Phase signal demodulation method based on space ellipse fitting

Abstract

The invention belongs to the technical field of optical fiber sensing signal processing, and particularly relates to a phase signal demodulation method based on space ellipse fitting. The method comprises the following steps: mapping three paths of output data of a 3 * 3 optical fiber coupler to a three-dimensional space, fitting through a least square method to obtain an ellipsoid expression in the three-dimensional space, and ensuring to obtain an ellipsoid solution by using a constraint matrix; fitting an expression of a plane where the ellipse is located by adopting a least square method; the section line of the fitting plane and the fitting ellipsoid is a space ellipse obtained by fitting; carrying out dimensionality reduction on a space ellipse to form an ellipse in a coordinate plane, carrying out affine transformation by utilizing a deflection angle and long and short axes of a plane ellipse, and mapping the ellipse into a circle; and mapping the data into corresponding affine transformation points by utilizing the transformation of mapping the ellipse into the circle, and carrying out phase resolving by utilizing transformed point coordinates based on a direct corresponding relationship between distribution points on the circle and phases. The method is less affected by system random noise, and can demodulate phase signals more accurately under the condition of small signals.

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 detection and pattern recognition, and have the characteristics of high sensitivity, anti-electromagnetic interference and large dynamic range. In order to obtain more effective information, it is often necessary to demodulate the phase signal from the light intensity signal. Using a 3×3 equal-division coupler to obtain an interference signal with a fixed phase relationship and then realize phase demodulation is one of the common means of interferometric sensing technology. The demodulation scheme will directly affect the performance of distributed fiber optic sensors. For an ideal 3×3 coupler, the demodulation algorithm uses the 1:1:1 splitting ratio and 2π / 3 additional phase of the 3×3 coupler to achieve demodulation. Therefore, the algorithm that regards the 3×3 coupler as an ideal device for demodulation has extremely high requirements for the process of the 3×3 coupler. For the couplers actually used, they usually deviate from the ideal splitting ratio of the 3×3 coupler and the additional phase of 2π / 3. The above demodulation method will cause demodulation errors due to the non-ideality of the coupler, and even cause serious harmonic distortion in the demodulation result; at the same time, this type of algorithm often relies on the maximum and minimum values ​​of the light intensity of the interference signal, which is greatly affected by 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 distortion, the present invention provides a phase signal demodulation method based on space ellipse fitting.

[0004] The phase signal demodulation method based on spatial ellipse fitting provided by the present invention uses three-way signals to perform 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 ellipse shape at the data point, that is, in the case of a small signal, it can demodulate the phase signal more accurately, and has broad application prospects.

[0005] The method for demodulating phase signals based on spatial ellipse fitting provided by the present invention takes three-way interference signals output by a 3×3 coupler as three-dimensional coordinates, fits the point set obtained by the changing interference signals by the least square method, obtains the ellipsoid expression in the three-dimensional space, and uses the constraint matrix to ensure the ellipse solution; then the least square method is used to fit the general expression of the plane where the ellipse is located; the phase intersection line between the fitting plane and the fitting ellipsoid is the spatial ellipse obtained by fitting; the spatial ellipse is reduced to a plane ellipse in the coordinate plane by projection or rotation, and the deflection angle and the major and minor axes of the plane ellipse are used to perform affine transformation, and the ellipse is mapped to a circle (especially, a unit circle) by translation, rotation and expansion; the data is mapped to corresponding affine mapping points by using the transformation of mapping the ellipse to the 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 the three output signals into a point set in a three-dimensional coordinate system;

[0008] Step 2: Use the least square method to fit the general ellipsoid equation in the 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 square method to fit the general equation of the plane where the ellipse is located in the three-dimensional space;

[0010] Step 4: Reduce the dimension of the three-dimensional ellipse; use the projection method to reduce the dimension, that is, substitute the general plane equation obtained by fitting into the ellipsoid equation, realize elimination, and obtain the plane ellipse equation in the two-dimensional coordinate system; or use the rotation method to reduce the dimension, that is, rotate the three-dimensional space ellipse to the coordinate plane or a plane parallel to the coordinate plane;

[0011] Step 5: According to the plane ellipse equation expression, 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 angle, 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] a 1 x 2 +a 2 y 2 +a 3 z 2 +2a 4 yz+2a 5 xz+2a 6 xy+2a 7 x+2a 8 y+2a 9 z+a 10 =0, (2)

[0018] α=(a 1 ,a 2 ,a 3 ,a 4 ,a 5 ,a 6 ,a 7 ,a 8 ,a 9 ,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] In the formula, D=(X 1 ,X 2 ,X 3 ,...,X n ), and we can get (DD T ) -1 C 0 The eigenvalue λ and eigenvector of , 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=b 1 x+b 2 y+b 3 , (6)

[0026] Among them, b 1 , b 2 , b 3 is the coefficient to be determined;

[0027] In step 4, the projection method is used to reduce the dimension to the coordinate plane. Taking the reduction to the XOY coordinate plane as an example, the expression parameters of z obtained by the plane equation can be directly substituted into the three-dimensional ellipse, and the parameter z can be eliminated to obtain the plane ellipse equation of the two-dimensional coordinate system expressed by the 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, [I 1 I 2 I 3 ] T is the original data point, [I 1jw0 I 2jw0 I 2jw0 ] T After rotation [I 1 I 2 I 3 ] 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 (x 0 ,y 0 )for:

[0036]

[0037] The lengths of the major and minor semi-axes are:

[0038]

[0039] The deflection angle of the ellipse is:

[0040]

[0041] In step 6, the ellipse 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 ellipse distribution through translation and rotation, with the major axis on the X-axis and the minor axis on the Y-axis. The corresponding data points are mapped as follows:

[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 ellipse affine transformation maps the corresponding data points to the circle. Finally, the phase signal is unwrapped using the inverse tangent function.

[0047]

[0048] Wherein, unwrap(·) represents the unwrapping algorithm, where k is the count of phase unwrapping. 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 value is less than -π, it is expressed as k=k+1, and the value of k is calculated starting from 0.

[0049] The specific principle of the present invention is analyzed as follows.

[0050] For the ideal interference structure of the equally divided 3×3 fiber coupler, ignoring non-ideal factors such as transmission loss, three interference signal outputs can be obtained, which are expressed as:

[0051]

[0052] Among them, φ 0 is the operating point introduced by the system, φ(t) is the phase change caused by the measured signal, A 0 is a parameter related to DC flow, B 0 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] Equation (18) and (19) jointly determine the graph formed by the ideal data points. Equation (18) is the graph passing through point (A 0 ,A 0 ,A 0 ) plane, equation (19) is the center point (A 0 ,A 0 ,A 0 ), the radius is 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 is circle.

[0057] If only the two-way interference signals of the 3×3 fiber coupler are used to form a two-dimensional coordinate system, an ellipse fitting is performed, such as using I 1 ,I 2 , 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, such as Figure 1 As shown in the figure, it can be seen that the curve obtained in three dimensions is a circle, and its curvature radius is the same everywhere. No matter where the curve formed by the data point set is, for the same phase change, its bending state is consistent. Therefore, in the presence of noise interference, the curve fitting effect is consistent. For the curve obtained in two dimensions, it is an ellipse, and the curvature radius is different at different positions. The curve formed by the data point set is at different positions and has different bending states. For the same phase change, in the presence of noise interference, the fitting effect will change, especially when the curve is located at a position with a large curvature radius of the ellipse, such as a2 b 2 It is more unfavorable for 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 signal of the 3×3 fiber coupler to form a three-dimensional coordinate 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 the ideal device parameters. Ignoring the transmission loss, the three-way interference output can be expressed as:

[0063]

[0064] Among them, A 1 , A 2 , A 3 is a parameter related to DC flow, B 1 , B 2 , B 3 is the amplitude of the interference term; Δα 1 , Δα 2 Deviation 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, z, we can get:

[0067] (xA 1 )+(yA 2 )+(zA 3 )=0 (23)

[0068] is the passing point (A 1 ,A 2 ,A 3 ) plane. Let k 1 , k 2 is a constant,

[0069]

[0070]

[0071] like Figure 2 As shown, from geometric knowledge, we know that there must be k 1 , k 2 So that:

[0072]

[0073] Then we have:

[0074]

[0075] Right now:

[0076]

[0077] Centered on (A 1 ,A 2 ,A 3 ) Ellipsoid surface equation. Combining equations (23) and (28), the curve formed by the intersection of the ellipsoid and the plane is centered at (A 1 ,A 2 ,A 3 ); since the parameters of the non-ideal 3×3 coupler are close to those of the 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, and the process of obtaining the fitted ellipse is: using the same set of point sets to fit the ellipsoid and the plane respectively, and then obtaining the intersecting curve by fitting the ellipsoid and the plane, that is, the fitted ellipse expression.

[0079] In order to facilitate calculation, the ellipse obtained in the three-dimensional space is subjected to dimensionality reduction processing to obtain an elliptic curve expression in the coordinate plane (XOY, YOZ or ZOX) or parallel to the coordinate plane. Taking dimensionality reduction to the XOY coordinate plane as an example, for example, the plane where the ellipse is located can be rotated to the XOY plane, or a plane parallel to the XOY plane; or the ellipse in the three-dimensional space can be projected to the XOY plane by projection.

[0080] The process of affine transformation of two-dimensional elliptic curve is as follows: Figure 3 As shown, it is essentially a scaling transformation in the direction of the major and minor axes of the ellipse, mapping the ellipse to 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 axis). 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; the ellipse can be transformed into a circle (which can be taken as a unit circle) by scaling along the coordinate axis (major and minor axis) to achieve the mapping from ellipse to circle. The data points obtained by 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 4As 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 transformation unwrapping method can be used. It should be noted that the phase solved by the affine transformation differs from the phase φ(t) in equation (1) by a constant. Since in actual measurement, the focus is on the change of the phase, that is, the AC component, it 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-way signal as a whole, constructs a three-dimensional ellipse model composed of the three-way signal, uses the spatial ellipsoid fitting algorithm and the plane fitting algorithm, obtains the three-dimensional ellipse curve, obtains the two-dimensional ellipse after dimensionality reduction, calculates the major and minor axes, deflection angle and ellipse center of the ellipse according to the expression of the two-dimensional ellipse, and then performs ellipse affine mapping on the two-way data signals after dimensionality reduction, and finally demodulates the phase signal. Different from the two-dimensional ellipse demodulation algorithm, the present invention is less affected by the random noise of the system. In the case of small signals, the demodulated signal is more accurate and has lower distortion. The present invention is also suitable for 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 Description 1 , k 2 existence.

[0086] Figure 3 It is the affine transformation process of mapping an ellipse to a circle.

[0087] Figure 4 Schematic diagram of phase solution after affine transformation.

[0088] Figure 5 It is a schematic diagram of the three-dimensional spatial distribution of three signals in an example of a 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 a 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 of the present invention.

[0091] Figure 8 A schematic diagram of an ellipse obtained by dimensionality reduction in an example of a small signal demodulation method based on spatial ellipse fitting of the present invention.

[0092] Fig. 9A 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] Fig.10 A schematic diagram of a unit circle obtained by signal affine in an example of a small signal demodulation method based on spatial ellipse fitting of the present invention.

[0094] Fig.11 A schematic diagram of the phase obtained by demodulation in an example of a small signal demodulation method based on spatial ellipse fitting of the present invention. DETAILED DESCRIPTION

[0095] In order to make the purpose, technical scheme and advantages of the embodiments of the present invention clearer, the specific implementation methods of the present method are further described in conjunction with the accompanying drawings and embodiments. The accompanying drawings in the present invention clearly and completely describe the technical scheme 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 creative work are within the scope of protection of the present invention. The present invention is further explained in conjunction with specific embodiments below. 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. Modifications to various equivalent forms of 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 optical 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 )}, the point set is an ellipse in an ideal situation and is located on a plane. Due to the influence of noise, its distribution appears as a spatial ellipse ring, which needs to be fitted. The general equation of the ellipsoid in three-dimensional space is fitted using the least squares method:

[0099] a 1 x 2 +a 2 y 2 +a 3 z 2 +2a 4 yz+2a 5 xz+2a 6xy+2a 7 x+2a 8 y+2a 9 z+a 10 =0 (30)

[0100] Get the ellipsoid. Use the least squares method to fit the plane where the point set is located and get the general equation of the plane:

[0101] z=b 1 x+b 2 y+b 3 (31)

[0102] Substitute the plane equation into the general equation of the ellipsoid and eliminate the parameter z to obtain the plane ellipse equation of 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] Calculate the lengths of the major and minor axes, the declination angle, the center of the ellipse, and the DC component.

[0105]

[0106]

[0107] The x of the formula 0 and 0 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. Then, the signal is affine transformed to obtain the unit circle distribution. First, the center of the ellipse is located at the origin O through coordinate translation transformation; then, the oblique ellipse is transformed into a regular ellipse (standard ellipse) through rotation, and the major axis is located at the horizontal axis (X axis) of the coordinate, and the minor axis is located at the vertical axis (Y axis) of the new coordinate. The transformation process is expressed as:

[0108]

[0109] Then scale the coordinate axes to change the ellipse into a circle:

[0110]

[0111] The final demodulation results in the phase signal:

[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 specifically described below with reference to a simulation example.

[0115] The simulated I 1 ,I 2 and I 3 As follows, the O-XYZ coordinate system is established, and the spatial point distribution is as follows Figure 5 shown.

[0116]

[0117] in, is the phase signal to be demodulated, t is the time, ε 1 , ε 2 and ε 3 is 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 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 equations (33)-(35), the ellipse center, the length of the major and minor semi-axis and the deflection angle of the ellipse are calculated.

[0124]

[0125] Perform an affine transformation on the signal and first obtain the standard elliptical distribution according to formula (36), as follows: Fig. 9 As shown, the unit circle distribution is obtained according to formula (37), as Fig.10 shown.

[0126] According to formula (38), the phase signal is demodulated and the result is as follows: Fig.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 the space 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 line between the fitting plane and the fitting ellipsoid is the fitted space ellipse; the space ellipse is reduced to a plane ellipse in the coordinate plane by projection or rotation, and the deflection angle and major and minor axes of the plane ellipse are used for affine transformation, and the ellipse is mapped to a 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.

2. The phase signal demodulation method according to claim 1, characterized in that: The specific steps are as follows: Step 1: Obtain three output signals from the 3×3 fiber coupler, and map the three output signals into a point set in a three-dimensional coordinate system; Step 2: Use the least square method to fit the general ellipsoid equation in the 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 square method to fit the general equation of the plane where the ellipse is located in the three-dimensional space; Step 4: Reduce the dimension of the three-dimensional ellipse; use the projection method to reduce the dimension, that is, substitute the general plane equation obtained by fitting into the ellipsoid equation, realize elimination, and obtain the plane ellipse equation in the two-dimensional coordinate system; or use the rotation method to reduce the dimension, that is, rotate the three-dimensional space ellipse to the coordinate plane or a plane parallel to the coordinate plane; Step 5: According to the plane ellipse equation expression, 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 angle, 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, characterized in that: 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 ), and we can 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, characterized in that: In step 3, the general equation of the plane where the ellipse in the three-dimensional space is located, which is 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 the 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 represented by x and y parameters, where: The rotation method is used to reduce the dimension. For the dimension 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 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: 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 deflection angle of the ellipse is:

7. The phase signal demodulation method according to claim 6, characterized in that: In step 6, the ellipse affine and demodulation phase, the specific process is as follows; First, the ellipse is mapped to a standard ellipse distribution through translation and rotation, with the major axis on the X-axis and the minor axis on the Y-axis. The corresponding data points are mapped as follows: 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 points are mapped as follows: 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 Wherein, unwrap(·) represents the unwrapping algorithm, where k is the count of phase unwrapping; if the phase difference between two consecutive calculation results of the inverse trigonometric function changes by more than π, the count k is expressed as k=k-1; when the phase difference change value is less than -π, it is expressed as k=k+1, and the value of k is calculated starting from 0.

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