A Phase Signal Demodulation Method Based on Spatial Ellipse Fitting
By using the methods of three-dimensional ellipse fitting and affine transformation, the problem that the 3×3 coupler demodulation is easily affected by noise is solved, high-precision phase demodulation is achieved in the case of small signals, and distortion is reduced.
Patent Information
- Application Number
- CN202510046832.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-13
- Publication Date
- 2025-09-30
- Estimated Expiration
- 2045-01-13
AI Technical Summary
The existing 3×3 coupler demodulation algorithm is easily affected by noise, especially in the case of small signals, and is prone to distortion. It also has extremely high requirements on the process of the coupler, resulting in severe harmonic distortion in the demodulation results.
A phase signal demodulation method based on spatial ellipse fitting is adopted. A three-dimensional ellipse model is constructed using three signals. The ellipsoid and plane equations are fitted by the least squares method. After dimensionality reduction, the ellipse is transformed into a circle by affine transformation. The phase relationship of the points on the circle is used to demodulate the phase signal.
Under small signal and noise interference, the demodulation result is more accurate and the distortion is lower. It is suitable for large signal demodulation and improves the demodulation accuracy of distributed optical fiber sensors.
Smart Images
Figure CN119984355B_ABST
Abstract
Description
Technical Field
[0001] The invention belongs to the technical field of optical fiber sensing signal processing, and in particular relates to a phase signal demodulation method based on space ellipse fitting. Background Art
[0002] Distributed fiber optic sensors are widely used in perimeter security, pipeline inspection, and pattern recognition, offering high sensitivity, immunity to electromagnetic interference, and a wide dynamic range. To obtain more effective information, it is often necessary to demodulate the phase signal from the light intensity signal. Using a 3×3 equally splitting coupler to obtain interference signals with a fixed phase relationship and subsequently achieve phase demodulation is a common approach in interferometric sensing technology. The demodulation scheme directly impacts the performance of distributed fiber optic sensors. For an ideal 3×3 coupler, the demodulation algorithm exploits the 1:1:1 splitting ratio and 2π / 3 additive phase of the 3×3 coupler to achieve demodulation. Therefore, algorithms that treat the 3×3 coupler as an ideal device for demodulation place extremely high demands on the 3×3 coupler's process technology. Actual couplers often deviate from the ideal splitting ratio and 2π / 3 additive phase. Such demodulation methods can introduce demodulation errors due to coupler non-ideality, and can even result in severe harmonic distortion in the demodulation results. Furthermore, such algorithms often rely on the maximum and minimum intensity values of the interference signal, making them significantly susceptible to intensity noise. Although the two-dimensional ellipse fitting demodulation method can overcome the above problems to a certain extent, the data points usually used for fitting need to approximately form a complete ellipse. In the case of small signals where the modulation signal amplitude is small and cannot form a complete ellipse, the algorithm is easily affected by noise and fails. Summary of the Invention
[0003] In order to solve the problems that small signal demodulation is easily affected by noise and prone to distortion, the present invention provides a phase signal demodulation method based on spatial ellipse fitting.
[0004] The phase signal demodulation method based on spatial ellipse fitting provided by the present invention uses three signals for spatial ellipse fitting according to the interference signal characteristics of a 3×3 coupler. It is less affected by system random noise and fails to form a complete elliptical shape at the data point. That is, in the case of small signals, it can more accurately demodulate the phase signal and has broad application prospects.
[0005] The present invention provides a method for demodulating a phase signal based on spatial ellipse fitting. The method uses three-way interference signals output by a 3×3 coupler as three-dimensional coordinates, fits a point set obtained from the changing interference signal by the least squares method, obtains an ellipsoid expression in the three-dimensional space, and uses a constraint matrix to ensure that an ellipse solution is obtained; then, the least squares method is used to fit a general expression of the plane where the ellipse is located; the intersecting line between the fitting plane and the fitting ellipsoid is the fitted spatial ellipse; the spatial ellipse is reduced to a plane ellipse in the coordinate plane by projection or rotation, and an affine transformation is performed using the deflection angle and the major and minor axes of the plane ellipse. The ellipse is mapped to a circle (particularly, a unit circle) by translation, rotation, and scaling; the data is mapped to corresponding affine mapping points by using the above-mentioned transformation of mapping the ellipse to a circle, and the phase is solved by using the coordinates of the affine mapping points based on the direct correspondence between the distribution points on the circle and the phase.
[0006] The specific steps are as follows:
[0007] Step 1: Obtain three output signals from the 3×3 fiber coupler and map them into a set of points in a three-dimensional coordinate system.
[0008] Step 2: Use the least squares method to fit the general ellipsoid equation in three-dimensional space, and use the constraint matrix to ensure that the quadratic surface equation obtained by fitting is an ellipsoid;
[0009] Step 3: Use the least squares method to fit the general equation of the plane where the ellipse is located in the three-dimensional space;
[0010] Step 4: Reduce the dimensionality of the three-dimensional ellipse. Use the projection method to reduce the dimensionality, that is, substitute the general plane equation obtained by fitting into the ellipsoid equation, eliminate the variables, and obtain the plane ellipse equation in the two-dimensional coordinate system; or use the rotation method to reduce the dimensionality, that is, rotate the three-dimensional ellipse to the coordinate plane or a plane parallel to the coordinate plane.
[0011] Step 5: According to the plane ellipse equation, calculate the length of the major and minor semi-axis, the deflection angle and the center of the ellipse respectively;
[0012] Step 6: Demodulate the phase through affine transformation. First, obtain the affine transformation matrix based on the ellipse center, deflection, and major and minor axis lengths of the ellipse expression in the two-dimensional coordinate system. This matrix can map the ellipse into a circle. Using this matrix, map the data points reduced to the two-dimensional coordinate system into corresponding affine points. Use the mathematical transformation of mapping the ellipse into a circle to map the original data points into corresponding points. Use the inverse tangent function according to the coordinates of the points and then unwrap them to complete the phase demodulation of the small signal.
[0013] Further:
[0014] In step 1, let the point set in the three-dimensional coordinate system be: {p i (x i,y i ,z i )};
[0015] X i =(x i 2 ,y i 2 ,z i 2 ,2y i z i ,2x i z i ,2x i y i ,2x i ,2y i ,2z i ,1) T , (1)
[0016] In step 2, the three-dimensional space ellipsoid that needs to be fitted has the general equation:
[0017] a1x 2 +a2y 2 +a3z 2 +2a4yz+2a5xz+2a6xy+2a7x+2a8y+2a9z+a 10 =0, (2)
[0018] α=(a1,a2,a3,a4,a5,a6,a7,a8,a9,a 10 ) T , (3)
[0019] a 1-10 is the equation parameter of the three-dimensional ellipsoid, α is the parameter matrix, and the expression of the constraint matrix is:
[0020]
[0021] Solve using the least squares method:
[0022]
[0023] Where, D=(X1,X2,X3,...,X n ), solve to get (DD T ) -1 The eigenvalue λ and eigenvector of C0, where the eigenvector corresponding to the largest positive eigenvalue is the three-dimensional space expression parameter of the ellipsoid.
[0024] In step 3, the general equation of the plane where the ellipse is located in the three-dimensional space is fitted using the least squares method:
[0025] z=b1x+b2y+b3, (6)
[0026] Among them, b1, b2, b3 are unknown coefficients;
[0027] In step 4, the projection method is used to reduce the dimension to the coordinate plane. Taking the dimension reduction to the XOY coordinate plane as an example, the expression parameters of z obtained from the plane equation can be directly substituted into the three-dimensional ellipse, and the parameter z can be eliminated to obtain the two-dimensional coordinate system plane ellipse equation expressed by x and y parameters, where:
[0028]
[0029] The rotation method is used for dimensionality reduction. Taking the dimensionality reduction to the parallel XOY coordinate plane as an example, the rotation matrix R is:
[0030]
[0031] The transformation of the corresponding data points is:
[0032]
[0033] In the above formula, [I1 I2 I3] T is the original data point, [I 1jw0 I 2jw0 I 2jw0 ] T After rotation [I1 I2 I3] T The new coordinates of the mapping. Suppose the plane ellipse equation reduced to a two-dimensional coordinate system is:
[0034] ax 2 +by 2 +cxy+dx+ey+f=0, (10)
[0035] In step 5, the calculated ellipse center (x0, y0) is:
[0036]
[0037] The lengths of the major and minor semi-axes are:
[0038]
[0039] The declination angle of the ellipse is:
[0040]
[0041] In step 6, the elliptical affine and demodulation phase are performed, and the specific process is as follows.
[0042] To facilitate mapping the ellipse to a circle, first map the ellipse to a standard elliptical distribution through translation and rotation, with the major axis on the X-axis and the minor axis on the Y-axis. The corresponding data point mapping is:
[0043]
[0044] Then perform a scaling transformation along the coordinate axis to transform the ellipse into a circle. Taking the mapping to the unit circle as an example, the corresponding data point mapping is:
[0045]
[0046] Above, I 1jw1 and I 2jw1 The elliptical affine transformation is mapped to the corresponding data points of the circle. Finally, the phase signal is unwrapped after using the inverse tangent function
[0047]
[0048] Here, unwrap(·) represents the unwrapping algorithm, where k is the phase unwrapping count. If the phase difference between two consecutive inverse trigonometric function calculations is greater than π, the count k is expressed as k = k - 1. If the phase difference is less than -π, it is expressed as k = k + 1, with the value of k starting from 0.
[0049] The specific principles of the present invention are analyzed as follows.
[0050] For the ideal equally divided 3×3 fiber coupler interferometer structure, ignoring non-ideal factors such as transmission loss, three interference signal outputs can be obtained, which are expressed as:
[0051]
[0052] Among them, φ0 is the operating point introduced by the system, φ(t) is the phase change caused by the measured signal, A0 is the parameter related to the DC quantity, and B0 is the amplitude of the interference term. ideal1 , I ideal2 , I ideal3 As the three-dimensional coordinate components x, y, and z, let From the above formula, we can get:
[0053]
[0054] and:
[0055]
[0056] Equations (18) and (19) together determine the graph formed by the ideal data points. Equation (18) is the plane passing through the point (A0, A0, A0), and Equation (19) is the plane with the center point (A0, A0, A0) and the radius Therefore, the figure formed is the curve intercepted by the sphere and the plane passing through the center of the sphere, that is, the diameter of circle.
[0057] If only the two interference signals of the 3×3 fiber coupler are used to form a two-dimensional coordinate system, and an ellipse is fitted, for example, using I1 and I2 as x and y, respectively, the obtained curve is:
[0058]
[0059] The above formula is an ellipse equation, that is, the two-dimensional curve formed is an ellipse, and its major and minor semi-axes are
[0060] Comparing the curves obtained by the three-dimensional and two-dimensional methods, e.g. Figure 1 As shown, the three-dimensional curve is a circle with the same radius of curvature at all locations. Regardless of the location of the curve formed by the data point set, the curvature is consistent for the same phase change. Therefore, the curve fitting effect is consistent in the presence of noise interference. The two-dimensional curve is an ellipse with different radii of curvature at different locations. The curve formed by the data point set is at different locations and has different curvature states. For the same phase change, the fitting effect will vary in the presence of noise interference, especially when the curve is located at a location with a large elliptical curvature radius, such as the a2b2 segment, which is more detrimental to the fitting effect of small signals that do not form a complete ellipse.
[0061] Based on the above analysis, it can be seen that the method of using the three-way interference signals of the 3×3 fiber coupler to form three-dimensional coordinates for curve fitting is more conducive to the solution of small signals.
[0062] When the 3×3 coupler is a non-ideal device, its parameters deviate from those of an ideal device. Ignoring transmission loss, the three-way interference output can be expressed as:
[0063]
[0064] Among them, A1, A2, and A3 are parameters related to the DC quantity, and B1, B2, and B3 are the amplitudes of the interference terms; Δα1 and Δα2 are deviations from From the nature of interference, we know that
[0065]
[0066] Therefore, I unideal1 , I unideal2 , I unideal3 As the three-dimensional coordinate components x, y, and z, we can get:
[0067] (x-A1)+(y-A2)+(z-A3)=0 (23)
[0068] is a plane passing through the point (A1, A2, A3). Let k1 and k2 be constants,
[0069]
[0070]
[0071] like Figure 2 As shown, from geometric knowledge, we know that there must exist k1 and k2 such that:
[0072]
[0073] Then we have:
[0074]
[0075] Right now:
[0076]
[0077] is the equation of the ellipsoid centered at (A1, A2, A3). Combining equations (23) and (28), the curve formed by the intersection of the ellipsoid and the plane is an ellipse centered at (A1, A2, A3); since the parameters of the non-ideal 3×3 coupler are close to those of an ideal device, the ellipse is close to a circle.
[0078] Based on the above analysis, in the present invention, a three-dimensional point set composed of three-way interference signals is used for ellipse fitting. The process of obtaining the fitted ellipse is as follows: using the same set of point sets, the ellipsoid and the plane are fitted respectively, and then the intersecting curves are obtained by fitting the ellipsoid and the plane, that is, the fitted ellipse expression.
[0079] To facilitate calculations, the ellipse obtained in three-dimensional space is subjected to dimensionality reduction to obtain an elliptic curve expression within or parallel to the coordinate plane (XOY, YOZ, or ZOX). Taking dimensionality reduction to the XOY coordinate plane as an example, for example, the plane containing the ellipse can be rotated to the XOY plane or a plane parallel to the XOY plane; or the ellipse in three-dimensional space can be projected onto the XOY plane.
[0080] The process of affine transformation of two-dimensional elliptic curve is as follows: Figure 3 As shown, it essentially performs a scaling transformation in the direction of the major and minor axes of the ellipse, mapping the ellipse into a circle. From a mathematical perspective, the transformation process can be decomposed into: first, translate the center of the ellipse to point O, and transform the ellipse into a regular ellipse (the major and minor axes are parallel to the coordinate axes). This step can be achieved by rotating the ellipse or rotating the coordinate system. Figure 3 The method used in this paper is to rotate the ellipse; by performing a scaling transformation along the coordinate axes (major and minor axes), the ellipse can be transformed into a circle (which can be taken as the unit circle), thus realizing the ellipse-to-circle mapping. The data points obtained by the affine transformation are distributed around the circle.
[0081] Since the central angle between any two points on a circle is the phase difference between these two points Therefore, if Figure 4 As shown, the phase can be obtained by calculating the angle of the data vector (the vector from the center of the circle to the data point). Usually, the inverse transform unwrapping method can be used. It should be noted that the phase solved by affine transformation differs from the phase φ(t) in equation (1) by a constant. Since the focus in actual measurement is the change in phase, that is, the AC component, this does not affect the final result.
[0082] The beneficial effects of the present invention are as follows.
[0083] The spatial ellipse fitting small signal phase demodulation method of the present invention considers the three signals as a whole, constructs a three-dimensional ellipse model composed of the three signals, uses a spatial ellipsoid fitting algorithm and a plane fitting algorithm to obtain a three-dimensional ellipse curve, and obtains a two-dimensional ellipse after dimensionality reduction. According to the expression of the two-dimensional ellipse, the major and minor axes, the deflection angle and the ellipse center of the ellipse are calculated, and then the two-dimensional data signals after dimensionality reduction are subjected to ellipse affine mapping, and finally the phase signal is demodulated. Different from the two-dimensional ellipse demodulation algorithm, the present invention is less affected by system random noise. In the case of small signals, the demodulated signal is more accurate and has lower distortion. The present invention is also applicable to large signal demodulation. BRIEF DESCRIPTION OF THE DRAWINGS
[0084] Figure 1 Comparison of the curves obtained in three dimensions with those obtained in two dimensions.
[0085] Figure 2 Explain the existence of k1 and k2.
[0086] Figure 3 The affine transformation process of mapping an ellipse to a circle.
[0087] Figure 4 Schematic diagram of phase calculation after affine transformation.
[0088] Figure 5 Schematic diagram of the three-dimensional spatial distribution of three signals in an example of the small signal demodulation method based on spatial ellipse fitting of the present invention.
[0089] Figure 6 Schematic diagram of the ellipsoid obtained by fitting in an example of the small signal demodulation method based on spatial ellipse fitting of the present invention.
[0090] Figure 7 A schematic diagram of a plane obtained by fitting in an example of a small signal demodulation method based on spatial ellipse fitting according to the present invention.
[0091] Figure 8Schematic diagram of an ellipse obtained by dimensionality reduction in an example of a small signal demodulation method based on spatial ellipse fitting in the present invention.
[0092] Figure 9 Schematic diagram of a standard ellipse obtained by signal affine in an example of a small signal demodulation method based on spatial ellipse fitting of the present invention.
[0093] Figure 10 Schematic diagram of the unit circle obtained by signal affine in an example of the small signal demodulation method based on spatial ellipse fitting of the present invention.
[0094] Figure 11 A schematic diagram of the phase obtained by demodulation in an example of the small signal demodulation method based on spatial ellipse fitting of the present invention. DETAILED DESCRIPTION
[0095] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the specific implementation methods of the present method are further described below in conjunction with the accompanying drawings and embodiments. The accompanying drawings clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without making creative efforts are within the scope of protection of the present invention. The present invention is further explained below in conjunction with specific embodiments. Those skilled in the art should understand that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. Various equivalent modifications to the present invention fall within the scope defined by the claims attached to this application.
[0096] In this embodiment, the spatial ellipse fitting small signal phase demodulation method of the present invention is used for phase demodulation, and the process includes: obtaining three-way output data of a 3×3 fiber coupler, and the output light intensity signals are respectively:
[0097]
[0098] like Figure 5 As shown, the three-way output data is mapped to the three-dimensional space coordinate system to form a point set {p i (x i ,y i ,z i )}, ideally, this point set is an ellipse and lies on a plane. Due to the influence of noise, its distribution appears as a spatial elliptical ring, which needs to be fitted. The least squares method is used to fit the general equation of the ellipsoid in three-dimensional space:
[0099] a1x 2 +a2y 2 +a3z 2+2a4yz+2a5xz+2a6xy+2a7x+2a8y+2a9z+a 10 =0 (30)
[0100] Obtain an ellipsoid. Use the least squares method to fit the plane where the point set is located and obtain the general equation of the plane:
[0101] z=b1x+b2y+b3 (31)
[0102] Substitute the plane equation into the general equation of the ellipsoid and eliminate the parameter z to obtain the plane ellipse equation in the two-dimensional coordinate system (taking the xoy coordinate system as an example).
[0103] ax 2 +bxy+cy 2 +dx+ey+f=0 (32)
[0104] Calculates the lengths of the major and minor axes, the declination angle, the center of the ellipse, and the DC component.
[0105]
[0106]
[0107] In the formula, x0 and y0 are the horizontal and vertical coordinates of the ellipse center, A and B are the lengths of the major and minor axes of the ellipse, and θ is the deflection angle of the ellipse. The signal is then affine transformed to obtain the unit circle distribution. First, the coordinate translation transformation is performed to position the ellipse center at the origin O. Then, the rotation is performed to transform the oblique ellipse into a regular ellipse (standard ellipse), with the major axis located on the horizontal axis (X-axis) and the minor axis located on the vertical axis (Y-axis) of the new coordinate. The transformation process is expressed as:
[0108]
[0109] Then perform a scaling transformation on the coordinate axis to turn the ellipse into a circle:
[0110]
[0111] The final demodulation phase signal is:
[0112]
[0113] The above-mentioned transformation processes such as translation and rotation can also be achieved through coordinate transformation.
[0114] The method of the present invention is described in detail below using a simulation example.
[0115] The I1, I2 and I3 obtained by simulation are as follows. The O-XYZ coordinate system is established, and the spatial point distribution is as follows Figure 5shown.
[0116]
[0117] in, is the phase signal to be demodulated, t is time, and ε1, ε2, and ε3 are Gaussian white noise. The ellipsoid is fitted using the least squares method, and the constraint equation is used to ensure that it is an ellipsoid solution. The general equation of the ellipsoid is:
[0118]
[0119] The ellipsoid is Figure 6 As shown. Using the least squares method, the plane where the ellipse is located is fitted, and the general equation of the plane is obtained as follows:
[0120] z=-1.166x-0.8852y+2.7737 (41)
[0121] The plane is Figure 7 As shown. Substituting the general equation of the plane into the ellipsoid equation, we get the general equation of the ellipse:
[0122] x 2 +0.7572xy+0.8510y 2 -2.6072x-2.1184y+0.9563=0 (42)
[0123] The ellipse is Figure 8 As shown, according to formulas (33)-(35), the ellipse center, the length of the major and minor axes and the deflection angle of the ellipse are calculated.
[0124]
[0125] Perform affine transformation on the signal and first obtain the standard elliptical distribution according to formula (36), as follows: Figure 9 As shown, the unit circle distribution is obtained according to formula (37), as Figure 10 shown.
[0126] According to formula (38), the phase signal is obtained by demodulation, and the result is as follows Figure 11 As shown, the result is Similarly, for a 1000Hz sinusoidal signal, the radian amplitude is approximately This shows that the algorithm can demodulate and obtain the phase signal in the presence of small signals and noise interference.
Claims
1. A phase signal demodulation method based on spatial ellipse fitting, characterized in that: According to the interference signal characteristics of the 3×3 coupler, three-way signals are used to fit a spatial ellipse, and the phase signal can be accurately demodulated in the case of small signals. Specifically, the three-way interference signals output by the 3×3 coupler are used as three-dimensional coordinates, and the point set obtained by the changing interference signal is fitted by the least squares method to obtain the ellipsoid expression in the three-dimensional space, and the constraint matrix is used to ensure that the ellipse solution is obtained. The least squares method is then used to fit the general expression of the plane where the ellipse is located. The phase intersection between the fitting plane and the fitting ellipsoid is the fitted spatial ellipse. The spatial ellipse is reduced to a plane ellipse in the coordinate plane through projection or rotation, and the deflection angle and major and minor axes of the plane ellipse are used for affine transformation. The ellipse is mapped to a circle through translation, rotation, and scaling. The data is mapped to corresponding affine mapping points using the above-mentioned transformation of mapping the ellipse to a circle. Based on the direct correspondence between the distribution points on the circle and the phase, the coordinates of the affine mapping points are used for phase solution.
2. The phase signal demodulation method according to claim 1, wherein: The specific steps are as follows: Step 1: Obtain three output signals from the 3×3 fiber coupler and map them into a set of points in a three-dimensional coordinate system. Step 2: Use the least squares method to fit the general ellipsoid equation in three-dimensional space, and use the constraint matrix to ensure that the quadratic surface equation obtained by fitting is an ellipsoid; Step 3: Use the least squares method to fit the general equation of the plane where the ellipse is located in the three-dimensional space; Step 4: Reduce the dimensionality of the three-dimensional ellipse. Use the projection method to reduce the dimensionality, that is, substitute the general plane equation obtained by fitting into the ellipsoid equation, eliminate the variables, and obtain the plane ellipse equation in the two-dimensional coordinate system; or use the rotation method to reduce the dimensionality, that is, rotate the three-dimensional ellipse to the coordinate plane or a plane parallel to the coordinate plane. Step 5: According to the plane ellipse equation, calculate the length of the major and minor semi-axis, the deflection angle and the center of the ellipse respectively; Step 6: Demodulate the phase through affine transformation. First, obtain the affine transformation matrix based on the ellipse center, deflection, and major and minor axis lengths of the ellipse expression in the two-dimensional coordinate system. This matrix can map the ellipse into a circle. Using this matrix, map the data points reduced to the two-dimensional coordinate system into corresponding affine points. Use the mathematical transformation of mapping the ellipse into a circle to map the original data points into corresponding points. Use the inverse tangent function according to the coordinates of the points and then unwrap them to complete the phase demodulation of the small signal.
3. The phase signal demodulation method according to claim 2, wherein: In step 1, let the point set in the three-dimensional coordinate system be: {p i (x i ,y i ,z i )}; In step 2, the general equation of the fitted three-dimensional space ellipsoid is: <h2 style=";text-align:left;direction:ltr">a1x<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> +a2y<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> +a3z<h2 style=";text-align:left;direction:ltr"> 2 <h2 style=";text-align:left;direction:ltr"> +2a4yz+2a5xz+2a6xy+2a7x+2a8y+2a9z+a<h2 style=";text-align:left;direction:ltr"> 10 <h2 style=";text-align:left;direction:ltr"> =0, (2) <h2 style=";text-align:left;direction:ltr">α=(a1,a2,a3,a4,a5,a6,a7,a8,a9,a<h2 style=";text-align:left;direction:ltr"> 10 <h2 style=";text-align:left;direction:ltr"> )<h2 style=";text-align:left;direction:ltr"> T <h2 style=";text-align:left;direction:ltr"> (3) The expression of the constraint matrix is: Solve using the least squares method: Where, D=(X1,X2,X3,...,X n ), solve to get (DD T ) -1 The eigenvalue λ and eigenvector of C0, where the eigenvector corresponding to the largest positive eigenvalue is the three-dimensional space expression parameter of the ellipsoid.
4. The phase signal demodulation method according to claim 3, wherein: In step 3, the general equation of the plane where the ellipse is located in the three-dimensional space fitted by the least squares method is specifically: z=b1x+b2y+b3, (6) Among them, b1, b2, and b3 are unknown coefficients.
5. The phase signal demodulation method according to claim 4, characterized in that: In step 4, the projection method is used for dimensionality reduction; for dimensionality reduction to the XOY coordinate plane, the expression parameters of z obtained by the plane equation are directly substituted into the general equation of the three-dimensional ellipse, and the parameter z can be eliminated to obtain the two-dimensional coordinate system plane ellipse equation expressed by x and y parameters, where: The rotation method is used for dimensionality reduction. For dimensionality reduction to the parallel XOY coordinate plane, the rotation matrix R is: The transformation of the corresponding data points is: In the above formula, [I1 I2 I3] T is the original data point, [I 1jw0 I 2jw0 I 3jw0 ] T After rotation [I1 I2 I3] T The new coordinates of the mapping; suppose the plane ellipse equation reduced to a two-dimensional coordinate system is: ax 2 +by 2 +cxy+dx+ey+f=0, (10)。 6. The phase signal demodulation method according to claim 5, characterized in that: In step 5, the calculated ellipse center (x0, y0) is: The lengths of the major and minor semi-axes are: The declination angle of the ellipse is:
7. The phase signal demodulation method according to claim 6, characterized in that: In step 6, the elliptical affine and demodulation phase, the specific process is as follows; First, the ellipse is mapped to a standard elliptical distribution through translation and rotation, with the major axis on the X-axis and the minor axis on the Y-axis. The corresponding data point mapping is: Then perform a scaling transformation along the coordinate axis to transform the ellipse into a circle. For the mapping to the unit circle, the corresponding data point mapping is: Above, I 1jw1 and I 2jw1 Map the elliptical affine transformation to the corresponding data points of the circle; finally, use the inverse tangent function to unwrap the phase signal Where unwrap(·) represents the unwrapping algorithm, where k is the phase unwrapping count. If the phase difference between two consecutive calculations of the inverse trigonometric function is greater than π, the count k is expressed as k = k - 1. When the phase difference change is less than -π, it is expressed as k = k + 1, and the value of k is counted starting from 0.