CN113625033B - Waveform uncertainty evaluation method - Google Patents
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Abstract
The invention discloses a waveform uncertainty evaluation method, which solves the problems of large calculation amount and difficult engineering realization of the existing method. The waveform uncertainty assessment method comprises the following steps: calculating a frequency domain covariance matrix for the frequency domain calibration result obtained by each calibration; rearranging the blocks of the frequency domain covariance matrix to obtain a block matrix, wherein the block principle is that the real part covariance matrix of the frequency domain calibration result is one block, the imaginary part covariance matrix is one block, the covariance matrix with the front real part and the rear imaginary part is one block, the covariance matrix with the front imaginary part and the rear imaginary part is one block, and the block matrix is a real symmetrical positive definite matrix; and carrying out dimension reduction processing on the block matrix to obtain a frequency domain covariance dimension reduction matrix. The invention can realize the rapid evaluation of waveform uncertainty.
Description
Technical Field
The invention belongs to the technical field of waveform measurement, and particularly relates to a waveform uncertainty evaluation method.
Background
The waveform measurement is essentially a set of discrete data points, which can be viewed as a vector, whose uncertainty can be expressed in the form of a covariance matrix. The diagonal elements of the covariance matrix represent the variance of each sampling point of the waveform measurement result, the square root of the covariance matrix is the corresponding standard deviation, namely the uncertainty of each sampling point of the waveform measurement result, and the off-diagonal elements represent the covariance between the sampling points of the waveform measurement result, namely the cross correlation between the sampling points. In the existing waveform uncertainty evaluation method, if a waveform measurement result has N sampling points, a corresponding covariance matrix is NxN, and the operation amount and N are 2 In proportion, N of the general waveform measurement result is relatively large, for example, a pulse waveform, so that the operation amount of the covariance matrix is large, and the pulse waveform measurement result uncertainty evaluation technology based on the covariance matrix is difficult to apply in engineering.
Disclosure of Invention
The invention provides a waveform uncertainty evaluation method, which solves the problems of large calculation amount and difficult engineering realization of the existing method.
In order to solve the problems, the invention is realized as follows:
the embodiment of the invention provides a waveform uncertainty evaluation method, which comprises the following steps: calculating a frequency domain covariance matrix for the frequency domain calibration result obtained by each calibration; rearranging the frequency domain covariance matrix blocks to obtain a block matrix, wherein the block principle is that the real part covariance matrix of the frequency domain calibration result is one block, the imaginary part covariance matrix is one block, the covariance matrix with the front imaginary part behind the real part is one block, and the block matrix is a real symmetric positive definite matrix; and carrying out dimension reduction processing on the block matrix to obtain a frequency domain covariance dimension reduction matrix.
Further, the method further comprises: and selecting different numbers of characteristic values to perform dimension reduction processing on the block matrixes to obtain corresponding different dimension reduction matrixes, and selecting the dimension reduction matrix with the minimum uncertainty information tolerance as the frequency domain covariance dimension reduction matrix.
Further, when the dimension reduction processing is performed on the block matrix to obtain the frequency domain covariance dimension reduction matrix, the tolerance of uncertainty information included in the frequency domain covariance dimension reduction matrix meets a preset tolerance requirement.
Further, the method further comprises: and carrying out positive square root operation on the block matrix, and carrying out dimension reduction on the obtained new block matrix.
Further, the method further comprises: subtracting the product of a dimensionality reduction eigenvalue decomposition matrix and a transposition matrix of the dimensionality reduction eigenvalue decomposition matrix from the frequency domain covariance decomposition matrix to obtain a comparison matrix, wherein the dimensionality reduction eigenvalue decomposition matrix is the front p columns of the eigenvalue decomposition matrix of the block matrix, and p is the maximum eigenvalue quantity of dimensionality reduction processing; and evaluating the accuracy of the frequency domain covariance dimension reduction matrix according to the comparison matrix.
Preferably, the method further comprises: and calculating the dimensionality reduction ratio according to the storage capacity of the frequency domain covariance dimensionality reduction matrix and the storage capacity of the frequency domain covariance matrix.
Preferably, the eigenvalues and eigenvectors of the matrix are calculated by the IRAM algorithm.
Preferably, the eigenvalues and eigenvectors are computed on the matrix by the ARPACK software.
The beneficial effects of the invention include: the invention provides a waveform uncertainty evaluation method, which reduces the storage capacity of a frequency covariance matrix through matrix rearrangement and a matrix dimension reduction mode, thereby effectively reducing the operation amount of the covariance matrix, improving the operation efficiency and having strong engineering applicability.
Drawings
The accompanying drawings, which 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 invention and do not limit the invention. In the drawings:
FIG. 1 is a flowchart of an embodiment of a waveform uncertainty assessment method;
FIG. 2 is a flowchart of an embodiment of a waveform uncertainty assessment method including a comparison matrix calculation.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the technical solutions of the present invention will be clearly and completely described below with reference to the specific embodiments of the present invention and the accompanying drawings. It is to be understood that the described embodiments are merely exemplary of the invention, and not restrictive of the full scope of the invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
The waveform measurement is essentially a set of discrete data points, which can be viewed as a vector, whose uncertainty can be expressed in the form of a covariance matrix. The diagonal elements of the covariance matrix represent the variance of each sampling point of the waveform measurement result, the square root of the covariance matrix is the corresponding standard deviation, namely the uncertainty of each sampling point of the waveform measurement result, and the off-diagonal elements represent the covariance between the sampling points of the waveform measurement result, namely the cross correlation between the sampling points. The technology considers the cross correlation between the waveform sampling point and the sampling point, and simultaneously can provide the uncertainty of a complete waveform measurement result instead of the uncertainty of a single waveform parameter. Therefore, the uncertainty of the waveform can be directly transmitted according to an uncertainty source, and then the specific waveform parameter can be transmitted according to the uncertainty of the waveform, and the technology separates the extraction of the waveform parameter from the extraction of the uncertainty of the waveform parameter, so that more different types of waveforms can be more conveniently represented, a user can calculate any waveform parameter according to a calibrated standard waveform, and the uncertainty of any waveform parameter can be obtained.
The waveform uncertainty evaluation method solves the problems existing in the traditional waveform parameter uncertainty evaluation method, utilizes the covariance matrix transfer technology of the waveform, considers the influence of related errors in the waveform, has robustness and universality, and can more accurately calculate the uncertainty of the waveform, but the method has certain problems, if the waveform measurement result has N sampling points, the corresponding covariance matrix is NxN, and the operation amount and N of the covariance matrix are N 2 Is in direct proportion. General waveform measuring junctionThe results N are all larger, for example, pulse waveforms, so that the computation amount of the covariance matrix is huge, and the pulse waveform measurement result uncertainty evaluation technology based on the covariance matrix is difficult to apply in engineering. In addition, in the field of radio parameter calibration, most of the situations are measurement in a time domain and a frequency domain, a frequency domain calibration result is obtained by calculation in the frequency domain, then the frequency domain calibration result is transformed to the time domain to obtain a time domain calibration result, a covariance matrix of frequency domain uncertainty is obtained in the whole process, and a covariance matrix of time domain uncertainty is obtained.
The innovation points of the invention are as follows: firstly, the frequency covariance matrix is subjected to block rearrangement, and the rearranged matrix is a symmetrical positive definite matrix, so that feature decomposition and matrix dimension reduction can be further performed; secondly, the frequency covariance matrix is subjected to dimensionality reduction, so that the operation amount of the original matrix is greatly reduced, and the operation efficiency is improved; and thirdly, the method is different from the traditional waveform evaluation method in that only diagonal variables of the frequency covariance matrix are processed, and the method processes all variables in the matrix, so that the evaluation method is higher in accuracy.
The technical solutions provided by the embodiments of the present invention are described in detail below with reference to the accompanying drawings.
Fig. 1 is a flow embodiment of a waveform uncertainty evaluation method, which can be used for engineering implementation of waveform uncertainty evaluation, and as an embodiment of the present invention, a waveform uncertainty evaluation method specifically includes the following steps 101 to 103:
In step 101, the obtained frequency domain calibration result is:
wherein,
for the frequency domain calibration result obtained from the kth calibration, k is calibrationThe standard numbers k =1,2,3 and … m, m is the total calibration number, and each calibration is an independent irrelevant event. n is the number of frequency components, the frequency domain calibration result contains n frequency components and does not contain direct current components, a 1k 、a 2k 、……、a nk The first, second, … …, the nth real part of the frequency component, b 1k 、b 2k 、……、b nk The imaginary part of the nth frequency component is the first, second, … … of the k-th time frequency domain calibration result.
In step 101, a frequency-domain covariance matrix is calculated for the frequency-domain calibration result as:
wherein,
for the frequency domain covariance matrix obtained for the kth calibration, the matrix dimension is 2n × 2n, e () represents the mathematical expectation of the calculation.
And 102, rearranging the frequency domain covariance matrix in blocks to obtain a block matrix.
In step 102, the blocking principle is that the real part covariance matrix of the frequency domain calibration result is one block, the imaginary part covariance matrix is one block, the covariance matrix with the real part preceding the imaginary part succeeding the real part is one block, and the covariance matrix with the imaginary part preceding the real part succeeding the real part is one block, and the blocking matrix is a real symmetric positive definite matrix.
Specifically, first, in order to analyze the structure of the frequency-domain covariance matrix more conveniently, the frequency-domain covariance matrix obtained by the kth calibration in formula (2) is simplified and expressed as:
wherein u (a) ik ,a jk ) Denotes a ik And a jk Covariance of (a), u (a) ik ,b jk ) Denotes a ik And b jk Covariance of (a), u (b) ik ,a jk ) Denotes b ik And a jk Covariance of (a), u (b) ik ,b jk ) Denotes b ik And b jk I and j are the first and second index of the frequency component, i, j =1,2, …, n, respectively.
In step 102, the n complex data of the frequency domain calibration result arranged in columns are split into real parts corresponding to the n complex data arranged in columns and imaginary parts corresponding to the n complex data, and the block matrix obtained after rearrangement is:
wherein,
the block matrix obtained for the k-th calibration.
Further, the blocking matrix may be represented as:
wherein,
respectively, a real part covariance matrix of the k-th calibration, a covariance matrix of the k-th calibration with its real part preceding the imaginary part, a covariance matrix of the k-th calibration with its imaginary part preceding the real part, and a covariance matrix of the k-th calibration with its imaginary part following the real part, respectively
The matrix dimensions are all n.
Wherein,
and satisfy
It is known that the block matrix is a real symmetric positive definite matrix, can be diagonalized, satisfies the condition of matrix eigen decomposition, and can be characterized by being expressed as
Wherein Q k Decomposing the matrix for the eigenvalues corresponding to the kth calibration, Q k ＝E k Λ k 1/2 ，E k The feature vector matrix corresponding to the k-th calibration is a 2n × 2 n-dimensional matrix, and each column vector is
Eigenvectors, Λ, corresponding to the eigenvalues k Is an eigenvalue matrix, is a 2n × 2n dimensional real diagonal matrix, and has diagonal elements of
The characteristic value of (2).
In step 102, since
Is a symmetric positive definite matrix, so k If the square root is more than 0, the positive square root can be taken in the matrix decomposition, namely, the positive square root operation can be carried out on the block matrix, and the dimension reduction processing is carried out on the obtained new block matrix.
In step 102, eigenvalues and eigenvectors of the matrix may be computed using the ARPACK software package, a large-scale matrix computation tool that provides eigenvalue computation services based on the implicit restart of the Arnoldi algorithm (IRAM), which is also part of the principal component analysis method.
And 103, performing dimension reduction processing on the block matrix to obtain a frequency domain covariance dimension reduction matrix.
In step 103, p largest pairs of eigenvalues are used to reduce the computation of the frequency-domain covariance matrix
Dimension reduction processing is carried out to obtain a corresponding dimension-reduced frequency domain covariance matrix
Further, when the dimension reduction processing is performed on the block matrix to obtain the frequency domain covariance dimension reduction matrix, the tolerance of uncertainty information included in the frequency domain covariance dimension reduction matrix meets a preset tolerance requirement.
Further, the method further comprises: and selecting different numbers of characteristic values to perform dimension reduction processing on the block matrixes to obtain corresponding different dimension reduction matrixes, and selecting the dimension reduction matrix with the minimum uncertainty information tolerance as the frequency domain covariance dimension reduction matrix.
That is, in order to ensure that uncertainty information contained in the reduced-dimension frequency domain covariance reduced-dimension matrix can meet the tolerance requirement, the principle of determining p is that when k > p, Λ is k Is approximately equal to 0. In practical operation, a plurality of different p values can be verified, and the p value is selected so that the uncertainty information included in the reduced-dimension frequency domain covariance matrix is as small as possible.
It should be noted that the preset tolerance requirement may be preset according to a calibration accuracy requirement, and the uncertainty information tolerance refers to a difference between a frequency domain covariance dimension reduction matrix and a frequency domain covariance matrix.
Further, the method further comprises: and calculating the dimensionality reduction ratio according to the storage capacity of the frequency domain covariance dimensionality reduction matrix and the storage capacity of the frequency domain covariance matrix.
In particular, for the rearranged frequency domain covariance matrix
Need to store
Element, and for the reduced-dimension frequency domain covariance matrix
Only need to store
The dimensionality reduction ratio is as follows:
wherein S is k For the k-th calibration of the corresponding dimensionality reduction ratio, due to
Therefore, when p < n, the operation amount of the frequency domain covariance matrix can be greatly reduced. At this point, no further calculations are required
Only need to calculate
The frequency domain covariance matrix which meets the acceptable uncertainty information can be obtained with less operation amount, and then uncertainty evaluation is carried out.
The invention provides a waveform uncertainty evaluation method, which reduces the operation amount and improves the operation efficiency by adopting a dimension reduction processing mode.
Fig. 2 is a flowchart of an embodiment of a waveform uncertainty evaluation method including a comparison matrix calculation, which can evaluate a dimensionality reduction effect, and as an embodiment of the present invention, the waveform uncertainty evaluation method specifically includes the following steps 101 to 105:
And 102, rearranging the blocks of the frequency domain covariance matrix to obtain a block matrix.
And 103, performing dimension reduction processing on the block matrix to obtain a frequency domain covariance dimension reduction matrix.
And 104, subtracting the product of the dimensionality reduction eigenvalue decomposition matrix and the transpose matrix of the dimensionality reduction eigenvalue decomposition matrix from the frequency domain covariance dimensionality reduction matrix to obtain a comparison matrix.
In step 104, the comparison matrix corresponding to the kth calibration is:
wherein D is k For said kth time calibrating the corresponding comparison matrix, Q 0k The k-th calibration of the corresponding reduced-dimension eigenvalue decomposition matrix, Q 0k Decomposing a matrix Q for eigenvalues of the blocking matrix k P is the maximum eigenvalue number of the dimensionality reduction process.
And 105, evaluating the accuracy of the frequency domain covariance dimension reduction matrix according to the comparison matrix.
In step 105, if D is found k So that
And
therefore, the variance and covariance related to the real part and the imaginary part of each frequency component complex data in the rearranged frequency domain covariance matrix (namely the block matrix) can be accurately reserved in the frequency domain covariance dimension reduction matrix after dimension reduction, and the accuracy of the frequency domain covariance dimension reduction matrix meets the requirement.
It is to be noted that the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising a … …" does not exclude the presence of another identical element in a process, method, article, or apparatus that comprises the element.
The above description is only an example of the present invention and is not intended to limit the present invention. Various modifications and alterations to this invention will become apparent to those skilled in the art. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the scope of the claims of the present invention.
Claims (7)
1. A method for evaluating waveform uncertainty, comprising the steps of:
calculating a frequency domain covariance matrix for the frequency domain calibration result obtained by each calibration;
rearranging the frequency domain covariance matrix in blocks to obtain a block matrix,
the blocking matrix is represented as:
wherein, V RRk 、V RIk 、V IRk 、V IIk Respectively are a real part covariance matrix of the k-th calibration, a covariance matrix of the k-th calibration with the real part at the front and the imaginary part at the back, a covariance matrix of the k-th calibration with the imaginary part at the front and the real part at the back, and a real part covariance matrix of the k-th calibration RRk 、V RIk 、V IRk 、V IIk The matrix dimensions are n × n;
and selecting different numbers of characteristic values to perform dimension reduction processing on the block matrixes to obtain corresponding different dimension reduction matrixes, and selecting the dimension reduction matrix with the minimum uncertainty information tolerance as a frequency domain covariance dimension reduction matrix.
2. The waveform uncertainty evaluation method according to claim 1, wherein when the block matrix is subjected to dimension reduction to obtain a frequency-domain covariance dimension reduction matrix, an uncertainty information tolerance included in the frequency-domain covariance dimension reduction matrix meets a preset tolerance requirement.
3. The method for waveform uncertainty assessment according to claim 1, further comprising:
and carrying out positive square root operation on the block matrixes, and carrying out dimension reduction on the obtained new block matrixes.
4. The method for waveform uncertainty assessment according to claim 1, further comprising:
subtracting a product of a dimension reduction eigenvalue decomposition matrix and a transpose matrix of the dimension reduction eigenvalue decomposition matrix from the frequency domain covariance dimension reduction matrix to obtain a comparison matrix, wherein the dimension reduction eigenvalue decomposition matrix is the front p columns of the eigenvalue decomposition matrix of the block matrix, and p is the maximum eigenvalue number of dimension reduction processing;
and evaluating the accuracy of the frequency domain covariance dimension reduction matrix according to the comparison matrix.
5. The method for waveform uncertainty assessment according to claim 1, further comprising: and calculating the dimensionality reduction ratio according to the storage capacity of the frequency domain covariance dimensionality reduction matrix and the storage capacity of the frequency domain covariance matrix.
6. The waveform uncertainty assessment method according to claim 1, characterized in that the eigenvalues and eigenvectors of the matrix are calculated by IRAM algorithm.
7. The waveform uncertainty assessment method according to claim 1, wherein the eigenvalue and eigenvector are calculated for the matrix by ARPACK software.
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