A method for multi-target vector fitting

Abstract

A method for multi-objective vector fitting includes the following steps: 1) processing a given dataset to construct a single-objective fitting dataset; 2) determining initial poles and using the initial poles and frequency sampling points to establish a fundamental matrix; 3) separating the real and imaginary parts of the fundamental matrix to construct a real fundamental matrix; 4) performing a simplified QR decomposition on the real fundamental matrix to obtain the first M columns of the complete Q matrix, denoted as Q. s Matrix, and for Q s The matrix is ​​stored; 5) Construct and solve the pole coefficient equation to obtain the pole coefficients; 6) Construct the pole characteristic equation using the pole coefficients and the initial poles to calculate the new poles; 7) Calculate the residues of each matrix element corresponding to the new poles to complete the vector fitting. This invention provides a method for multi-objective vector fitting that significantly reduces the computational complexity and improves computational efficiency.

Description

Technical Field

[0001] This invention relates to the technical field of passive device modeling, and in particular to a method for multi-objective vector fitting. Background Technology

[0002] Vector fitting is widely used in the modeling of passive devices. Especially in the modeling of transmission lines, the W-element or S-parameter method is used to give the physical description of the transmission line. After the signal passes through the transmission line, in order to correctly calculate the output signal, vector fitting is usually required to obtain the impulse response of the transmission line in the frequency domain.

[0003] In the process of implementing vector fitting, the user needs to provide data on the spectrum at discrete points. The user provides the dataset (ω). i ,Y i ), i = 1, ..., N, where ω i It is the i-th frequency point, Y i It is ω i The corresponding physical quantity value, which can be an S-parameter, a Y-parameter, etc. In the single-port case, Y... i It is a single number; however, in multi-port scenarios, Y i It is a matrix. And Y i Unlike single-objective vector fitting, which involves a single number of targets, this vector fitting is now multi-objective vector fitting.

[0004] When Y i When Y is a matrix, two methods are typically used for vector fitting. The first method is global fitting, which is Y... i Each matrix element has the same poles, but the residues may differ. The second method is local fitting, i.e., Y... i The residues and poles of each matrix element are different, and these poles and residues are calculated through multiple single-objective vector fitting operations.

[0005] Assume the system has N ports. p For the local fitting method, it is equivalent to performing... Vector fitting of the second single port, that is, solving The equation is a 2N×(2M+1) linear equation, where M is the number of poles in the vector fitting. It offers higher accuracy than global fitting, but is computationally more expensive in transient simulations. Global fitting methods, due to the use of common poles, have the advantage of lower computational cost in subsequent transient simulations. However, because all matrix elements use the same poles, the fitting accuracy is lower than local fitting methods; however, in many problems, its accuracy is sufficient. Furthermore, in traditional methods, global fitting requires solving... The time complexity of a system of linear equations for a dense matrix is ​​O(n log n). For the local fitting method, the time complexity is O(n log n). The ratio of the two is approximately If M >> 1, The ratio of the two can be simplified to Obviously, when N p When the value is large, the computation time of the global fitting method increases significantly.

[0006] Therefore, there is an urgent need for a new vector fitting method that can significantly reduce computational complexity and improve computational efficiency while ensuring computational accuracy. Summary of the Invention

[0007] To address the shortcomings of existing technologies, the present invention aims to provide a new method for vector fitting of poles using a global fitting method for multi-port transmission lines. This method significantly reduces computational complexity and improves computational efficiency while maintaining the same accuracy as traditional methods.

[0008] To achieve the above objectives, the present invention provides a method for multi-objective vector fitting, comprising the following steps:

[0009] 1) Process the given dataset to construct a single-objective fitting dataset;

[0010] 2) Determine the initial poles, and use the initial poles and frequency sampling points to establish the basic matrix;

[0011] 3) Separate the real and imaginary parts of the fundamental matrix to construct a real fundamental matrix;

[0012] 4) Perform a simplified QR decomposition on the real fundamental matrix to obtain the first M columns of the complete Q matrix, denoted as Q. s Matrix, and for Q s Stored in matrices;

[0013] 5) Construct and solve the equations for finding the pole coefficients to obtain the pole coefficients;

[0014] 6) Construct poles using pole coefficients and initial poles, solve the characteristic equation, and calculate the new poles;

[0015] 7) Calculate the residues of each matrix element corresponding to the new pole and complete the vector fitting.

[0016] Furthermore, step 1) also includes, given a dataset denoted as (ω) i ,Y i ), i = 1, ..., N, at each frequency ω i Extract Y i The matrix elements constitute A single-objective fitted dataset, represented as:

[0017] (ω i ,Y i (m,n)), i=1,…,N

[0018] m = 1, ..., N p n = 1, ..., N p

[0019] Among them, Y i It is N p ×N p The physical quantity matrix has a dimension N. p Equals the number of ports in a passive system, where N is the number of elements in the dataset, and Y... i (m,n) represents Y i The matrix element in the m-th row and n-th column of the matrix.

[0020] Further, the basic matrix described in step 2) is represented as A:

[0021]

[0022] Wherein, the fundamental matrix A is an N×(M+1) matrix; s i =jω i ; i = 1, ..., N; {a l Let {l = 1, ..., M} be the initial poles; M is the number of poles.

[0023] Furthermore, the real fundamental matrix is ​​represented as matrix B:

[0024]

[0025] Wherein, the real fundamental matrix B is a 2N×(M+1) matrix, Re(A) represents the matrix formed by taking the real part of each element of matrix A; Im(A) represents the matrix formed by taking the imaginary part of each element of matrix A.

[0026] Furthermore, step 6) further includes fitting the dataset {(ω} to each single-target vector. i ,Y i (m,n)), i=1,…,N}, m=1,…,N p n=1,…,N p ,conduct In the next iteration, solve for R. k Matrix and b k Vector use R k Matrix and b k Vector construction of the pole coefficient solution equation is expressed as follows: Among them, {cl ,l=1,…,M} are the obtained pole coefficients.

[0027] Furthermore, solve R k Matrix and b k The vector step also includes the following steps:

[0028] a) Establish D mn Matrix, the D mn The matrix is ​​represented as:

[0029]

[0030] b) D mn Separate the real and imaginary parts to construct a real matrix F. mn The F mn The matrix is ​​represented as:

[0031]

[0032] c) Calculate the G matrix, which is represented as:

[0033] G = F mn -Q s (Q s T F mn );

[0034] d) Perform a simplified QR decomposition on the G matrix to obtain Q mn and R mn ;

[0035] e)Use {Y i Construct a vector f from the group (m,n), i=1,…,N}. mn , is represented as:

[0036]

[0037] f) Calculate b mn The b mn The calculation formula is:

[0038] b mn =Q mn T f mn

[0039] g) According to k = N p m+n, R mn and b mn Rewrite in R k and b k .

[0040] Furthermore, step 6) also includes, based on the pole coefficients {cl {a, l = 1, ..., M} and the first pole {a} l Construct poles {a, l = 1, ..., M}, solve the characteristic equation, calculate the eigenvalues ​​by calculating the characteristic equation, and obtain the new poles {a}. l ',l=1,…,M}.

[0041] Furthermore, it also includes determining the new pole {a} l After ',l=1,…,M}, it is necessary to determine whether convergence has occurred:

[0042] If the convergence condition is not met, let a l =a l ',l=1,…,M, set the calculated new poles as the initial poles, and repeat steps 2) to 6) until the convergence condition is met;

[0043] If the convergence condition is met, proceed to step 7.

[0044] To achieve the above objectives, the present invention also provides an apparatus for multi-objective vector fitting, comprising a memory and a processor, wherein the memory stores a program that runs on the processor, and the processor executes the steps of the above-described method for multi-objective vector fitting when running the program.

[0045] To achieve the above objectives, the present invention also provides a computer-readable storage medium having computer instructions stored thereon, which, when executed, perform the steps of the above-described method for multi-objective vector fitting.

[0046] Beneficial effects: This invention will transform the traditional Solving the linear equation transforms it into a simplified QR decomposition of 2N×(M+1), plus... The computational complexity involves the simplified QR decomposition of a 2N×M matrix, plus solving a system of NM×M linear equations. The time complexity can be calculated as: Usually N >> M, And M>>1. The ratio of the time complexity of the old method to the new method is approximately Obviously, when N p When the value is large, the computational efficiency of the calculation method provided by this invention will be greatly improved, and the computation time will be lower than that of the local fitting method.

[0047] Other features and advantages of the invention will be set forth in the description which follows, and will be apparent in part from the description, or may be learned by practicing the invention. Attached Figure Description

[0048] The accompanying drawings are provided to further illustrate the invention and form part of the specification. They are used together with embodiments of the invention to explain the invention and do not constitute a limitation thereof. In the drawings:

[0049] Figure 1 This is a flowchart of the method for multi-objective vector fitting according to the present invention. Detailed Implementation

[0050] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0051] Figure 1 The following is a flowchart of the method for multi-objective vector fitting according to the present invention, with reference to... Figure 1 The present invention provides a detailed description of the method for multi-objective vector fitting.

[0052] In step 11, the given dataset is processed to construct a single-objective fitting dataset.

[0053] First, the user provides the dataset, denoted as (ω). i ,Y i ),i=1,...,N. Among them, Y i It is N p ×N p The physical quantity matrix has a dimension N. p This is equal to the number of ports in a passive system, where N is the number of frequency samples ω and the number of physical quantity matrices Y in the dataset. In some passive systems, N... p It can be in the tens or even hundreds.

[0054] At each frequency point ω i Extract Y i The matrix elements constitute A single-objective fitted dataset is represented as follows:

[0055] (ω i ,Y i (m,n)), i=1,…,N (1)

[0056] m = 1, ..., N p n = 1, ..., N p

[0057] In this step, it's equivalent to treating each matrix element of Y at different frequencies as a separate vector fitting dataset for a single target. Since the matrix elements of Y have... Therefore, the single-objective vector fitting dataset is also [number], Here, Y is a general term for a physical quantity; it can be an S-parameter, a Y-parameter, or other physical quantities.

[0058] In step 22, the initial poles are determined.

[0059] In this step, the initial poles are determined using a traditional method, as shown below:

[0060] {a l ,l=1,…,M} (2)

[0061] Here, M is the number of poles. Poles can be real or complex numbers; if complex, they appear as complex conjugates. To maintain the stability and causality of the system, the real parts of these poles are negative.

[0062] In step 33, the basic matrix is ​​established using the initial poles and frequency sampling points.

[0063] In this step, the fundamental matrix is ​​established using the initial poles and frequency sampling points. The fundamental matrix is ​​denoted as matrix A, and matrix A is represented as follows:

[0064]

[0065] Where matrix A is an N×(M+1) matrix, s i =jω i , i = 1, ..., N.

[0066] It should be noted that, due to s i and a l All of them could be complex numbers, and matrix A is a complex matrix.

[0067] In step 44, the real and imaginary parts of the fundamental matrix are separated to construct the real fundamental matrix.

[0068] In this step, the real and imaginary parts of the fundamental matrix A established in step 33 are separated to construct a real fundamental matrix, which is represented as matrix B. Matrix B is represented as follows:

[0069]

[0070] Here, matrix B is a 2N×(M+1) matrix, and Re(·) represents the matrix formed by taking the real part of each matrix element of the independent variable. Similarly, Im(·) represents the matrix formed by taking the imaginary part of each matrix element of the independent variable. Matrix B is equivalent to separating the real and imaginary parts of the complex matrix A and then combining them vertically to form B. The number of rows in matrix B is twice that of matrix A.

[0071] In step 55, the real fundamental matrix is ​​reduced to QR decomposition to obtain the first M columns of the complete Q matrix.

[0072] In this step, the fundamental real matrix is ​​subjected to simplified QR decomposition, i.e., orthogonal triangular decomposition, to obtain the first M columns of the Q matrix, denoted as Q. s Matrix, and for Q s Store, where Q s It is a 2N×(M+1) matrix.

[0073] In step 66, the pole coefficients are constructed and the equations are solved.

[0074] In this step, it is necessary to first fit the dataset {(ω) to each single-target vector. i ,Y i (m,n)), i=1,…,N}, m=1,…,N p n = 1, ..., N p Solve for R k Matrix and bk vector. Where R... k It is an M×M matrix, and b k For an M×1 vector, k=N p *m+n. Since m and n each have N p k can take several values. One, meaning it needs to be solved. R k Matrix and b k Vector. The purpose of setting k is to convert the two-dimensional variables m and n into one-dimensional vectors, facilitating subsequent matrix construction. This step requires... This is the second iteration. Specifically, this step includes the following sub-steps:

[0075] In step a), using matrix A and {Y} i Establish D (m,n),i=1,…,N} to construct D mn Matrix. D mn It is an N×M matrix, represented as follows:

[0076]

[0077] In step b), D mn Separate the real and imaginary parts to construct a real matrix F. mn F mn It is a 2N×M matrix, represented as follows:

[0078]

[0079] In step c), the G matrix is ​​calculated, and its expression is as follows:

[0080] G = F mn -Q s (Q s T Fmn (7)

[0081] In step d), the G matrix is ​​reduced to QR decomposition to obtain Q. mn and R mn .

[0082] In step e), using {Y i Construct a vector f from the group (m,n), i=1,…,N}. mn .

[0083]

[0084] In step f), calculate b mn b mn The calculation formula is as follows:

[0085] b mn =Q mn T f mn (9)

[0086] In step g), according to the definition of k, i.e., k = N p m+n, R mn and b mn Rewrite in R k and b k Solving for R yields... k and b k .

[0087] In step p), using R k Matrix and b k Vector construction of pole coefficients to solve equations.

[0088]

[0089] This step is equivalent to... R k Matrix and b k Vectors are stacked vertically to form a new matrix.

[0090] In step 77, the pole coefficients are solved by solving the equation.

[0091] In this step, the pole coefficient equation constructed in step 66) is solved to obtain the pole coefficients {c}. l ,l=1,…,M}.

[0092] In step 88, the poles are constructed, the characteristic equation is solved, and the new poles are calculated.

[0093] In this step, based on {c l{a, l = 1, ..., M} and the old pole {a} l For the region {a, l=1,…,M}, the poles are constructed using traditional methods to solve the characteristic equation. By calculating the characteristic equation, the eigenvalues ​​are obtained, and the new poles {a} can be found. l ',l=1,…,M}. The specific construction method is the same as the traditional vector fitting method.

[0094] After determining the new poles, it is necessary to determine whether convergence has occurred. If the convergence condition is not met, let a... l =a l Let l = 1, ..., M. Return to step 33 and continue iterating, repeating steps 33 to 88 until the convergence condition is met. If the convergence condition is met, proceed to the next step, i.e., step 99.

[0095] In step 99, the residues of each matrix element are calculated to complete the vector fitting.

[0096] Once the poles are determined in step 88, the dataset {(ω} is fitted based on a single-objective vector. i ,Y i (m,n)), i=1,…,N}, m=1,…,N p n = 1, ..., N p Vector fitting can be completed by calculating the residues of each matrix element.

[0097] The present invention also provides an apparatus for multi-objective vector fitting, including a memory and a processor, wherein the memory stores a program that runs on the processor, and the processor executes the steps of the method for multi-objective vector fitting described above when running the program.

[0098] The present invention also provides a computer-readable storage medium storing computer instructions thereon, which, when executed, perform the steps of the above-described method for multi-objective vector fitting. The method for multi-objective vector fitting is described in the foregoing section and will not be repeated here.

[0099] It will be understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method for multi-objective vector fitting for modeling passive devices, characterized in that, Includes the following steps: 1) Process the given dataset to construct a single-objective fitting dataset; 2) Determine the initial poles, and use the initial poles and frequency sampling points to establish the fundamental matrix, where the initial poles are represented as { a l , l =1,…, M },pole a l The real part is negative; 3) Separate the real and imaginary parts of the fundamental matrix to construct a real fundamental matrix; 4) Perform simplified QR decomposition on the real fundamental matrix to obtain the first part of the complete Q matrix. M Column, denoted as Q s Matrix, and on Q s Stored in matrices; 5) Construct and solve the equations for finding the pole coefficients to obtain the pole coefficients; 6) Construct poles using pole coefficients and initial poles, solve the characteristic equation, and calculate the new poles; Step 6) further includes fitting the dataset {( ω i , Y i ( m , n )), i =1,…, N }, m =1,…, N p , n =1,…, N p ,conduct N p 2 The second iteration solves the problem. R k Matrix and b k Vector k =1,…, N p 2 ,use N p 2 indivual R k Matrix and N p 2 indivual b k Vector construction of the pole coefficient solution equation is expressed as follows: ,in,{ c l , l =1,…, M } represents the obtained pole coefficients. ω i It is the first i One frequency point, Y i ( m , n ) corresponds to the i physical quantity matrix Y i The m Line 1 n Column matrix elements, N p The number of ports in a passive system; Solve R k Matrix and b k The vector step also includes the following steps: a) Establish D mn Matrix, the D mn The matrix is ​​represented as: here s i = j ω i ,in ω i It is the i-th frequency point; i =1,…, N ;{ a l , l =1,…, M } represents the initial pole; b) D mn Construct a real matrix by separating the real and imaginary parts. F mn The F mn The matrix is ​​represented as: c) Calculation G Matrix, the G The matrix is ​​represented as: G = F mn - Q s ( Q s T F mn ) d) For G The matrix is ​​reduced to QR decomposition to obtain Q mn and R mn ; e) Using { Y i ( m , n ), i =1,…, N Construct vectors f mn , represented as: f) Calculation b mn The b mn The calculation formula is: b mn = Q mn T f mn g) According to k = N p m+n ,Bundle R mn and b mn Rewritten R k and b k ; 7) Calculate the residues of each matrix element corresponding to the new pole and complete the vector fitting.

2. The method for multi-objective vector fitting for passive device modeling according to claim 1, characterized in that, Step 1) also includes, given a dataset denoted as ( ω i , Y i ), i =1,…, N At each frequency point ω i ,extract Y i The matrix elements constitute N p 2 A single-objective fitted dataset, represented as: ( ω i , Y i ( m , n ))， i =1,…, N m =1,…, N p ， n =1,…, N p in, Y i yes N p × N p The dimension of the physical quantity matrix N p Equal to the number of ports in a passive system. N It is the number of elements in the dataset. Y i ( m , n )express Y i The first of the matrix m row and number n Columns of matrix elements.

3. The method for multi-objective vector fitting for passive device modeling according to claim 1, characterized in that, Step 2) The basic matrix is ​​represented as follows A : Among them, the fundamental matrix A yes N ×( M +1) matrix; s i = j ω i ,in ω i It is the i-th frequency point; i =1,…, N ;{ a l , l =1,…, M } represents the initial pole; M It is the number of poles.

4. The method for multi-objective vector fitting for passive device modeling according to claim 3, characterized in that, The real number fundamental matrix is ​​represented as a matrix B : Among them, the real fundamental matrix B It is 2 N ×( M+ 1) Matrix, here N It is the number of data points. M It is the number of poles; Re( A ) represents the A The matrix formed by taking the real parts of each matrix element; Im( A ) represents the matrix A The matrix is ​​formed by taking the imaginary part of each matrix element.

5. The method for multi-objective vector fitting for passive device modeling according to claim 1, characterized in that, Step 6) further includes, based on the pole coefficients { c l , l =1,…, M } and initial poles { a l , l =1,…, M }, construct poles, solve the characteristic equation, calculate the eigenvalues ​​by calculating the characteristic equation, and obtain the new poles { a l ', l =1, …, M } 6. The method for multi-objective vector fitting for passive device modeling according to claim 5, characterized in that, This also includes determining the new poles { a l ', l =1, …, M After that, it is necessary to determine whether convergence has occurred: If the convergence condition is not met, let a l = a l ' , l =1,…, M Set the calculated new pole as the initial pole, and repeat steps 2) to 6) until the convergence condition is met; If the convergence condition is met, proceed to step 7.

7. A device for multi-objective vector fitting for modeling passive devices, characterized in that, It includes a memory and a processor, wherein the memory stores a program that runs on the processor, and the processor, when running the program, performs the steps of the method for multi-objective vector fitting for modeling passive devices as described in any one of claims 1-6.

8. A computer-readable storage medium storing computer instructions thereon, characterized in that, When the computer instructions are executed, they perform the steps of the multi-objective vector fitting method for modeling passive devices as described in any one of claims 1-6.

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