CN113536567A - Method for multi-target vector fitting - Google Patents

Method for multi-target vector fitting Download PDF

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CN113536567A
CN113536567A CN202110797338.4A CN202110797338A CN113536567A CN 113536567 A CN113536567 A CN 113536567A CN 202110797338 A CN202110797338 A CN 202110797338A CN 113536567 A CN113536567 A CN 113536567A
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吴大可
周振亚
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Beijing Empyrean Technology Co Ltd
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    • G06F—ELECTRIC DIGITAL DATA PROCESSING
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Abstract

A method for multi-objective vector fitting, comprising the steps of: 1) processing a given data set to construct a single-target fitting data set; 2) determining an initial pole, and establishing a basic matrix by using the initial pole and a frequency sampling point; 3) separating the real part and the imaginary part of the basic matrix to construct a real basic matrix; 4) simplifying the real number basic matrix to obtain the first M rows of the complete Q matrix and recording the first M rows as QsMatrix, and pair QsStoring the matrix; 5) constructing and solving a pole coefficient solving equation to obtain a pole coefficient; 6) constructing a pole by using the pole coefficient and the initial pole to solve a characteristic equation, and calculating to obtain a new pole; 7) and calculating the residue number of each matrix element corresponding to the new pole to complete vector fitting. The method for multi-target vector fitting greatly reduces the multi-target vector fittingThe calculation complexity improves the calculation efficiency.

Description

Method for multi-target vector fitting
Technical Field
The invention relates to the technical field of passive device modeling, in particular to a method for multi-target vector fitting.
Background
Vector fitting is widely used in the modeling process of passive devices. Particularly, in modeling of a transmission line, people adopt a W-element or S-parameter method to give a physical description of the transmission line, and after a signal passes through the transmission line, in order to correctly calculate an output signal, vector fitting is generally required to obtain an impulse response of the transmission line in a frequency domain.
In implementing the vector fitting process, the user needs to provide data of the spectrum at discrete points, and the user provides a data set (ω)i,Yi) I is 1, …, N, where ω isiIs the ith frequency point, YiIs omegaiThe value of the corresponding physical quantity, which may be an S parameter, a Y parameter, etc. In the single port case, YiIs a number; while in multiport, YiIs a matrix. And YiThe fitting of a number of single target vectors is different, and the vector fitting at the moment is the fitting of a multi-target vector.
When Y isiWhen it is a matrix, two methods are usually used for vector fitting. The first method is a global fitting method, i.e. YiHave the same poles but the residue may be different. The second method is a local fitting method, namely YiThe residue numbers and poles of each matrix element are different, and the poles and the residue numbers are obtained through multiple times of single target vector fitting calculation.
Suppose the number of system ports is NpFor the local fitting method, it is equivalent to
Figure BDA0003163317720000011
Vector fitting, i.e. solving, of a single port at a time
Figure BDA0003163317720000012
Second 2N (2M +1) linear equation, where M is the number of vector fitting poles. It is more accurate than global fitting, but the computational cost in transient simulation is higher. For the global fitting method, due to the adoption of the common pole, the method has the advantage of low calculation cost in subsequent transient simulation. However, since all matrix elements use the same poles, the fitting accuracy is not as good as that of the local fitting method, but in many cases, the accuracy can meet the requirement of the problem. Furthermore, in the conventional method, the calculation of the global fitting method requires solving
Figure BDA0003163317720000021
A linear system of equations of a dense matrix with a time complexity of
Figure BDA0003163317720000022
Figure BDA0003163317720000023
For the local fitting method, the time complexity is
Figure BDA0003163317720000024
The ratio of the two is about
Figure BDA0003163317720000025
Figure BDA0003163317720000026
If M is>>1，
Figure BDA0003163317720000027
The ratio of the two can be simplified to
Figure BDA0003163317720000028
Obviously, when N ispWhen the size is larger, the calculation time of the global fitting method is greatly increased.
Therefore, a new vector fitting method is needed to greatly reduce the complexity of calculation and improve the calculation efficiency on the premise of ensuring the calculation accuracy.
Disclosure of Invention
In order to solve the defects of the prior art, the invention aims to provide a new method for vector fitting of a multi-port transmission line by adopting a global fitting method pole, greatly reduces the complexity of calculation and improves the calculation efficiency on the premise of ensuring the same precision as that of the traditional method.
In order to achieve the above object, the present invention provides a method for multi-target vector fitting, comprising the following steps:
1) processing a given data set to construct a single-target fitting data set;
2) determining an initial pole, and establishing a basic matrix by using the initial pole and a frequency sampling point;
3) separating the real part and the imaginary part of the basic matrix to construct a real basic matrix;
4) simplifying the real number basic matrix to obtain the first M rows of the complete Q matrix and recording the first M rows as Q sMatrix, and pair QsStoring the matrix;
5) constructing and solving a pole coefficient solving equation to obtain a pole coefficient;
6) constructing a pole by using the pole coefficient and the initial pole to solve a characteristic equation, and calculating to obtain a new pole;
7) and calculating the residue number of each matrix element corresponding to the new pole to complete vector fitting.
Further, the step 1) further comprises the step of assigning a given data set as (ω)i,Yi) I is 1, …, N, at each frequency point ωiExtracting YiElement of (2) matrix, composition
Figure BDA0003163317720000029
A single target fitting data set, represented as:
(ωi,Yi(m,n))，i＝1,…,N
m＝1,…,Np，n＝1,…,Np
wherein, YiIs Np×NpMatrix of physical quantities of dimension NpEqual to the number of ports of the passive system, N being the number of elements in the data set, Yi(m, n) represents YiThe m-th row and the n-th column of the matrix.
Further, step 2) the fundamental matrix is represented as a:
Figure BDA0003163317720000031
wherein the basic matrix A is an N (M +1) matrix; si＝jωi；i＝1,…,N；{alL is 1, …, M is the initial pole; m is the number of poles.
Further, the real fundamental matrix is represented as matrix B:
Figure BDA0003163317720000032
wherein, the real number basic matrix B is a 2N (M +1) matrix, and Re (A) represents a matrix formed by taking a real part of each matrix element of the A matrix; im (A) represents a matrix formed by taking the imaginary part of each matrix element of the matrix A.
Further, the step 6) further comprises fitting the data set to each single target vector { (ω) } i,Yi(m,n))，i＝1,…,N}，m＝1,…,Np,n＝1,…,NpTo proceed with
Figure BDA0003163317720000033
Secondary loop, solving for RkMatrix sum bkThe vector of the vector is then calculated,
Figure BDA0003163317720000034
by using
Figure BDA0003163317720000035
R iskMatrix sum
Figure BDA0003163317720000036
A bkThe vector constructs the pole coefficient solution equation expressed as
Figure BDA0003163317720000037
Wherein, { clAnd l is 1, …, and M is the pole coefficient obtained.
Further, solve for RkMatrix sum bkThe vector step further comprises the following steps:
a) establishment of DmnMatrix, said DmnThe matrix is represented as:
Figure BDA0003163317720000038
b) will DmnAre separated into real and imaginary partsMaking a matrix F of real numbersmnSaid FmnThe matrix is represented as:
Figure BDA0003163317720000039
c) computing a G matrix, the G matrix being represented as:
G＝Fmn-Qs(Qs T Fmn)；
d) carrying out simplified QR decomposition on the G matrix to obtain QmnAnd Rmn；
e) Using { Yi(m, N), i ═ 1, …, N } construction vector fmnExpressed as:
Figure BDA0003163317720000041
f) calculation of bmnSaid b ismnThe calculation formula is as follows:
bmn＝Qmn Tfmn
g) according to k being equal to Npm + n, RmnAnd bmnIs modified to RkAnd bk。
Further, the step 6) further comprises the step of basing on the pole coefficient { c }l1, …, M and a first pole alAnd l is 1, …, M, constructing a pole and solving a characteristic equation, calculating a characteristic value by calculating the characteristic equation, and solving to obtain a new pole { a }l’,l＝1,…,M}。
Further, the method also comprises the step of determining a new pole { a }lAfter', l is 1, …, M }, it is necessary to determine whether convergence occurs:
if the convergence condition is not satisfied, let al＝al' l is 1, …, M, setting the new pole obtained by calculation as the initial pole, and repeating the steps 2) to 6) until the convergence condition is satisfied;
Go to step 7) if the convergence condition is satisfied.
In order to achieve the above object, the present invention further provides an apparatus for multi-target vector fitting, comprising a memory and a processor, wherein the memory stores a program running on the processor, and the processor executes the program to perform the steps of the method for multi-target vector fitting.
To achieve the above object, the present invention also provides a computer readable storage medium having stored thereon computer instructions which, when executed, perform the steps of the above method for multi-target vector fitting.
Has the advantages that: the invention is to be conventional
Figure BDA0003163317720000051
Is solved and converted into a 2N x (M +1) reduced QR decomposition, plus
Figure BDA0003163317720000052
A simplified QR decomposition of 2 nxm dimensional matrices, plus a solution to a NM × M system of linear equations. The computational complexity of which can be calculated as
Figure BDA0003163317720000053
Figure BDA0003163317720000054
In general N>>M，
Figure BDA0003163317720000055
And M>>1. The ratio of the time complexity of the old method to the new method is about
Figure BDA0003163317720000056
Obviously, when N ispWhen the calculation time is longer, the calculation efficiency of the calculation method provided by the invention is greatly improved, and the calculation time is lower than that of a local fitting method.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.
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The accompanying drawings are included to provide a further understanding of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the principles of the invention and not to limit the invention. In the drawings:
FIG. 1 is a flow chart of a method for multi-target vector fitting of the present invention.
Detailed Description
The preferred embodiments of the present invention will be described in conjunction with the accompanying drawings, and it will be understood that they are described herein for the purpose of illustration and explanation and not limitation.
Fig. 1 is a flowchart of a method for multi-target vector fitting according to the present invention, and the method for multi-target vector fitting of the present invention will be described in detail with reference to fig. 1.
At step 11, a given data set is processed to construct a single target fitting data set.
First, a data set is provided by a user, and the data set is denoted as (ω)i,Yi) I is 1, …, N. Wherein, YiIs Np×NpMatrix of physical quantities of dimension NpEqual to the number of ports of the passive system, and N is the number of frequency ω samples and physical quantity matrix Y in the data set. In some passive systems, NpAnd can be tens or even hundreds.
At each frequency point omega iExtracting YiElement of (2) matrix, composition
Figure BDA0003163317720000057
A single target fitting dataset, represented as follows:
(ωi,Yi(m,n))，i＝1,…,N (1)
m＝1,…,Np，n＝1,…,Np
in this step, it is equivalent to fitting each matrix element of Y at different frequencies individually as a single target vector to the data set. Since the matrix element of Y is
Figure BDA0003163317720000058
An, standFitting the data set with the constructed single target vectors is also
Figure BDA0003163317720000059
And (4) respectively. Here, Y is a general term for a physical quantity, and may be an S parameter, a Y parameter or other physical quantities.
At step 22, an initial pole is determined.
In this step, the initial pole is determined using conventional methods, as follows:
{al,l＝1,…,M} (2)
where M is the number of poles. The poles may be real or complex, and if complex, appear as complex conjugates. To maintain the stability and causality of the system, the real parts of these poles are negative numbers.
At step 33, a fundamental matrix is established using the initial poles and frequency sampling points.
In the step, an initial pole and a frequency sampling point are utilized to establish a basic matrix, the basic matrix is recorded as an A matrix, and the A matrix is expressed as follows:
Figure BDA0003163317720000061
wherein the matrix A is an N × (M +1) matrix, si＝jωi，i＝1,…,N。
It should be noted that s is due toiAnd alIt is possible that the matrix a is a complex matrix.
At step 44, the real and imaginary parts of the base matrix are separated to construct a real base matrix.
In this step, the real and imaginary parts of the basic matrix a established in step 33 are separated, and a real basic matrix is constructed and expressed as matrix B, which is expressed as follows:
Figure BDA0003163317720000062
where matrix B is a 2N (M +1) matrix and Re (-) represents a matrix with real parts for each element of the argument. Like Im (-) represents a matrix with imaginary parts for each element of the argument. The matrix B is equivalent to combining the real part and the imaginary part of the complex matrix A separately and vertically into B, and the number of rows of the matrix B is twice that of the matrix A.
In step 55, the real number basic matrix is subjected to simplified QR decomposition to obtain the first M columns of the complete Q matrix.
In this step, a real number basic matrix is subjected to a simplified QR decomposition, i.e., an orthogonal triangular decomposition, to obtain the first M columns of a Q matrix, which is denoted as QsMatrix, and pair QsPerforming storage, wherein QsIs a 2N × (M +1) matrix.
At step 66, a pole coefficient solution equation is constructed.
In this step, it is necessary to first fit the data set to each single target vector { (ω) }i,Yi(m,n))，i＝1,…,N}，m＝1,…,Np，n＝1,…,NpSolving for RkMatrices and bk vectors. Wherein R iskIs an M × M matrix, and bkFor an Mx 1 vector, k ═ NpM + n. Since m and N each have NpEach value of k has
Figure BDA0003163317720000071
That is to say need to be solved for
Figure BDA0003163317720000072
R is kMatrix sum
Figure BDA0003163317720000073
A bkAnd (4) vectors. The purpose of setting k is to convert m and n two-dimensional variables into one dimension, so that a matrix can be conveniently constructed subsequently. In this step, it is necessary to perform
Figure BDA0003163317720000074
And (5) performing secondary circulation. In particular, this step in turn comprises the following sub-steps:
in step a), using the matrices A and Yi(m, N), i ═ 1, …, N } establishmentDmnAnd (4) matrix. DmnIs an N × M matrix, represented as follows:
Figure BDA0003163317720000075
in step b), D) ismnIs separated from the imaginary part to construct a real matrix Fmn。FmnIs a 2N × M matrix, represented as follows
Figure BDA0003163317720000076
In step c), a G matrix is calculated, the expression of which is as follows:
G＝Fmn-Qs(Qs T Fmn) (7)
in step d), carrying out simplified QR decomposition on the G matrix to obtain QmnAnd Rmn。
In step e), using { Yi(m, N), i ═ 1, …, N } construction vector fmn。
Figure BDA0003163317720000077
In step f), b) is calculatedmn，bmnThe calculation formula is as follows:
bmn＝Qmn Tfmn (9)
in step g), according to the definition of k, i.e. k ═ Npm + n, RmnAnd bmnIs modified to RkAnd bkSolving to obtain RkAnd bk。
In step p), using
Figure BDA0003163317720000078
R iskMatrix sum
Figure BDA0003163317720000079
A bkAnd constructing a pole coefficient solving equation by using the vector.
Figure BDA0003163317720000081
This step is equivalent to the step of
Figure BDA0003163317720000082
R iskMatrix sum bkThe vectors are vertically stacked to form a new matrix.
In step 77, the pole coefficient solving equation is solved to obtain the pole coefficient.
In this step, the pole coefficient solving equation constructed in step 66) is solved to obtain the pole coefficient { c } l,l＝1,…,M}。
In step 88, the poles are constructed to solve the characteristic equation, and new poles are calculated.
In this step, based on { c }l1, …, M and old pole alAnd l is 1, …, M, constructing a pole by using a traditional method, solving a characteristic equation, calculating a characteristic value by calculating the characteristic equation, and solving a new pole { a }l', l-1, …, M }. The specific construction method is the same as the traditional vector fitting method.
After determining the new pole, judging whether to converge, if not, letting al＝al', l-1, …, M. The iteration continues back to step 33, and steps 33 to 88 are repeated until the convergence condition is satisfied. If the convergence condition is satisfied, the process proceeds to the next step, step 99.
In step 99, the residue of each matrix element is calculated to complete the vector fitting.
Once the poles are determined in step 88, the data set is fitted based on the single target vector { (ω)i,Yi(m,n))，i＝1,…,N}，m＝1,…,Np，n＝1,…,NpAnd the residue of each matrix element can be calculated to complete vector fitting.
The invention also provides a device for multi-target vector fitting, which comprises a memory and a processor, wherein the memory is stored with a program running on the processor, and the processor executes the steps of the method for multi-target vector fitting when running the program.
The present invention further provides a computer-readable storage medium, on which computer instructions are stored, and when the computer instructions are executed, the steps of the method for multi-target vector fitting are performed, and the method for multi-target vector fitting is described in the foregoing section and is not described in detail again.
Those of ordinary skill in the art will understand that: although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that changes may be made in the embodiments and/or equivalents thereof without departing from the spirit and scope of the invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (10)

1. A method for multi-objective vector fitting, comprising the steps of:
1) processing a given data set to construct a single-target fitting data set;
2) determining an initial pole, and establishing a basic matrix by using the initial pole and a frequency sampling point;
3) separating the real part and the imaginary part of the basic matrix to construct a real basic matrix;
4) simplifying the real number basic matrix to obtain the first M rows of the complete Q matrix and recording the first M rows as QsMatrix, and pair QsStoring the matrix;
5) constructing and solving a pole coefficient solving equation to obtain a pole coefficient;
6) constructing a pole by using the pole coefficient and the initial pole to solve a characteristic equation, and calculating to obtain a new pole;
7) And calculating the residue number of each matrix element corresponding to the new pole to complete vector fitting.
2. Method for multi-target vector fitting according to claim 1, wherein step 1) further comprises the notation of a given dataset as (ω)i,Yi) I is 1, …, N, at each frequency point ωiExtracting YiElement of (2) forming Np 2A single target fitting data set, represented as:
(ωi,Yi(m,n))，i＝1,…,N
m＝1,…,Np，n＝1,…,Np
wherein, YiIs Np×NpMatrix of physical quantities of dimension NpEqual to the number of ports of the passive system, N being the number of elements in the data set, Yi(m, n) represents YiThe m-th row and the n-th column of the matrix.
3. Method for multi-objective vector fitting according to claim 1, characterized in that step 2) the basis matrix is represented by a:
Figure FDA0003163317710000011
wherein the basic matrix A is an N (M +1) matrix; si＝jωi；i＝1,…,N；{alL is 1, …, M is the initial pole; m is the number of poles.
4. Method for multi-objective vector fitting according to claim 1, characterized in that the real basis matrix is represented as matrix B:
Figure FDA0003163317710000021
wherein, the real number basic matrix B is a 2N (M +1) matrix, and Re (A) represents a matrix formed by taking a real part of each matrix element of the A matrix; im (A) represents a matrix formed by taking the imaginary part of each matrix element of the matrix A.
5. The method for multi-target vector fitting of claim 1, wherein step 6) further comprises fitting a dataset { (ω) to each single target vectori,Yi(m,n))，i＝1,…,N}，m＝1,…,Np,n＝1,…,NpCarry out Np 2Secondary loop, solving for RkMatrix sum bkVector, k 1, …, Np 2. By using Np 2R iskMatrix sum Np 2A bkThe vector constructs the pole coefficient solution equation expressed as
Figure FDA0003163317710000022
Wherein, { clAnd l is 1, …, and M is the pole coefficient obtained.
6. The method for multi-objective vector fitting of claim 5, wherein solving for RkMatrix sum bkThe vector step further comprises the following steps:
a) establishment of DmnMatrix, said DmnThe matrix is represented as:
Figure FDA0003163317710000023
b) will DmnIs separated from the imaginary part to construct a real matrix FmnSaid FmnThe matrix is represented as:
Figure FDA0003163317710000024
c) computing a G matrix, the G matrix being represented as:
G＝Fmn-Qs(Qs T Fmn)
d) carrying out simplified QR decomposition on the G matrix to obtain QmnAnd Rmn；
e) Using { Yi(m, N), i ═ 1, …, N } construction vector fmnExpressed as:
Figure FDA0003163317710000031
f) calculation of bmnSaid b ismnThe calculation formula is as follows:
bmn＝Qmn Tfmn
g) according to k being equal to Npm + n, RmnAnd bmnIs modified to RkAnd bk。
7. The method for multi-target vector fitting according to claim 1, wherein step 6) further comprises basing the pole coefficients { c } onl1, …, M and an initial pole a lAnd l is 1, …, M, constructing a pole and solving a characteristic equation, calculating a characteristic value by calculating the characteristic equation, and solving to obtain a new pole { a }l’,l＝1,…,M}。
8. The method for multi-target vector fitting according to claim 7, further comprising determining a new pole { a }lAfter', l is 1, …, M }, it is necessary to determine whether convergence occurs:
if the convergence condition is not satisfied, let al＝al' l is 1, …, M, setting the new pole obtained by calculation as the initial pole, and repeating the steps 2) to 6) until the convergence condition is satisfied;
go to step 7) if the convergence condition is satisfied.
9. An apparatus for multi-target vector fitting, comprising a memory and a processor, the memory having stored thereon a program for execution on the processor, the processor when executing the program performing the steps of the method for multi-target vector fitting of any of claims 1-8.
10. A computer readable storage medium having stored thereon computer instructions, which when executed perform the steps of the method for multi-target vector fitting of any of claims 1-8.
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CN116226728A (en) * 2023-05-09 2023-06-06 中国海洋大学 Floating structure frequency response function identification method based on single regular wave excitation
CN116226728B (en) * 2023-05-09 2023-08-01 中国海洋大学 Floating structure frequency response function identification method based on single regular wave excitation

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