Sine frequency modulation harmonic error compensation method based on Bessel function modulation characteristics

Abstract

The invention discloses a sine frequency modulation harmonic error compensation method based on Bessel function modulation characteristics, and relates to the technical field of sine frequency modulation nonlinear parameter compensation and high-precision measurement of target absolute distance based on multi-wavelength interference. The problem that the constant phase demodulation precision is insufficient due to the fact that carrier harmonic waves, accompanying amplitude modulation and modulation depth errors cannot be effectively suppressed due to laser modulation nonlinearity in existing sine frequency modulation distance measurement is solved. According to the method, after sine frequency modulation interference signals are collected, high-pass filtering is carried out to remove low-frequency components, Hilbert transform is carried out to remove envelope suppression associated amplitude modulation, window function frequency mixing is carried out to extract modulation depth, a matched carrier containing second harmonics is constructed, and frequency modulation harmonic parameter optimization is achieved through baseband amplitude maximization. And finally, carrying out orthogonal demodulation on the compensated interference signal to obtain a high-precision constant phase. The method is suitable for high-precision multi-wavelength interference ranging, precision displacement measurement and optical metering systems.

Description

Technical Field

[0001] This involves the fields of sinusoidal frequency modulation nonlinear parameter compensation and high-precision measurement of target absolute distance based on multi-wavelength interferometry, specifically sinusoidal frequency modulation harmonic error compensation based on Bessel function modulation characteristics. Background Technology

[0002] Multiwavelength interferometric ranging technology is an important means of achieving long-range, high-precision absolute distance measurement. After more than a century of development, it has formed a systematic theoretical framework and engineering methods. While the traditional fractional-number repetition method can achieve step-by-step distance amplification by synthesizing wavelength chains of different wavelengths, its reliable ranging range is significantly limited by the coherence length of early light sources. With the development of lasers, multiwavelength ranging technology has broken through the range bottleneck. By constructing multi-level synthesized wavelength chains, it can achieve sub-micron or even nanometer-level measurement accuracy over a wide range, and is therefore widely used in high-precision measurement scenarios such as precision manufacturing, optical metrology, and aerospace.

[0003] In multi-wavelength ranging signal demodulation methods, sinusoidal frequency modulation-based phase demodulation methods are widely used due to their simple optical path structure, high system stability, and low sensitivity to interference. For example, existing methods typically inject a sinusoidal radio frequency signal into a laser, causing the laser frequency to change sinusoidally over time. The interference signal is then mixed, filtered, and quadrature demodulated to extract a constant phase, thereby completing the absolute distance measurement. However, the actual performance of sinusoidal frequency modulation methods is highly dependent on the linear response of the laser to the applied modulation voltage. Real lasers generally exhibit significant nonlinear modulation effects, leading to the following problems.

[0004] First, the modulation response of a laser is not an ideal sinusoidal function; the actual frequency output usually contains multiple higher-order harmonic components of the modulation frequency. These harmonic terms, after entering the interference signal, are mixed into the baseband region during mixing and filtering, causing systematic errors in constant-phase demodulation. Second, when a laser performs sinusoidal frequency modulation, it is often accompanied by intensity modulation, resulting in significant envelope fluctuations in the interference signal. Existing technologies mostly suppress intensity modulation by improving hardware or adding feedback control methods, but this increases system complexity and is not suitable for measurement devices requiring high stability and compactness. Third, existing harmonic error compensation methods generally rely on additional calibration optical paths or auxiliary devices, such as reference interferometers, bypass detectors, or high-precision RF calibration links. This makes the system structure complex, difficult to debug, and degrades long-term stability, and is also unfavorable for the synchronous use of dual-wavelength or multi-wavelength ranging systems.

[0005] Furthermore, regarding harmonic suppression of sinusoidal frequency-modulated signals, some studies have attempted to reduce nonlinear errors using methods such as trapezoidal frequency modulation, digital predistortion compensation, and sinusoidal fitting correction. However, these methods either require real-time calculations with high-bandwidth hardware or are highly sensitive to laser parameters, temperature stability, and driving conditions, making it difficult to achieve long-term stable operation in general-purpose ranging systems. Especially in multi-wavelength systems, due to the inconsistent modulation nonlinearities of lasers of different wavelengths, existing compensation strategies struggle to achieve simultaneous calibration and unified compensation, further limiting demodulation accuracy and wavelength chain connection accuracy.

[0006] In summary, existing technologies suffer from several drawbacks, including the inability to accurately calibrate and effectively compensate for nonlinear parameters of sinusoidal frequency modulation through simple structures, difficulty in suppressing harmonic terms and associated amplitude modulation, and insufficient demodulation accuracy of the phase constant of the interference signal. Summary of the Invention

[0007] To address the shortcomings of existing technologies, such as the inability to accurately calibrate and effectively compensate for nonlinear parameters of sinusoidal frequency modulation through simple structures, the difficulty in suppressing harmonic terms and associated amplitude modulation, and the resulting insufficient demodulation accuracy of the phase constant of the interference signal, the technical solution provided by this invention is as follows: A sinusoidal frequency modulation harmonic error compensation method based on Bessel function modulation characteristics includes: The steps include acquiring sinusoidal frequency-modulated interference signals and using them as inputs for interference signal preprocessing; The steps include high-pass filtering the interference signal to remove low-frequency modulation components and outputting a high-frequency interference signal as the input for accompanying amplitude modulation suppression; The steps include performing Hilbert transform on the high-frequency interference signal to obtain the envelope, performing envelope removal processing to obtain an amplitude-stable interference signal, and outputting it as the input for modulation depth extraction. The steps include mixing the amplitude-stabilized interference signal with a periodic window function, extracting the carrier fundamental frequency modulation depth based on the position of the maximum spectral value, and outputting it as the input for constructing the matched carrier. The steps are as follows: constructing a matched carrier containing second harmonic components based on carrier fundamental frequency modulation depth and outputting it as the input for frequency modulation harmonic compensation; The steps include mixing the interference signal, the periodic window function, and the matched carrier, determining the second harmonic amplitude and phase through a baseband amplitude maximization optimization method to obtain a harmonic-compensated interference signal, and outputting it as a constant phase demodulation input. The steps for orthogonally demodulating the harmonic compensation interference signal to extract the phase constant of the interference signal.

[0008] Furthermore, in a preferred embodiment, the interference signal preprocessing includes using the sinusoidal frequency-modulated interference signal output from the polarization-maintaining optical path structure as input and suppressing the low-frequency AC component converted from the DC quantity modulated by the laser through high-pass filtering.

[0009] Furthermore, in a preferred embodiment, the accompanying amplitude modulation suppression includes obtaining the amplitude envelope using Hilbert transform and dividing the original interference signal by the envelope.

[0010] Furthermore, in a preferred embodiment, modulation depth extraction includes mixing the amplitude-stabilized interference signal with a periodic window function and determining the carrier fundamental frequency modulation depth based on the lateral position corresponding to the maximum amplitude of the mixed signal spectrum.

[0011] Furthermore, in a preferred embodiment, the matched carrier construction includes constructing a matched carrier containing adjustable second harmonic amplitude and second harmonic phase based on the carrier fundamental frequency modulation depth.

[0012] Furthermore, in a preferred embodiment, frequency modulation harmonic compensation includes mixing the interference signal, the periodic window function, and the matched carrier, and optimizing the baseband signal amplitude by utilizing the variation of the second harmonic parameters.

[0013] Based on the same inventive concept, this invention also provides a sinusoidal frequency modulation harmonic error compensation device based on Bessel function modulation characteristics, comprising: A module that acquires sinusoidal frequency-modulated interference signals and uses them as inputs for interference signal preprocessing; A module that performs high-pass filtering on the interference signal to remove low-frequency modulation components and outputs a high-frequency interference signal as the input for accompanying amplitude modulation suppression; A module that performs Hilbert transform on high-frequency interference signals to obtain the envelope and performs envelope removal processing to obtain amplitude-stabilized interference signals and outputs them as input for modulation depth extraction; This module mixes the amplitude-stabilized interference signal with a periodic window function, extracts the carrier fundamental frequency modulation depth based on the position of the maximum spectral value, and outputs it as the input for constructing the matched carrier. A module that constructs a matched carrier containing second harmonic components based on carrier fundamental frequency modulation depth and outputs it as the input for frequency modulation harmonic compensation. The module mixes the interference signal, the periodic window function, and the matched carrier, and determines the second harmonic amplitude and phase through a baseband amplitude maximization optimization method to obtain the harmonic compensation interference signal and output it as a constant phase demodulation input. A module for orthogonally demodulating harmonic-compensated interference signals to extract the phase constant of the interference signal.

[0014] Based on the same inventive concept, the present invention also provides a computer storage medium for storing a computer program, wherein when the computer program is read by a computer, the computer executes the method described thereon.

[0015] Based on the same inventive concept, the present invention also provides a computer, including a processor and a storage medium, wherein when the processor reads a computer program stored in the storage medium, the computer executes the method described thereon.

[0016] Based on the same inventive concept, the present invention also provides a computer program product, which, when executed, implements the method described.

[0017] Compared with the prior art, the advantages of the technical solution provided by the present invention are as follows: The effect of using high-pass filtering to suppress the modulated DC component in the interference signal is to effectively eliminate low-frequency interference terms generated by the nonlinear modulation of the laser, so that the subsequent baseband phase information is no longer masked by DC drift and low-frequency fluctuations. Existing sinusoidal frequency modulation ranging methods generally rely on hardware amplitude stabilization or optical path compensation to suppress DC disturbances, while high-pass filtering directly removes low-frequency components at the signal level, which can obtain a cleaner interference signal basis without adding any optical components, thereby improving the stability of constant phase demodulation.

[0018] The effect of extracting the envelope using Hilbert transform and removing it from the interference signal is a significant reduction in the accompanying amplitude modulation caused by laser intensity modulation, thus stabilizing the amplitude of the interference signal. Traditional methods often reduce the AM effect by adding linearization circuits to the light source drive or modulation feedback links, but these hardware methods are complex and susceptible to temperature drift. Accurately obtaining the envelope and removing it from the signal using the Hilbert algorithm enables digital suppression of the accompanying amplitude modulation, which helps avoid systematic errors caused by envelope fluctuations in phase demodulation.

[0019] The effect of mixing the periodic window function with the interference signal and extracting the modulation depth of the carrier fundamental frequency term is to accurately identify the true modulation depth of the sinusoidal frequency-modulated interference signal, so that parameter solving no longer depends on the linear assumption of the laser. Existing studies mostly estimate the modulation depth through hardware calibration or manufacturer parameters, but the actual modulation depth will drift significantly with factors such as temperature and device aging. This scheme utilizes the expansion characteristics of Bessel functions to directly reflect the modulation depth through the position of the mixing peak, thereby achieving real-time, online calibration without additional optical paths and improving the robustness of the overall system.

[0020] Based on the characteristic that the carrier term is completely eliminated and the energy is concentrated in the baseband when the independent variable of the Bessel function is zero, optimizing the matched carrier parameters can accurately compensate for the high-order harmonic effects generated by laser modulation, thus achieving optimal purity in the baseband signal. Compared to existing methods that suppress high-order harmonics through feedforward correction or digital predistortion, this scheme utilizes the inherent mathematical properties of the Bessel function to optimize the matched carrier, allowing the harmonic terms to be gradually modulated and canceled out. This enables the acquisition of an equivalent signal close to ideal sinusoidal frequency modulation without the need for additional hardware.

[0021] The method of constructing a matched carrier wave containing second harmonic components and iteratively determining the relative amplitude and phase of harmonics by maximizing the baseband amplitude allows for precise calibration and compensation of the second-mode frequency-modulated harmonics, the dominant source of error, thereby significantly reducing the impact of harmonics on the constant phase demodulation accuracy. Traditional compensation methods rely on model fitting or external calibration, while this scheme utilizes the sensitivity of the mixing baseband energy to obtain the harmonic parameters that best match the actual laser characteristics through numerical optimization, making the compensation results closer to the true physical output and improving the accuracy and consistency of phase demodulation.

[0022] After harmonic compensation, the effect of extracting the constant phase of the interference signal using quadrature demodulation is to achieve high-precision phase recovery on the purified baseband signal, significantly reducing the nonlinear deviation caused by the accumulation of harmonic, amplitude modulation, and modulation depth errors. Compared with the demodulation offset that often occurs in traditional sinusoidal frequency modulation methods when nonlinear errors are not compensated, this scheme, through the combined effects of pre-stage filtering, envelope removal, modulation depth extraction, and harmonic compensation, enables quadrature demodulation to operate stably under near-ideal conditions, thereby obtaining a highly reliable constant phase and improving the overall accuracy and connectivity of multi-wavelength ranging chains.

[0023] It is suitable for precision optical interferometric ranging scenarios that require high-precision absolute distance measurement over a large area. Attached Figure Description

[0024] Figure 1 This is a schematic diagram of the optical path in sinusoidal frequency modulation ranging. Figure 2 To compensate for the accompanying amplitude modulation interference signal; Figure 3 To compensate for the accompanying amplitude modulation interference signal; Figure 4 This is a diagram illustrating the extraction process of second harmonic parameters. Figure 5 To compensate for the previous constant phase demodulation error; Figure 6 This is to compensate for the constant phase demodulation error. Detailed Implementation

[0025] To make the advantages and benefits of the technical solution provided by the present invention clearer, the technical solution provided by the present invention will now be described in further detail with reference to the accompanying drawings, specifically: Implementation Method 1: This implementation method provides a sinusoidal frequency modulation harmonic error compensation method based on Bessel function modulation characteristics, including: The steps include acquiring sinusoidal frequency-modulated interference signals and using them as inputs for interference signal preprocessing; The steps include high-pass filtering the interference signal to remove low-frequency modulation components and outputting a high-frequency interference signal as the input for accompanying amplitude modulation suppression; The steps include performing Hilbert transform on the high-frequency interference signal to obtain the envelope, performing envelope removal processing to obtain an amplitude-stable interference signal, and outputting it as the input for modulation depth extraction. The steps include mixing the amplitude-stabilized interference signal with a periodic window function, extracting the carrier fundamental frequency modulation depth based on the position of the maximum spectral value, and outputting it as the input for constructing the matched carrier. The steps are as follows: constructing a matched carrier containing second harmonic components based on carrier fundamental frequency modulation depth and outputting it as the input for frequency modulation harmonic compensation; The steps include mixing the interference signal, the periodic window function, and the matched carrier, determining the second harmonic amplitude and phase through a baseband amplitude maximization optimization method to obtain a harmonic-compensated interference signal, and outputting it as a constant phase demodulation input. The steps for orthogonally demodulating the harmonic compensation interference signal to extract the phase constant of the interference signal.

[0026] Interference signal preprocessing includes using the sinusoidal frequency-modulated interference signal output from the polarization-maintaining optical path structure as input and suppressing the low-frequency AC component converted from the DC quantity modulated by the laser through high-pass filtering.

[0027] Accompanying amplitude modulation suppression involves using the Hilbert transform to obtain the amplitude envelope and then dividing the original interference signal by the envelope.

[0028] Modulation depth extraction involves mixing the amplitude-stable interference signal with a periodic window function and determining the carrier fundamental frequency modulation depth based on the lateral position corresponding to the maximum amplitude of the mixed signal spectrum.

[0029] The matching carrier construction involves building a matching carrier with adjustable second harmonic amplitude and phase based on the carrier fundamental frequency modulation depth.

[0030] Frequency modulation harmonic compensation involves mixing the interference signal, a periodic window function, and a matched carrier, and then optimizing the baseband signal amplitude by utilizing the variation of the second harmonic parameters.

[0031] A sinusoidal frequency modulation harmonic error compensation device based on Bessel function modulation characteristics is also provided, comprising: A module that acquires sinusoidal frequency-modulated interference signals and uses them as inputs for interference signal preprocessing; A module that performs high-pass filtering on the interference signal to remove low-frequency modulation components and outputs a high-frequency interference signal as the input for accompanying amplitude modulation suppression; A module that performs Hilbert transform on high-frequency interference signals to obtain the envelope and performs envelope removal processing to obtain amplitude-stabilized interference signals and outputs them as input for modulation depth extraction; This module mixes the amplitude-stabilized interference signal with a periodic window function, extracts the carrier fundamental frequency modulation depth based on the position of the maximum spectral value, and outputs it as the input for constructing the matched carrier. A module that constructs a matched carrier containing second harmonic components based on carrier fundamental frequency modulation depth and outputs it as the input for frequency modulation harmonic compensation. The module mixes the interference signal, the periodic window function, and the matched carrier, and determines the second harmonic amplitude and phase through a baseband amplitude maximization optimization method to obtain the harmonic compensation interference signal and output it as a constant phase demodulation input. A module for orthogonally demodulating harmonic-compensated interference signals to extract the phase constant of the interference signal.

[0032] A computer storage medium is also provided for storing a computer program, which, when read by the computer, executes the method.

[0033] A computer is also provided, including a processor and a storage medium, wherein the computer executes the method when the processor reads a computer program stored in the storage medium.

[0034] A computer program product is also provided, which, when executed, implements the method described.

[0035] Implementation Method Two: This implementation method is a further detailed description of the technical solution provided in Implementation Method One, specifically: To achieve high-precision extraction of the constant phase in sinusoidal frequency-modulated interference signals, this embodiment provides a sinusoidal frequency-modulated harmonic error compensation method based on the modulation characteristics of Bessel functions. This method, starting from interference signal acquisition, sequentially performs interference signal preprocessing, accompanying amplitude modulation suppression, modulation depth extraction, matched carrier construction, harmonic parameter optimization compensation, and final phase demodulation.

[0036] First, the sinusoidal frequency-modulated interference signal output by the measurement interferometer is acquired. This interference signal is generated by the laser being driven by sinusoidal frequency modulation and interfering with the reference optical path. It includes a carrier term for ranging, a constant phase term, and accompanying amplitude modulation and multi-order frequency modulation harmonic terms introduced by the nonlinear response of the laser. This acquired signal serves as the input for subsequent signal preprocessing steps.

[0037] The input interference signal is then preprocessed by using a high-pass filter to suppress the low-frequency modulation DC component. Since the original DC intensity of the laser is converted into a low-frequency AC term that varies with the modulation frequency after modulation, these low-frequency terms will cover the baseband information required for deconvolution. Therefore, a high-pass filter is used to retain only the high-frequency content, so that the output preprocessed signal contains only the carrier-related frequency domain component, which can then be used as the input for subsequent amplitude modulation suppression.

[0038] Next, the amplitude envelope of the high-pass filtered signal is obtained using the Hilbert transform, and this envelope is removed from the signal to suppress the accompanying amplitude modulation effect caused by laser intensity modulation. The envelope obtained by the Hilbert transform reflects the periodic change of amplitude over time. Dividing the original signal by the envelope yields an amplitude-stable normalized interferometric signal. This normalized signal serves as the input for subsequent modulation depth estimation.

[0039] The normalized interferometric signal is then mixed with a periodic window function to extract the modulation depth of the carrier term. According to Bessel function expansion, the modulation depth determines the peak position of the mixed spectrum. By detecting the position of maximum amplitude in the mixed signal spectrum, the fundamental frequency modulation depth parameter of the interferometric signal is obtained. This modulation depth serves as the input for the next step of constructing the matched carrier.

[0040] Then, based on the modulation depth obtained in the previous step, a matching carrier is constructed to compensate for the harmonic terms in the interference signal. The initial form of this matching carrier includes a fundamental frequency component consistent with the main carrier of the interference signal, and additionally contains a second harmonic component, the second term of which is given adjustable amplitude and phase for optimization in subsequent processes. The constructed matching carrier serves as the input to the harmonic compensation optimization step.

[0041] Next, the interference signal, periodic window function, and matched carrier are mixed again, and the change in the amplitude of the baseband component after mixing with the matching carrier parameters is monitored. Utilizing the characteristic that baseband energy reaches its maximum when the carrier is completely canceled, an optimization algorithm continuously adjusts the phase and relative amplitude of the second harmonic in the matched carrier until the mixed baseband amplitude reaches its maximum. When the maximum is reached, it is considered that the main carrier term and the second harmonic term in the interference signal are effectively canceled, thus obtaining a harmonic-compensated pure baseband signal. This pure signal serves as the input for the final phase demodulation.

[0042] Finally, the harmonic-compensated baseband signal is quadrature demodulated, and the constant phase is recovered by extracting the real and imaginary parts of the signal. Since the previous steps have eliminated the modulation DC term, the associated amplitude modulation term, the modulation depth error, and the second frequency modulation harmonic, the current signal has an approximately ideal sinusoidal frequency modulation structure, enabling quadrature demodulation to accurately reflect the true constant phase of the interference signal, thereby achieving high-precision absolute distance measurement.

[0043] Implementation Method 3: This implementation method is described in detail with reference to the accompanying drawings. Specific embodiments are provided to further illustrate the technical solutions offered above. Specifically: Figure 1 This embodiment uses an optical path structure, in which the optical path of the measuring interferometer is mainly used for measuring the absolute distance to the target, and the balanced detector 1 receives the ranging interferometric signal. The expression is as follows:

[0044] In the formula, Indicates the modulation bandwidth of the laser. It is the frequency of sinusoidal modulation. It is the time delay for measuring the difference in arm lengths of the interferometer. It is the carrier phase delay of the interference signal. This represents the constant phase of the interference signal. Since the interference signal processing methods for both lasers are the same, for ease of explanation, the following section will only describe in detail the processing method for the laser generating wavelength 1. The product of is much less than 1, therefore The above formula can be rewritten as follows:

[0045] in To represent the carrier modulation depth, construct a periodic window function and a matched carrier, and achieve constant-phase demodulation of the measured interference signal, the following equation is required:

[0046]

[0047] In the formula This represents the delay of the matched carrier phase delay. This indicates the matched carrier modulation depth. Mixing yields...

[0048] when , At that time, it becomes:

[0049] Applying a low-pass filter to the above equation to retain only the baseband signal, we get:

[0050] By taking the real and imaginary parts of Equation 7 and performing orthogonal demodulation, the constant phase of the interference signal can be obtained. However, due to the nonlinear response of the laser to the applied modulation voltage, the output laser frequency is not a standard sine curve, but rather exhibits harmonic terms of the modulation frequency. In addition, there is also an accompanying amplitude modulation phenomenon. Therefore, considering the nonlinear error of the laser, the interference signal can be expressed as:

[0051] in This represents the original DC term being modulated. Indicates the relative intensity of the accompanying amplitude modulation. This represents the phase delay of intensity modulation. This represents the relative intensity of each harmonic term. This represents the phase delay of each harmonic term. When the interference signal is then directly mixed with the window function and the matched carrier to obtain the initial phase, the result will differ significantly from the standard deviation. Therefore, compensation is needed to address the nonlinear error of the laser, achieving high-precision phase resolution.

[0052] First, the original DC current is modulated into a frequency. The AC input, which is additive with the target signal, is located in the low-frequency band. Since low-frequency information is useless to our demodulation algorithm, it can be directly filtered out using a high-pass filter. Then, the intensity modulation term can be obtained directly using the Hilbert transform to calculate the envelope, and then removed from the original signal. This suppresses the influence of accompanying amplitude modulation.

[0053] Since the sinusoidal frequency-modulated interference signal can be expanded using a Bessel function, with the modulation depth as the independent variable... As can be seen from the properties of the Bessel function, the Bessel function reaches its maximum value when the independent variable is approximately equal to the order. Therefore, when the window function is mixed with the interference signal, the horizontal axis corresponding to the maximum value of the mixed signal spectrum is the modulation depth of the carrier fundamental frequency term.

[0054] Finally, the parameters of the higher-order harmonic terms are calibrated and compensated, as can be seen from the Bessel function.

[0055] Therefore, when the modulation depth is 0, that is, when the carrier term is completely eliminated, the energy of the baseband signal reaches its maximum value. Combining the carrier fundamental frequency term parameters obtained above, a matched carrier is constructed. Since the amplitudes of the third-order and higher-order carrier terms are very small, we only consider the second-order harmonic term:

[0056] in This represents the relative strength of the quadratic term in the matched carrier. This represents the phase delay of the quadratic term in the matched carrier. The interferometric signal, the matched carrier, and the window function are mixed, and the amplitude of the baseband of the mixed signal is monitored. The second harmonic parameters are continuously adjusted until the baseband amplitude reaches its maximum. At this point, it can be considered that the carrier term of the interferometric signal has been completely eliminated, and finally, the demodulation of the constant phase is completed.

[0057] The above algorithm was simulated, assuming a laser modulation frequency of 20kHz, a modulation bandwidth of 15GHz, an initial phase of 1.1rad for the interference signal, a carrier modulation depth of 100rad, a system delay of 2.5μs, a second harmonic phase delay of 110°, and an amplitude of 5% of the carrier term. Finally, the intensity coefficient of the DC term was set to 0.4, with a phase delay of 60°, and the intensity coefficient of the AC term was set to 0.2, with a phase delay of 15°. Phase demodulation was then performed on the interference signal. The comparison of the interference signal before and after amplitude modulation suppression is shown in the figure below. Figure 2 , 3 As shown, the optimization process for second harmonic parameter calibration is as follows: Figure 4 As shown, a set of parameters is found that maximizes the amplitude of the mixing signal. A matched carrier is constructed to eliminate the carrier term in the interference signal, and finally, the phase of the interference signal is obtained. Figure 5 , 6 The demodulation results of the phase before and after compensation are given.

[0058] The simulation results demonstrate that this method can compensate for sinusoidal frequency modulation harmonic errors.

[0059] Advantages of the invention: This implementation proposes a sinusoidal frequency modulation (FM) harmonic error compensation method based on the modulation characteristics of Bessel functions. First, high-pass filtering and the Hilbert de-envelope algorithm are used to suppress the influence of accompanying amplitude modulation (AM). Then, a periodic window function is mixed with the interferometric signal to solve for the modulation depth of the fundamental frequency term. Next, a matched carrier is constructed, and based on the property of the Bessel function that signal energy is concentrated in the baseband when the independent variable is zero, the parameters of the matched carrier are continuously optimized to ultimately suppress the influence of FM harmonics and achieve high-precision phase decoupling. No additional active devices are introduced into the measurement system of this method; the calibration and compensation of the nonlinear parameters of sinusoidal FM can be achieved solely through the algorithm, thus making the system more concise. In a specific embodiment: This embodiment uses Figure 1 Based on the optical path structure, two sinusoidally frequency-modulated lasers are used to generate interference signals, and the wavelength signal generated by one of the lasers is explained. The measurement optical path consists of a laser, a polarization-maintaining isolator, a circulator, and an external optical system. Under the action of an applied radio frequency modulation voltage, the laser outputs a frequency-modulated laser signal whose frequency varies sinusoidally with time. After being transmitted unidirectionally by the polarization-maintaining isolator, it enters the first port of the circulator, then exits from the second port and enters the optical system. After illuminating the target, the signal is reflected back to the optical system, returns to the second port of the circulator via the same path, and is output from the third port to a balanced detector. The balanced detector receives and outputs the sinusoidal frequency-modulated interference signal corresponding to the absolute distance to the target. This interference signal is the initial input of the entire compensation method in this embodiment.

[0060] The interference signal includes modulation bandwidth, modulation frequency, time delay caused by the difference in the length of the measured interferometer arms, carrier phase delay, and a constant phase to be extracted. Since the product of modulation frequency and time delay is much less than one, the interference signal can be simplified to a sinusoidal frequency-modulated structure controlled by the carrier modulation depth under approximate conditions, laying the signal model foundation for subsequent processing.

[0061] In practical systems, because the laser's response to the applied modulation voltage is nonlinear, the actual output frequency is no longer a single fundamental frequency sine function, but rather includes several higher-order harmonic components corresponding to the modulation frequency. Simultaneously, modulation introduces an intensity modulation effect, transforming the original DC light intensity term into an AC term of the modulation frequency, superimposed with envelope ripple. The underlying material represents the nonlinear interference signal as a combination of DC modulation terms, accompanying amplitude modulation terms, and multiple frequency-modulated harmonic terms. These terms all introduce additional errors during mixing and demodulation, causing deviations in constant phase demodulation. To ensure the accuracy of subsequent phase extraction, these nonlinear error terms need to be suppressed and compensated for one by one.

[0062] First, the detected interference signal is high-pass filtered. Since the DC term becomes a low-frequency AC term with the modulation frequency as the main component after modulation, and this part is irrelevant to the demodulation target, this term will cover the effective quadrature components of the baseband region. Therefore, high-pass filtering directly removes all low-frequency components, resulting in a high-frequency interference signal containing only carrier and harmonic related components. This output serves as the input for accompanying amplitude modulation suppression.

[0063] Next, a Hilbert transform is performed on the high-pass processed signal to obtain the amplitude envelope. According to the disclosed information, the intensity modulation of the laser causes significant envelope fluctuations in the interference signal, and the envelope obtained through the Hilbert transform reflects the amplitude modulation changes. Dividing the original signal by the envelope restores the amplitude to a stable form, eliminating the influence of the accompanying amplitude modulation. This amplitude-stabilized signal serves as the input for solving the modulation depth.

[0064] Next, the amplitude-stabilized interference signal is mixed with a periodic window function. The documentation clearly states that since the sinusoidal frequency-modulated interference signal can be expanded into a Bessel function, the modulation depth of its fundamental frequency term can be determined by the abscissa of the point where the amplitude of the mixed signal's spectrum is maximum. Therefore, in this step, spectral analysis is performed on the mixed signal to find the point of maximum amplitude, and its abscissa position is used as the modulation depth of the fundamental frequency term. This modulation depth serves as the input parameter for subsequently constructing the matched carrier.

[0065] Subsequently, a matched carrier is constructed based on the obtained modulation depth. The matched carrier includes a portion consistent with the fundamental frequency term of the interference signal. Furthermore, according to the disclosed materials, since the second harmonic is the most prevalent among higher-order harmonics, a separate second harmonic term is added to the matched carrier. This second harmonic term includes two free parameters: adjustable relative amplitude and relative phase. These adjustable parameters will be continuously adjusted during the subsequent optimization process to achieve the cancellation of frequency-modulated harmonics. This constructed matched carrier serves as the input for frequency-modulated harmonic compensation.

[0066] Next, the interference signal, the constructed matched carrier, and the periodic window function are remixed. The amplitude change of the baseband component of the mixed signal is then monitored. According to the properties of the Bessel function explained in the documentation, when the modulation depth completely cancels out the carrier term, the signal energy is concentrated in the baseband region, thus the baseband amplitude reaches its maximum. Based on this characteristic, by continuously adjusting the phase delay and relative intensity of the second harmonic in the matched carrier, the amplitude of the baseband component of the mixed signal is maximized, thereby obtaining the optimal second harmonic parameters. This process corresponds to the optimization trajectory illustrated in the documentation. By scanning the baseband amplitude point by point, harmonic parameters consistent with the nonlinear characteristics of the actual laser are finally obtained, significantly reducing the influence of the second harmonic. The output at this point is the harmonic-compensated interference signal.

[0067] Finally, constant-phase demodulation is performed on the harmonic-compensated interference signal. As described in the documentation, the real and imaginary parts of the compensated signal are processed using orthogonal component extraction, allowing the constant phase to be restored to a high-precision phase value. Because the preceding high-pass filter removes the DC modulation term, the Hilbert transform removes the accompanying amplitude modulation, the modulation depth extraction corrects the carrier modulation depth error, and the matched carrier construction combined with baseband maximization optimization accurately compensates for the second-order frequency modulation harmonic, the final baseband signal approximates an ideal single-frequency sinusoidal frequency modulation form, significantly reducing the constant-phase demodulation error. The documentation demonstrates the phase curves before and after compensation through simulation examples, proving that this method can significantly reduce harmonic errors and achieve high-precision phase demodulation.

[0068] The above description of several specific embodiments further details the technical solution provided by the present invention in order to highlight the advantages and benefits of the technical solution provided by the present invention. However, the above-described specific embodiments are not intended to limit the present invention. Any reasonable modifications and improvements to the present invention, combinations of embodiments, and equivalent substitutions based on the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for compensating sinusoidal frequency modulation harmonic error based on Bessel function modulation characteristic, characterized in that, It comprises: a step of collecting a sinusoidal frequency-modulated interference signal and taking it as an interference signal preprocessing input; a step of high-pass filtering the interference signal to remove modulated low-frequency components and outputting a high-frequency interference signal as an associated amplitude modulation suppression input; a step of performing Hilbert transform on the high-frequency interference signal to obtain an envelope and completing envelope removal processing to obtain an amplitude-stable interference signal and outputting it as a modulation depth extraction input; a step of mixing the amplitude-stable interference signal with a periodic window function and extracting a carrier base frequency modulation depth according to the position of the maximum value of the frequency spectrum and outputting it as a matching carrier construction input; a step of constructing a matching carrier containing a second harmonic component based on the carrier base frequency modulation depth and outputting it as a frequency-modulated harmonic compensation input; a step of mixing the interference signal, the periodic window function, and the matching carrier and determining the second harmonic amplitude and phase by a baseband amplitude maximization optimization method to obtain a harmonic compensation interference signal and outputting it as a constant phase demodulation input; a step of quadrature demodulating the harmonic compensation interference signal to extract the constant phase of the interference signal.

2. The method according to claim 1, wherein, The interference signal preprocessing includes taking the sinusoidal frequency-modulated interference signal output by the polarization-maintaining optical path structure as the input and suppressing the low-frequency alternating current component converted by the laser modulation direct current through a high-pass filtering method.

3. The method of claim 1, wherein the method is based on a Bessel function modulation characteristic. The associated amplitude modulation suppression includes using Hilbert transform to obtain the amplitude envelope and dividing the original interference signal by the envelope.

4. The method of claim 1, wherein the method is based on a Bessel function modulation characteristic. The modulation depth extraction includes mixing the amplitude-stable interference signal with a periodic window function and determining the carrier base frequency modulation depth according to the transverse position corresponding to the maximum amplitude of the mixed signal frequency spectrum.

5. The method of claim 1, wherein the method is based on a Bessel function modulation characteristic. The matching carrier construction includes constructing a matching carrier containing an adjustable second harmonic amplitude and a second harmonic phase based on the carrier base frequency modulation depth.

6. The method of claim 1, wherein the method is based on a Bessel function modulation property. The frequency-modulated harmonic compensation includes mixing the interference signal, the periodic window function, and the matching carrier and performing optimization using the rule that the baseband signal amplitude changes with the second harmonic parameters.

7. A sinusoidal frequency modulation harmonic error compensation device based on the Bessel function modulation characteristic, characterized by, It comprises: a module for collecting a sinusoidal frequency-modulated interference signal and taking it as an interference signal preprocessing input; a module for high-pass filtering the interference signal to remove modulated low-frequency components and outputting a high-frequency interference signal as an associated amplitude modulation suppression input; a module for performing Hilbert transform on the high-frequency interference signal to obtain an envelope and completing envelope removal processing to obtain an amplitude-stable interference signal and outputting it as a modulation depth extraction input; a module for mixing the amplitude-stable interference signal with a periodic window function and extracting a carrier base frequency modulation depth according to the position of the maximum value of the frequency spectrum and outputting it as a matching carrier construction input; a module for constructing a matching carrier containing a second harmonic component based on the carrier base frequency modulation depth and outputting it as a frequency-modulated harmonic compensation input; a module for mixing the interference signal, the periodic window function, and the matching carrier and determining the second harmonic amplitude and phase by a baseband amplitude maximization optimization method to obtain a harmonic compensation interference signal and outputting it as a constant phase demodulation input; a module for quadrature demodulating the harmonic compensation interference signal to extract the constant phase of the interference signal.

8. Computer storage medium for storing a computer program, characterized in that When the computer program is read by the computer, the computer executes the method of claim 1.

9. A computer comprising a processor and a storage medium, characterized in that When the processor reads the computer program stored in the storage medium, the computer executes the method of claim 1.

10. Computer program product as computer program, characterized in that The computer program is embodied as computer program code configured to, when executed, implement the method of claim 1.

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