Low-noise digital bandwidth compensation method for high-order frequency response of sensor

Abstract

The invention provides a low-noise digital bandwidth compensation method for high-order frequency response of a sensor. The signal-to-noise ratio in the compensation process is remarkably improved while the signal bandwidth is expanded. The method comprises the following steps: S1, constructing a lumped parameter model based on a sensor actual measurement physical process, and deducing and calculating an overall high-order transfer function of the lumped parameter model; s2, acquiring actual sensor frequency response data and fitting the actual sensor frequency response data to determine model parameters in the high-order transfer function; s3, constructing an ideal expectation response model according to the target compensation bandwidth, and dividing the transfer function of the ideal expectation response model by the fitting transfer function obtained in the step S2 to obtain a continuous time transfer function of the compensation filter; s4, discretizing the continuous time transfer function of the compensation filter to obtain an infinite impulse response filter; and S5, cascading the infinite impulse response filter with a low-pass digital filter to form a final compensation filter to carry out bandwidth compensation on the output signal of the sensor.

Description

Technical Field

[0001] This invention belongs to the field of sensor signal processing technology, and relates to digital signal processing, sensor bandwidth compensation technology and adaptive filter technology for MEMS (Micro-Electro-Mechanical Systems) sensors. Specifically, it is a low-noise digital bandwidth compensation method for the high-order frequency response of capacitive MEMS accelerometers. Background Technology

[0002] Capacitive MEMS accelerometers, with their advantages of small size, low cost, and high integration, have been widely used in consumer electronics, navigation systems, and autonomous driving. These sensors typically operate within a bandwidth of several hundred hertz, with high-performance models reaching several thousand hertz. However, with the emergence of new applications such as acoustic sensing, high-impact motion detection, and industrial condition monitoring, the industry urgently needs MEMS accelerometers with superior performance. Among various performance indicators, bandwidth shows the most significant difference, making bandwidth improvement a key technological bottleneck for expanding the applications of MEMS accelerometers.

[0003] Bandwidth compensation technology, with its flexibility, universal model, and low implementation cost, has received widespread attention in recent years to address the bandwidth enhancement problem of sensors. Bandwidth compensation technology enhances frequency response by introducing a dynamic characteristic compensator at the sensor output and cascading it with the sensor. Currently, research has applied bandwidth compensation technology to capacitive MEMS accelerometers. References [1,2] introduce a digital bandwidth compensation method based on a second-order system model, designing a digital notch filter to compensate for the frequency band of the capacitive MEMS accelerometer near its resonant frequency. Although a capacitive MEMS accelerometer can theoretically be simplified to a second-order system, in practical applications, the complex motion modes of the MEMS comb structure and the modal response of the circuit board supporting the MEMS chip result in a complex high-order frequency response. For bandwidth compensation of high-order complex frequency responses, research on bandwidth compensation for other types of sensors can be referenced, mainly divided into two categories: one is a transfer function-based modeling method, and the other is a data-driven system identification method.

[0004] The transfer function-based modeling method first identifies the transfer function of the sensor, and then designs a dynamic characteristic compensator based on the transfer function using the pole-zero placement method. For the high-order frequency response of the sensor, references [3, 4, 5] propose a method for fitting the discrete-time transfer function to identify the sensor's transfer function. This method traverses from the low-order form of the transfer function to the high-order form, uses the least squares method to fit the parameters of the transfer function, selects the optimal transfer function order by minimizing the mean square error of the fitting result, and designs a digital compensation filter as a dynamic characteristic compensator accordingly. In reference [6], a study proposes to fit the detailed frequency response by using a cascaded second-order error model, which extends the structure and order of the traditional second-order notch filter as a dynamic characteristic compensator. The order can be adjusted as needed, and the higher the order, the smaller the bandwidth compensation error.

[0005] Data-driven system identification methods do not employ explicit sensor modeling, but instead directly optimize the dynamic characteristic compensator using the output of a high-precision reference sensor. In reference [7], researchers used a swarm intelligence algorithm to iteratively optimize the coefficients of the digital compensation filter. In addition, reference [8] introduced sparse decomposition techniques to reduce the complexity of the compensator and improve the optimization accuracy. The latest research references [9, 10, 11] use deep learning models such as convolutional neural networks (CNN), generative adversarial networks (GAN), and long short-term memory networks (LSTM) to extend the compensator to nonlinear systems.

[0006] While system identification methods may theoretically achieve better compensation results, their high dependence on massive amounts of high-quality training data and the need for multiple optimization iterations lead to high deployment costs. Furthermore, existing system identification methods often employ time-domain mean square error as the optimization strategy for the compensator. This approach often fails to capture weak frequency response characteristics within specific frequency bands and lacks the ability to interpret sensor frequency responses, thus limiting the effectiveness of their localized compensation. In contrast, transfer function-based compensation methods offer a solid mathematical foundation, simple compensation parameter deployment, and stable application, making them particularly suitable for target compensation within specific frequency bands.

[0007] Despite significant advancements in existing sensor bandwidth compensation techniques, there are still considerable shortcomings in addressing the bandwidth compensation problem for the high-order frequency response of capacitive MEMS accelerometers.

[0008] 1. Currently, the only bandwidth compensation technology applied to capacitive MEMS accelerometers is the digital bandwidth compensation method based on a second-order system model. Compensation filters based on the second-order system transfer function can only compensate for local frequency responses (near the resonant frequency) with a gain greater than 0dB, limiting the bandwidth compensation range. While higher-order transfer functions can theoretically extend the compensable bandwidth, existing higher-order transfer functions rely entirely on fitting mathematical models in the z-domain. This approach easily leads to overfitting of the frequency response, resulting in pole-zero distributions in the parameter fitting results that do not conform to physical laws. Therefore, it is necessary to map the fitted poles and zeros to the unit circle in the z-domain for additional stabilization post-processing. However, this post-processing may introduce distortion into the fitted frequency response, increasing the bandwidth compensation error.

[0009] 2. Bandwidth compensation technology enhances frequency response by introducing a dynamic characteristic compensator at the sensor output and cascading it with the sensor. However, current bandwidth compensation technology does not consider the impact of cascading with the sensor on the original sensor output noise. When the compensator extends the bandwidth for a frequency band where the sensor's original gain is below 0dB, it will inevitably amplify the sensor's noise, which is a defect ignored by traditional bandwidth compensation. In some high-frequency bands, the compensator's gain can even reach 15dB (e.g., reference [3]), which will significantly amplify the sensor's noise and reduce the signal-to-noise ratio of the compensated signal.

[0010] The aforementioned literature information is as follows: [1] Lin J, Pham L, Tao R. A Low-Power, Wide-Bandwidth, Three-AxisMEMS Accelerometer ASIC Using Beyond-Resonant-Frequency Sensing[J / OL]. IEEEjournal of solid-state circuits, 2024, 59(3): 774-783. [2] Rossi GB , Crenna F .Dynamic Compensation for a Second OrderMeasuring Device[C] / / 2020 IEEE International Instrumentation and MeasurementTechnology Conference (I2MTC).IEEE, 2020. [3] Fan Y , Deren K , Lin K .Accurate measurement of high-frequencyblast waves through dynamic compensation of miniature piezoelectric pressuresensors[J].Sensors & Actuators A Physical, 2018, 280: [4] Yang W , Li W , Lu H ,et al.Dynamic Compensation Method forHumidity Sensors Based on Temperature and Humidity Decoupling[J].Sensors(14248220), 2022, 22(19). [5] Paniagua G ,R. Dénos.Digital compensation of pressure sensors inthe time domain[J].Experiments in Fluids, 2002, 32(4):417-424. [6] Yang S L , Yang R , Zha F Y ,et al.Dynamic compensation methodbased on system identification and error-overrun mode correction for strainforce sensor[J].Mechanical Systems and Signal Processing, 2020, 140:106649-. [7] Juan W , Zhijie Z , Wenjie Y ,et al.The Dynamic CompensationMethod Based on Improving QR-PSO Algorithm for Pressure Sensor[J].ChineseJournal of Sensors and Actuators, 2017, 30(4):550-554. [8] Xu B , Han T , Liu H ,et al.Dynamic Compensation ofPiezoresistive Pressure Sensor Based on Sparse Domain[J].Journal of Sensors,2020, 2020(1):1-8. [9] Zhao B, Shi W, Sun D,et al.Extending Velocity Sensor Bandwidthby Compensating Temperature Dependency Based on BP Neural Network[J].IEEEAccess, 2019, 7:1-1.

[10] Wang

[11] Zhao B, Shi W, Sun D, ​​et al. Compensation Network for Temperature Dependency and Bandwidth Expanding of Magnetoelectric VelocitySensor Based on Phase Prediction Compensator[J]. IEEE Sensors Journal, 2020, PP(99):1-1. Summary of the Invention

[0011] To address the shortcomings of existing bandwidth compensation techniques, this invention provides a low-noise digital bandwidth compensation method for the high-order frequency response of capacitive MEMS accelerometers. By establishing a lumped parameter model based on the actual physical processes of the sensor, a highly robust digital compensation filter is designed, and adaptive filtering technology is introduced to suppress noise introduced by the compensation. This achieves the dual advantages of wide bandwidth and low-noise output, effectively correcting the frequency response and suppressing noise interference. It also significantly improves the signal-to-noise ratio (SNR) during the compensation process while expanding the signal bandwidth.

[0012] The technical solution adopted by this invention to solve its technical problem is as follows: A method for low-noise digital bandwidth compensation of the high-order frequency response of a sensor, comprising at least the following steps: S1: Construct a lumped parameter model based on the actual physical process measured by the sensor, and derive the global higher-order transfer function for calculating the lumped parameter model. ; S2: Acquire actual sensor frequency response data and fit it to determine the higher-order transfer function. The model parameters are used to obtain the fitted transfer function; S3: Construct an ideal expected response model based on the target compensation bandwidth, and divide its transfer function by the fitted transfer function obtained in step S2 to obtain the continuous-time transfer function of the compensation filter. S4: Discretize the continuous-time transfer function of the compensation filter to obtain an infinite impulse response filter; S5: The infinite impulse response filter is cascaded with a low-pass digital filter to form the final compensation filter to compensate the bandwidth of the sensor output signal.

[0013] Furthermore, this method is applied to capacitive MEMS accelerometers.

[0014] Furthermore, the lumped parameter model mentioned in step S1 includes a mechanical response structure model, a readout circuit model, and a low-pass filter circuit model.

[0015] Furthermore, the derivation and calculation of the global higher-order transfer function of the lumped parameter model in step S1... ,include: S101. Based on the aforementioned mechanical response structural model, derive and calculate the first differential equation: ; in, It represents the acceleration of the object being measured; Indicates the displacement of the PCB board; , and These represent the equivalent mass, stiffness, and damping at the connection between the PCB board and the object under test, respectively. This indicates the displacement of the sensor chip housing relative to the PCB board. Indicates the quality of the sensor chip casing; and These represent the stiffness and damping of the pad, respectively. This indicates the displacement of the comb-shaped mass block inside the sensor chip used for sensing relative to the sensor chip casing. , and These represent the mass, stiffness, and damping of the comb tooth mass block, respectively. Applying the Laplace transform to the first differential equation yields the transfer function. ; S102. Based on the readout circuit model, derive and calculate the second differential equation: ; in, This is the output voltage of the capacitor-voltage readout circuit; This represents the displacement of the comb tooth mass block; To read the circuit gain; Applying the Laplace transform to the second differential equation yields the transfer function. ; S103. Based on the aforementioned low-pass filter circuit model, derive and calculate the third differential equation: ; in: The voltage after filtering is RC; RC is the time constant of the low-pass filter circuit. Performing a Laplace transform on the third differential equation yields the transfer function. ; S104. Will , as well as Multiply the three together and then... By normalizing the sensor sensitivity and simplifying the model parameter processing, the overall high-order transfer function of the lumped parameter model is obtained. .

[0016] Furthermore, the overall higher-order transfer function The expression is as follows: ; in, , , and Indicates the natural frequency; , , and Indicates the damping ratio; is the time constant, and s is the complex variable of the transfer function, representing the system's response to input signals of different frequencies.

[0017] Furthermore, in step S2, the frequency response data of the actual sensor is obtained through dynamic calibration, using the overall higher-order transfer function. As a model, the least squares method is used to fit the frequency response data of the actual sensor to determine the overall higher-order transfer function. The specific values ​​of the natural frequency, damping ratio, and time constant.

[0018] Furthermore, according to the low-noise digital bandwidth compensation method for high-order frequency response as described in claim 1, the pole frequency of the ideal desired response model in step S3 is set at a position not less than three times the target compensation bandwidth.

[0019] Furthermore, it also includes: S6: Perform adaptive noise suppression on the signal after final compensation filter processing. The adaptive noise suppression is implemented using an adaptive Kalman filter based on an autoregressive model.

[0020] Furthermore, the adaptive noise suppression step includes: The noise floor after sensor compensation is measured offline, and its autocorrelation function is calculated and recorded. ; When performing bandwidth compensation online, the output signal is calculated in real time. autocorrelation function Then the autocorrelation function of the ideal compensation output is... Approximately ; based on The Levinson-Durbin recursive method was used to estimate the coefficients of the autoregressive model and white noise. The variance; A Kalman filter state-space model is constructed with the ideal compensation output as the system state, the autoregressive model as the state transition model, and the actual compensation signal as the observation value to achieve noise suppression.

[0021] Furthermore, the state-space model of the Kalman filter is expressed as: ; ; in: Represents the state variable matrix; Represents the state transition matrix; This represents the state variable matrix from the previous time step; Represents the observation matrix; Indicates process noise; This represents the sensor noise after amplification by the final compensation filter; This represents the ideal compensation signal output by the final compensation filter.

[0022] The beneficial effects of this invention include: 1. This invention proposes a physics-based digital compensation filter design method. By analyzing the actual measurement physics of a capacitive MEMS accelerometer, a lumped parameter model is established. Based on this model, a higher-order transfer function is derived, enabling robust fitting of the higher-order frequency response. The digital compensation filter designed using the fitted higher-order transfer function and the zero-pole placement method has strong physical interpretability and structural robustness, avoiding the overfitting and zero-pole distortion problems commonly found in pure mathematical models of discrete-time transfer functions, and effectively reducing the error of bandwidth compensation over a wide frequency range.

[0023] 2. This invention proposes a bandwidth compensation method that combines adaptive noise suppression with a digital compensation filter. Considering the characteristics of the noise floor after bandwidth compensation, the autocorrelation of the compensated signal is calculated in real time. An adaptive Kalman filter based on an autoregressive model is used, which can effectively suppress noise interference in non-signal frequency bands. In particular, it has a significant suppression effect on high-frequency noise amplified by the compensation filter. This solves the problem that when the gain of the digital compensation filter exceeds 0dB, it will significantly amplify the noise of the sensor and reduce the signal-to-noise ratio of the compensated signal. Thus, the bandwidth compensation method proposed in this invention can improve the signal-to-noise ratio of the compensated signal while maintaining the extended compensation bandwidth.

[0024] In summary, the bandwidth compensation method proposed in this invention enables capacitive MEMS accelerometers to achieve the dual advantages of wide bandwidth and low noise output. This method can effectively correct the frequency response and suppress noise interference, while significantly improving the signal-to-noise ratio (SNR) during the compensation process while expanding the signal bandwidth. Attached Figure Description

[0025] Figure 1 This is a flowchart of the overall bandwidth compensation scheme in this invention; Figure 2 A lumped parameter model for each link in the signal transmission path of a capacitive MEMS sensor. Figure 3 The flowchart shows the specific implementation scheme of the adaptive noise suppression method (AR model update is achieved through the Levinson-Durbin recursive method, and the estimation of the compensation output is achieved through Kalman filtering). Figure 4 The fitting results of the higher-order transfer function of the lumped parameter model, and the amplitude-frequency and phase-frequency characteristics of the corresponding designed digital compensation filter; Figure 5 The experimental results for adaptive noise suppression in the time and frequency domains are shown (when AR-Kalman adaptive filtering is enabled, the signal-to-noise ratio (SNR) increases from 7.42 dB to 12.48 dB). Figure 6This is a comparison chart of the present invention and other bandwidth compensation methods (due to the reduction of bandwidth compensation error, the bandwidth can be extended to a maximum of 6000Hz, and the phase linear regression coefficient is the highest at 0.9995). Detailed Implementation

[0026] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0027] Furthermore, the technical features involved in the different embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0028] This invention provides a low-noise digital bandwidth compensation method for the high-order frequency response of capacitive MEMS accelerometers. The key technology of this method is: 1. High-order transfer function model for measuring physical processes based on capacitive MEMS accelerometer: The lumped parameter model established by this method combines the coupled dynamic characteristics of mechanical structure and electrical interface. The high-order transfer function derived from this model has strong physical interpretability and structural robustness, avoiding the overfitting and zero-pole distortion problems commonly found in pure mathematical models of amplitude-frequency and phase-frequency characteristics, and effectively reducing the error of bandwidth compensation in a wide frequency range.

[0029] 2. A digital bandwidth compensation method combining adaptive noise suppression and bandwidth compensation filter: By combining an adaptive Kalman filter based on an autoregressive model with a bandwidth compensation filter, noise interference in non-signal frequency bands is effectively suppressed, especially the high-frequency noise amplified by the compensation filter. This allows the bandwidth compensation method proposed in this invention to maintain the extended compensation bandwidth while improving the signal-to-noise ratio of the compensated signal.

[0030] Example 1: Reference Figure 1 The specific implementation steps of this method are as follows: 1) Physics-based compensation filter design.

[0031] MEMS devices are typically soldered onto printed circuit boards (PCBs), providing both electrical connection and structural support. Therefore, the signal transmission path of a capacitive MEMS sensor involves multiple stages: first, the external acceleration signal arrives at the PCB and is connected to the MEMS chip housing via solder pads; then, the signal is transmitted to the sensing mass block inside the chip, and finally converted into a voltage signal by the readout circuit and low-pass filter, such as... Figure 2 As shown.

[0032] The lumped parameter model used for high-order frequency response compensation consists of three parts: the mechanical response structure model, the readout circuit model, and the low-pass filter circuit model. Based on the lumped parameter model, differential equations describing the system characteristics can be derived.

[0033] The response characteristics of the measured object to acceleration are analyzed in the reference coordinate system. Due to the small size of the MEMS devices, the influence of the interaction forces during the cascading process can be ignored. The flexible connection between the PCB and the measured object is modeled as a mass-spring-damped system. The MEMS chip casing is soldered to the PCB via pads and is modeled as another mass-spring-damped system. The behavior of the comb-shaped mass block inside the MEMS chip used for sensing is described using a second-order dynamic system. Based on the above mechanical response structural model, the dynamic differential equation (displacement change of the comb-shaped mass block relative to the input acceleration) can be derived as follows: ; in, It represents the acceleration of the object being measured; Indicates the displacement of the PCB board; , and These represent the equivalent mass, stiffness, and damping at the connection between the PCB board and the object under test, respectively. This indicates the displacement of the sensor chip housing relative to the PCB board. Indicates the quality of the sensor chip casing; and These represent the stiffness and damping of the pad, respectively. This indicates the displacement of the comb mass block relative to the sensor chip housing; , and These represent the mass, stiffness, and damping of the comb tooth mass block, respectively.

[0034] Taking the Laplace transform of the above dynamic differential equations yields the transfer function describing the system's output-input relationship. The input is acceleration. The output is the displacement of the comb-shaped mass block. .

[0035] Similarly, differential equations were derived for the readout circuit model and the low-pass filter circuit model, and the transfer function was obtained after Laplace transform. and : When the displacement of the comb mass block is sufficiently small, the capacitance change is approximately linearly related to the displacement. At this point, the on-chip readout circuit converts this change into a voltage output. Over a wide frequency band, this readout circuit can be modeled as a constant gain transfer function. .

[0036] Specifically, the input to the readout circuit model is the displacement of the comb mass block. After passing through the capacitor-voltage readout circuit, the output voltage is... The differential equation is derived as follows: ; Then perform a Laplace transform on it to obtain the transfer function. ; At the end of the on-chip readout circuit, a first-order RC low-pass filter is typically used, whose transfer function characteristics are derived from... Characterization.

[0037] Specifically, the input to the low-pass filter circuit model is the output voltage of the capacitor-voltage readout circuit. The output is the filtered voltage. The differential equation is derived as follows: ; The transfer function can be obtained by performing a Laplace transform on it. : .

[0038] The subsystems in the measurement chain are connected in series. Therefore, the overall transfer function of the lumped parameter model is the product of the transfer functions of each subsystem, denoted as . .analyze The structure reveals that it contains a pair of complex zeros, three pairs of complex poles, and one real pole. By normalizing the sensor sensitivity and simplifying the model parameters, the simplified zero-pole form of the transfer function can be derived: ; in, , , and Indicates the natural frequency; , , and Indicates the damping ratio; is the time constant, and s is the complex variable of the transfer function, representing the system's response to input signals of different frequencies.

[0039] Analyzing the zeros and poles of the transfer function: The natural frequency of the comb mass block inside the MEMS chip corresponds to a pair of complex poles, the low-pass filter contributes a real pole, and the remaining complex zeros and pole pairs characterize the coupling mechanical response between the PCB and the chip. By obtaining actual sensor frequency response data through dynamic calibration and fitting it using the least squares method, these model parameters can be determined.

[0040] The design of the compensation filter employs the pole-zero placement method. First, the pole locations of the desired ideal model are determined based on the required compensation bandwidth. These poles are set at frequencies at least three times the target bandwidth to ensure sufficient dynamic recovery capability. Next, the transfer function of the desired ideal response is constructed and divided by the transfer function of the fitted sensor model to obtain the continuous-time transfer function of the compensation filter. Then, this continuous-time transfer function is discretized to form a causal infinite impulse response (IIR) filter. To ensure frequency domain consistency and system stability in digital implementation, the discrete poles and zeros must be matched as precisely as possible to their corresponding continuous-time points. Finally, a low-pass digital filter is cascaded with the aforementioned IIR filter to form the final compensation filter. ,like Figure 4 As shown in the figure. The cutoff frequency of the low-pass digital filter determines the compensated 3dB bandwidth.

[0041] 2) Adaptive noise suppression.

[0042] Under noise-free conditions, the ideal compensation signal output by the compensation filter designed above... It can accurately reproduce the real sensor input signal. However, in practical applications, the output signal... It will also include sensor noise amplified by a compensation filter. Although this noise is random, its power spectral density and autocorrelation characteristics exhibit near-stationary features. Based on these statistical properties, Figure 3 This demonstrates a specific implementation scheme for the adaptive noise suppression method: In the offline experiment, the noise floor after sensor compensation was first measured, and its autocorrelation function was calculated and recorded. When performing bandwidth compensation online, real-time calculation is performed. autocorrelation function Then the autocorrelation function of the ideal compensation output is... Approximately Next, the Levinson-Durbin recursive method is applied to estimate the coefficients of the AR model (i.e., the autoregressive model) and white noise. The variance. Assuming the system uses a p-order AR model, the ideal compensation output is considered as the system state, the AR model is used as the state transition model, and the actual compensation signal is used as the observation value, thus forming the state-space expression of the Kalman filter: ; ; in: Represents the state variable matrix; Represents the state transition matrix; This represents the state variable matrix from the previous time step; Represents the observation matrix; Indicates process noise; This represents the sensor noise after amplification by the final compensation filter; This represents the ideal compensation signal output by the final compensation filter.

[0043] ; Among them, a1, a2...a p These are the AR model coefficients estimated using the Levinson-Durbin recursive method; ; .

[0044] Based on the above steps, such as Figure 6 As shown, the bandwidth compensation method proposed in this invention enables capacitive MEMS accelerometers to achieve the dual advantages of wide bandwidth and low noise output. This method can effectively correct the frequency response and suppress noise interference, while significantly improving the signal-to-noise ratio (SNR) during the compensation process while expanding the signal bandwidth.

[0045] Obviously, the above embodiments are merely illustrative examples for clear explanation and are not intended to limit the implementation. Those skilled in the art will recognize that other variations or modifications can be made based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations here. However, obvious variations or modifications derived therefrom are still within the scope of protection of this invention.

Claims

1. A method of low-noise digital bandwidth compensation of a sensor high-order frequency response, characterized by the steps of At least comprising: S1 : Construct lumped parameter model based on actual sensor measurements of physical process and derive overall high order transfer function of lumped parameter model for computation ; S2: Obtain actual sensor frequency response data and perform fitting to determine the model parameters in the high order transfer function to obtain a fitted transfer function; S3: Constructing an ideal desired response model according to the target compensation bandwidth, and dividing the transfer function of the ideal desired response model by the fitted transfer function obtained in step S2 to obtain a continuous-time transfer function of the compensation filter; S4: Discretizing the continuous-time transfer function of the compensation filter to obtain an infinite impulse response filter; S5: Cascading the infinite impulse response filter with a low-pass digital filter to form a final compensation filter for compensating the bandwidth of the sensor output signal.

2. The method of claim 1, wherein, The method is applied to a capacitive MEMS acceleration sensor.

3. The method of claim 2, wherein, The lumped parameter model in step S1 includes a mechanical response structure model, a readout circuit model, and a low-pass filter circuit model.

4. The method of claim 3, wherein, deriving in step S1 an overall high order transfer function of the lumped parameter model comprising: S101. Deriving and calculating a first differential equation based on the mechanical response structure model: ； wherein, represents the acceleration of the measured object; represents the displacement of the PCB board; , and respectively represent the equivalent mass, stiffness and damping of the connection between the PCB board and the measured object; represents the displacement of the sensor chip housing relative to the PCB board; represents the mass of the sensor chip housing; and respectively represent the stiffness and damping of the pad; represents the displacement of the comb mass inside the sensor chip for sensing relative to the sensor chip housing; , and respectively represent the mass, stiffness and damping of the comb mass. performing a Laplace transform on the first differential equation to obtain a transfer function ; S102. Deriving and calculating a second differential equation based on the readout circuit model: ； wherein, is an output voltage of the capacitance-voltage readout circuit; is a displacement of the comb mass; is a readout circuit gain; performing a Laplace transform on the second differential equation to obtain a transfer function ; S103. Deriving and calculating a third differential equation based on the low-pass filter circuit model: ； wherein: Vf is the filtered voltage; RC is the time constant of the low pass filter circuit; performing a Laplace transform on the third differential equation to obtain a transfer function ; S104. multiplying , and each other, and processing the three multiplication results by normalizing the sensor sensitivity and simplifying the model parameters to obtain an overall high-order transfer function of the lumped parameter model .

5. The method of claim 4, wherein, The overall high order transfer function Is expressed as follows: ； wherein , , and represent the natural frequency; , , and represent the damping ratio; is the time constant, s is the complex variable of the transfer function, representing the system's response to different frequency input signals.

6. The method of claim 5, wherein, In step S2, frequency response data of the actual sensor is obtained by dynamic calibration, to the overall high order transfer function is modeled, and the frequency response data of the actual sensor is fitted using least squares to determine the overall high order transfer function with specific values of the natural frequencies, damping ratios and time constants.

7. The method of claim 2, wherein, The high-order frequency response low-noise digital bandwidth compensation method according to claim 1, wherein the pole frequency of the ideal desired response model in step S3 is set to be not less than three times the target compensation bandwidth.

8. The method of claim 1-7, wherein, Further comprising: S6: Performing adaptive noise suppression on the signal processed by the final compensation filter, and the adaptive noise suppression is realized by using a Kalman filter adaptive filter based on an autoregressive model.

9. The method of claim 8, wherein, The adaptive noise suppression step includes: The noise floor of the compensated sensor is measured offline and the autocorrelation function is calculated ; When performing bandwidth compensation online, the output signal is calculated in real time. autocorrelation function Then the autocorrelation function of the ideal compensation output is... Approximately ; based on The Levinson-Durbin recursive method was used to estimate the coefficients of the autoregressive model and white noise. The variance; Constructing a Kalman filter state space model with an ideal compensation output as a system state, an autoregressive model as a state transition model, and an actual compensation signal as an observation value to realize noise suppression.

10. The method of claim 9, wherein, The Kalman filter state space model is expressed as: ； ； wherein: represents a state variable matrix; represents a state transition matrix; represents a state variable matrix of a previous time; represents an observation matrix; represents process noise; represents sensor noise amplified by a final compensation filter; represents an ideal compensation signal output by the final compensation filter.

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