Multi-parameter automatic test method for APD device
By introducing a multi-parameter automatic testing method of upper-machine driver in the APD device test system, the problem of single parameters and low efficiency of the existing test system is solved, and high-precision and high-efficiency testing of multiple parameters of the APD device is achieved.
Patent Information
- Application Number
- CN202311207493.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2023-09-19
- Publication Date
- 2025-03-21
AI Technical Summary
The existing APD device testing system has the problem of single test parameters and inconvenient testing, and the instruments produced by domestic and foreign manufacturers do not mainly test APD, resulting in incomplete test parameters and insufficient testing accuracy.
It provides a multi-parameter automatic testing method for APD devices, and drives instruments and equipment through the host computer program to realize automatic and accurate measurement of various photoelectric parameters such as optical power calibration, IV test, CV test, rise time and pulse optical test.
Automatic testing of multiple parameters of APD devices is realized, testing accuracy and efficiency is improved, APD performance can be comprehensively and accurately analyzed, and the complexity and error rate of manual operation are reduced.
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Figure CN119667421A_ABST
Abstract
Description
Technical Field
[0001] The invention relates to the technical field of avalanche photodiode testing, in particular to a multi-parameter automatic testing method for APD devices. Background Art
[0002] APD (avalanche photodiode) is a photoelectric detection device that uses the avalanche effect generated by carrier movement to rapidly multiply the device's photocurrent in a very short time, thereby obtaining photocurrent gain. Compared with other photoelectric conversion devices, APD has the advantages of large current gain, high responsiveness, high power consumption, and low noise. Based on this, APD is suitable for receiving and detecting weak signals, and is widely used in laser radar, wavefront sensing, photon counting, space optical communications and other fields.
[0003] In view of the application field of APD, high-performance and high-sensitivity devices need to be used. Therefore, it is necessary to build an integrated system to conduct comprehensive and accurate testing. The multi-parameter, high-precision automatic test system can accurately and comprehensively analyze the performance of APD, and the detection capability can also provide feedback and evaluation on the quality of the test system. There is not a long history of development for the APD test system. The instruments produced by foreign manufacturers do not mainly test APD, resulting in incomplete measured parameters; domestic manufacturers have developed comprehensive parameter testing equipment based on APD, but there is a problem of insufficient test accuracy. The lack of pertinence of the equipment and the complex system structure are the main factors restricting the comprehensive and efficient testing of APD.
[0004] A variety of parameters can be used to comprehensively analyze the performance of APD. The IV characteristic characterizes the relationship between the output current and the reverse bias voltage. The dark current measured in the absence of light reflects the noise characteristics of the device, and the current measured in the light reflects the device's responsiveness to light. The CV characteristic characterizes the relationship between the junction capacitance and the reverse bias voltage. The terminal capacitance reflects the response speed of the device and has an impact on the timeliness of the output signal. The rise time t r With the cut-off frequency f c There is a conversion relationship f c (GHz)=0.35 / t r (ns), so the rise time can characterize the limit where the output signal energy begins to drop significantly. For APD, there has been the development of automatic IV performance test systems in related research; there are also APDs integrated into modules for automatic testing. The above research has the problem of low efficiency due to single test parameters and inconvenient testing. Summary of the invention
[0005] The content of the present invention is to address the existing deficiencies and provide a multi-parameter automatic testing method for APD devices. The method improves the degree of automation by driving instruments and equipment through a host computer program. It has high integration, high testing efficiency, and high precision. It can realize automatic and accurate measurement of multiple photoelectric parameters, which is conducive to comprehensive and accurate analysis of the performance of APD devices.
[0006] To solve the above problems, the technical solution proposed by the present invention is: a multi-parameter automatic testing method for APD devices, comprising the following steps:
[0007] Step 1: Use the fitting parameters to perform optical power calibration, which includes optical attenuation calibration and optical branch calibration. The optical attenuation calibration uses an optical attenuator to calibrate the attenuation relationship of each wavelength; the optical branch calibration refers to the calibration of the optical power irradiated to the device surface and the monitoring end;
[0008] Step 2: The host computer drives the illumination light source to adjust the visual brightness, and uses semi-automatic equipment to align the position of the microscopic coaxial module and the APD device, so that the output light of the microscopic coaxial module is vertically irradiated on the center of the photosensitive surface of the APD device;
[0009] Step 3: Perform corresponding tests on the wafer-type APD devices distributed in the array according to the needs, specifically, control the source meter, impedance analyzer, and oscilloscope through the host computer to complete the test of the corresponding photoelectric parameters;
[0010] Step 4: Without ending the test, choose whether to perform relevant photoelectric parameter tests on other APD test points. If you choose to continue testing other APD test points, drive the semi-automatic device to align the corresponding APD test point with the microscope coaxial module position, and then return to step 3; otherwise, end the test process.
[0011] Preferably, the specific method of light attenuation calibration is:
[0012] The pulsed laser emits a light source with a wavelength of λ, driving the optical attenuator to move step by step from the initial position S0 to the maximum attenuation position S z , and at the same time drive the optical power meter to record the optical power value P corresponding to each step S w , and the optical power value P w Converted to the optical power value P corresponding to dBm dBm ;
[0013] Unattenuated optical power value P dBm (0) Subtract P dBm Get the positive value of light attenuation P A ;
[0014] The light attenuation value P A The slope k is obtained by linear fitting with the decay step position S as the independent variable x and the dependent variable y.λ and intercept b λ , and save it to the attenuation parameter configuration.
[0015] Preferably, the specific method of optical splitting calibration is:
[0016] The monochromator and the pulsed laser were used to emit light with wavelength λ, and two optical power meters were used to collect the stable power P at the surface of the APD device and the monitoring end. S and P t , and calculate the ratio m = P S / P t , and save the parameter m representing the optical branch power relationship at the wavelength λ to the power parameter configuration.
[0017] Preferably, the test of the photoelectric parameters in step 3 includes: IV test, CV test, and rise time and pulse light test, which are respectively:
[0018] IV test: turn off the illumination source and turn on the monochromator or pulse laser;
[0019] The host computer drives the optical attenuator to attenuate the test light to the set value P according to the calibrated optical attenuation relationship;
[0020] The optical power meter at the monitoring end performs real-time monitoring of optical power according to the calibrated optical power relationship m between the monitoring end and the device surface;
[0021] CV test: turn off the lighting source and the test light source; control the impedance analyzer through the host computer to complete the voltage sweep test with AC component, and complete the impedance R within the voltage range of ±40V and the frequency range of 20Hz-30MHz p , Series capacitor C s , parallel capacitor C p Measurement of
[0022] Rise time and pulse light test: turn off the illumination source, turn on the pulse laser, and adjust the laser to the appropriate pulse width and repetition rate;
[0023] The host computer drives the optical attenuator to attenuate the test light to a suitable value P according to the calibrated optical attenuation relationship;
[0024] The optical power meter at the monitoring end monitors the optical power in real time according to the calibrated optical power relationship between the monitoring end and the device surface; the source meter is adjusted so that the APD works under the voltage corresponding to the specified multiplication M;
[0025] Control the oscilloscope to complete the corresponding rise time and pulse light waveform test after filtering and noise reduction.
[0026] Preferably, an IV test is performed at an optical power P to complete the photocurrent I under voltage scanning. p and dark current Id The measurement is based on the photocurrent I p and dark current I d Complete APD device reverse breakdown voltage V br , multiplication factor M, and response R measurement; wherein the multiplication factor is specifically:
[0027]
[0028] Where V r is the reverse bias voltage, V br is the reverse breakdown voltage, n is a constant greater than 1;
[0029] The specific responsiveness is:
[0030]
[0031] In the formula, I p is the photocurrent, I d is the dark current, and P is the optical power on the surface of the APD device.
[0032] Compared with the prior art, the present invention has the following significant advantages: the automatic testing method of the present invention can realize the automatic testing of multiple parameters of a single or multiple test points of an APD, thereby being able to comprehensively and accurately characterize and analyze the performance of the APD; the test process uses a host computer to drive the required instruments and equipment, automatically execute the test logic, and store data, which can reduce the error rate and complexity of manual operation, thereby effectively improving the test accuracy and test efficiency of the APD parameter test; the pulse light test module is combined with algorithm filtering for processing, which can effectively reduce signal noise, smooth the pulse signal, enhance the signal's anti-interference ability, improve the test accuracy, and better analyze the device performance; in addition, the present invention can comprehensively test optoelectronic devices other than APD, and is convenient for transplantation on different types of optoelectronic device test systems. BRIEF DESCRIPTION OF THE DRAWINGS
[0033] Figure 1 It is a schematic diagram of the upper computer program driving the instrumentation of the present invention.
[0034] Figure 2 This is a flow chart of the multi-parameter automatic testing method of the present invention.
[0035] Figure 3 This is a schematic diagram of the IV test principle of the present invention.
[0036] Figure 4 It is the CV test principle diagram of the present invention.
[0037] Figure 5 This is a schematic diagram of the rise time and pulse light testing principle of the present invention.
[0038] Figure 6It is the EMD decomposition flow chart of the present invention.
[0039] Figure 7 This is a pulse signal filtering flow chart of the present invention.
[0040] Figure 8 Schematic diagram of the initial waveform (a) and filtered waveform (b) of the pulse light signal of the present invention. DETAILED DESCRIPTION
[0041] In order to further explain the technical solution of the present invention, the technical details of the present invention will be described in detail below with reference to the accompanying drawings and embodiments.
[0042] like Figure 1 , Figure 2 As shown, the present invention proposes a multi-parameter automatic testing method for APD devices, which is based on an APD photoelectric parameter testing system, and the APD photoelectric parameter testing system includes a bromine tungsten lamp, a monochromator, a pulse laser, an optical attenuator, an optical power meter, a microscopic coaxial module, a power supply voltage, a semi-automatic device, a source meter, an impedance analyzer, and an oscilloscope. The bromine tungsten lamp, the monochromator, and the pulse laser are connected to the 90% branch of the optical attenuator and connected to the microscopic coaxial module, so that the test light is vertically irradiated on the center of the APD photosensitive surface; the optical power meter is connected to the 10% branch of the optical attenuator, and the real-time power of the APD surface can be monitored based on power calibration; the source meter, the impedance analyzer, and the oscilloscope are connected to the APD test circuit, and the corresponding photoelectric parameters of the APD can be selected for real-time testing according to the needs. The above-mentioned instruments are connected to the host computer through RS232 and USB serial port lines, and the semi-automatic equipment is connected to the host computer through a network cable based on the TCP protocol. The host computer program automatically controls the entire instrument and equipment according to the test logic, improves system integration, reduces the complexity of manual testing, enhances test continuity, and improves test speed and comprehensiveness. It includes the following steps:
[0043] Before the formal test, check the status of the instrument. The host computer program sends the initialization operation command parameters to the instrument through the corresponding control protocol, and receives the status feedback parameters of the instrument, and then judges the working status of each instrument and reads the configuration file of the relevant parameter information. If all the instruments are working normally and the configuration file is read correctly, proceed to the next step, otherwise terminate the current test;
[0044] Specifically, the basic parameter configuration of the device test read includes the configuration files of the following parameter information: parameter configuration information of the repetition frequency and pulse width of the pulse light source, the wavelength of the monochromatic light source, the attenuation relationship of the optical attenuator to light sources of different wavelengths, and the relationship between optical monitoring and the surface optical power of the device; parameter information of the power supply voltage and current limit; parameter information of the maximum attenuation value allowed by the attenuator for light of different wavelengths and the attenuation accuracy.
[0045] Step 1: Use the fitting parameters to perform optical power calibration, which is divided into optical attenuation calibration and optical splitting calibration;
[0046] Light attenuation calibration is the calibration of the attenuation relationship of the optical attenuator to each wavelength. Select the light source with wavelength λ. The pulse light of the pulse laser can be selected, or the monochromatic light emitted by the combination of bromine tungsten lamp and monochromator can be selected. Drive the optical attenuator to move from the initial position S0 to the maximum attenuation position S in steps. z , and at the same time drive the optical power meter to record the optical power value P corresponding to each step S w , the optical power value P w Converted to the optical power value P corresponding to dBm dBm ,Right now Light attenuation positive value P A Then it is the unattenuated optical power value P dBm (0) Subtract P dBm , that is, P A =P dBm (0)-P dBm , the light attenuation value P A As the independent variable x, the attenuation position step S is used as the dependent variable y for linear fitting to obtain the slope k λ and intercept b λ , and save it to the attenuation parameter configuration; the fitting formula is S=k λ P A +b λ ;
[0047] Optical splitter calibration is to measure the optical power P on the device surface. s With monitoring terminal P t For calibration, select a light source with a wavelength of λ. You can choose pulsed light emitted by a pulsed laser or monochromatic light emitted by a bromine tungsten lamp and a monochromator. Use two optical power meters, one to replace the device and place it on the original surface of the device, and one to place it on the monitoring end to collect the stable power P S and P t , we get m = P S / P t , and save it to the power parameter configuration;
[0048] Step 2: The host computer drives the illumination light source to adjust the visual brightness. The point light source controller can adjust the illumination brightness and switch under the drive of the host computer. The semi-automatic equipment is used to configure the vertical position of the microscopic coaxial module and the APD so that the test light source is vertically irradiated to the center of the APD photosensitive surface. The host computer uses the TCP protocol to control the semi-automatic equipment;
[0049] Step 3: Through the host computer's control of the source meter, impedance analyzer, oscilloscope and other test instruments in the system module, the following corresponding optoelectronic parameters are tested for the array-distributed wafer-type APDs:
[0050] (1) IV test: Turn off the lighting source to ensure a stray light-free environment, drive the optical power meter and light source to work. The light source can be a pulsed laser or a bromine tungsten lamp and monochromatic light emitted by a monochromator. Drive the optical attenuator to attenuate the light to an appropriate value, and drive the source meter to complete the IV test. The test wiring method uses the four-wire method to reduce the impact of noise. The IV test schematic diagram is as follows: Figure 3 As shown;
[0051] Perform IV test at optical power P, complete photocurrent I under voltage sweep p and dark current I d The measurement is based on the photocurrent I p and dark current I d Complete APD device reverse breakdown voltage V br , multiplication factor M, and response R measurement; wherein the multiplication factor is specifically:
[0052]
[0053] Where V r is the reverse bias voltage, V br is the reverse breakdown voltage, n is a constant greater than 1;
[0054] The specific responsiveness is:
[0055]
[0056] In the formula, I p is the photocurrent, I d is the dark current, and P is the optical power on the surface of the APD device.
[0057] (2) CV test: Turn off the lighting source to ensure a dark environment, drive the impedance analyzer to perform CV test, and the host computer can perform linear frequency sweep test under specified bias voltage and linear voltage sweep test under specified voltage frequency. The test wiring method uses the four-wire method to reduce the impact of noise. The CV test schematic diagram is as follows: Figure 4 As shown;
[0058] This test uses the host computer to control the impedance analyzer to complete the voltage sweep test with AC components, and complete the impedance R within the voltage range of ±40V and the frequency range of 20Hz-30MHz p , Series capacitor C s , parallel capacitor C p Measurement of
[0059] (3) Rise time and pulse light test: Turn off the illumination source to ensure a stray light-free environment, drive the optical power meter, pulse laser, and source meter to work, drive the optical attenuator to attenuate the light to an appropriate value, drive the oscilloscope to complete the rise time and pulse light test, and use the source meter to set the reverse bias voltage of M at a specified multiplication. Figure 5 The test is carried out by the schematic diagram method shown in the figure. The rise time and pulse light signal are measured at the oscilloscope end through the control of the host computer, and the pulse light signal is measured according to Figure 6 and Figure 7 The method for filtering is:
[0060] The specific process of filtering is:
[0061] Performing empirical mode decomposition on the collected pulse light waveform data to obtain each intrinsic mode function component and residual component;
[0062] Extract high-frequency intrinsic mode function components and perform Gaussian filtering;
[0063] The processed high-frequency intrinsic mode function component replaces the initial component and is reconstructed with other components to obtain a filtered pulse waveform.
[0064] The relationship between the signal and the decomposed components is:
[0065]
[0066] In the formula, IMF i is the eigenmode component at each level, r n is the final residual component.
[0067] Step 4: For wafer-type APDs, which are distributed in an array, choose whether to test other APD test points for related photoelectric parameters without ending the test. If you choose to continue testing other APD test points, drive the semi-automatic equipment to align the corresponding APD test point with the position of the microscopic coaxial module and return to step 3; otherwise, complete the test. When selecting modules in a loop, you can select other modules, or you can select the current module to adjust the light intensity and continue testing under different experimental conditions; when selecting other APD test points, the host computer uses the TCP protocol to control the semi-automatic equipment to select other APD test points;
[0068] Repeat steps 3 to 4 to select and test the modules. During this process, the test conditions can be adjusted as needed to complete the parameter diversity test until the host computer terminates the test process when it is needed.
[0069] The key methods of the above steps 1 to 4 are further described below in conjunction with the parameters of the embodiment:
[0070] In this embodiment, the object of the test is a silicon-based APD with a reverse breakdown voltage of 150V to 180V, and a 1064nm pulse laser with a repetition rate of 30kHz and a pulse width of 20ns is used, and a microscopic coaxial module equipped with a 10X objective lens is used for comprehensive verification testing;
[0071] In this embodiment, optical power calibration is performed using fitting parameters, and the calibration is divided into optical attenuation calibration and optical branching calibration;
[0072] Optical attenuation calibration is the calibration of the attenuation relationship of the optical attenuator to each wavelength. The wavelength 1064nm is selected to drive the optical attenuator to move step by step from the initial position S0 to the maximum attenuation position S z , and at the same time drive the optical power meter to record the optical power value P corresponding to each step S w , the optical power value P w Converted to dBm corresponding to the optical power value P dBm , the light attenuation is positive value P A Then it is the unattenuated optical power value P dBm (0) Subtract P dBm , the light attenuation value P A A linear fit is performed using x as the independent variable and S as the dependent variable y to obtain a slope of 5772.020 and an intercept of -306.844, which are then saved in the attenuation parameter configuration;
[0073] Optical splitter calibration is to measure the optical power P on the device surface. s With monitoring terminal P t For calibration, select the pulse light of the pulse laser with a wavelength of 1064nm, use two optical power meters, one is placed on the surface of the device instead of the device, and the other is placed on the monitoring end. Collect the stable power of 18.6uw and 52.2uw at both ends, and get m=2.806, which is saved in the power parameter configuration;
[0074] In this embodiment, one of the test modules is entered to test the module corresponding to the APD test point;
[0075] (1) If you enter the IV test module, turn off the lighting source to ensure a stray light-free environment, drive the optical power meter and light source to work, select pulse laser as the light source, drive the optical attenuator to attenuate the light to 1μw, use the four-wire method for test wiring to reduce the impact of noise, drive the source meter to complete the IV test, and complete the linear voltage scan from 0 to 160V with a step of 1V. The IV test schematic diagram is as follows: Figure 3 As shown;
[0076] (2) If you enter the CV test module, turn off the lighting source to ensure a dark environment, drive the impedance analyzer to perform CV testing, and the host computer can perform linear frequency sweep testing under a specified bias voltage and linear voltage sweep testing under a specified voltage frequency. The test wiring method uses the four-wire method to reduce the impact of noise. The CV test schematic diagram is shown in the figure. Figure 4 As shown;
[0077] (3) If you enter the rise time and pulse light test module, turn off the illumination source to ensure a stray light-free environment, drive the optical power meter, pulse laser, and source meter to work, drive the optical attenuator to attenuate the light to an appropriate value, drive the oscilloscope to complete the rise time and pulse light test, and the source meter is given a reverse bias voltage of 150V under a specified multiplier M = 100. Figure 5 The test is carried out by the schematic diagram method shown in the figure. The rise time and pulse light signal are measured at the oscilloscope end through the control of the host computer, and the pulse light signal is measured according to Figure 6 and Figure 7 The filtering method is used. Figure 8 Comparison diagram of collected pulse light signal (a) and filtered pulse signal (b);
[0078] The above is only an embodiment of the present invention and does not limit the present invention in any form. Therefore, any equivalent structure made by using the contents of the present invention and the drawings, directly or indirectly applied to other related technical fields, is also within the patent protection scope of the present invention.
Claims
1. A multi-parameter automatic testing method for APD devices, characterized in that: The steps include: Step 1: Use the fitting parameters to perform optical power calibration, which includes optical attenuation calibration and optical branch calibration. The optical attenuation calibration uses an optical attenuator to calibrate the attenuation relationship of each wavelength; the optical branch calibration refers to the calibration of the optical power irradiated to the device surface and the monitoring end; Step 2: The host computer drives the illumination light source to adjust the visual brightness, and uses semi-automatic equipment to align the position of the microscopic coaxial module and the APD device, so that the output light of the microscopic coaxial module is vertically irradiated on the center of the photosensitive surface of the APD device; Step 3: Perform corresponding tests on the wafer-type APD devices distributed in the array according to the needs, specifically, control the source meter, impedance analyzer, and oscilloscope through the host computer to complete the test of the corresponding photoelectric parameters; Step 4: Without ending the test, choose whether to perform relevant photoelectric parameter tests on other APD test points. If you choose to continue testing other APD test points, drive the semi-automatic device to align the corresponding APD test point with the microscope coaxial module position, and then return to step 3; otherwise, end the test process.
2. The multi-parameter automatic testing method for APD devices according to claim 1, characterized in that: The specific method of light attenuation calibration is: The pulsed laser emits a light source with a wavelength of λ, driving the optical attenuator to move step by step from the initial position S0 to the maximum attenuation position S z , and at the same time drive the optical power meter to record the optical power value P corresponding to each step S w , and the optical power value P w Converted to the optical power value P corresponding to dBm dBm ; Unattenuated optical power value P dBm (0) Subtract P dBm Get the positive value of light attenuation P A ; The light attenuation value P A The slope k is obtained by linear fitting with the decay step position S as the independent variable x and the dependent variable y. λ and intercept b λ , and save it to the attenuation parameter configuration.
3. The multi-parameter automatic testing method for APD devices according to claim 1, characterized in that: The specific method of optical splitter calibration is: The monochromator and the pulsed laser were used to emit light with wavelength λ, and two optical power meters were used to collect the stable power P at the surface of the APD device and the monitoring end. S and P t , and calculate the ratio m = P S / P t , and save the parameter m representing the optical branch power relationship at the wavelength λ to the power parameter configuration.
4. The multi-parameter automatic testing method for APD devices according to claim 1, characterized in that: The photoelectric parameter tests in step 3 include: IV test, CV test, rise time and pulse light test, which are as follows: IV test: turn off the illumination source and turn on the monochromator or pulse laser; The host computer drives the optical attenuator to attenuate the test light to the set value P according to the calibrated optical attenuation relationship; The optical power meter at the monitoring end performs real-time monitoring of optical power according to the calibrated optical power relationship m between the monitoring end and the device surface; CV test: turn off the lighting source and the test light source; control the impedance analyzer through the host computer to complete the voltage sweep test with AC component, and complete the impedance R within the voltage range of ±40V and the frequency range of 20Hz-30MHz p , Series capacitor C s , parallel capacitor C p Measurement of Rise time and pulse light test: turn off the illumination source, turn on the pulse laser, and adjust the laser to the appropriate pulse width and repetition rate; The host computer drives the optical attenuator to attenuate the test light to a suitable value P according to the calibrated optical attenuation relationship; The optical power meter at the monitoring end monitors the optical power in real time according to the calibrated optical power relationship between the monitoring end and the device surface; the source meter is adjusted so that the APD works under the voltage corresponding to the specified multiplication M; Control the oscilloscope to complete the corresponding rise time and pulse light waveform test after filtering and noise reduction.
5. The multi-parameter automatic testing method for APD devices according to claim 4, characterized in that: Perform IV test at optical power P, complete photocurrent I under voltage sweep p and dark current I d The measurement is based on the photocurrent I p and dark current I d Complete APD device reverse breakdown voltage V br , multiplication factor M, and response R measurement; wherein the multiplication factor is specifically: Where V r is the reverse bias voltage, V br is the reverse breakdown voltage, n is a constant greater than 1; The specific responsiveness is: In the formula, I p is the photocurrent, I d is the dark current, and P is the optical power on the surface of the APD device.
6. The multi-parameter automatic testing method for APD devices according to claim 1, characterized in that: The specific process of filtering is: Performing empirical mode decomposition on the collected pulse light waveform data to obtain each intrinsic mode function component and residual component; Extract high-frequency intrinsic mode function components and perform Gaussian filtering; The processed high-frequency intrinsic mode function component replaces the initial component and is reconstructed with other components to obtain a filtered pulse waveform.
7. The multi-parameter automatic testing method for APD devices according to claim 1, characterized in that: The relationship between the signal and the decomposed components is: In the formula, IMF i is the eigenmode component at each level, r n is the final residual component.