPA is used to transmit the data to be calculated to the access point AP. The movable receiving antenna MA of the ISAC sensor can dynamically adjust its position within a preset spatial range to receive the reflected signal of the sensing target. The access point (AP) receives the superimposed signal of the data to be calculated transmitted by multiple ISAC sensors through the fixed transmitting antenna (FPA) via its movable receiving antenna (MA), and performs data aggregation calculation at the physical layer based on the AirComp principle. The access point (AP) is also used to integrate the reflected signals received by the movable receiving antennas (MA) of each of the ISAC sensors to achieve joint sensing, and to construct an optimization model with the goal of minimizing the mean square error of aerial computation based on the Cramer-Rao boundary (CRB) of sensing accuracy, the transmit power of each of the ISAC sensors and the position constraints of the movable receiving antennas (MA). The access point (AP) solves the optimization model using the block coordinate descent (BCD) algorithm, iteratively optimizing the transmit beamforming matrix of each of the ISAC sensors, its own receive beamforming matrix, the position of the movable receiving antenna (MA) of each of the ISAC sensors, and its own movable receiving antenna (MA) position, in order to adapt to the uplink multi-sensor joint sensing and computing scenario.

2. The integrated sensing communication and over-the-air computing system based on a movable antenna as described in claim 1, characterized in that, Both the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor are equipped with a position driving module and a position feedback module. The position driving module is used to drive the antenna to translate or rotate within a preset spatial range according to the optimized position parameters output by the access point (AP). The location feedback module is used to collect the actual location information of the antenna in real time and feed it back to the access point (AP) to correct the location constraint parameters in the optimization model.

3. The integrated sensing communication and over-the-air computing system based on a movable antenna as described in claim 1, characterized in that, In the optimization model constructed by the access point (AP), the Cramé-Rao bound constraint for sensing accuracy is specifically as follows: Based on the reflected signals received by the movable receiving antenna MA of each of the ISAC sensors, the theoretical lower limit of the positioning error of the perceived target is calculated, and the lower limit does not exceed a preset threshold. Specifically, the positional constraint of the movable receiving antenna MA is as follows: the distance between any two movable receiving antennas MA is not less than a preset safety distance, and the antenna position does not exceed the preset physical space boundary.

4. The integrated sensing communication and over-the-air computing system based on a movable antenna as described in claim 1, characterized in that, The ISAC sensor is also equipped with a signal preprocessing module; The signal preprocessing module is used to perform noise reduction and modulation processing on the data to be transmitted by the fixed transmitting antenna FPA, and to adjust the signal transmission direction according to the transmit beamforming matrix issued by the access point AP. The access point (AP) is also equipped with a calculation result verification module, which compares the data aggregation calculation result obtained based on the AirComp principle with a preset accuracy threshold. If the threshold is exceeded, the block coordinate descent (BCD) algorithm is triggered to iterate and optimize again.

5. An aerial calculation method, characterized in that, The aforementioned aerial computing method is applied to an integrated sensing communication and aerial computing system based on a movable antenna, and the aerial computing method includes: The multiple ISAC sensors of the integrated sensing communication and over-the-air computing system based on movable antennas transmit the data to be calculated to the access point (AP) of the base station through their respective fixed transmitting antennas (FPAs) and receive the reflected signals of the sensing targets through their respective movable receiving antennas (MAs). The access point (AP) receives a superimposed signal formed by the superimposed data transmitted by multiple ISAC sensors through its configured movable receiving antenna (MA), and performs data aggregation calculation on the superimposed signal at the physical layer based on the AirComp principle. The access point (AP) integrates the reflected signals received by the movable receiving antennas (MA) of each of the ISAC sensors to achieve joint sensing. Based on the Cramer-Rao boundary (CRB) of sensing accuracy, the transmit power of each ISAC sensor, and the positional constraints of the movable receiving antennas (MA), an optimization model is constructed with the goal of minimizing the mean square error of aerial computation (MSE). The access point (AP) solves the optimization model using the block coordinate descent (BCD) algorithm, iteratively optimizing the transmit beamforming matrix of each of the ISAC sensors, its own receive beamforming matrix, the position of the movable receiving antenna (MA) of each of the ISAC sensors, and the position of its own movable receiving antenna (MA).

6. The aerial calculation method according to claim 5, characterized in that, Before the multiple ISAC sensors transmit the data to be computed to the access point (AP), the over-the-air computing method further includes: Construct an integrated sensing communication and over-the-air computing system based on movable antenna assistance, which includes the sensing target, the access point (AP), and multiple ISAC sensors; The access point (AP) is configured with multiple movable receiving antennas (MA), and each ISAC sensor is synchronously configured with a fixed transmitting antenna (FPA) and a movable receiving antenna (MA). The transmitting direction of the FPA is fixed to be towards the AP. The movement range of the movable receiving antennas (MA) of the access point AP and the movable receiving antennas (MA) of the ISAC sensor is limited to a preset physical space area. The position vectors representing the movement range of the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor, which are limited to a preset physical space area, are expressed as follows: ）； Among them, the The position vectors of the movable receiving antenna MA of the access point AP and the movable receiving antenna MA of the ISAC sensor are respectively represented as follows: ； 。 7. The aerial calculation method according to claim 5, characterized in that, The steps of the multiple ISAC sensors transmitting data to be calculated to the access point (AP) of the base station through their respective fixed transmitting antennas (FPAs) and receiving reflected signals from the sensed target through their respective movable receiving antennas (MAs) include: The ISAC sensor first acquires the locally collected data to be calculated and the reflected signal of the sensed target. Then, based on the current AP's MA position parameters and its own MA position parameters, it adjusts the signal transmission power of the FPA and transmits the data to be calculated to the AP in a directional manner through the FPA. Each of the aforementioned ISAC sensors captures reflected signals through its own MA, filters out signal noise, and then transmits the signals to the AP to achieve multi-sensor joint sensing.

8. The aerial calculation method according to claim 5, characterized in that, The steps of receiving the superimposed signal formed by the superimposed data transmitted by multiple ISAC sensors through its configured movable receiving antenna MA, and performing data aggregation calculation on the superimposed signal at the physical layer based on the AirComp principle, include: In time slot t, each sensor sends a data symbol vector to the access point (AP). , used for AirComp aggregation, where Let be the number of aggregation functions, with signs following an independent Gaussian distribution of zero mean and unit variance, satisfying . and hour ; set up For the transmit beamforming matrix, the first The transmitted signals from each sensor are transmitted to the AP via a channel. The channel matrix between the AP and the sensors is as follows: ； in This is the AP receive antenna response matrix. This is the response matrix of the sensor's transmitting antenna. This is the path gain matrix; The aggregated signal received by the AP is: ； in To receive the beamforming matrix, It is Additive White Gaussian Noise (AWGN). Air Comp performance for receiving signals With launch symbols and Mean squared error (MSE) measurement: ； In the MA-ISCCO integrated sensing communication and airborne computing system based on a movable antenna, the data symbols transmitted by the sensors are simultaneously used to detect far-field point targets. The radar channel adopts a line-of-sight (LoS) propagation model, expressed as: ； in, and These are the transmit and receive steering vectors, respectively, and the position of the sensor's fixed transmit antenna. and the location of the movable receiving antenna Related; For complex channel coefficients, For the first The angle of the sensor to the target; After matched filtering, the first The target signal received by each sensor is: ； in, For AWGN; Target positioning accuracy is measured by CRB, and the expression is: ； in, for right The derivative, For the emission covariance matrix, Let be the number of coherent time slots. The trace term in the denominator can be simplified to... , These functions facilitate optimization and solution. ； ； ； in: , , ; The target position estimates from each sensor are transmitted to the AP via AirComp, and the final positioning result is obtained after aggregation. The optimization objective is to minimize the MSE of the AirComp by jointly designing the transmit and receive beamforming and MA position, while satisfying the constraints of CRB, transmit power and antenna movement area. The issues related to optimization variables are described as follows: ； ； ； ； ； ； ； The above formula ensures that the estimation of each sensor meets the required accuracy; limits the transmit power of each sensor; restricts the received MA to a specific spatial region; and enforces a minimum spacing to prevent antenna coupling. Due to variable coupling and non-convex CRB constraints, this problem is NP-hard and difficult to solve directly.

9. The aerial calculation method according to claim 5, characterized in that, Solving the optimization model using the block coordinate descent BCD algorithm, and iteratively optimizing the transmit beamforming matrix, the receive beamforming matrix, the movable receiving antenna MA position, and the movable receiving antenna MA position of each ISAC sensor, includes the following steps: The first iteration is performed, fixing the MA position of each ISAC sensor and the MA position of the AP, and optimizing the transmit beamforming matrix of each ISAC sensor; The second iteration is performed, fixing the transmit beamforming matrix and MA position, and optimizing the receive beamforming matrix of the AP; The third iteration is performed, with the beamforming matrix fixed, to optimize the MA position of each of the ISAC sensors; Perform the fourth iteration, fix the sensor MA position and beamforming matrix, and optimize the AP MA position; repeat the above four iterations until the difference between the mean square error (MSE) of the aerial computation between two adjacent iterations is less than the preset convergence threshold, stop the iteration and output the current optimization parameters.

10. The aerial calculation method according to claim 9, characterized in that, The step of solving the optimization model using the block coordinate descent BCD algorithm and iteratively optimizing the transmit beamforming matrix, the receive beamforming matrix, the movable receiving antenna MA position, and the movable receiving antenna MA position of each ISAC sensor includes: The non-convex optimization problem in step 102 is solved based on the BCD method. The optimization variables are divided into four blocks: transmit beamforming matrix, receive beamforming matrix, position of movable antenna (MA) at the sensor, and position of MA at the AP. The optimization is iteratively performed. 1) Update the transmit beamforming matrix: Optimize the transmit beamforming matrix while keeping other variables fixed. The subproblems are: ； Introducing auxiliary variables And by employing semidefinite relaxation (SDR), the subproblem is reconstructed as follows: ； ； ； ； Transform it into a convex optimization problem using Schur complement: ； 2) Update the transmit beamforming matrix: With other variables fixed, the optimization of the receiving beamforming matrix Z becomes an unconstrained convex problem: ； The optimal solution is MMSE beamforming. ； 3) Update the MA position at the sensor. With other variables fixed, the MA position at the sensor is optimized by maximizing the denominator of the CRB. : ； in, Given a set of positional constraints, transform it into its equivalent form: ； The SCA method is used to linearize the nonconvex terms, and the update formula is as follows: ； in , , , The parameters are known. 4) Update the MA position at AP. Alternating optimization is used, optimizing the position of a single antenna each time. The subproblem is: ； in, , ; Transform into: ； in, , ; The surrogate function is constructed using the MM method, and the update formula is as follows: ； in, , , Given parameters, For projection operators; The algorithm flow is as follows: initialize variables, iteratively update transmit beamforming, receive beamforming, sensor MA position, and AP-MA position until the MSE decreases below a threshold. The computational complexity mainly comes from each step, namely... , and .

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