Resonator phase noise suppression method based on Kalman filtering

Abstract

The invention provides a resonator phase noise suppression method based on Kalman filtering. The resonator phase noise suppression method comprises the following implementation steps: sampling displacement of a resonator; calculating a priori error matrix and a priori estimation value of each sampling; calculating the Kalman gain of each sampling; calculating an error matrix of each sampling; and obtaining a resonator phase noise suppression result. According to the method, suppression of the resonator phase noise in the mechanical resonant device is realized based on Kalman filtering, and the method is suitable for various noise types including high-frequency noise, can flexibly cope with different working environments and noise types, and has relatively good adaptability; based on the recursive calculation characteristic of Kalman filtering, state estimation can be updated every time new data arrives, processing of all historical data is not needed, meanwhile, dynamic estimation and optimization are carried out on input signals in combination with a prediction model and noise observation, a filter can rapidly respond to the change of signal frequency, and the accuracy and the reliability of the method are improved. And an efficient real-time adjustment capability is provided.

Description

Technical Field

[0001] The present invention belongs to the technical field of control circuits and relates to a method for suppressing phase noise of a resonator, and specifically to a method for suppressing phase noise of a resonator based on Kalman filtering, which can be applied to signal processing in communication systems, precision measuring instruments, navigation systems and various electronic devices. Background Art

[0002] Resonant devices are divided into mechanical resonant devices, electric resonant devices, optical resonant devices, acoustic resonant devices, magnetic resonant devices, thermal resonant devices, microwave resonant devices, etc. The resonator in the mechanical resonant device is based on the resonance principle and uses the resonant frequency change of the sensitive element resonator to measure other physical quantities. Since only the frequency needs to be measured to detect the change of physical quantity, it is not easily affected by amplitude gain. Therefore, the resonator has the advantages of quasi-digital output, strong anti-interference ability and high precision.

[0003] However, the output of the resonator is frequency, so it is easily interfered by phase noise. Since the frequency of the phase noise is the same as the output signal frequency, it is difficult to suppress it through low-pass filters, high-pass filters and band-pass filters, which causes its output signal waveform to become uneven, interfering with the frequency measurement of the frequency measurement module and seriously affecting the output accuracy of the resonator. Therefore, suppressing phase noise and smoothing the output signal into an ideal sinusoidal signal are of great significance to improving the overall performance of the resonator.

[0004] For example, the patent application with the application publication number CN117544115A and the name "A low phase noise vibration-resistant crystal oscillator" discloses a method for suppressing phase noise of mechanical resonant devices. The method first receives a stable reference signal from a high-precision frequency source through a phase-locked loop, and the internal phase comparator compares the reference signal with the phase of the oscillator output signal to generate an error signal proportional to the phase difference. The signal is passed through a low-pass filter to remove high-frequency noise and generate a control voltage. Finally, the control voltage is sent to the voltage-controlled oscillator to gradually approach the reference signal, thereby achieving frequency locking. The invention has limited ability to suppress high-frequency noise, especially in the presence of strong interference or high-frequency noise, which may cause the output frequency to be unstable. When the frequency changes greatly, the dynamic response in the locking process is slow and requires a certain amount of time to adjust. In addition, the accuracy of the locked frequency depends on the stability of the reference signal and the performance of the phase comparator, which may be insufficient in low-phase noise applications. Summary of the invention

[0005] The purpose of the present invention is to overcome the defects of the above-mentioned prior art and propose a resonator phase noise suppression method based on Kalman filtering to solve the technical problems of poor adaptability and real-time performance in the prior art.

[0006] To achieve the above object, the technical solution adopted by the present invention includes the following steps:

[0007] (1) Sampling the displacement of the resonator:

[0008] The displacement of the resonator is sampled I times at a time interval of T, and the displacement set Z = {z i |0<i≤I}, where T>0, z i represents the displacement of the i-th sampling, I ≥ 10;

[0009] (2) Calculate the a priori error matrix and a priori estimate of the i-th sampling:

[0010] Let i = 1, construct the state transfer matrix A through the state transfer model under the resonant condition of the resonator, and calculate the prior error matrix P of the i-th sampling through A i ' and the prior estimate

[0011] (3) Calculate the Kalman gain of the i-th sampling:

[0012] The prior error matrix P of the i-th sampling i 'Calculate the Kalman gain K of the i-th sampling i ;

[0013] (4) Calculate the error matrix of the i-th sampling:

[0014] The prior error matrix P of the i-th sampling i ' and Kalman gain K i Calculate the error matrix P of the i-th sampling i ;

[0015] (5) Obtain the resonator phase noise suppression results:

[0016] The Kalman gain K of the i-th sampling i Calculate the displacement z of the i-th sampling i The corresponding filtered signal x i ′, and determine whether i=I is true. If so, I filtered signals are obtained. Otherwise, let i=i+1, P i '=P i , and execute step (2).

[0017] Compared with the prior art, the present invention has the following advantages:

[0018] (1) The present invention suppresses the phase noise of the resonator in a mechanical resonant device based on Kalman filtering, is applicable to a variety of noise types including high-frequency noise, and can maintain the stability of the output frequency. Compared with the prior art, it can flexibly cope with different working environments and noise types and has better adaptability.

[0019] (2) Based on the recursive calculation characteristics of Kalman filtering, the present invention can update the state estimation every time new data arrives without processing all historical data. At the same time, it combines the prediction model and noise observation to dynamically estimate and optimize the input signal, so that the filter can quickly respond to changes in signal frequency and provide efficient real-time adjustment capabilities. BRIEF DESCRIPTION OF THE DRAWINGS

[0020] Figure 1 It is a flow chart for implementing the present invention.

[0021] Figure 2 is a schematic diagram of a resonant model of a resonator of the present invention;

[0022] Figure 3 It is a comparison diagram of the output signal before and after filtering of the present invention. DETAILED DESCRIPTION

[0023] The present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0024] Reference Figure 1 , the present invention comprises the following steps:

[0025] Step 1) Sample the displacement of the resonator:

[0026] When the resonator is disturbed by phase noise, T is 4.4×10 -7 s is used as the interval time to sample the displacement of the resonator I times, and the displacement set Z = {z i |0≤i≤I}, where T>0, z i represents the displacement of the i-th sampling, I ≥ 10,

[0027] Reference Figure 2 , the ideal resonator can be equivalent to a mass block-spring-damper second-order system, m, c and k represent the equivalent mass, motion damping and equivalent stiffness of the resonator respectively. Under the action of external force F, the mass block performs periodic motion at the equilibrium position, and its displacement is expressed as x i , when the driving force applied to the resonator is sinusoidal excitation, its kinematic equation can be expressed as:

[0028] ma+cv+kx i =Fcos(ω e iT)

[0029] When the resonator is in resonance, the frequency ω0 of the resonator is equal to the driving force angular frequency ω e , the expression of the resonance model is:

[0030] ma+cv+kx i =Fcos(ω0iT);

[0031] From the above formula, the theoretical displacement of the resonator can be deduced as:

[0032] x i =A m cos(ω0iT-90°)

[0033] The amplitude of the resonator In the formula, the damping ratio of the resonator is

[0034] z i The expression is:

[0035] z i =A m cos(ω0iT-90°)+A ξ cos(ω0iT+φ ξ );

[0036] Among them, a and v represent the acceleration and velocity of the resonator, respectively, F represents the amplitude of the driving force, and A ξ ,φ ξ are the amplitude and phase of the phase noise, both of which conform to the Gaussian distribution.

[0037] The resonator output amplitude is taken as 6200, and the specific values ​​of other mechanical parameters of the above resonator can be shown in Table 1:

[0038] Table 1

[0039]

[0040] Step 2) Obtain the state transition model under the resonant condition of the resonator:

[0041] Through the motion principle of the resonator, the state transfer model under resonance conditions is obtained, and the expression is:

[0042]

[0043] In the formula, x i 、v i and a i denote the theoretical displacement, velocity and acceleration of the resonator sampled at the i-th time, respectively, and x i-1 、v i-1 and a i-1 are the theoretical values ​​of the resonator displacement, velocity and acceleration of the i-1th sampling respectively.

[0044] Step 3) The state transfer matrix A is constructed through the state transfer model under the resonant condition of the resonator, and its expression is:

[0045]

[0046] In this embodiment, substituting specific mechanical parameters into:

[0047]

[0048] Step 4) The prior error matrix P of the i-th sampling can be calculated by the known A i ', the calculation formula is:

[0049] P i '=AP i-1 A T +Q

[0050] According to the state transfer matrix A and the estimated noise level in the resonator measurement and control circuit, the predicted noise covariance matrix Q can be taken as:

[0051]

[0052] To ensure the convergence of Kalman filtering, the error matrix P0 of the 0th sampling is taken as:

[0053]

[0054] Where P i-1 represents the error matrix of the i-1th sampling, Q represents the prediction noise covariance matrix, A T represents the transpose operation of A, and P0 represents the error matrix of the 0th sampling.

[0055] Step 5) Calculate the prior estimate of the i-th sampling The calculation formula is:

[0056]

[0057] x i-1 ′ represents the filtered signal of the i-1th sampling.

[0058] Step 6) Calculate the Kalman gain of the i-th sampling:

[0059] The prior error matrix P of the i-th sampling i 'Calculate the Kalman gain K of the i-th sampling i , and its calculation formula is:

[0060] K i =P i 'H T(HP i 'H T +R) -1

[0061] In the formula, H, H T and R represent the measurement matrix, the transpose operation of H, and the observation noise covariance matrix, respectively.

[0062] Since the resonator can only observe displacement but not velocity directly, based on the observed value Z i In the ideal model, it corresponds to displacement, so the observation matrix H is given as:

[0063] H=[1 0 0]

[0064] Based on the observed value Z i The phase noise level of the observation noise covariance matrix R is given as:

[0065] R=1

[0066] Step 7) Calculate the error matrix of the i-th sampling:

[0067] The prior error matrix P of the i-th sampling i ' and the Kalman gain K of the i-th sampling i Calculate the error matrix P of the i-th sampling i , the calculation formula is:

[0068] P i =(I3-K i H)P i '

[0069] Where I3 is the identity matrix with dimension 3.

[0070] Step 8) Obtain the resonator phase noise suppression result:

[0071] The displacement z of the i-th sampling i The corresponding filtered signal x i ′, the calculation formula is:

[0072]

[0073] Since the frequency of phase noise is consistent with the frequency of the output signal, it is difficult to filter it out using traditional filtering methods. However, Kalman filtering has a good inhibitory effect on Gaussian distributed noise. Therefore, the Kalman filtering algorithm of the resonance model can be burned into the digital circuit to filter the output signal z. i Sampling to obtain the observation set Z i , phase noise is suppressed by Kalman filtering to obtain the optimal estimate x i, the output signal of the circuit is filtered into an ideal sine wave after Kalman filtering. Based on the parameters in step 1, the square comparison between the original signal and the signal after Kalman filtering can be obtained. Figure 3 shown.

[0074] The above are only specific steps of the present invention and do not constitute any limitation to the protection scope of the present invention; any technical solutions formed by equivalent transformation or equivalent replacement shall fall within the protection scope of the present invention; some parts not elaborated in detail in the present invention belong to the known technologies of those skilled in the art.

[0075] The following is a further description of the technical effects of the present invention in conjunction with the simulation results:

[0076] 1. Experimental conditions and contents:

[0077] The simulation software is MATLAB 2023b, and the output signals before and after filtering of the present invention are compared and simulated. The results are as follows: Figure 3 shown.

[0078] 2. Experimental results analysis:

[0079] Reference Figure 3 , phase noise will cause the phase of the signal to fluctuate randomly, causing irregular changes in the waveform, especially at the peak and valley of each cycle, the waveform curve becomes uneven. After Kalman filtering the signal after resonator filtering, the influence of phase noise is eliminated and it can be observed that the output signal curve becomes smooth.

Claims

1. A resonator phase noise suppression method based on Kalman filtering, characterized in that: The steps include: (1) Sampling the displacement of the resonator: The displacement of the resonator is sampled I times at a time interval of T, and the displacement set Z = {z i |0<i≤I}, where T>0, z i represents the displacement of the i-th sampling, I ≥ 10; (2) Calculate the a priori error matrix and a priori estimate of the i-th sampling: Let i = 1, construct the state transfer matrix A through the state transfer model under the resonant condition of the resonator, and calculate the prior error matrix P of the i-th sampling through A i ' and the prior estimate (3) Calculate the Kalman gain of the i-th sampling: The prior error matrix P of the i-th sampling i 'Calculate the Kalman gain K of the i-th sampling i ; (4) Calculate the error matrix of the i-th sampling: The prior error matrix P of the i-th sampling i ' and Kalman gain K i Calculate the error matrix P of the i-th sampling i ; (5) Obtain the resonator phase noise suppression results: The Kalman gain K of the i-th sampling i Calculate the displacement z of the i-th sampling i The corresponding filtered signal x i ′, and determine whether i=I is true. If so, I filtered signals are obtained. Otherwise, let i=i+1, P i '=P i , and execute step (2).

2. The method according to claim 1, characterized in that The displacement z of the i-th sampling described in step (1) i , whose expression is: z i =A m cos(ω0iT-90°)+A ξ cos(ω0iT+φ ξ ) Among them, A m , ω0 represent the amplitude and frequency of the resonator respectively, A ξ ,φ ξ Represent the amplitude and phase of phase noise respectively.

3. The method according to claim 1, characterized in that The state transition model under the resonant condition of the resonator described in step (2) is expressed as: Among them, x i 、v i and a i denote the theoretical displacement, velocity and acceleration of the resonator sampled at the i-th time, respectively, and x i-1 、v i-1 and a i-1 are the theoretical displacement, velocity and acceleration of the resonator sampled at the i-1th time, respectively; m, c, k represent the equivalent mass, motion damping and equivalent stiffness of the resonator, respectively.

4. The method according to claim 3, characterized in that The state transfer matrix A described in step (2) is expressed as:

5. The method according to claim 4, characterized in that The prior error matrix P of the i-th sampling described in step (2) i ', the calculation formula is: P i '=AP i-1 From T +Q Among them, P i-1 represents the error matrix of the i-1th sampling, Q represents the prediction noise covariance matrix, A T represents the transpose operation of A, and P0 represents the error matrix of the 0th sampling.

6. The method according to claim 5, characterized in that The prior estimate of the i-th sampling described in step (2) The calculation formula is: Among them, x i-1 ′ represents the filtered signal of the i-1th sampling.

7. The method according to claim 6, characterized in that The Kalman gain K of the i-th sampling described in step (3) i , the calculation formula is: K i =P i 'H T (HP i 'H T +R) -1 H＝[100] R＝1 Among them, H, H T and R represent the measurement matrix, the transpose operation of H, and the observation noise covariance matrix, respectively.

8. The method according to claim 7, characterized in that The error matrix P of the i-th sampling described in step (4) i , the calculation formula is: P i =(I3-K i H)P i ' Among them, I3 is the identity matrix with dimension 3.

9. The method according to claim 1, characterized in that: The displacement z of the i-th sampling described in step (5) i The corresponding filtered signal x i ′, the calculation formula is:

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