Grating interference signal phase extraction method based on multi-iteration ICEEMDAN-HT

Abstract

The present application relates to the technical field of precision displacement measurement and signal processing, and discloses a grating interference signal phase extraction method based on multi-iteration ICEEMDAN-HT, which comprises the following steps: obtaining an actual noisy grating interference signal sequence, and generating multi-order intrinsic mode components by using a multi-iteration improved adaptive noise complete ensemble empirical mode decomposition algorithm; evaluating the component complexity by using a sample entropy algorithm, identifying effective phase components in combination with an energy proportion criterion, and storing the effective phase components in an independent set; adding and reconstructing non-feature components, eliminating trend item components with a sample entropy value lower than a preset threshold, and obtaining noise mode components; determining the noise energy convergence state through a recursive mechanism, feeding back signals not reaching the threshold to the decomposition step for further processing, and stopping until the iteration termination condition is met. The present application extracts and accumulates weak features through multiple iterations, suppresses the influence of environmental interference and direct current fluctuation on measurement, and improves the phase calculation precision and system robustness.

Description

Technical Field

[0001] This invention relates to the field of precision displacement measurement and signal processing technology, specifically to a phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT. Background Technology

[0002] In the fields of ultra-precision manufacturing and semiconductor inspection, grating interferometers are the core means to achieve nanometer-level displacement measurement. The phase extraction accuracy of the grating interferometer signal directly determines the measurement resolution of the system. However, in actual measurement environments, the signal acquisition process is inevitably affected by fluctuations in laser source power, environmental vibrations, and electronic thermal noise.

[0003] Existing grating signal processing methods mostly employ adaptive decomposition algorithms to denoise the original signal. While these methods have certain advantages in processing non-stationary signals, they still face several technical limitations. Because the injection parameters for the masking signal during decomposition are often set empirically, the algorithms are prone to mode aliasing when faced with strong noise interference. This causes the effective phase information and noise components to overlap in the frequency space, making accurate separation difficult.

[0004] Furthermore, traditional processing schemes typically perform only a single decomposition. Under low signal-to-noise ratio conditions, some weak phase feature components are often misclassified as random noise and discarded, resulting in the loss of feature information. Simultaneously, low-frequency trend terms generated by laser drift or mechanical stress release, along with dynamic DC bias, alter the symmetry of the interference signal, leading to nonlinear errors in subsequent Hilbert transform-based phase calculations. Existing algorithms lack quantifiable complexity evaluation criteria when dealing with such non-stationary trend terms, blurring the boundary between the trend term and the effective low-frequency signal, thus limiting the system's measurement robustness and long-term stability at the sub-nanometer scale. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a grating interference signal phase extraction method based on multi-iteration ICEEMDAN-HT, which solves the problems mentioned in the background section.

[0006] This invention provides a phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT.

[0007] This method addresses the environmental noise interference, DC bias fluctuations, and mode aliasing issues present in the acquisition of grating interference signals. It achieves phase information extraction through a multi-round iterative adaptive decomposition and complexity determination mechanism.

[0008] The specific plan is as follows:

[0009] The actual noisy grating interference signal is acquired and characterized as a discrete time-domain sequence. The analog signal is converted into a digital sequence using a grating measurement device and an analog-to-digital converter.

[0010] An improved adaptive noise-complete set empirical mode decomposition algorithm (EMD) is used to adaptively decompose the discrete-time sequence. During the decomposition process, white noise with a specific standard deviation coefficient is introduced as a masking signal at each order. Cubic spline interpolation is used to fit the upper and lower envelopes and extract the local mean. The residual components are obtained by calculating the arithmetic mean of multiple experimental results and are then sequentially separated from the sequence to be decomposed to obtain the multi-order intrinsic mode components.

[0011] The sample entropy algorithm is used to evaluate the complexity of the obtained intrinsic mode components (EMS). By setting specific embedding dimensions and similarity tolerances, the Chebyshev distance between reconstructed vectors is calculated to obtain the sample entropy value of each component. Combining the energy proportion of the EMS in the original signal, the components that meet the preset entropy range and energy threshold are identified as valid phase components and stored in a preset independent set for iterative accumulation.

[0012] For the remaining intrinsic mode components other than the effective phase components, they are defined as non-feature components and reconstructed by summation. During the reconstruction process, components with sample entropy values ​​lower than a preset complexity threshold are identified and removed. These components correspond to trend terms generated by laser power fluctuations or environmental drift. After removing trend terms, the remaining high-frequency random fluctuation components are reconstructed as noise mode components.

[0013] A recursive processing mechanism is executed, feeding the noise mode components as the input sequence for the next round to the decomposition step, and re-executing the adaptive decomposition. This recursive process continues until the iteration termination condition is met, i.e., the preset maximum number of iterations is reached, or the relative rate of change of the energy of the noise mode components generated in two adjacent iterations is less than a preset convergence threshold.

[0014] A global reconstruction is performed on all valid phase components accumulated in the independent set during each iteration. The statistical mean of the reconstructed signal within the sampling window is calculated. By subtracting this mean, the dynamic DC bias of the signal is eliminated, bringing the signal centroid to zero, resulting in the DC-free displacement signal. A Hilbert transform is performed on the DC-free displacement signal to generate an orthogonal imaginary part sequence and construct an analytic signal. The wrapped phase is solved using the four-quadrant arctangent function. During the solution process, phase expansion is performed by monitoring the phase jump characteristics, and the instantaneous displacement value in physical space is calculated based on the ratio of the total phase to the grating pitch constant.

[0015] The technical solution disclosed in this invention extracts and accumulates the phase feature components remaining in the noise through multiple rounds of iterative decomposition. Sample entropy is used to quantify the complexity of the components, distinguishing between effective phase signals, high-frequency noise, and low-frequency environmental trend terms. This solution solves the feature loss problem caused by traditional decomposition methods in low signal-to-noise ratio environments, improving the phase extraction accuracy and system robustness of grating interferometers in displacement measurement.

[0016] This invention provides a phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT. It has the following advantages:

[0017] 1. This invention employs a multi-round iterative decomposition and recursive processing mechanism to adaptively decompose the noise components after initial decomposition as input sequences, thereby extracting residual phase feature components from the noisy background. This iterative accumulation method solves the feature loss problem that traditional single-step decomposition methods are prone to in strong noise environments, ensuring the complete acquisition of effective phase information.

[0018] 2. This invention utilizes a sample entropy algorithm to perform complex quantitative evaluation of eigenmode components of each order, and combines this with an energy proportion criterion to classify effective signals, random high-frequency noise, and low-frequency environmental trend terms. This method filters data based on its own statistical characteristics, eliminating interference from laser power fluctuations and environmental drift on phase extraction, and improving the system's operational stability under complex conditions.

[0019] 3. This invention eliminates the impact of signal drift on phase calculation accuracy by globally reconstructing and dynamically eliminating the accumulated effective phase components. Combining Hilbert transform and phase expansion algorithms, the instantaneous phase is mapped to physical spatial displacement. With an amplitude protection mechanism, this improves the phase resolution of the grating interference signal and the accuracy of displacement measurement. Attached Figure Description

[0020] Figure 1 This is a flowchart of the method steps of the present invention;

[0021] Figure 2 This is a schematic diagram illustrating the process of the present invention;

[0022] Figure 3 This is a schematic diagram of the ICEEMDAN method of the present invention;

[0023] Figure 4 This is the time-frequency diagram of the noisy original interference signal STFT of the present invention;

[0024] Figure 5 This is the STFT time-frequency diagram after multiple iterations of ICEEMDAN processing according to the present invention;

[0025] Figure 6The diagram shows the phase expansion and a magnified view of the original signal after Hilbert transform in this invention.

[0026] Figure 7 The diagram shows the phase expansion and a magnified view of the processed signal after Hilbert transform according to the present invention. Detailed Implementation

[0027] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. 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.

[0028] Please see the appendix Figure 1 Please refer to the appendix for a flowchart illustrating the method steps of this invention. Figure 2 For a process comparison diagram of the present invention, please refer to the appendix. Figure 3 As a schematic diagram of the ICEEMDAN method of the present invention, an embodiment of the present invention provides a grating interference signal phase extraction method based on multi-iteration ICEEMDAN-HT, including the following steps:

[0029] Step S1 establishes a precise mapping from the physical domain signal to the digital computation domain, thus providing a high-quality original characterization for subsequent nonlinear decomposition. Based on the general physical principle of photoelectric conversion, the intensity changes of the interference fringes need to be converted into charge carriers by a semiconductor photosensitive surface, and then converted into a voltage signal by a transimpedance amplifier circuit. The grating measurement device captures and quantizes the original physical quantities output by the grating interferometer through its high-speed signal acquisition circuit at the front end, thereby inputting the actual noisy grating interference signal. In the single-channel processing logic, in order to provide a consistent data input standard for subsequent algorithm units, the signal processing module of the grating measurement device uniformly characterizes the acquired first or second interference signal into a discrete time-domain sequence to be processed. .

[0030] Step S11: In a preferred embodiment, the grating measurement device uses a configured photodetector to receive the intensity of the fringes formed by grating interference. After pre-amplification and low-pass filtering, the analog voltage signal is converted into a digital sequence with a bit depth of 12 or 16 bits via an analog-to-digital converter (ADC). A quantization bit depth of 12 bits or more is chosen to ensure, at the physical level, that the dynamic range of the signal covers the complete contrast of the interference fringes, thereby controlling the quantization noise within the system's allowable tolerance range. In the specific hardware implementation, to ensure the dynamic performance and real-time nature of phase extraction, the ADC sampling frequency must strictly satisfy the Nyquist sampling theorem. In this embodiment, considering the instantaneous Doppler frequency generated when the grating moves at high speed and the group delay compensation requirements in the algorithm processing, the sampling frequency is typically set to 5 to 10 times the highest frequency component generated by the grating movement. At this time, the input actual noisy grating interference signal contains characteristic components reflecting displacement information as well as nonlinear components introduced due to the external environment and inherent system characteristics.

[0031] Step S12: The grating measurement device establishes a mathematical analysis model for the input noisy grating interference signal, expressing it as a function containing DC components, AC modulation components, and random noise terms. The technical purpose of this modeling method is to explicitly decouple the ideal physical motion process from unavoidable hardware interference terms using mathematical language. For the two orthogonal interference signals synchronously acquired by the grating measurement device, the specific mathematical expression is as follows:

[0032] ;

[0033] ;

[0034] In the above formula, and These represent the first and second discrete sampling signals acquired by the grating measurement device, respectively. For discrete-time indexes, it satisfies ,in For the sampling sequence index, The sampling period; and The DC bias component present in the signal is physically represented by a real constant greater than 0, and its value is mainly determined by the power fluctuation at the center of the laser source and the dark current response level of the photoelectric conversion device. and The AC amplitude component of the signal, whose value ranges from 0 to 1, reflects the modulation depth of the interference fringes and is directly constrained by the reflectivity of the grating surface and the collimation of the optical path. and This represents random additive Gaussian white noise caused by circuit thermal noise, quantum noise, and mechanical vibration of the displacement stage during the measurement process. Its mathematical expectation tends to 0, but its standard deviation increases with the increase of environmental electromagnetic interference. and It is the instantaneous phase change that evolves over time, and its rate of change is modulated in real time by the mechanical displacement velocity of the measured object.

[0035] Step S13: The grating measurement device, through analysis of the above mathematical model, knows that the phases of the two orthogonal signals have a fixed phase difference relationship, i.e. .in, This represents the initial phase delay. In this embodiment, due to limited processing precision of the optical elements or installation stress causing inaccurate waveplate delay, Usually at the standard value Small offsets occur in the vicinity, and these offsets constitute the phase orthogonality error of the grating measurement.

[0036] Step S14: Based on the physical causal logic of grating interference, the grating measurement device identifies that the energy attenuation of the laser as the working time increases, local contamination on the surface of the grating ruler, or alignment deviation will cause the AC amplitude component to... and Fluctuations occur with the displacement stroke, manifesting as envelope distortion in the signal. Simultaneously, the electromagnetic pulses introduced by the stage power supply cause noise spikes with distinct abrupt changes to be superimposed on the signal. The actual noisy grating interference signal input in step S1 has the following time-frequency characterization: Figure 4 As shown, significant frequency spread and background noise are evident. These nonlinear factors constitute the background disturbance source for phase extraction. If traditional methods such as arctangent are used directly without noise reduction processing, phase transitions will be triggered in the computational domain due to the excessively low signal-to-noise ratio.

[0037] Regarding the various error terms defined in step S1 above, the actual noisy grating interference signal in this embodiment not only includes low-frequency baseline drift caused by grating patterning errors, but also high-frequency broadband electrical noise. For analog gain control, common-mode rejection ratio adjustment, and quantization error correction in signal acquisition, those skilled in the art can perform parameterization configuration according to specific sensor manuals. The specific circuit design and register initialization operations are well-known technologies in the field and will not be elaborated here. By executing step S1 above, the grating measurement device provides a basic dataset with original physical characteristics for subsequent multi-iteration ICEEMDAN decomposition. After completing step S1, the grating measurement device acquires discrete sequences... The data is temporarily stored in a cache. This caching mechanism is designed to ensure logical consistency of memory addressing in multiple loop computations, thereby facilitating the subsequent step S2 to perform multi-scale adaptive noise decomposition operations based on a specific sampling length.

[0038] In this embodiment, step S2 aims to construct an adaptive recursive decomposition framework to transform the complex time-domain signal input in step S1. The intrinsic mode functions (EMFs) are extracted into EEMFs with different physical scales. Unlike traditional Empirical Mode Decomposition (EMD), which is prone to mode aliasing, this invention employs multi-iterative ICEEMDAN (Improved Adaptive Noise Complete Set Empirical Mode Decomposition), which utilizes the statistically significant spectral uniformity of white noise as a masking benchmark. By introducing specific residual components in each decomposition, the completeness and reconstruction accuracy of the decomposition process are ensured. The grating measurement device performs this step to obtain vector representations of the first to higher order EEMFs.

[0039] ;

[0040] Step S21: Based on the general principle of ensemble averaging in signal processing, the grating measurement device first initializes the key operators and hyperparameters required by the algorithm. Definitions An envelope operator for extracting the local mean of a signal identifies all local maxima and minima in the signal sequence and fits an upper and lower envelope using a cubic spline interpolation function. Finally, the arithmetic mean of the two envelopes at each time point is taken. In this embodiment, if the number of extreme points in the sequence to be processed is less than three, making it impossible to construct an effective cubic spline curve, then the operator... The original signal will be output directly to avoid numerical singularities caused by interpolation exceeding limits. Definition The first one with a mean of 0 and a standard deviation of 1 The white noise sequence introduced in this experiment, among which The total number of ensemble experiments, typically an integer between 50 and 100, representing a trade-off between computational real-time performance and statistical convergence accuracy. Definition For the first The standard deviation control coefficient for the injected noise is used to adjust the masking strength of the auxiliary noise on the current residual signal. Its value is set to 0.1 to 0.2 times the standard deviation of the current residual signal. Definition The first standard EMD decomposition obtained Step Extraction operator, the symbol {:} represents the extraction operator. The arithmetic mean of the experimental results is calculated.

[0041] In step S22, the grating measurement device constructs an initial sequence set with added first-order masking noise. The technical objective of this process is to utilize the reference frequency provided by white noise to assist the algorithm in accurately locating the highest-frequency component in the interference signal against a complex interference background. Specifically, the noisy sequence is constructed using the following formula: The local mean of the ensemble average is calculated to obtain the first-order residual components.

[0042] ;

[0043] ;

[0044] In the above formula, The original discrete sampled signal input in step S1; The noise gain coefficient in the initial stage has the physical function of exciting the hidden high-frequency small fluctuations in the signal. Indicates the first The first eigenmode extracted after performing a standard EMD decomposition on a white noise sequence. In the calculation... During the division operation, the denominator is the integrated intersection number. ,because Since it is a preset positive integer constant, there is no risk of calculation anomalies due to a denominator of 0.

[0045] Step S23: After obtaining the first-order background residual, the grating measurement device extracts the first-order intrinsic mode components by removing the low-frequency mean component from the original input signal. :

[0046] ;

[0047] Please see the appendix Figure 4 In this embodiment, the noisy original interference signal STFT time-frequency diagram of the present invention is used. The components mainly capture high-frequency random jitter introduced by circuit thermal noise during grating displacement, as well as some subtle dynamic phase change information.

[0048] In step S24, the grating measurement device recursively executes higher-order decomposition logic. This is based on the previous-order residual signal. The energy distribution is used to construct subsequent masking signals and calculate the first... order residual Its general recursive formula is as follows:

[0049] ;

[0050] Thus, the first First-order intrinsic modal components:

[0051] ;

[0052] In the above formula, Indicates the order index of the modal components; The residuals left over from the previous decomposition; The noise weights are dynamically adjusted with the decomposition depth, and change with the order. As the signal energy gradually shifts to lower frequencies, The absolute value will be reduced accordingly to maintain the stability of the signal-to-noise ratio. This step-by-step stripping process ensures that each... It has good narrowband characteristics in the frequency domain.

[0053] In step S25, the grating measurement device determines whether to terminate the iteration by monitoring the state variables of the decomposition process. As a preferred implementation, the logical judgment for algorithm termination is based on a multi-dimensional criterion: when the residual signal... When the number of extreme points is reduced to no more than 2, or when the mean square error (MSE) between the residual signals generated by two adjacent iterations is less than a preset threshold. At this point, it is determined that the signal can no longer be decomposed. At this point, the last remaining... Defined as the trend term of the signal, it reflects the ultra-low frequency line drift caused by laser power fluctuations. Ultimately, the grating measurement device outputs a vector composed of components of various orders. This vector, serving as a feature source for subsequent phase information extraction, has achieved preliminary decoupling between the physical disturbance term and the effective displacement information.

[0054] Please see the appendix Figure 5 In this embodiment, as the STFT time-frequency diagram processed by multiple iterations of ICEEMDAN, step S3 utilizes the sample entropy algorithm from information entropy theory to quantitatively evaluate the nonlinear dynamic characteristics of each order of intrinsic mode components obtained from step S2 decomposition. Based on the statistical physics principle of signal complexity, sample entropy characterizes the complexity of a signal sequence by measuring the probability of generating new modes. For grating interference signals, effective displacement phase components exhibit high self-similarity and regularity in the time domain, and their corresponding sample entropy values ​​are usually in the middle frequency band. High-frequency electronic noise, due to its randomness, generates unpredictable modes, resulting in high entropy values; conversely, DC drift or slowly varying trend terms exhibit extremely strong determinism, with extremely low entropy values. Based on this physical difference, by setting multi-dimensional logical judgment criteria, effective components carrying phase information can be accurately extracted from the decomposition products. The grating measurement device determines whether each component is within the effective feature range by setting a sample entropy threshold and energy ratio criterion.

[0055] Step S31: Based on the technical principle of phase space reconstruction, the grating measurement device measures each component in the vector. Perform feature mapping. Let the current component sequence to be evaluated be... ,in , This represents the total number of discrete sampling points for this component. In this embodiment, the embedding dimension is defined. Define the similarity tolerance as 2. It is 0.2 times the standard deviation of that component, that is The reason for choosing an embedding dimension of 2 is to minimize computational complexity and improve the evaluation accuracy for short sequences while still capturing signal dynamics features; and to reduce similarity tolerance. The standard deviation of the component itself The hook, a technique designed to eliminate the impact of signal amplitude scaling on complexity assessment, ensures physical consistency of the assessment benchmark across different measurement ranges. The grating measurement device reconstructs the sequence into a set. dimensional vector .

[0056] Step S32: The grating measurement device calculates the distance metric and pattern matching number between the reconstructed vectors. In this embodiment, the following is defined: For vectors and The maximum absolute value of the difference between corresponding elements is the Chebyshev distance. For each Value, statistically satisfied of The quantity, and requirements To exclude self-matching. Based on probabilistic statistical logic, the grating measurement device calculates the dimensions respectively. and When the tolerance condition is satisfied, the proportion of vector pairs that satisfy the tolerance condition to the total number of logarithms is denoted as . and These two parameters represent lengths of... and The self-similarity probability of a subsequence at a given similarity threshold.

[0057] Step S33: The grating measurement device calculates the sample entropy value of this component based on the above statistical proportions. Its specific mathematical analytical expression is defined as follows:

[0058] ;

[0059] In practical engineering calculations, when the signal length is limited or the signal is extremely irregular, problems are likely to occur. or When the numerator or denominator approaches zero, division or logarithmic operations become invalid. To ensure the completeness of the algorithm logic, this embodiment introduces a Laplace smoothing term during calculation; that is, when the numerator or denominator is zero, it is replaced with a very small positive number. This ensures the robustness of numerical computation.

[0060] Step S34: Based on the spectral characteristics and energy distribution of the grating interference signal, the grating measurement device presets a sample entropy threshold range of [range to be specified]. As a preferred implementation, this embodiment sets... and The basis for setting this range is that when the entropy value exceeds 0.3, the component is dominated by broadband random noise, and the waveform exhibits disordered oscillations; when the entropy value is below 0.1, the component exhibits a monotonic waveform or ultra-low frequency disturbances. Furthermore, to avoid the one-sidedness of judging solely by the entropy value, this embodiment also introduces an energy percentage factor. :

[0061] ;

[0062] In the formula, For the first First-order components, This is the original signal. Only when the energy percentage of this component exceeds a preset energy threshold (e.g., 1%) will it participate in the entropy value determination. This multi-dimensional determination logic aims to exclude false modes that, although their entropy values ​​are within a reasonable range, have too little physical energy and are not valuable for extraction.

[0063] Step S35, the grating measurement device performs measurements on all gratings according to the above criteria. The system performs logical checks. If a component simultaneously satisfies the condition that its sample entropy is between 0.1 and 0.3 and its energy percentage meets the threshold requirement, it is identified as a valid phase component. The grating measurement device then linearly superimposes all components that meet the conditions to reconstruct a pure grating interference signal. Its mathematical expression is:

[0064] ;

[0065] In the formula, This is the set of valid component indices that have passed the determination. By executing step S3, the grating measurement device achieves precise noise reduction of the original signal. This screening method based on nonlinear dynamic characteristics, compared with traditional frequency domain hard filtering, can more completely preserve the instantaneous frequency components of the signal during dynamic acceleration and deceleration, thus providing data support for the accuracy of subsequent phase calculation.

[0066] In this embodiment, step S4 aims to physically isolate and reassemble the feature components selected in step S3. By constructing an independent logical space, it ensures that the stored data sequence contains only phase features directly related to the physical displacement of the grating. Based on the principle of storage consistency in digital signal processing, due to the decomposition generated in step S2... The component vectors contain a large number of noise terms and low-frequency trend terms. Direct global computation would introduce accumulated errors and consume unnecessary computational resources. Therefore, by reserving a specific storage address range in the high-speed cache of the signal processing module and initializing it as an independent empty set, the effective components determined by dynamics can be stored in a structured manner, thus providing a clean input source for the subsequent Hilbert transform. The signal processing module of the grating measurement device sets an independent empty set and saves n components that meet the threshold standard. This completes the logical transformation from feature recognition to data aggregation.

[0067] Step S41: Based on the memory allocation strategy in computer organization principles, the grating measurement device initializes an independent set variable in local memory to temporarily store valid feature mode components. As a specific lower-level implementation, this independent empty set is represented at the underlying hardware level as a pre-allocated linear buffer or dynamic array. Its initial element count is set to 0, and the memory address it occupies does not overlap with the original noisy address, thus achieving physical data isolation. Define the initial set. And set a counter Used to count the number of valid components that eventually enter the set.

[0068] Step S42: Based on the Boolean decision result generated in step S3, the grating measurement device executes the component extraction and storage logic. In this embodiment, the signal processing module traverses all... The order intrinsic mode components are given if and only if the sample entropy value of a certain component is... When the energy percentage within the preset range [0.1, 0.3] exceeds a threshold, a storage instruction is triggered. The mapping logic can be represented as follows:

[0069] ;

[0070] ;

[0071] In the above formula, This represents the target set for storing valid components; The first step obtained by decomposing in step S2 eigenmode components; An index sequence that satisfies specific physical characteristic constraints; For set The total number of elements in the decomposition is between 0 and N, where N is the total order of the ICEEMDAN decomposition. This index-based storage method aims to establish a logical index between the original decomposition sequence and the effective feature sequence, avoiding frequent physical movement of the original large data volume and improving processing efficiency.

[0072] Step S43: Considering the robustness of the algorithm in extremely low signal-to-noise ratio environments, the number of effective components in the grating measurement device is adjusted. Perform boundary checks. If a counter is detected after the traversal is complete... If no component meets the threshold criterion, the currently acquired signal is determined to be an invalid signal or a signal consisting entirely of noise. In this case, as a preferred implementation, the algorithm will trigger an exception handling mechanism, sending an error message to the grating measurement system and stopping subsequent phase calculations, thereby preventing numerical calculation overflow or program crashes caused by the participation of an empty set in the operation. When >0, the grating measurement equipment will be assembled All components are time-aligned to ensure that each path... The initial sampling time maintains a strict synchronous causal relationship with the original signal.

[0073] Step S44, the grating measurement device will collect In The effective components are stored in physical order of their frequencies from high to low. Since ICEEMDAN decomposition inherently possesses frequency reduction characteristics—that is, the larger the order λ, the lower the dominant frequency of the component—this arrangement essentially reflects the energy distribution law of the grating interference signal in the frequency domain. After storage, the signal processing module will... and its dimensional information The parameters are packaged into the input interface parameters for the subsequent step S5. In this embodiment, by setting an independent empty set and filling it as needed, the grating measurement device accurately extracts the phase calculation factor reflecting physical displacement against a complex noise background. For memory addressing, pointer offset calculation, and memory reclamation operations of dynamic arrays, those skilled in the art can perform register-level configuration according to the specific microprocessor (such as DSP or FPGA) architecture. The specific underlying driver writing is a well-known technology in the art and will not be described in detail here. Through the logical scheduling of the above sub-steps S41 to S44, the grating measurement device achieves efficient organization of effective information.

[0074] In this embodiment, step S5 involves a secondary recombination and refined stripping of the remaining non-feature components screened in step S3. This aims to extract the high random noise floor from complex background interference and eliminate extremely low-frequency quasi-static drift terms. Based on the energy completeness principle of signal multi-scale decomposition, although step S4 has locked the core phase components, the remaining modes still contain crucial information reflecting the environmental state of the measurement system. Specifically, components with a sample threshold greater than 0.3 typically correspond to broadband high-frequency noise generated by the grating readhead circuit, while components with extremely low sample thresholds correspond to DC trend terms introduced by laser source power fluctuations, ambient temperature drift, or guide rail mechanical vibration. By classifying these non-feature components, accurate physical parameters can be provided for subsequent phase calculation DC compensation and noise suppression. The signal processing module of the grating measurement device extracts the remaining non-feature components from the remaining threshold range. By summing and reconstructing, and removing the extremely low components whose amplitudes approach zero in p samples, deep decoupling of the interference term is achieved.

[0075] S51. Based on the complement mapping logic of set theory, a grating measurement device determines the set of ineffective indices that do not belong to the phase feature subspace. In this embodiment, the set consists of all component indices generated in step S2, excluding those already stored in the independent set in step S4. index The subsequent construction. The complementary index sequence is defined as follows: ,in Let be the total order of the ICEEMDAN decomposition. This is the index set of the effective components. Based on this, the signal processing module allocates a temporary superposition register space in memory for processing the set. The complexity of each corresponding intrinsic mode component is compared by secondary quantization.

[0076] Step S52, based on the physical discrimination criterion for the degree of determinism of the signal, the grating measurement device sets the complement set. The system identifies and filters out extremely low-frequency components whose sample entropy values ​​approach zero. The threshold for determining extremely low entropy is defined as follows: In this embodiment, The value range is set between 0.01 and 0.05, preferably 0.05. This range is chosen because in grating displacement measurement, when the sample entropy is below 0.05, the signal sequence exhibits extremely strong regularity or quasi-monotonicity. The corresponding physical phenomena are usually linear drift of laser power or ultra-low frequency displacement caused by environmental thermal expansion. The grating measurement equipment statistically meets the conditions. The number of components and denoted as The extraction process is shown in the following formula: This adaptive determination through real-time entropy value statistics The value-based approach enables the algorithm to automatically remove trend terms in different measurement scenarios, avoiding low-frequency interference residues or excessive filtering of useful information caused by a fixed truncation order.

[0077] Step S53, the grating measurement device obtains the complement set Remove the above The extremely low components yield the final set of indices for the noise acoustic modes. Under this logical framework, sets The retained components are all located in the high-frequency random fluctuation band in the frequency domain. The grating measurement equipment performs an addition and reconstruction operation to calculate the reconstructed residual signal. Its mathematical formula is defined as follows: In the above formula, For the first The sequence of high-frequency noise mode components is preserved. In this embodiment, by stripping that... The reconstructed DC component with sample entropy approaching zero It exhibits random oscillation characteristics with a zero mean in the time domain. This zero-mean characteristic is of significant engineering importance for subsequent evaluation of the system's low-noise energy and for correcting the analytic signal offset of the Hilbert transform.

[0078] Step S54: Considering the numerical singularity of the actual discrete sampling data, the grating measurement device reconstructs the data. Perform energy consumption ratio verification. As a preferred implementation, the system calculates... With the original signal The energy ratio. If this ratio exceeds a preset abnormal threshold (e.g., 5%), the system determines that there is severe electromagnetic interference in the current measurement environment and outputs an environmental warning signal; conversely, if the energy ratio is within the normal range, it will... The variance of the system's dynamic noise is called the standard deviation, which is used to assist in the subsequent sub-nanometer phase compensation algorithm. For overflow protection in the addition operation and the index rearrangement operation of the dynamic array in memory, those skilled in the art can perform conventional low-level optimizations based on the specific embedded development platform (such as ARM or DSP). The specific code implementation logic is well-known in the art and will not be elaborated here. Through the orderly execution of the above steps S51 to S54, the grating measurement device completes the classification, decoupling, and environmental modeling of non-phase feature components.

[0079] In this embodiment, step S6 introduces a recursive processing mechanism based on the energy convergence criterion to perform closed-loop dynamic optimization of the extraction process in steps S2 to S5, ensuring that the eigenmode components stored in the independent empty set can completely cover the effective phase characteristics of the grating interference signal. Based on the completeness principle of non-stationary signal processing, a single decomposition and screening often fails to completely remove nonlinear interference components coupled with the real displacement, especially under conditions where there is local geometric distortion in the grating pair or a wide distribution of environmental vibration spectrum, where weak useful features still remain in the residual signal. By establishing a cyclic iterative logic, the residual signal reconstructed in step S5 is remapped to the original input of the next round of decomposition, enabling layer-by-layer deep extraction of phase information. The technical purpose of this recursive operation is to use multi-round sample entropy dynamic discrimination to reduce the final remaining interference energy to below a preset physical noise threshold, thereby providing a high-fidelity feature source for sub-nanometer fine interpolation. The signal processing module of the grating measurement device drives the automatic evolution of the entire processing chain by setting an iteration termination criterion.

[0080] Step S61: Based on the convergence design requirements of the iterative algorithm, the grating measurement device initializes the iteration counter before starting the loop. and the maximum number of iterations limit In this embodiment, the initial value of the iteration counter is set to... =1, maximum number of iterations The value range is set to 3 to 5 times. The reason for this is... The upper limit is limited to 5 iterations to take into account the real-time requirements of the grating displacement measurement system, avoid excessive iterations that could cause the calculation delay to exceed the servo control cycle, and prevent excessive decomposition from generating physically spurious components. Simultaneously, the signal processing module synchronously sets the residual energy change rate threshold ξ, the value of which is typically based on the quantization noise level of the reference system's numerical control converter; in this embodiment, it is preferably 10. -4 .

[0081] Step S62: Based on the quantitative analysis principle of signal energy conservation, the grating measurement device calculates the residual signal energy generated in the current iteration. Define the... The residual energy measurement after the next iteration is Its specific mathematical expression is as follows:

[0082] ;

[0083] In the above formula, Indicates the first The reconstructed residual sequence generated in step S5 after the next iteration For discrete-time sampling point index, This represents the total number of sampling points. Based on this, the grating measurement equipment further calculates the relative energy decay rate between two adjacent iterations. This is used to evaluate the convergence progress of the algorithm, and its calculation formula is defined as follows:

[0084] ;

[0085] Considering the completeness of numerical calculations, when the original input signal is close to zero or the previous iteration has approached the ideal silent state, the denominator... It tends to approach 0. To avoid logic crashes caused by dividing floating-point numbers by zero, this embodiment presets a very small decision constant. If detected Then the grating measurement equipment will force The value is assigned to 0, and the current iteration process is terminated prematurely.

[0086] Step S63: Based on the logical determination result of the above dual constraint conditions, the grating measurement device executes a loop control command. In this embodiment, the determination criterion follows a weighted priority logic: if the current iteration represents... And the relative rate of change of energy satisfies If the residual signal still contains extractable phase features, then the grating measurement device will determine the current residual signal. The feedback is sent to the input of step S2 as a new sequence to be processed, and the counter increment logic is executed synchronously. This cycle triggers steps S2 through S5.

[0087] Step S64, when the logical judgment satisfies or At this point, the grating measurement device exits the loop. This means that the current residual signal has converged into stationary random white noise, or the computational overhead has reached the system's preset threshold limit. As the data aggregation stage after the loop ends, the signal processing module refines the complete displacement information from each iteration. This recursive aggregation-based processing method not only effectively suppresses mode aliasing caused by single decomposition, but also ensures that the extracted signal still possesses strict sinusoidality under the frequency shift caused by the high-speed movement of the grating pair, laying a high signal-to-noise ratio data foundation for subsequent instantaneous phase extraction based on Hilbert transform. For memory stack management within the loop body and the calculation of pointer offset during iteration, those skilled in the art can use general digital signal processing instruction sets for optimization. The specific memory allocation strategy is a well-known technology in the field and will not be elaborated here. Through the recursive scheduling of the above sub-steps S5 to S64, the grating measurement device achieves robust extraction of non-stationary interference signal features.

[0088] ;

[0089] In the formula, This is the final iteration number when the loop exits; For the first The set of valid component indices determined by sample entropy in the next iteration; Indicates the first The first one extracted in the second loop The first-order characteristic components. Through this multi-round cumulative reconstruction, the grating measurement equipment demonstrates in its manual how to refine complete displacement information from massive decomposition results.

[0090] In this embodiment, step S7 involves time-domain aggregation of the effective eigenstate components that have been recursively filtered and temporarily stored in the independent set in the previous steps, and by executing a global reconstruction algorithm, constructing a high signal-to-noise ratio displacement signal that can accurately characterize the grating displacement features. From a physical perspective of signal integrity restoration, due to the energy conservation constraint followed by the ICEEMDAN decomposition process, the original signal is deconstructed into a series of orthogonal primitives with different local feature scales. After undergoing the dynamic feature identification in steps S3 and S6, the independent set... Uncorrelated random noise and broadband interference have been removed. At this point, by coherently superimposing these physically complementary characteristic modes, the autocorrelation characteristics of the useful signal in spatial location can be utilized to suppress residual noise while maximizing the recovery of the waveform contrast of the interference fringes. This reconstruction is not merely a simple merging of data dimensions, but also provides a high-fidelity input source for subsequent construction of an analytic signal via Hilbert transform (HT). The signal processing module of the grating measurement equipment implements the physical reconstruction of the signal through the following logic.

[0091] Step S71, based on the memory mapping and pointer synchronization indexing mechanism, the grating measurement device retrieves data from the independent set maintained in step S4. The system extracts all eigenstate sequences that have been determined to be valid through multiple iterations. In this embodiment, the signal processing module accesses the starting address of the cache and performs a rigorous comparison based on the sampling timestamps of each valid component during the decomposition process. Considering the sampling jitter that exists under different hardware clocks, the system needs to ensure that each order involved in the reconstruction... The components satisfy the following on the time axis Consistency constraints, where Represents the order of the components. This represents the sampling point index. If a sequence length inconsistency is detected, the least common length is used to truncate the end, thereby avoiding waveform distortion caused by phase misalignment.

[0092] Step S72: Based on the general principle of linear superposition, the grating measurement device performs a point-by-point summation operation on all extracted valid eigenstate components. As a specific implementation, the system uses an accumulator register to sum the values ​​at each time step. Linear aggregation of the effective modal amplitudes is performed. The displacement signal is then reset. The mathematical analytical expression is defined as follows:

[0093] ;

[0094] In the above formula, This represents the reconstructed high signal-to-noise ratio displacement signal; This refers to the final iteration round determined in step S6; For the first The effective component index sequence selected in the next iteration; For the first The first one extracted in the second loop The effective eigenstate components. The technical objective of this step is to regroup the phase information scattered across different frequency bands through the logical merging of multi-scale energy, forming a single analytical signal unit with a single rotational characteristic, thereby eliminating the amplitude modulation error present in the original grating signal.

[0095] Step S73: Considering the stringent physical requirements of the Hilbert transform on the signal's central symmetry, the grating measurement device measures the reconstructed signal. Dynamic DC bias rejection is performed. In this embodiment, if the reconstructed signal contains a non-zero mean component, it will directly cause the evolution trajectory of the analytic signal in the complex plane to deviate from the origin, thereby introducing a periodic nonlinear phase deviation. Therefore, the system calculates... The statistical mean within the full sampling window is then used, and the following centering correction is applied:

[0096] ;

[0097] In the formula, This is the DC displacement signal ultimately used for phase calculation; here This represents the total number of points within the current sampling period. The system first verifies this before performing the division operation. The value is checked for being greater than zero to avoid logical collapse caused by a denominator of zero. This mean drift compensation forces the centroid of the displacement signal to return to the zero-level reference line, ensuring that the subsequently obtained instantaneous phase is within the range specified in the standard function. The accuracy of linear mapping within the interval.

[0098] Step S74, the grating measurement device measures the generated... A multi-dimensional joint assessment of amplitude normalization and signal quality is performed. As a preferred implementation, the validity of the output no longer relies solely on single peak detection, but is based on a weighted logic of amplitude stability and signal-to-noise ratio (SNR). The system calculates the peak value of the reconstructed signal. and residual noise variance When satisfied and When this occurs, the currently reconstructed signal is determined to be a reliable displacement carrier. The voltage threshold range is specified. Typically, the dynamic range of the analog front end of the grating sensor is preset as follows: If the judgment result does not meet the above conditions, the system will automatically backtrack to step S2 to adjust the decomposition parameters, or send a strong anomaly warning to the main control unit. This multi-dimensional judgment logic ensures that the data output to the phase calculation unit always maintains high physical fidelity in complex industrial environments. For floating-point overflow protection during the reconstruction process and the sliding average window configuration in the DC-DC removal operation, those skilled in the art can optimize the algorithm according to the specific embedded processing architecture (such as DSP or FPGA). The specific logic gate-level implementation is a well-known technology in the field and will not be elaborated here. The grating measurement device completes the physical mapping from the original stray signal to the clean displacement guide by executing the above steps S71 to S74.

[0099] Please see the appendix Figure 6 For the phase expansion diagram and local magnified diagram of the original signal after Hilbert transform in this invention, please refer to the appendix. Figure 7 As shown in the attached diagram, the phase expansion diagram and local magnified diagram of the processed signal after Hilbert transform according to the present invention are provided. Figure 6 and 7 The horizontal axis represents time in milliseconds, and the vertical axis represents phase in radians. In this embodiment, step S8 reconstructs the high signal-to-noise ratio signal output from step S7. The Hilbert transform is performed to construct an analytic signal from which instantaneous phase information that can directly map the physical displacement of the grating is extracted. Based on the physical construction principle of analytic signals, the real-valued signal generated by the grating interferometer system only exhibits amplitude fluctuations in the time domain, making it difficult to directly obtain the phase angle with direction recognition capability from a single signal. By performing the Hilbert transform, a sequence of imaginary parts that is strictly orthogonal to the phase of the original signal can be generated, thereby transforming the real-valued oscillation into a rotating vector in the complex plane. The technical purpose of this transformation is to achieve complete decoupling of phase and amplitude using the mathematical properties of the complex envelope, ensuring that the extracted phase value maintains extremely high instantaneous linearity even when the grating ruler undergoes acceleration or deceleration or when the ambient light intensity fluctuates slowly. The phase calculation module of the grating measurement equipment performs the calculation according to the following logic.

[0100] Step S81: Based on the orthogonalization principle of integral transform, the grating measurement device processes the displacement signal after DC removal. Perform convolution operations to generate corresponding orthogonal components. In this embodiment, the transformation process is defined in the time domain as the convolution of the signal with a specific operator sequence, and its specific mathematical integral expression is as follows:

[0101] ;

[0102] In the above formula, for The corresponding Hilbert transform sequence; Cauchy principal value; For integration variables; These are discrete sampling time points. From a frequency domain perspective, the physical significance of this step lies in processing the positive frequency components of the signal. The fixed phase shift, while performing phase shift on the negative frequency components. Phase shifting. Considering the efficiency of practical digital signal processing, those skilled in the art can typically implement the above process using a frequency domain phase shifting method based on FFT (Fast Fourier Transform). The specific window function truncation and zero-padding operations are well-known techniques in the field and will not be elaborated here. This processing ensures that the generated imaginary part sequence maintains a strictly orthogonal relationship with the real part across the entire frequency band.

[0103] Step S82, based on the vector synthesis logic of the analytic signal, the grating measurement device will convert the real part sequence... With imaginary part sequence Perform complex number aggregation to construct analytic signals And calculate the package phase accordingly. The dynamic evolution characteristics of an analytic signal in the complex plane can be described by the following equation:

[0104] ;

[0105] In the formula, The imaginary unit; The instantaneous amplitude of the analytical signal is used to characterize the modulation index of the interference signal; This is the required instantaneous phase angle. The grating measurement device uses the following four-quadrant arctangent function to obtain the phase value:

[0106] ;

[0107] When executing the above division operation logic, considering the numerical singularity induced by the denominator approaching zero, the system simultaneously performs an instantaneous energy check. In this embodiment, a preset amplitude protection threshold is used. Its value is usually set to 1% of the signal standard deviation (e.g., 10). -5V). If an instantaneous modulus is detected. The system determines that the current sampling point is a signal failure point and forcibly skips the phase update of that point, using the phase value from the previous moment. This multi-dimensional anomaly detection logic avoids phase calculation crashes caused by the instantaneous extinguishing of the interference signal.

[0108] Step S83: Based on the abrupt change characteristics of the phase angle within the principal value range, the grating measurement device measures the wrapped phase obtained in step S82. Perform phase expansion. Because the output of the arctangent operation is limited to... Within the interval, when the grating displacement exceeds one grating pitch period, discontinuous phase jumps will occur. Therefore, the system establishes a phase accumulation... By monitoring the phase difference between adjacent sampling times To identify boundary violations, the specific entanglement logic is defined as follows: In the above formula, The total phase after continuous expansion; This represents the floor operation. During this process, the system... The slope limit is applied to determine the magnitude of the change. If the phase jump between single samples exceeds... If the radian value is determined to be a pseudo-phase caused by electromagnetic pulse interference, the point is smoothed and corrected using a linear interpolation algorithm.

[0109] Step S84, the grating measurement device will measure the total phase after unfolding. Convert to physical displacement values This conversion process is based on the physical causal relationship between the grating pitch and the phase period, and its calculation formula is as follows: In the formula, This refers to the instantaneous displacement output by the grating measurement system. This is the grating pitch constant of the grating sensor. In this embodiment, The typical range of values ​​is to Preferred The value is determined based on the physical line density of the moiré fringe subdivision plate used. This scaling factor is chosen because each rotation of the analytical signal in the complex plane corresponds precisely to the reading head sweeping through a complete grating pitch cycle in physical space. Through the above calculations, the grating measurement device ultimately achieves a precise mapping from mode decomposition features to sub-nanometer physical displacement. For the accumulator clearing mechanism in phase expansion and the multi-rate output configuration of displacement data, those skilled in the art can use general FPGA logic units or real-time interrupt handling mechanisms for design. The specific circuit implementation logic is well-known in the field and will not be elaborated here. The grating measurement device completes the phase demodulation of the interference signal through the sequential execution of sub-steps S81 to S84.

Claims

1. A phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT, characterized in that, Includes the following steps: Acquire the actual noisy grating interference signal and characterize it as a discrete time-domain sequence; The discrete-time sequence is adaptively decomposed using a multi-iteration improved adaptive noise complete set empirical mode decomposition algorithm to obtain multi-order intrinsic mode components. The complexity of the multi-order intrinsic mode components is evaluated using the sample entropy algorithm, and the effective phase components are identified by combining the energy proportion criterion. The effective phase components are stored in a preset independent set for iterative accumulation, and the remaining intrinsic mode components other than the effective phase components are summed and reconstructed. The remaining intrinsic mode components are non-feature components. After removing the trend terms in the non-feature components whose sample entropy values ​​are lower than a preset complexity threshold, the noise mode components are obtained. A recursive processing mechanism is executed to determine whether the energy convergence state of the noise mode component meets the preset convergence threshold. If the convergence threshold is not met, the noise mode component is used as the input sequence for the next round to re-execute the adaptive decomposition until the iteration termination condition is met. All valid phase components accumulated and stored in the independent set during each iteration are globally reconstructed, dynamic DC bias elimination and Hilbert transformation are performed, instantaneous phase is extracted and mapped to physical displacement values.

2. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 1, characterized in that, When acquiring the actual noisy grating interference signal, a grating measurement device is used to convert the analog voltage signal into a digital sequence with a bit depth of 12 or 16 bits through an analog-to-digital converter, and the sampling frequency of the analog-to-digital converter is set to 5 to 10 times the highest frequency component generated by the grating motion.

3. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 1, characterized in that, When performing the adaptive decomposition, the decomposition process includes: introducing a white noise sequence with standard deviation coefficients as a masking signal in each decomposition, fitting the upper and lower envelopes of the noisy sequence using a cubic spline interpolation function and extracting the local mean, obtaining the residual component by calculating the arithmetic mean of multiple experimental results, and removing the residual component from the sequence to be decomposed to extract the eigenmode component of the current order.

4. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 1, characterized in that, When using the sample entropy algorithm for complexity evaluation, the evaluation process includes: setting the embedding dimension to 2, defining the similarity tolerance as 0.2 times the standard deviation of the current intrinsic mode component to be evaluated, calculating the Chebyshev distance between the reconstructed vectors, counting the proportion of vector pairs that satisfy the similarity tolerance condition at dimensions m and m+1, and taking the negative natural logarithm of the ratio of the proportions as the sample entropy value.

5. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 4, characterized in that, The logical criterion for identifying the effective phase component is as follows: determine whether the sample entropy value is within the range of 0.1 to 0.3, and at the same time determine whether the proportion of the energy of the intrinsic mode component in the actual noisy grating interference signal exceeds a preset energy threshold; if both of the above criteria are met, it is determined to be an effective phase component.

6. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 1, characterized in that, When removing the trend term, the removal process includes: identifying components in the non-feature components whose sample entropy value is lower than the preset complexity threshold, defining them as trend terms that reflect laser power fluctuations or environmental drift and removing them, and linearly summing the remaining high-frequency random fluctuation components to reconstruct the noise mode components.

7. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 1, characterized in that, When executing the recursive processing mechanism, the iteration termination conditions include: the current iteration round reaches the preset maximum iteration limit, or the relative rate of change of energy of the noise mode components generated by two adjacent iterations is less than the convergence threshold.

8. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 1, characterized in that, After global reconstruction, the dynamic DC bias removal includes: calculating the statistical mean of the reconstructed displacement signal within the full sampling window, subtracting the statistical mean from the displacement signal, so that the centroid of the displacement signal returns to the zero-level reference line, and obtaining the DC-free displacement signal.

9. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 8, characterized in that, The method for extracting the instantaneous phase includes: using Hilbert transform to perform full-band fixed-rate phase shift on the displacement signal after DC removal to generate an orthogonal imaginary part sequence, constructing an analytical signal, and solving the wrapping phase using a four-quadrant arctangent function; during the solution process, if the instantaneous amplitude of the analytical signal is lower than a preset amplitude protection threshold, the phase value of the previous moment is used.

10. The phase extraction method for grating interference signals based on multi-iteration ICEEMDAN-HT according to claim 9, characterized in that, The process of mapping the instantaneous phase to a physical displacement value includes: monitoring the phase difference between adjacent sampling times, performing phase expansion based on the jump characteristics of the difference within the main value range to obtain the total phase, and calculating the instantaneous displacement in physical space based on the ratio of the total phase to the grating pitch constant.

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