Online phase correction method based on Wi-Fi perception

Abstract

The invention provides an online phase correction method based on Wi-Fi (Wireless Fidelity) perception. Firstly, a transmitter equipped with a transmitting antenna is placed at a known position, that is, the included angle between the normal direction of an array antenna and the direction of a perpendicular incidence path is known; secondly, collecting a channel state information (CSI) data packet in an environment with a high signal to noise ratio (SNR); secondly, extracting phase differences among different channels by using antenna conjugate multiplication, and screening available data packets; thirdly, the phase of the direct incidence path and the initial phase deviation are extracted by adopting a VMD (Variation Mode Decomposition) and Hilbert-Huang Transform (HHT) time-frequency analysis method, and the influence of the phase of the reflection path is eliminated, and then, the phase of the direct incidence path and the initial phase deviation are extracted by adopting the Hilbert-Huang Transform (Hilbert-Huang Transform) time-frequency analysis method; and finally, averaging the estimated initial phase differences of the subcarriers on different antennas in the plurality of data packets to obtain fine initial phase offset. According to the method, extra measuring equipment is not needed, the method can be achieved in the normal communication process, the reliability of error elimination and the accuracy of (Angle of Arrival, AoA) estimation are guaranteed, and wide application of commercial Wi-Fi equipment in the positioning field is promoted.

Description

Technical Field

[0001] The present invention belongs to the technical field of using commercial Wi-Fi devices to achieve positioning, and specifically relates to an online phase correction method based on Wi-Fi perception. This method is used to solve the problem that offline phase calibration relies on additional measurement devices and interrupts communication. It implements online phase correction based on a linear array, synchronizes the initial phase of each radio frequency channel, and serves the positioning method based on the angle of arrival (AoA). Background Art

[0002] With the continuous advancement of wireless communication technology, indoor sensing technology based on Wi-Fi signals has become a hot topic in current research and application. Wi-Fi Channel State Information (CSI) provides fine-grained wireless channel characteristics, including signal amplitude and phase information. It is widely used in fields such as human perception, activity recognition, respiratory monitoring, and indoor positioning. Indoor positioning plays a crucial role in scenarios such as navigation in large shopping malls and finding cars in indoor parking lots, improving user experience and operational efficiency.

[0003] In the field of indoor positioning, most work uses linear antenna arrays with half-wavelength array spacing to collect CSI. Mainstream indoor positioning models can be divided into neural network-based fingerprint positioning methods and parameter estimation-based geometric spatial positioning methods. However, due to the shortcomings of fingerprint positioning, such as environmental dependence and high time deployment costs, geometric spatial positioning methods based on parameter estimation, which have low computational costs, have been widely studied. Typically, these methods integrate multipath parameters such as AoA, time of flight, and Doppler shift to achieve target positioning and tracking. Because AoA estimation methods require initial phase synchronization between channels, and synchronization equipment is expensive and difficult to achieve economic benefits, researchers have gradually ported algorithms to commercial Wi-Fi devices to achieve low-cost positioning systems. However, CSI data collected by commercial Wi-Fi devices is often affected by various hardware errors, such as carrier frequency offset (CFO), sampling frequency offset (SFO), packet detection delay (PDD), and initial phase error caused by the phase-locked loop (PLL). CFO, SFO, and PDD have little impact on the active positioning method using AoA estimation, while the initial phase error will cause a significant loss of positioning accuracy, thereby affecting the accuracy of phase-based perception tasks.

[0004] Currently, phase error calibration primarily involves offline and online calibration. Offline calibration typically relies on external high-precision measurement equipment, pre-measuring the phase difference between each antenna link to achieve error correction. While highly accurate, these methods suffer from high cost, system complexity, and the need to interrupt communications, making them difficult to adapt to the large-scale deployment requirements of commercial Wi-Fi devices. Online calibration, on the other hand, is computationally complex and may not meet real-time requirements. It is also sensitive to noise and interference, which can affect calibration accuracy.

[0005] In response to the above problems, the present invention proposes an online phase correction method based on Wi-Fi perception. First, a transmitter equipped with a transmitting antenna is placed at a known position, that is, the angle between the normal direction of the array antenna and the direction of the direct path is known; secondly, CSI data packets are collected in a high signal-to-noise ratio environment; then, the phase difference between different channels is extracted by antenna conjugate multiplication and the available data packets are screened; thirdly, the time-frequency analysis method of variational mode decomposition (VMD) and Hilbert-Huang Transform (HHT) is combined to extract the phase of the direct path and the initial phase offset, eliminating the influence of the reflection path phase; finally, the initial phase difference estimated by the subcarriers on different antennas in multiple data packets is averaged to obtain a fine initial phase offset. The method of the present invention does not require additional measurement equipment and can be implemented during normal communication, ensuring the reliability of error elimination and the accuracy of AoA estimation, and promoting the widespread application of commercial Wi-Fi equipment in the field of positioning. Summary of the Invention

[0006] The purpose of the present invention is to solve the problem that offline phase calibration relies on additional measurement devices and interrupts communication, and to propose a novel online phase error correction algorithm.

[0007] The online phase correction method based on Wi-Fi perception described in the present invention specifically includes the following steps:

[0008] Step 1: Use the Wi-Fi device to obtain the CSI of a single data packet;

[0009] In the WiFi sensor system, the transmitting device uses the Orthogonal Frequency Division Multiplexing (OFDM) technology to modulate the RF signal onto multiple subcarriers, and then transmits it to the receiving device through the wireless channel, achieving parallel transmission and improving the bandwidth utilization. Consider a uniform linear array in space, the distance between the array elements is d = λ / 2, and the center frequency of the transmitted signal is f cThe CSI of a single data packet on the i-th (1≤i≤I)th antenna and the k-th (1≤k≤K)th subcarrier at the receiving end is expressed as

[0010]

[0011] Where j represents the imaginary number symbol, L represents the number of paths, α l represents the complex attenuation of the lth path, θ l and τ l are the AoA and ToF of the signal, Δf is the subcarrier spacing, c is the speed of light, σ i,k represents additive noise.

[0012] In fact, commercial Wi-Fi devices are affected by the initial phase offset caused by the PLL. When the receiver is turned on, the WiFi chipset will introduce a fixed initial phase offset on different RF channels. And each time the device is turned on, the offset value will change randomly. In addition, due to hardware differences between RF channels, the attenuation and time delay experienced by the signal during transmission are also different, which results in a different initial phase offset for each RF channel. Due to defects such as CFO, SFO, and PDD in commercial Wi-Fi transmission pairs, the collected CSI amplitude and phase are distorted, which can be expressed as

[0013]

[0014] Among them, θ pll (i) represents the initial phase offset of the i-th RF channel, θ offset Random phase offset θ including CFO, SFO and PDD offset =2π[kΔfτ to (i)+iΔtf cfo ], τ to and f cfo They represent timing offset and CFO respectively, and Δt represents the time interval between data packets.

[0015] Step 2: Use antenna conjugate multiplication to extract the phase difference between different channels;

[0016] The CSI change caused by the direct path is The CSI changes caused by the remaining reflection paths are: Then the CSI with initial phase offset is expressed as

[0017]

[0018] Wherein, l=1 represents the direct path, and θ1 represents the angle between the normal direction of the array antenna and the direction of the direct path.

[0019] In an OFDM-MIMO system, when the distance between the transmitter and receiver is close, the direct path becomes the main signal propagation path, while the impact of the reflected path and noise is small. Since the direct path has the smallest propagation loss, its received power is much higher than the reflected signal after the additional path loss, making the impact of the multipath effect relatively insignificant. Secondly, in a high signal-to-noise ratio environment, the impact of noise on system performance is relatively small. Therefore, without considering thermal noise, when the transmitter distance meets the far-field condition, the wavefront can be approximated as a plane wave, and the CSI of antenna i and i+Δi are conjugate multiplied.

[0020]

[0021] Where Δi represents the interval of antenna numbers, Represents H′ i+Δi,k The conjugate of mix Represents the term containing the reflection path. Due to the time delay τ of the path l Same for all antennas, with subcarrier phase rotation At the same time, based on the similarity of the random phase offsets of each antenna in the Wi-Fi chip at the same time, the random phase offset θ in the received signal is eliminated by the conjugate multiplication operation. offset .

[0022] Step 3: Screen reliable data packets;

[0023] First, the CSI of the pth (1≤p≤P)th data packet on the kth subcarrier at the i-th antenna of the receiving end is expressed as H i,k,p , then the initial phase difference between antenna i and i+Δi is roughly estimated to be

[0024]

[0025] Here, arg(·) represents the phase part of the complex number.

[0026] Secondly, a scoring mechanism based on a multi-attribute decision-making method is proposed, which can screen out reliable data packets based on signal strength (Received Signal Strength Indicator, RSSI) and phase stability. The main idea of this method is that when the signal of the direct path is dominant, the phase of the CSI measurement on the subcarrier should be smooth. In addition, the RSSI of each channel also indirectly characterizes the degree of signal fading. Therefore, the higher the RSSI of the received signal, the higher the reliability of the data packet. In order to obtain the RSSI index, the average received power of all antennas and subcarriers in the data packet p is calculated.

[0027]

[0028] Then, the fluctuation degree of the subcarrier phase difference between adjacent antennas is counted

[0029]

[0030] Among them, μ p (1,1+Δi) represents the average phase difference between the reference antenna and the antennas with a spacing of Δi. Secondly, the score is obtained by integrating the two indicators.

[0031]

[0032] Among them, max(·) and min(·) represent the maximum and minimum values in the set, respectively.

[0033] Finally, the threshold is set using the median principle, that is, the score F p ≥0.5(max(F p )+min(F p )) is the filtered reliable data packet.

[0034] Step 4: Extract the phase of the direct path and the initial phase offset based on the VMD-HHT time-frequency analysis method to eliminate the influence of the reflected path phase.

[0035] First, considering the time dimension, a Savitzky-Golay filter is used to smooth the phase, removing outliers and small amounts of high-frequency noise. The phase of the direct path typically varies slightly from the initial phase and has a very narrow bandwidth, making it considered a low-frequency component. The reflected path, on the other hand, is easily overwhelmed by noise, varies dramatically at different sampling times, and has a wide bandwidth, making it considered a high-frequency component.

[0036] In order to extract the phase and initial phase offset of the direct path from the residual high-frequency components, the present invention adopts the VMD-HHT time-frequency analysis method. First, the initial phase difference estimated for each subcarrier is decomposed into multiple modal components by the VMD method.

[0037]

[0038] Among them, A k (i,i+Δi)=[Δθ k,1 (i,i+Δi),Δθ k,2 (i,i+Δi),...,Δθ k,P (i,i+Δi)] is the initial phase difference between the kth subcarrier antenna (i,i+Δi), M is the number of modes, is the mth modal component.

[0039] Secondly, we construct a variational problem and decompose the original signal into M modal components. The goal is to minimize the sum of the estimated bandwidths of each mode, with the constraint that the sum of all modes is equal to the original signal. The VMD constrained variational model is then

[0040]

[0041] Where Δ[·] represents the difference of n. represents the Hilbert transform, f m is the center frequency of the mth modal component. In order to calculate and f m , transform the constrained variational problem into an unconstrained variational problem, take advantage of the quadratic penalty term and the Lagrange multiplier method, and then combine the augmented Lagrangian to solve the variational problem to find the optimal solution

[0042]

[0043] Where α is the penalty factor and λ is the Lagrange multiplier. Finally, the Alternating Direction Method of Multipliers (ADMM) is used to solve the problem and obtain M modal components.

[0044] Then, the HHT calculation is applied to obtain the time-frequency information of each mode, and an analytical signal based on the original signal and its Hilbert transform is constructed to extract the instantaneous frequency and instantaneous energy, thereby obtaining the energy distribution of each mode in different frequency ranges. The zero-frequency component obtained by decomposing the initial phase difference on the kth subcarrier is The remaining high-frequency components are

[0045] Finally, based on the characteristics that the direct path and the initial phase offset have strong stability in the channel environment, and the reflected path phase shows large fluctuations under the multipath effect, the zero-frequency component is extracted. As the phase and initial phase offset characteristics of the direct path.

[0046] Step 5: Average the estimated initial phase differences of subcarriers on different antennas in multiple data packets to obtain a fine initial phase offset;

[0047] Since single measurement is susceptible to additive noise and frequency selective fading and has a certain degree of randomness, in order to improve the estimation stability and robustness, a multi-data packet statistical fusion method is used to effectively suppress the errors caused by noise and time-varying channel disturbances. Finally, the phase difference estimation is completed through the following expression

[0048]

[0049] in, Express Take the average of all elements in .

[0050] Beneficial effects

[0051] This method first places a transmitter equipped with a single transmitting antenna at a specific location, where the angle between the normal direction of the array antenna and the direction of the direct path is known. Secondly, CSI data packets are collected in a high signal-to-noise ratio environment. Antenna conjugate multiplication is then used to extract the phase differences between different channels and filter available data packets. Thirdly, the VMD-HHT time-frequency analysis method is used to extract the direct path phase and initial phase offset, eliminating the influence of the reflected path phase. Finally, the estimated initial phase differences of subcarriers on different antennas within multiple data packets are averaged to obtain a refined initial phase offset.

[0052] The present invention provides an online phase correction method based on Wi-Fi perception. It ensures the reliability of error elimination and the accuracy of arrival angle estimation without interrupting communication, promoting the widespread application of commercial Wi-Fi devices in the field of positioning. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 This is a system block diagram of the online phase correction method based on Wi-Fi perception of the present invention Specific implementation plan

[0054] The purpose of the present invention is to solve the problem that offline phase calibration relies on additional measurement devices and interrupts communication, and to propose a novel online phase error correction algorithm.

[0055] The online phase correction method based on Wi-Fi perception described in the present invention specifically includes the following steps:

[0056] Step 1: Use the Wi-Fi device to obtain the CSI of a single data packet;

[0057] In the WiFi sensor system, the transmitting device uses the Orthogonal Frequency Division Multiplexing (OFDM) technology to modulate the RF signal onto multiple subcarriers, and then transmits it to the receiving device through the wireless channel, achieving parallel transmission and improving the bandwidth utilization. Consider a uniform linear array in space, the distance between the array elements is d = λ / 2, and the center frequency of the transmitted signal is f c The CSI of a single data packet on the i-th (1≤i≤I)th antenna and the k-th (1≤k≤K)th subcarrier at the receiving end is expressed as

[0058]

[0059] Where j represents the imaginary number symbol, L represents the number of paths, α l represents the complex attenuation of the lth path, θ l and τ l are the AoA and ToF of the signal, Δf is the subcarrier spacing, c is the speed of light, σ i,k represents additive noise.

[0060] In fact, commercial Wi-Fi devices are affected by the initial phase offset caused by the PLL. When the receiver is turned on, the WiFi chipset will introduce a fixed initial phase offset on different RF channels. And each time the device is turned on, the offset value will change randomly. In addition, due to hardware differences between RF channels, the attenuation and time delay experienced by the signal during transmission are also different, which results in a different initial phase offset for each RF channel. Due to defects such as CFO, SFO, and PDD in commercial Wi-Fi transmission pairs, the collected CSI amplitude and phase are distorted, which can be expressed as

[0061]

[0062] Among them, θ pll (i) represents the initial phase offset of the i-th RF channel, θ offset Random phase offset θ including CFO, SFO and PDD offset =2π[kΔfτ to (i)+iΔtf cfo ], τ to and f cfo They represent timing offset and CFO respectively, and Δt represents the time interval between data packets.

[0063] Step 2: Use antenna conjugate multiplication to extract the phase difference between different channels;

[0064] The CSI change caused by the direct path is The CSI changes caused by the remaining reflection paths are: Then the CSI with initial phase offset is expressed as

[0065]

[0066] Wherein, l=1 represents the direct path, and θ1 represents the angle between the normal direction of the array antenna and the direction of the direct path.

[0067] In an OFDM-MIMO system, when the distance between the transmitter and receiver is close, the direct path becomes the main signal propagation path, while the impact of the reflected path and noise is small. Since the direct path has the smallest propagation loss, its received power is much higher than the reflected signal after the additional path loss, making the impact of the multipath effect relatively insignificant. Secondly, in a high signal-to-noise ratio environment, the impact of noise on system performance is relatively small. Therefore, without considering thermal noise, when the transmitter distance meets the far-field condition, the wavefront can be approximated as a plane wave, and the CSI of antenna i and i+Δi are conjugate multiplied.

[0068]

[0069] Where Δi represents the interval of antenna numbers, Represents H′ i+Δi,k The conjugate of mix Represents the term containing the reflection path. Due to the time delay τ of the path l Same for all antennas, with subcarrier phase rotation At the same time, based on the similarity of the random phase offsets of each antenna in the Wi-Fi chip at the same time, the random phase offset θ in the received signal is eliminated by the conjugate multiplication operation. offset .

[0070] Step 3: Screen reliable data packets;

[0071] First, the CSI of the pth (1≤p≤P)th data packet on the kth subcarrier at the i-th antenna of the receiving end is expressed as H i,k,p , then the initial phase difference between antenna i and i+Δi is roughly estimated to be

[0072]

[0073] Here, arg(·) represents the phase part of the complex number.

[0074] Secondly, a scoring mechanism based on a multi-attribute decision-making method is proposed, which can screen out reliable data packets based on signal strength (Received Signal Strength Indicator, RSSI) and phase stability. The main idea of this method is that when the signal of the direct path is dominant, the phase of the CSI measurement on the subcarrier should be smooth. In addition, the RSSI of each channel also indirectly characterizes the degree of signal fading. Therefore, the higher the RSSI of the received signal, the higher the reliability of the data packet. In order to obtain the RSSI index, the average received power of all antennas and subcarriers in the data packet p is calculated.

[0075]

[0076] Then, the fluctuation degree of the subcarrier phase difference between adjacent antennas is counted

[0077]

[0078] Among them, μ p (1,1+Δi) represents the average phase difference between the reference antenna and the antennas with a spacing of Δi. Secondly, the score is obtained by integrating the two indicators.

[0079]

[0080] Among them, max(·) and min(·) represent the maximum and minimum values in the set, respectively.

[0081] Finally, the threshold is set using the median principle, that is, the score F p ≥0.5(max(F p )+min(F p )) is the filtered reliable data packet.

[0082] Step 4: Extract the phase of the direct path and the initial phase offset based on the VMD-HHT time-frequency analysis method to eliminate the influence of the reflected path phase.

[0083] First, considering the time dimension, a Savitzky-Golay filter is used to smooth the phase, removing outliers and small amounts of high-frequency noise. The phase of the direct path typically varies slightly from the initial phase and has a very narrow bandwidth, making it considered a low-frequency component. The reflected path, on the other hand, is easily overwhelmed by noise, varies dramatically at different sampling times, and has a wide bandwidth, making it considered a high-frequency component.

[0084] In order to extract the phase and initial phase offset of the direct path from the residual high-frequency components, the present invention adopts the VMD-HHT time-frequency analysis method. First, the initial phase difference estimated for each subcarrier is decomposed into multiple modal components by the VMD method.

[0085]

[0086] Among them, A k (i,i+Δi)=[Δθ k,1 (i,i+Δi),Δθ k,2 (i,i+Δi),...,Δθ k,P (i,i+Δi)] is the initial phase difference between the kth subcarrier antenna (i,i+Δi), M is the number of modes, is the mth modal component.

[0087] Secondly, we construct a variational problem and decompose the original signal into M modal components. The goal is to minimize the sum of the estimated bandwidths of each mode, with the constraint that the sum of all modes is equal to the original signal. The VMD constrained variational model is then

[0088]

[0089] Where Δ[·] represents the difference of n. represents the Hilbert transform, f m is the center frequency of the mth modal component. In order to calculate and f m , transform the constrained variational problem into an unconstrained variational problem, take advantage of the quadratic penalty term and the Lagrange multiplier method, and then combine the augmented Lagrangian to solve the variational problem to find the optimal solution

[0090]

[0091] Where α is the penalty factor and λ is the Lagrange multiplier. Finally, the Alternating Direction Method of Multipliers (ADMM) is used to solve the problem and obtain M modal components.

[0092] Then, the HHT calculation is applied to obtain the time-frequency information of each mode, and an analytical signal based on the original signal and its Hilbert transform is constructed to extract the instantaneous frequency and instantaneous energy, thereby obtaining the energy distribution of each mode in different frequency ranges. The zero-frequency component obtained by decomposing the initial phase difference on the kth subcarrier is The remaining high-frequency components are

[0093] Finally, based on the characteristics that the direct path and the initial phase offset have strong stability in the channel environment, and the reflected path phase shows large fluctuations under the multipath effect, the zero-frequency component is extracted. As the phase and initial phase offset characteristics of the direct path.

[0094] Step 5: Average the estimated initial phase differences of subcarriers on different antennas in multiple data packets to obtain a fine initial phase offset;

[0095] Since single measurement is susceptible to additive noise and frequency selective fading and has a certain degree of randomness, in order to improve the estimation stability and robustness, a multi-data packet statistical fusion method is used to effectively suppress the errors caused by noise and time-varying channel disturbances. Finally, the phase difference estimation is completed through the following expression

[0096]

[0097] in, Express Take the average of all elements in .

Claims

1. An online phase correction method based on Wi-Fi perception, comprising the following steps: Step 1: Use the Wi-Fi device to obtain the CSI of a single data packet; A uniform linear array in space, where the distance between array elements is d = λ / 2, where λ is the wavelength and the center frequency of the transmitted signal is f c ; The CSI of a single data packet on the 1≤i≤I antennas and 1≤k≤K subcarriers at the receiving end is expressed as Where j represents the imaginary number symbol, L represents the number of paths, α l represents the complex attenuation of the lth path, θ l and τ l are the AoA and ToF of the signal, Δf is the subcarrier spacing, c is the speed of light, σ i,k represents additive noise; Due to the defects of CFO, SFO, and PDD in commercial Wi-Fi transmission, the collected CSI amplitude and phase are distorted, which can be expressed as Among them, θ pll (i) represents the initial phase offset of the i-th RF channel, θ offset Random phase offset θ including CFO, SFO and PDD offset =2π[kΔfτ to (i)+iΔtf cfo ], τ to and f cfo denote timing offset and CFO respectively, Δt denotes the time interval between data packets; Step 2: Use antenna conjugate multiplication to extract the phase difference between different channels; The CSI change caused by the direct path is The CSI changes caused by the remaining reflection paths are: Then the CSI with initial phase offset is expressed as Where l = 1 represents the direct path, and θ1 represents the angle between the normal direction of the array antenna and the direction of the direct path; Without considering thermal noise, when the transmitter distance meets the far-field condition, the wavefront can be approximated as a plane wave, and the CSI of antenna i and i+Δi are conjugate multiplied. Where Δi represents the interval of antenna numbers, Represents H′ i+Δi,k The conjugate of mix Represents the term containing the reflection path; due to the path delay τ l Same for all antennas, with subcarrier phase rotation It is canceled in the conjugate multiplication; at the same time, based on the similarity of the random phase offsets of each antenna in the Wi-Fi chip at the same time, the random phase offset θ in the received signal is eliminated by the conjugate multiplication operation. offset ; Step 3: Filter reliable data packets; First, consider the time dimension of CSI, collect multiple CSI data packets, and express the CSI of the 1≤p≤Pth data packet on the kth subcarrier of the i-th antenna at the receiving end as H i,k,p , then the initial phase difference between antenna i and i+Δi is roughly estimated to be Where arg(·) represents the phase part of the complex number; Secondly, reliable data packets are screened out based on the signal strength (Received Signal Strength Indicator, RSSI) and phase stability; in order to obtain the RSSI index, the average received power of all antennas and subcarriers in the data packet p is calculated. Then, the fluctuation degree of the subcarrier phase difference between adjacent antennas is counted Among them, μ p (1,1+Δi) represents the average phase difference between the reference antenna and the antenna with a spacing of Δi; again, the score is obtained by integrating the two indicators. Among them, max(·) and min(·) represent the maximum and minimum values in the set respectively; Finally, the threshold is set using the median principle, that is, the score F p ≥0.5(max(F p )+min(F p )) is the filtered reliable data packet; Step 4: Extract the phase of the direct path and the initial phase offset based on the VMD-HHT time-frequency analysis method to eliminate the influence of the reflected path phase; Considering the time dimension, the Savitzky-Golay filter is used to smooth the phase and filter out outliers and a small amount of high-frequency noise. The VMD-HHT time-frequency analysis method is used to extract the phase and initial phase offset of the direct path from the residual high-frequency components. First, the initial phase difference estimated for each subcarrier is decomposed into multiple modal components by the VMD method. Among them, A k (i,i+Δi)=[Δθ k,1 (i,i+Δi),Δθ k,2 (i,i+Δi),...,Δθ k,P (i,i+Δi)] is the initial phase difference between the kth subcarrier antenna (i,i+Δi), M is the number of modes, is the mth modal component; Secondly, a variational problem is constructed to decompose the original signal into M modal components. The goal is to minimize the sum of the estimated bandwidths of each mode. The constraint is that the sum of all modes is equal to the original signal. Then the VMD constrained variational model is Where Δ[·] represents the difference of n. represents the Hilbert transform, f m is the center frequency of the mth modal component; in order to calculate and f m , transform the constrained variational problem into an unconstrained variational problem, take advantage of the quadratic penalty term and the Lagrange multiplier method, and then combine the augmented Lagrangian to solve the variational problem to find the optimal solution Where α is the penalty factor and λ is the Lagrange multiplier. Finally, the Alternating Direction Method of Multipliers (ADMM) is used to solve the problem and obtain M modal components. Then, the HHT calculation is applied to obtain the time-frequency information of each mode, and an analytical signal based on the original signal and its Hilbert transform is constructed to extract the instantaneous frequency and instantaneous energy, thereby obtaining the energy distribution of each mode in different frequency ranges; the zero-frequency component obtained by decomposing the initial phase difference on the kth subcarrier is The remaining high-frequency components are Finally, based on the characteristics that the direct path and the initial phase offset have strong stability in the channel environment, and the reflected path phase shows large fluctuations under the multipath effect, only the zero-frequency component is extracted. As the phase and initial phase offset characteristics of the direct path; Step 5: Average the initial phase differences estimated for the subcarriers on different antennas in multiple data packets to obtain a fine initial phase offset. in, Express Take the average of all elements in .

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