CN115081390B

CN115081390B - Non-uniform sampling optimization method and device for integrated circuit Hankel transform filter

Info

Publication number: CN115081390B
Application number: CN202210744579.7A
Authority: CN
Inventors: 王芬
Original assignee: Beijing Wisechip Simulation Technology Co Ltd
Current assignee: Beijing Wisechip Simulation Technology Co Ltd
Priority date: 2022-06-28
Filing date: 2022-06-28
Publication date: 2022-12-13
Anticipated expiration: 2042-06-28
Also published as: CN115081390A

Abstract

The invention belongs to the technical field of integrated circuits, in particular to a non-uniform sampling optimization method and a non-uniform sampling optimization device for a Hankel transform filter of an integrated circuit, wherein the method comprises the following steps: researching errors calculated by uniform sampling points by taking the sampling intervals as variables, and obtaining a series of sampling intervals with local minimum errors as fixed sampling intervals; obtaining a filter coefficient of Bessel integral-Hankel transformation for calculating a Green function of an electromagnetic field of the integrated circuit at a minimum fixed sampling interval, calculating the error influence of abandonment of each sampling point on the calculation of the whole Bessel integral in a Bessel integral-Hankel transformation pair, determining whether the sampling point is reserved, updating the position of the reserved sampling point, and calculating the Bessel integral based on the reserved non-uniform sampling point; according to the invention, the sampling interval is set to be the fixed sampling interval with the local minimum error, so that the calculation amount of Bessel integral is greatly reduced under the condition of meeting the precision requirement, and the electromagnetic field calculation speed of the integrated circuit is improved.

Description

Non-uniform sampling optimization method and device for integrated circuit Hankel transform filter

Technical Field

The invention relates to the technical field of integrated circuits, in particular to a non-uniform sampling optimization method and device for a Hankel transform filter of an integrated circuit.

Background

When the integrated circuit works, a high-frequency alternating electromagnetic field can be formed on a multilayer layout of the integrated circuit due to the transmission of high-speed signals, and meanwhile, in order to improve the performance of electronic equipment, reduce the volume and reduce the cost, transistors, other components and circuits are integrated on a small semiconductor substrate. In order to realize more functions, the ultra-large scale integrated circuit has a structure from tens of layers to hundreds of layers, each layer of structure is extremely complex, millions or even tens of millions of transistors are integrated, and the ultra-large scale integrated circuit has a multi-scale structure from a centimeter level to the latest nanometer level at present. In order to ensure that the integrated circuit can normally work and realize the function designed in advance, the power integrity and the signal integrity of the integrated circuit need to be ensured firstly, so that the power integrity and the signal integrity of the integrated circuit with a multi-scale structure of tens of layers and hundreds of layers need to be accurately analyzed by adopting an electromagnetic field analysis means, which is a great problem in the electromagnetic field analysis of the ultra-large scale integrated circuit.

The method comprises the steps of calculating fields generated by a point current source at any spatial position based on a Green function, calculating the fields generated by a surface current source at the same position by using a Gaussian integration method based on the linear superposition property of the fields generated by the source, and further calculating the fields generated by currents on a multilayer metal plate with a complex shape of an integrated circuit at different spatial positions.

But integration problems with bezier functions are often encountered when using green's functions to quickly calculate the electromagnetic field problems of very large scale integrated circuits. Due to the characteristics of high oscillation, slow attenuation and the like, the Bessel function has the hotspot problem of research on the fast calculation and high precision of the integral. For the integration of the Bessel function, a method of segmented integration is initially adopted, namely segmentation is carried out in an integration area, and each segment is accumulated after numerical integration is adopted. The cost of this approach is a substantial increase in computation time. In rapidly evolving computational physics, computation time is very important, directly determining computational efficiency.

In the early 70 s, a digital linear filter was introduced into the fast green function calculation, so that the integration problem about the bessel function was better solved. The linear filtering method is convenient and simple to calculate, and the calculating speed is at least one order of magnitude higher than that of the traditional method.

The filter coefficients are generally obtained by means of uniform sampling, an optimal sampling interval is determined according to the sampling theorem, and then a limited number of sampling points are selected based on the sampling interval to form the filter. However, in actual calculation, taking large-density uniform samples at equal intervals in different integral kernel functions is not optimal in calculation time, and unnecessary calculation amount and calculation time are sometimes wasted. And because the final integral is the accumulation of different sampling points and different weights, the contribution of different sampling points to the whole integral in the Bessel integral-Hankel transformation pair is different, and the contribution of some points can be almost ignored.

The method for discarding the sampling points with small contribution determines whether the sampling points are reserved or not according to the contribution of each sampling point simply and intuitively, and then a new filter is formed directly based on the reserved sampling points and the corresponding filter coefficients. However, in more complex computational models of multi-layer very large scale integrated circuits, the method of discarding contributing small sample points starts to become inaccurate because the contribution of this sample point is calculated only according to a certain specific bezier integral with analytical expressions, whereas the bezier integral in the actual computational model of multi-layer very large scale integrated circuits is not completely such a bezier integral; further analysis of the method of discarding samples with small contributions has found that this method is only an asymmetric uniformly sampled filter that reduces the number of filter points, this asymmetry being determined by the contributions of the filter samples, which is applicable to some bezier integrals, but if the kernel function is changed, the contribution of this sample point changes, this simple discarding will fail because it essentially reduces the total sampling time of the filter, and for the slowly decaying bezier integrals, the simple discarding will lead to large truncation errors, and furthermore, the method simply compares the term of the convolution for each point with the total convolution (i.e. the integration of the bezier function by the filter) and takes it as the contribution of this point, rather than strictly updating the sampling time of the sample point after this point is removed, and then recalculates the filter after this point is removed and calculates the bezier integral, which is compared with the bezier integral before this point is removed. In fact, the difference between the Bessel integrals before and after the sampling point is strictly compared and removed, and the contribution of the Bessel integral to the whole Bessel integral in the Bessel integral-Hankel transformation pair is more accurate than the previous judgment by taking the Bessel integral as the sampling point, so that whether the sampling point reserves a calculation model which cannot accurately process a more complex multilayer super-large-scale integrated circuit is determined only according to the contribution of the sampling point, and a more accurate calculation method is required for the calculation model with the requirement of accurate calculation.

Disclosure of Invention

In view of the above-mentioned deficiencies of the prior art, the present invention aims to provide a non-uniform sampling optimization method and device for a hankel transform filter of an integrated circuit, so as to analyze sampling points more accurately and to perform homogenization processing on the sampling points after discarding the sampling points, so as to obtain the analysis condition of an electromagnetic field rapidly and accurately.

In order to solve the problems, the invention adopts the following technical scheme:

in a first aspect, the present invention provides a non-uniform sampling optimization method for a hankel transform filter of an integrated circuit, including:

s100, acquiring all Bezier integrals contained in a dyadic Green function based on the dyadic Green function of a field generated by a current source at any point on a copper-clad region of any layer of the integrated circuit on other layers, and determining the order of the Bezier function used in the dyadic Green function; the Bezier integral is an infinite integral of an integrand containing a Bezier function;

s200, carrying out Hankel transformation on the Bessel integral according to an integrand function formed by multiplying the integral kernel function and the Bessel function; the integral kernel function is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from a source point to a field point, wherein the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad area of the integrated circuit to the field point acting on other layers;

s300, calculating errors of a numerical solution of the Bezier integral and an accurate solution of the Bezier integral by using a Hankel transformation method based on uniform sampling points by taking the sampling intervals as variables, obtaining a relation curve of the sampling intervals and the calculated errors of the Bezier integral, and obtaining a series of sampling intervals with local minimum errors as a fixed sampling interval sequence based on the relation curve;

s400, discretizing the Hankel transformation by uniform sampling by taking the minimum fixed sampling interval in the fixed sampling interval sequence as a sampling interval to obtain an expression of an infinite length filter;

s500, according to the layer thicknesses of different layers of the integrated circuit and the attenuation speed of an integrand determined by the characteristics of an interlayer medium, the infinite length filter is cut off within a preset precision control range to obtain an expression of the finite length filter, and according to the expression of the finite length filter, an equation set of the finite length filter is obtained;

s600, constructing a Bessel integral-Hankel transformation pair by adopting a known Bessel integral analytic expression, or constructing a Bessel integral which is closest to an actual kernel function by adopting typical parameters suitable for an integrated circuit aiming at the Bessel integral which cannot be actually analyzed to form the Bessel integral-Hankel transformation pair, and calculating the Bessel integral in the Bessel integral-Hankel transformation pair at high precision;

s700, substituting the Bessel integral-Hankel transformation pair into an expression of the finite length filter based on the constructed Bessel integral-Hankel transformation pair to obtain a filter coefficient matrix equation corresponding to a filter equation set, and improving the filter coefficient matrix equation;

s800, solving the improved filter coefficient matrix equation to obtain a filter coefficient;

s900, for sampling points obtained by uniform sampling with the minimum fixed sampling interval as the sampling interval, screening preliminarily determined abandoned sampling points according to the contribution values of the sampling points, and judging whether the sampling points are abandoned or not according to the error influence of the abandonment of the preliminarily determined abandoned sampling points on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair; the method comprises the following steps:

s910, determining the sampling point corresponding to the sampling point with the contribution value smaller than the reserved threshold value as a sampling point which is determined to be abandoned preliminarily by calculating the contribution value of each sampling point except the first sampling point and the last sampling point; if the preliminarily determined abandoned sampling points do not exist, the step S1000 is carried out;

s920, based on the relation between the current sampling interval and the sampling interval in the fixed sampling interval sequence, trying to abandon the ith sampling point in the preliminarily determined abandoned sampling points, trying to update the positions of the reserved sampling points to form non-uniform sampling points, and calculating the error influence of abandonment of the ith sampling point in the preliminarily determined abandoned sampling points on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair;

s930, resuming the attempting to discard the ith sampling point of the preliminarily determined discarded sampling points and resuming the attempting to perform position updating on the remaining sampling points, letting i = i +1, and going to the step S920, obtaining an error influence calculated on the whole bezier integral in the bezier integral-hankerr transformation pair after all the preliminarily determined discarded sampling points are discarded individually, and finding out an individually discarded sampling point with a minimum error influence; if the minimum error influence is less than or equal to the preset error threshold, discarding the sampling point corresponding to the minimum error influence, updating the position of the reserved sampling point, and turning to step S910; if the minimum error influence result is larger than the preset error threshold, the step S1000 is executed;

s1000, calculating corresponding Bessel integrals based on the reserved sampling points, the corresponding filter coefficients and the integral kernel functions.

As an implementation manner, the S900 is specifically:

s911, setting a retention threshold value of the preliminary abandoned sampling point based on the contribution value and an error threshold value of the abandoned sampling point based on the error according to the layer thickness of different layers of the integrated circuit and the attenuation speed of the integrand determined by the characteristics of the dielectric medium between the layers;

s912, calculating the contribution value of each sampling point except the first sampling point and the last sampling point according to the filter coefficient and the sampling point of the filter;

s913, regarding all the sampling points with the contribution values smaller than the retention threshold values of the sampling points as preliminarily determined abandoned sampling points, and forming a preliminarily determined abandoned sampling point set; if the discarded sampling point set is determined to be an empty set preliminarily, the step S1000 is carried out, otherwise, the step S921 is carried out;

s921, trying to discard the ith sampling point in the preliminarily determined discarded sampling point set, updating the positions of the sampling points around the ith sampling point, and calculating the filter coefficients of the rest sampling points based on the sampling points after trying to discard the ith sampling point to obtain the filter coefficients of the sampling points after trying to discard the ith sampling point;

s922, calculating a Bezier function integral based on the filter coefficient, and obtaining a numerical solution of the Bezier function integral in the Bezier integral-Hankel transformation pair;

s923, calculating an accurate solution obtained by integrating the Bessel function in the Bessel integral-Hankel transformation pair by adopting an analytical expression or a high-accuracy calculation method, and calculating a relative error between the numerical solution and the accurate solution to obtain an error influence;

s931, restoring the trying to discard the ith sampling point in the preliminarily determined discarded sampling point set and restoring the trying to update the positions of the sampling points around the ith sampling point, setting i = i +1, and turning to the S921 to obtain a relative error set after trying to discard each preliminarily determined discarded sampling point independently;

s932, obtaining a minimum relative error according to the error magnitude of the relative error set;

s933, if the minimum relative error is smaller than or equal to the error threshold, discarding the sampling point and the filter coefficient corresponding to the minimum relative error, updating the positions of the sampling points around the sampling point corresponding to the minimum relative error, and turning to S912; if the minimum relative error is greater than the error threshold, go to step S1000.

As an implementable mode, the attempting to discard an ith sampling point of the preliminary determination discarded sampling points and attempting to perform position update on sampling points around the ith sampling point based on the relation between the sampling interval and the fixed sampling interval sequence includes:

after the ith sampling point is tried to be abandoned, the position of the sampling points around the ith sampling point is tried to be updated, so that the movement or non-movement of the (i-1) th sampling point and/or the (i + 1) th sampling point is met, and the deviation between the sampling interval between the (i-2) th sampling point and the (i-1) th sampling point, the (i-1) th sampling point and the (i + 2) th sampling point and one fixed sampling interval in the fixed sampling interval sequence is minimum; if all the deviations are smaller than a preset sampling interval deviation threshold value, the ith sampling point can try to be removed, and the sampling points around the ith sampling point are subjected to position updating according to the position updating attempt; if any deviation is larger than or equal to a preset sampling interval deviation threshold value, the ith sampling point cannot try to remove the deviation, and the position is not updated.

As an implementable mode, the attempting to discard an ith sampling point of the preliminarily determined discarded sampling points and attempting to perform position update on sampling points around the ith sampling point based on a relation between a sampling interval and the fixed sampling interval sequence includes: adding corresponding deviation to each fixed sampling interval except the minimum fixed sampling interval in the fixed sampling interval sequence, and adjusting the deviation to be integral multiple of the minimum fixed sampling interval in the fixed sampling interval sequence; if the deviation added to the fixed sampling interval in the fixed sampling interval sequence is smaller than the preset fixed sampling interval rounding deviation, and the error value corresponding to the adjusted fixed sampling interval is smaller than the sampling interval error threshold, determining the fixed sampling interval corresponding to the deviation as a new fixed sampling interval sequence; if the deviation is greater than or equal to the preset fixed sampling interval rounding deviation, or the error value corresponding to the adjusted fixed sampling interval is greater than or equal to the sampling interval error threshold, removing the fixed sampling interval corresponding to the deviation;

based on the new fixed sampling interval sequence, if the position updating of the sampling points around the ith sampling point is attempted after the ith sampling point is tried to be abandoned, the (i-1) th sampling point and/or the (i + 1) th sampling point are/is not moved, so that the sampling intervals between the (i-2) th sampling point and the (i-1) th sampling point, the (i-1) th sampling point and the (i + 1) th sampling point, the (i + 1) th sampling point and the (i + 2) th sampling point are one or more in the new fixed sampling interval sequence, and the moving distance square sum of the (i-1) th sampling point and the (i + 1) th sampling point is the minimum, the ith sampling point can be tried to be removed, and the position updating of the sampling points around the ith sampling point is attempted according to the position updating.

As an implementation, the contribution value is calculated by the following formula:

wherein, C _i Is the contribution of the ith sample point, r is the spatial distance acted by the Green function, h _i The filter coefficient of the ith sampling point is determined by the Bessel function order contained in the dyadic Green function of the field generated by the current source at any point on the copper-clad region of any layer of the integrated circuit on other layers, and lambda is _i G (-) is an input function determined by the electromagnetic parameters of the materials of the layers of the integrated circuit, the thickness of the layers, the operating frequency of the integrated circuit, and the distance from the source point to the field point, G (-) is an output function, and 2L < -1 > is the filter length.

In another aspect, the present invention provides an apparatus for non-uniform sampling optimization of a hank transform filter of an integrated circuit, comprising:

the system comprises a Bessel integral acquisition module, a Hankel transformation module, a fixed sampling interval sequence screening module, an infinite filter module, a finite filter module, a transformation pair construction module, a matrix equation improvement module, a filter coefficient calculation module, a sampling point rejection module and a Bessel integral calculation module;

the Bessel integral obtaining module is used for obtaining all Bessel integrals contained in the dyadic Green function based on the dyadic Green function of the field generated by a current source at any point on a copper-clad region of any layer of the integrated circuit on other layers, so that the order of the Bessel function used in the dyadic Green function is determined; the Bezier integral is an infinite integral of an integrand containing a Bezier function;

the Hankel transformation module is used for carrying out Hankel transformation on the Bessel integral according to an integrand function formed by the product of the integral kernel function and the Bessel function; the integral kernel function is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from a source point to a field point, wherein the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad area of the integrated circuit to the field point acting on other layers;

the fixed sampling interval sequence screening module is used for calculating errors of numerical solutions of Bezier integrals and accurate solutions of the Bezier integrals by taking a sampling interval as a variable based on a Hankel transformation method of uniform sampling points, obtaining a relation curve of the sampling interval and the errors of the calculated Bezier integrals, and obtaining a series of sampling intervals with local minimum errors based on the relation curve to serve as a fixed sampling interval sequence;

the infinite filter expression module is used for discretizing the Hankel transformation by uniform sampling by taking the minimum fixed sampling interval in the fixed sampling interval sequence as a sampling interval to obtain an expression of an infinite length filter;

the finite filter expression module is used for truncating the infinite length filter within a preset precision control range according to the attenuation speed of an integrand determined by the layer thickness of different layers of the integrated circuit and the characteristics of an interlayer medium to obtain an expression of the finite length filter, and obtaining an equation set of the finite length filter according to the expression of the finite length filter;

the transformation pair construction module is used for constructing a Bessel integral-Hankel transformation pair by adopting a known Bessel integral analysis expression or constructing the Bessel integral which is closest to an actual kernel function by adopting typical parameters suitable for an integrated circuit aiming at the Bessel integral which cannot be analyzed actually to form the Bessel integral-Hankel transformation pair and calculating the Bessel integral in the Bessel integral-Hankel transformation pair at high precision;

the matrix equation improvement module is used for substituting the expression of the finite length filter into the Bessel integral-Hankel transformation pair based on the structure to obtain a filter coefficient matrix equation corresponding to a filter equation set and improving the filter coefficient matrix equation;

the filter coefficient calculation module is used for solving the improved filter coefficient matrix equation to obtain a filter coefficient;

the sampling point abandoning module is used for screening and preliminarily determining abandoned sampling points according to the contribution values of the sampling points for the sampling points obtained by uniformly sampling with the minimum fixed sampling interval as the sampling interval, and judging whether the sampling points are abandoned or not according to the error influence of the abandonment of the preliminarily determined abandoned sampling points on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair;

the Bezier integral calculation module is used for calculating corresponding Bezier integrals based on reserved sampling points, corresponding filter coefficients and integral kernel functions;

the sampling point abandoning module comprises a preliminary determination abandoned sampling point screening unit, a Bessel integral error influence calculation unit and a point cut updating unit;

the preliminary determined abandoned sampling point screening unit is used for calculating the contribution value of each sampling point except the first sampling point and the last sampling point, and determining the sampling point corresponding to the contribution value smaller than the reserved threshold value as the preliminary determined abandoned sampling point; if the sampling points which are discarded in the preliminary determination do not exist, switching to the Bessel integral calculation module;

the Bessel integral error influence calculation unit is used for trying to abandon the ith sampling point in the preliminarily determined abandoned sampling points and trying to update the position of the reserved sampling point to form a non-uniform sampling point based on the relation between the current sampling interval and the sampling interval in the fixed sampling interval sequence, and calculating the error influence of the abandonment of the ith sampling point in the preliminarily determined abandoned sampling point on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair;

the point-cut updating unit is used for recovering the attempt to discard the ith sampling point in the preliminarily determined discarded sampling points and recovering the attempt to update the positions of the reserved sampling points, and the step S920 is carried out to obtain the error influence calculated on the whole Bessel integral in the Bessel integral-Hankel transformation pair after all the preliminarily determined discarded sampling points are discarded separately, and find out the separately discarded sampling point with the smallest error influence; if the minimum error influence is less than or equal to a preset error threshold, discarding the sampling point corresponding to the minimum error influence, updating the position of the reserved sampling point, and switching to the primary determination discarded sampling point screening unit; and if the minimum error influence result is larger than a preset error threshold value, switching to the Bessel integral calculation module.

As an implementation manner, the preliminary determination abandoned sampling point screening unit includes an error threshold setting subunit, a contribution value calculating subunit and a preliminary determination abandoned sampling point determining subunit;

the error threshold setting subunit is configured to set the retention threshold based on the preliminary discarded sampling points of the contribution value and the error threshold based on the error discarded sampling points according to the layer thicknesses of different layers of the integrated circuit and the attenuation speed of the integrand determined by the characteristics of the interlayer medium;

the contribution value operator unit is used for calculating the contribution value of each sampling point except the first sampling point and the last sampling point according to the filter coefficient and the sampling point of the filter;

the preliminary determination abandoned sampling point determination subunit is used for determining all the sampling points of which the contribution values are smaller than the retention threshold values of the sampling points as the preliminary determination abandoned sampling points to form a preliminary determination abandoned sampling point set; if the sampling point set which is preliminarily determined to be abandoned is an empty set, switching to the Bessel integral calculation module, and otherwise, switching to a filter coefficient calculation subunit;

the Bessel integral error influence calculating unit comprises a filter coefficient calculating subunit, a numerical solution calculating subunit and an error influence calculating subunit;

the filter coefficient calculation subunit is configured to try to discard an ith sampling point in the preliminarily determined discarded sampling point set, update positions of sampling points around the ith sampling point, calculate filter coefficients of remaining sampling points based on the sampling points after the ith sampling point is tried to be discarded, and obtain the filter coefficients of the sampling points after the ith sampling point is tried to be discarded;

the numerical solution calculating subunit is configured to calculate a bezier function integral based on the filter coefficient, and obtain a numerical solution of the bezier function integral in the bezier integral-hankerr transformation pair;

the error influence calculating subunit is configured to calculate a relative error between the numerical solution and the precise solution based on a precise solution obtained by calculating the bezier function integral in the bezier integral-hankerr transformation pair by using an analytic expression or a high-precision calculation method, so as to obtain an error influence;

the point-cut updating unit comprises a relative error set calculation subunit, a minimum relative error calculation subunit and a point-cut updating subunit;

the relative error set calculating subunit is configured to restore the trying to discard an ith sampling point in the preliminarily determined discarded sampling point set and restore the trying to update positions of sampling points around the ith sampling point, set i = i +1, and turn to the filter coefficient calculating subunit to obtain a relative error set after trying to discard each preliminarily determined discarded sampling point individually;

the minimum relative error calculating subunit is configured to obtain a minimum relative error according to the error magnitude of the relative error set;

the point-cutting updating subunit is configured to discard the sampling point and the filter coefficient corresponding to the minimum relative error if the minimum relative error is smaller than or equal to the error threshold, perform position updating on the sampling points around the sampling point corresponding to the minimum relative error, and shift the sampling points to the contribution value operator unit; and if the minimum relative error is larger than the error threshold value, switching to the Bessel integral calculation module.

As an implementable mode, the attempting to discard an ith sampling point of the preliminarily determined discarded sampling points and attempting to perform position update on sampling points around the ith sampling point based on a relation between a sampling interval and the fixed sampling interval sequence includes:

based on the new fixed sampling interval sequence, if trying to abandon the ith sampling point, trying to update the positions of the sampling points around the ith sampling point to meet the requirement of moving or not moving the (i-1) th sampling point and/or the (i + 1) th sampling point, so that the sampling intervals between the (i-2) th sampling point and the (i-1) th sampling point, the (i-1) th sampling point and the (i + 2) th sampling point are one or more in the new fixed sampling interval sequence, and the moving distance square sum of the (i-1) th sampling point and the (i + 1) th sampling point is the minimum, then the (i) th sampling point can try to remove, and try to update the positions of the sampling points around the (i) th sampling point according to the position update.

As an implementation manner, the contribution value in the contribution value operator unit is calculated by the following formula:

The invention has the beneficial effects that: the invention provides a non-uniform sampling method for fast Hankel transformation, which strictly calculates the error influence of abandoning of each sampling point on the whole integral calculation in a Bessel integral-Hankel transformation pair on the premise of ensuring the total sampling time, renews the sampling time of the sampling point near the sampling point when trying to abandon each sampling point, reestablishes a solving matrix of a filter coefficient based on the renewed sampling time to form a set of new filter coefficient, calculates the Bessel integral based on the new filter coefficient, compares the calculated integral with an analytic solution, and determines whether the sampling point is abandoned or not through the error. Compared with the method of simply abandoning sampling points with small contribution, the method does not change the total sampling time and evaluates the abandoning points more accurately, so the method is a complete non-uniform sampling method and is suitable for more complex models.

Drawings

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention will be described in further detail with reference to the accompanying drawings, in which:

FIG. 1 is a flow chart of a non-uniform sampling optimization method for a Hankel transform filter of an integrated circuit according to the present invention;

FIG. 2 is a schematic diagram of a fixed sampling interval according to an embodiment of the present invention;

FIG. 3 is a flowchart illustrating a step S900 according to an embodiment of the present invention;

FIG. 4 is a schematic diagram of an apparatus for non-uniform sampling optimization of a Hankel transform filter of an integrated circuit according to the present invention.

Detailed Description

The present invention will be described in further detail with reference to specific examples.

It should be noted that these examples are only for illustrating the present invention, and not for limiting the present invention, and the simple modification of the method based on the idea of the present invention is within the protection scope of the present invention.

Referring to fig. 1, a non-uniform sampling optimization method for a hankel transform filter of an integrated circuit includes:

s100, acquiring all Bezier integrals contained in a dyadic Green function based on a dyadic Green function of a field generated by a current source at any point on a copper-clad region of any layer of the integrated circuit on other layers, and determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function;

s200, according to an integrand function formed by the product of the integral kernel function and the Bessel function, carrying out Hankel transformation on the Bessel integral; the integral kernel function is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from a source point to a field point, wherein the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-coated region of the integrated circuit to the field point acting on other layers;

s300, calculating errors of a numerical solution of the Bezier integral and an accurate solution of the Bezier integral by using a Hankel transformation method based on uniform sampling points by taking the sampling intervals as variables, obtaining a relation curve of the sampling intervals and the calculated errors of the Bezier integral, and obtaining a series of sampling intervals with local minimum errors based on the relation curve to serve as a fixed sampling interval sequence;

s400, discretizing the Hankel transformation by uniform sampling by taking the minimum fixed sampling interval in the fixed sampling interval sequence as a sampling interval to obtain an expression of the infinite length filter;

s500, according to the layer thicknesses of different layers of the integrated circuit and the attenuation speed of an integrand function determined by the characteristics of an interlayer medium, the infinite length filter is cut off within a preset precision control range to obtain an expression of the finite length filter, and according to the expression of the finite length filter, an equation set of the finite length filter is obtained;

s700, substituting an expression of a filter with a finite length into a Bessel integral-Hankel transformation pair based on the structure to obtain a filter coefficient matrix equation corresponding to a filter equation set, and improving the filter coefficient matrix equation;

s900, for sampling points obtained by uniform sampling with the minimum fixed sampling interval as the sampling interval, screening and preliminarily determining abandoned sampling points according to the contribution values of the sampling points, and judging whether the sampling points are abandoned or not according to the error influence of the abandonment of the preliminarily determined abandoned sampling points on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair; the method comprises the following steps:

s910, determining the sampling point corresponding to the sampling point with the contribution value smaller than the reserved threshold value as a sampling point which is determined to be abandoned preliminarily by calculating the contribution value of each sampling point except the first sampling point and the last sampling point; if the sampling point which is determined to be abandoned preliminarily does not exist, the step S1000 is carried out;

s920, based on the relation between the current sampling interval and the sampling interval in the fixed sampling interval sequence, trying to abandon the ith sampling point in the preliminarily determined abandoned sampling points, trying to update the position of the reserved sampling point to form a non-uniform sampling point, and calculating the error influence of abandoning of the ith sampling point in the preliminarily determined abandoned sampling points on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair;

s930, recovering to try to discard the ith sampling point in the preliminarily determined discarded sampling points and recovering to try to update the positions of the reserved sampling points, enabling i = i +1, turning to the step S920, obtaining the error influence of the discarded sampling points after being discarded independently on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair, and finding out the singly discarded sampling point with the minimum error influence; if the minimum error influence is less than or equal to the preset error threshold, discarding the sampling point corresponding to the minimum error influence, updating the position of the reserved sampling point, and turning to step S910; if the minimum error influence result is larger than the preset error threshold, the step S1000 is executed;

and S1000, calculating corresponding Bessel integrals based on reserved sampling points, corresponding filter coefficients and integral kernel functions.

As an implementation manner, step S100 obtains all bezier integrals included in the dyadic green function based on a dyadic green function of fields generated by current sources at any point on a copper-clad region of an arbitrary layer of the integrated circuit at other layers, thereby determining the order of the bezier function used in the dyadic green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function; the specific process is as follows:

based on the calculation of the dyadic Green function of the field generated by the current source at any point on the copper-clad region of any layer of the integrated circuit on other layers, all Bessel integrals contained in the dyadic Green function are obtained:

all bezier integrals contained by the dyadic green function of the fields generated by the current source at any point on the copper clad regions of any layer of the integrated circuit on the other layers generally have the following form:

g (r) is a Green function to be integrated, r is a space distance acted by the Green function and is a distance between a source point and a field point of the currently calculated integrated circuit; g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of all layers of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the point current source point on the copper-clad area of the integrated circuit to the field point of other layers acting on the integrated circuit, J _v And calculating a Bessel function used in a dyadic Green function of a current source at any point on a copper-clad region of any layer of the integrated circuit on fields generated by other layers for the v order, wherein v is the order of the Bessel function, and lambda is an integral variable.

Specifically, the whole integrated circuit is provided with n layers, the serial numbers of the layers are l =0,1,2, … and n-1, the source is on the jth layer, and the electromagnetic parameters of the layers are mu _l ，ε _v，l ，ε _h，l Thickness t of layer _l ＝z _l -z _l-1 (l =1,2, … n-1), then located at (x) _T ，y _T ，z _T ) The field formed by the source of point current to the field point located at (x, y, z) can be represented by the following green's function:

wherein, the nine elements of the dyadic Green function are respectively

The calculation sequence does not affect the final calculation result.

Nine elements of the dyadic green function comprise six Bessel integrals R1-R6, and the specific acquisition process refers to patent CN112989750B.

If R1-R6 are calculated, nine components of the whole dyadic Green function can be calculated

And the key to calculating R1 to R6 is to calculate the bezier integral including the integrated circuit information, including: electromagnetic parameters of materials of each layer of the integrated circuit, thickness of each layer, working frequency of the integrated circuit, and distance from a point current source on a copper-clad area of the integrated circuit to a field point of other layers. For example, in R1, the order of the kernel function and the bezier function of the integrand determined by the integrated circuit information is:

the Bessel integral is an infinite integral of an integrand containing a Bessel function;

as can be seen from the expressions of the infinite integrals R1 to R6 of the Bezier functions, the orders of the Bezier functions are clearly shown in the expressions, wherein R1, R4 and R5 need to calculate the Bezier integrals of 1 order, and R2, R3 and R6 need to calculate the Bezier integrals of 0 order.

Aiming at the step S200, according to an integrand function formed by the product of the integral kernel function and the Bessel function, carrying out Hankel transformation on the Bessel integral; the integral kernel function is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from a source point to a field point, wherein the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-coated region of the integrated circuit to the field point acting on other layers; the hankel transformation for this example is as follows:

the variables r and λ in equation (1) are exponentially transformed:

r＝e ^x ，λ＝e ^-y (2)

wherein x and y are new variables of exponential transformation. Since the spatial distance between the source point and the field point of the currently computed integrated circuit and the integral variable are both greater than 0, the range of the exponentially transformed variables x, y is both- ∞ to + ∞.

Substituting the formula (2) into the formula (1) to obtain a Bessel integral transformation formula:

equation (3) is rewritten as:

for step S300: calculating errors of a numerical solution of the Bezier integral and an accurate solution of the Bezier integral by using a Hankel transformation method based on uniform sampling points by taking the sampling intervals as variables, obtaining a relation curve of the sampling intervals and the calculated errors of the Bezier integral, and obtaining a series of sampling intervals with local minimum errors based on the relation curve to serve as a fixed sampling interval sequence;

referring to fig. 2, when the calculated errors of the uniform sampling points are researched by taking different sampling intervals as variables, the error is the smallest when the sampling interval of the uniform sampling points is 0.05 for the classical bezier integral, and besides the first time, the error is the second time as at the black point in the graph; i.e. as a sequence of fixed sampling intervals.

If the sampling intervals of all the non-uniform sampling points fall at the positions in the fixed sampling interval sequence, the fast hankel transform designed based on the non-uniform sampling points has higher calculation precision.

For step S400: discretizing the Hankel transformation by uniform sampling by taking the minimum fixed sampling interval in the fixed sampling interval sequence as a sampling interval to obtain an expression of the infinite length filter; the specific process is as follows:

discretizing equation (4) using convolution and letting x = sn, y = sm, i.e. replacing the original continuous spatial distance and integral variable with the spatial distance from the source point to the field point and the discrete integral variable of the discrete integrated circuit, yields:

wherein s is a sampling interval; m and n are discrete sequence numbers, J _v Is a Bessel function of order v.

For ease of writing, the terms in equation (5) are labeled as follows:

equation (5) is reduced to the following equation, resulting in an infinite length filter expression:

wherein G (-) is an input function, which is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance between a source point and a field point, h (-) is a filter function, which is determined by Bezier function orders contained in a dyadic Green function of fields generated by current sources at any point on a copper-clad region of any layer of the integrated circuit on other layers, and G (-) is an output function, which is a finally calculated Bezier integral.

As an implementable manner, step S500, truncating the infinite length filter within a preset precision control range according to the layer thickness of different layers of the integrated circuit and the attenuation speed of an integrand determined by the characteristics of an interlayer medium to obtain an expression of the finite length filter, and obtaining an equation set of the finite length filter according to the expression of the finite length filter;

the length of the ideal filter is infinite, it is noted that an integral kernel function determined by electromagnetic parameters of materials of all layers of an integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance between a source point and a field point is an exponential decay function, a Bessel function contained in a dyadic Green function of a field generated by a current source at any point on a copper-clad region of any layer of the integrated circuit on other layers is an oscillation decay function, and the filter with the finite length can be arranged in a precision control range, so that the precision of an output function reaches the precision requirement specified in advance. If the filter length is 2L +1, equation (7) becomes:

as can be seen from equation (8), for each value of the output function G (n), 2L +1 value of the input function G (n + m) can be found, thereby obtaining 1 system of equations. To solve this filter coefficient, i.e. to solve the response problem of the kernel function, 2l +1 equation sets relating to the filter coefficient h need to be written in sequence; if a function pair with an analytical solution is selected, and the input function G (-) and the output function G (-) are known, then a system of equations relating G (-) to h (-) can be established to solve for the filter coefficient h. However, an analytic solution can not be found at all times, and when the analytic solution cannot be found, an inaccurate bessel function integral or even a wrong bessel function integral can be obtained by using the method, so that the output of the finite-length filter is obtained by using the improved adaptive piecewise integration method, that is, the right-end term of formula (9) is obtained, as shown in the following formula:

obtaining the equation system of the finite length filter according to the expression of the finite length filter, namely obtaining the equation system of the finite length filter through the formula (8):

in specific application, the improved filter coefficient matrix equation can be correctly solved according to the Hankel transformation pair of the known analytic expression to obtain a filter coefficient vector h, and finally, the Bessel function integral is solved according to the filter coefficient and the sampling point of the filter. The correct solution of Bessel function integral is the basis of the correct analysis of the electromagnetic field, and whether the electromagnetic field can be analyzed correctly or not is related. However, when the electromagnetic field from a source point formed by a continuous current discrete point current source to a field point acting on other layers on a copper-clad area of a multilayer complex integrated circuit is analyzed, for most Bessel integrals, an analytical expression of the integrals cannot be obtained, and the integration containing the Bessel function as the common function is accurately obtained by an improved adaptive piecewise integration method according to a set error threshold.

As an implementation, step S600: constructing a Bessel integral-Hankel transformation pair by adopting a known Bessel integral analysis expression, or constructing a Bessel integral which is closest to an actual kernel function by adopting typical parameters suitable for an integrated circuit aiming at the Bessel integral which cannot be analyzed actually to form the Bessel integral-Hankel transformation pair, and calculating the Bessel integral in the Bessel integral-Hankel transformation pair at high precision; the method specifically comprises the following steps:

s610, constructing a corresponding Bessel integral-Hankel transformation pair by adopting a known Bessel integral analytic expression: according to the category and the order of the Bezier function of which the integral is to be calculated in the dyadic Green function, the Bezier integral capable of providing an analytical expression of the integral is found, and a Bezier integral-Hankel transformation pair of the category and the order of the specified Bezier integral is constructed;

the selection of the input and output functions plays an important role in the calculation accuracy of the filter coefficient, and h is essentially a vector of the linear filter coefficient, can be determined in advance according to the orders of the integral kernel function and the Bessel function, and is then applied to calculating the field generated by the current source at any point on the copper-clad area of any layer of the integrated circuit on other layers. The method constructs a Bessel integral-Hankel transformation pair of the specified class and order of the Bessel integral as follows:

wherein a and b are constants, the closer the constant is to the values determined by the electromagnetic parameters of materials of each layer of the integrated circuit, the thickness of each layer and the working frequency of the integrated circuit, the more accurate the Bessel integral calculated by using the transformation to carry out the Hankel transformation on the calculated filter coefficient is.

S620, aiming at the Bessel integral which cannot be actually analyzed, constructing the Bessel integral closest to an actual kernel function by adopting typical parameters suitable for an integrated circuit to form a Bessel integral-Hankel transformation pair; the method comprises the following steps:

the output of the finite length filter is obtained by adopting an improved self-adaptive segmented integration method for Bessel integration corresponding to the integrand of the Hankel transform filter, and the specific implementation process is as follows:

step S621.1: extracting a Bezier function of an integrand of the Hankel transform filter, and calculating a zero point of the Bezier function by adopting an iteration method; determining the range or the number of zero points according to the convergence speed of an integrand formed by the product of the integral kernel function and the Bessel function;

calculating zero of Bezier function by iteration method, i.e. determining that zero a of Bezier function is the solution of Bezier function value 0, i.e. J _v (a) A solution of =0, calculated by the following Halley algorithm:

step S621.1, setting p =1;

step S621.2, setting the initial guess value of the p-th zero point

Step S621.3, calculating by the following iterative formula

Nearby Bessel function J _v (a) P th zero point of (c):

and (3) iteration termination conditions:

where δ is a predefined threshold. Wherein, J' _v Representing Bessel function J _v First derivative of (A), J ″ _v Representing Bessel function J _v V is the order of the Bessel function; q denotes the qth iteration, and the value at q =0 is the initial guess value.

Step S621.4, if the calculated zero point reaches the specified interval range, the zero point calculation is completed; otherwise, let p = p +1 and proceed to step S6.1.2.

According to the zero point of the Bessel function, the sectional point forming the integral interval obtained based on the zero point of the Bessel function can be determined as

Wherein r' is the space distance acted by the green function, and the integral interval is equivalent to the value range of the integral variable.

Step S622: performing self-adaptive integration on an integrand formed by the product of an integral kernel function and a Bessel function, and setting an integral subinterval as m =1;

step S623: adaptively segmenting the mth subinterval of the Bessel integration according to an integrand function formed by the product of the integral kernel function and the Bessel function and a zero point, performing Bessel integration and accumulation on the next-stage subinterval after the mth subinterval segmentation to obtain an accumulated integration result after the mth subinterval segmentation, and accumulating the accumulated integration result to the Bessel integration of the whole interval;

step S624: judging whether the accumulated integral result after the mth subinterval segmentation of the Bessel integral is smaller than a first threshold value or not; if the accumulated integral result after the mth subinterval division of the Bessel integration is smaller than the first threshold, the Bessel integration of the whole integral subinterval is the final integral result of the Bessel integration corresponding to the integrand function of the Hankel transform filter calculated by the adaptive segmentation method, and the step S625 is carried out; otherwise, if m = m +1, the step S623 is executed;

step S625: and obtaining the output of the finite length filter based on the Bessel integral corresponding to the integrand of the Henkel transform filter.

The self-adaptive method for segmenting the mth subinterval of the Bezier integral and accumulating the segmented next-level subinterval of the mth subinterval to obtain the accumulated integral result of the segmented mth subinterval comprises the following steps:

step S623.1: setting the number j of segmentation, wherein j is equal to zero before segmentation;

step S623.2: setting j = j +1, and calculating a division point when the division frequency is j

Comprises a starting point and an end point;

step S623.3: using division points

Dividing the integral of the mth subinterval into the accumulation of j +1 integral of next subintervals;

step S623.4: integration for each next level subinterval

Calculating the integral value of each next-stage subinterval by adopting a Gaussian integration method and accumulating the integral value to be used as the integral of the mth subinterval

Step S623.5: comparing the integral of the mth subinterval when the number of times of division is j

Integral with mth subinterval at previous division

Step S623.6: if it is satisfied with

If the value is less than the second threshold value, the division is finished to obtain the integral result after the m subinterval division, namely

Otherwise, go to step S623.2.

Calculating the division point when the division times is j

The formula is as follows:

wherein d is the d next level subinterval, λ _m Is the piecewise integral variable of the mth subinterval.

Using division points

Dividing the integral of the mth subinterval into the accumulation of j +1 integral of next-level subintervals, and the formula is as follows:

wherein g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of each layer of the integrated circuit, the thickness of each layer, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable; j. the design is a square _v In order to calculate a v-order Bessel function used in a dyadic Green function of a current source at any point on a copper-clad region of any layer of an integrated circuit in fields generated by other layers, v is the order of the Bessel function,

of the d-th next-level sub-interval of the m-th sub-interval divided by jAnd f, a green function to be integrated, wherein r is the currently calculated space distance between the source point and the field point of the integrated circuit.

And (3) calculating the integral value of each next-stage subinterval by adopting a Gaussian integration method, wherein the formula is as follows:

wherein r is the spatial distance acted by the green function and the distance between a source point and a field point of the integrated circuit which is calculated at present, and K is the total number of Gaussian integration points; d is each next level subinterval [ lambda ] _d-1 ，λ _d ]Conversion to standard Gaussian integration interval [ -1,1]Of Jacobian transformation of D ^-1 An inverse transform of D; x is the number of _k Is the kth Gaussian point, w _k Is the weight corresponding to the kth gaussian point; g (D (x) _k ) A function g (λ) is taken to be D (x) at λ _k ) The time value g (lambda) is an integral kernel function determined according to the currently calculated electromagnetic parameters of the source point and the field point of the integrated circuit and materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from the source point to the field point of the integrated circuit, and lambda is an integral variable; j. the design is a square _v In order to calculate a v-order Bessel function used in a dyadic Green function of a current source at any point on a copper-clad region of any layer of an integrated circuit in fields generated by other layers, v is the order of the Bessel function.

S620, aiming at the Bessel integral which cannot be actually analyzed, constructing the Bessel integral closest to an actual kernel function by adopting typical parameters suitable for an integrated circuit to form a Bessel integral-Hankel transformation pair; further comprising:

from the output of the finite length filter, we construct pairs of bezier integral-hankerr transforms corresponding to different components of the dyadic green function, as follows:

for the infinite integral R1 of a Bezier function corresponding to 9 components of a dyadic Green function of a current source on a copper-clad area of any layer of an integrated circuit, wherein the fields are generated by other layers at any point, the kernel function of the product function is as follows:

having a Bessel function of

According to the method, a common function of the integrand formed by the product of the kernel function of the integrand and the Bessel function is obtained, and the kernel function g (lambda) and the Bessel function of the integrand are obtained

And substituting the integration result G of Bezier infinite integration expressed by the common function and calculated by the self-adaptive segmented integration method into the formula (6) to form a Bezier integration-Hankel transformation pair corresponding to the infinite integration R1 of the Bezier function corresponding to 9 components of the parallel vector Green function of the field generated by the current source at any point on the copper-clad region of any layer of the integrated circuit on other layers.

For step S700: substituting the constructed Bessel integral-Hankel transformation pair into an expression of a filter with a limited length to obtain a filter coefficient matrix equation corresponding to a filter equation set, and improving the filter coefficient matrix equation; the specific implementation steps are as follows:

step S701: substituting the Bessel integral-Hankel transformation pair based on the structure into an expression of a finite length filter:

for Bessel function J shown in equation (10) ₀ The Bessel integral-Hankel transform pair of (B) has:

for Bessel function J shown in equation (11) ₁ The Bessel integral-Hankel transform pair of (B) has:

for the case where a corresponding bezier integral-hankerr transformation pair is constructed using a known bezier integral analysis expression, the above equations (12) and (14) are calculated using step S610; the bezier integral that cannot be actually analyzed is calculated by the above equations (12) and (14) in step S620.

Step S702: obtaining a filter coefficient matrix equation corresponding to the filter equation set;

in this embodiment, the matrix equation can be solved by substituting equations (12) and (13) into equation (9) to obtain h ₀ (ii) a Substituting the equations (14) and (15) into the equation (9), the matrix equation can be solved to obtain h ₁ To obtain the corresponding Bessel function J ₀ And J ₁ The filter coefficients of (1).

Step S703: improving a filter coefficient matrix equation;

step S702 is found in the actual calculation of the present embodiment: the filter coefficient matrix equation corresponding to the obtained filter equation set is seriously ill, so that the filter coefficient is incorrectly solved. For example, the designed filter length is 201, and the size of the formed matrix is 2l +1=201, where L represents the value range of the finite length filter, that is, the value range of the finite length filter is [ -L, L [ -L + =201]With a condition number of up to 10 ³⁰ ～10 ¹⁰⁰ Is caused by a constant value of a and a filter length of 2l +1 in equations (13) and (15); due to the fact that

When one of n and m is not changed, the other is in [ -L, L]In the middle variation, if the sampling interval s is small or L is large, the corresponding elements of adjacent rows or columns (corresponding n or m only differs by 1) in the matrix are not very different, that is, the adjacent row or column elements in the matrix g have high repeatability, within the identification accuracy of a computer, the determinant corresponding to the matrix is close to zero, the matrix is close to singular, and the direct solution of the matrix can cause the incorrect solved filter coefficient. The improved formula (19) is calculated in the precision range, the condition number of the filter coefficient matrix can be obviously improved, and the improved matrix equation can be correctly solved. The specific derivation process is as follows:

step S703.1: establishing a filter matrix equation according to an equation set of the finite length filter;

writing the original equation set (9) in matrix form:

gh＝G (16)

g is matrix representation of an input function, which is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance between a source point and a field point, h is matrix representation of a filter coefficient, the Bessel function order contained in a parallel vector Green function of fields generated by current sources at any point on a copper-clad region of any layer of the integrated circuit on other layers is determined, and G is matrix representation of an output function, namely an analytic value of a selected Bessel integral with an analytic expression under each corresponding n;

step S703.2: adding a column vector delta h to both sides of a filter matrix equation; this example calculates as follows:

gh+δh＝G+δh (17)

step S703.3: delta is taken to be a value small enough that the ratio between the modulus of the column vector and the modulus of the output vector is less than or equal to a third threshold;

in the formula, | | · | | represents the modulus of the vector, TH3 represents the secondThree thresholds, such as delta can be set to machine accuracy 10- ¹⁶ Such that the ratio between the modulus of the column vector and the modulus of the output vector is equal to or less than a third threshold.

Step S703.4: and neglecting the column vector delta h of the output end in the filter matrix equation to obtain the improved filter coefficient matrix equation.

Since δ is small enough that δ h is negligible compared to G, equation (17) can be simplified as:

(g+δI)h＝G (19)

δ is a sufficiently small constant and I is an identity matrix.

The matrix of the equation shown in equation (19) is changed from g to g + δ I, which is shown in that each diagonal element of the matrix g is increased by a small amount, but the condition number of the matrix g is greatly reduced as a result, because the relationship between the matrix condition number and the matrix eigenvalue shows that:

in the formula

And

respectively representing the maximum and minimum values of the eigenvalues of matrix g, and cond (g) representing the condition number of matrix g.

By definition of matrix eigenvalues:

gx＝Λx (21)

the matrix g + δ I satisfies:

(g+δI)x＝(Λ+δ)x (22)

it can be seen that the eigenvalues of matrix g + δ I become Λ + δ, so that the condition number of matrix g + δ I is:

obviously, a small number of delta pairs

Has little influence but on

Has a large influence, for example, up to 10 in cond (g) ¹⁰⁰ When the temperature of the water is higher than the set temperature,

up to 10 ^-100 Of order, if δ is set to machine accuracy, e.g. 10 ^-16 Then cond (g + δ I) can be improved to 10 ¹⁶ The order of the filter coefficients is improved obviously, so that the condition number of the filter coefficient matrix is improved, and the improved matrix equation can be solved correctly.

As an implementation, step S900: for sampling points obtained by uniform sampling with the minimum fixed sampling interval as the sampling interval, screening and preliminarily determining abandoned sampling points according to the contribution values of the sampling points, and judging whether the sampling points are abandoned or not according to the error influence of the abandonment of the preliminarily determined abandoned sampling points on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair; the method comprises the following steps:

s930, recovering to try to discard the ith sampling point in the preliminarily determined discarded sampling points and recovering to try to update the positions of the reserved sampling points, enabling i = i +1, turning to the step S920, obtaining the error influence of the discarded sampling points after being discarded independently on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair, and finding out the singly discarded sampling point with the minimum error influence; if the minimum error influence is less than or equal to the preset error threshold, discarding the sampling point corresponding to the minimum error influence, updating the position of the reserved sampling point, and turning to step S910; if the minimum error influence result is greater than the preset error threshold, go to step S1000.

Referring to fig. 3, step S900 specifically includes:

s911, setting a retention threshold value based on the preliminary abandoned sampling point of the contribution value and an error threshold value based on the error abandoned sampling point according to the layer thickness of different layers of the integrated circuit and the attenuation speed of the integrand determined by the characteristics of the medium between the layers;

the contribution is calculated by:

wherein, C _i Is the contribution of the ith sample point, r is the spatial distance acted by the Green function, h _i The filter coefficient of the ith sampling point is determined by the Bessel function order contained in the dyadic Green function of the field generated by the current source at any point on the copper-clad area of any layer of the integrated circuit on other layers, and lambda is _i G (-) is an input function determined by electromagnetic parameters of materials of all layers of the integrated circuit, thicknesses of all layers, working frequency of the integrated circuit and the distance between a source point and a field point, G (-) is an output function, and 2L < -1 > is the length of the filter;

s913, identifying all the sampling points with the contribution values smaller than the retention threshold of the sampling points as the preliminarily determined abandoned sampling points to form a preliminarily determined abandoned sampling point set; if the abandoned sampling point set is preliminarily determined to be an empty set, the step S1000 is carried out, otherwise, the step S921 is carried out;

as a determination and update method: after the ith sampling point is tried to be abandoned, the position updating of the sampling points around the ith sampling point is tried to meet the requirement that the ith-1 sampling point and/or the (i + 1) th sampling point are moved or are not moved, so that the deviation between the sampling interval between the (i-2) th sampling point and the (i-1) th sampling point, the (i-1) th sampling point and the (i + 2) th sampling point and one of fixed sampling intervals in the fixed sampling interval sequence is minimum (namely, the sampling points are moved to enable the sampling interval between the sampling points to be the closest sampling interval of the fixed sampling interval sequence, and therefore, the + deviation of the sampling intervals between the sampling points = one of the fixed sampling intervals in the fixed sampling interval sequence); further judging the size of the deviation threshold value of the deviation and the preset sampling interval, namely, the deviation can not be discarded if the deviation is too large; if all the deviations are smaller than a preset sampling interval deviation threshold value, the ith sampling point can try to be removed, and the sampling points around the ith sampling point are subjected to position updating according to the position updating attempt; if any deviation is more than or equal to a preset sampling interval deviation threshold value, the ith sampling point cannot try to remove the deviation, and the position is not updated;

as a determination and update method: and adding corresponding deviation to each fixed sampling interval except the minimum fixed sampling interval in the fixed sampling interval sequence, and adjusting the deviation to be integral multiple of the minimum fixed sampling interval in the fixed sampling interval sequence. In fig. 2, the fixed sampling interval sequences obtained according to the error curves are 0.05, 0.08, 0.11, 0.136, 0.149 and 0.16, the corresponding deviations can be added to 0.08, 0.11 and 0.149 to become 0.10, 0.10 and 0.15, and finally, the repeated adjusted sampling sequences are removed, and the fixed sampling interval sequences are adjusted to 0.05, 0.10 and 0.15 to form new samplesA fixed sequence of sampling intervals; the rule for adjusting the original fixed sampling interval sequence is: if the added deviation of the fixed sampling interval in the fixed sampling interval sequence is smaller than the rounding deviation of the preset fixed sampling interval, and the error value corresponding to the adjusted fixed sampling interval is smaller than the error threshold of the sampling interval, determining the fixed sampling interval corresponding to the deviation as a new fixed sampling interval sequence (if the rounding deviation of the preset fixed sampling interval is 0.02, the error threshold of the preset sampling interval is 10) ^-11 Firstly, according to the rounding deviation of 0.02, the original fixed sampling intervals of 0.08 and 0.11 are adjusted to be 0.10, the original fixed sampling intervals of 0.136 and 0.16 do not meet the rounding deviation, the fixed sampling intervals are removed, and the original fixed sampling interval of 0.149 is adjusted to be 0.15; secondly, whether the error values corresponding to the adjusted sampling intervals 0.10 and 0.15 meet the error threshold value 10 smaller than the sampling interval or not is judged ^-11 It can be seen in fig. 2 that it is satisfied that a new sequence of fixed sampling intervals of 0.05, 0.10, 0.15) is finally formed); if the deviation is greater than or equal to the preset fixed sampling interval rounding deviation, or the error value corresponding to the adjusted fixed sampling interval is greater than or equal to the sampling interval error threshold, the fixed sampling interval corresponding to the deviation is removed;

based on a new fixed sampling interval sequence, if trying to abandon an ith sampling point, trying to update the positions of the sampling points around the ith sampling point to meet the requirement that the ith-1 sampling point and/or the (i + 1) th sampling point are moved or not moved, so that the sampling intervals between the (i-2) th sampling point and the (i-1) th sampling point, the (i-1) th sampling point and the (i + 1) th sampling point, the (i + 1) th sampling point and the (i + 2) th sampling point are one or more in the new fixed sampling interval sequence, and the moving distance square sum of the (i-1) th sampling point and the (i + 1) th sampling point is the minimum, then the ith sampling point can be tried to be removed, and the positions of the sampling points around the ith sampling point are tried to be updated according to the position update;

s922, calculating a Bezier function integral based on the filter coefficient to obtain a numerical solution of the Bezier function integral in a Bezier integral-Hankel transformation pair;

s923, calculating an accurate solution obtained by integrating the Bezier function in the Bezier integral-Hankel transformation pair by adopting an analytical expression or a high-accuracy calculation method, and calculating a relative error between a numerical solution and the accurate solution to obtain an error influence;

s931, recovering to try to discard the ith sampling point in the preliminarily determined discarded sampling point set, and recovering to try to update the positions of the sampling points around the ith sampling point, setting i = i +1, and turning to S921 to obtain a relative error set after trying to discard each preliminarily determined discarded sampling point independently;

s932, obtaining a minimum relative error according to the error size of the relative error set;

s933, if the minimum relative error is smaller than or equal to a preset error threshold, discarding a sampling point and a filter coefficient corresponding to the minimum relative error, updating positions of sampling points around the sampling point corresponding to the minimum relative error, and turning to S912; if the minimum relative error is greater than the predetermined error threshold, go to step S1000.

For step S1000, based on the retained sample points and the corresponding filter coefficients and the integral kernel function, the corresponding bezier integral is calculated:

and (3) according to the reserved sampling points lambda ' and the reserved filter coefficients h ', numbering the reserved sampling points from 1 to L ', and calculating the Bessel integral shown in the formula by using the following formula:

referring to fig. 4, a non-uniform sampling optimization apparatus for a hankel transform filter of an integrated circuit includes:

the device comprises a Bessel integral acquisition module 100, a Hankel transformation module 200, a fixed sampling interval sequence screening module 300, an infinite filter module 400, a finite filter module 500, a transformation pair construction module 600, a matrix equation improvement module 700, a filter coefficient calculation module 800, a sampling point rejection module 900 and a Bessel integral calculation module 1000;

the Bezier integral obtaining module 100 is used for obtaining all Bezier integrals contained in a dyadic Green function based on a dyadic Green function of fields generated by current sources at any point on a copper-clad region of any layer of the integrated circuit on other layers, and accordingly determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function;

the hankerr transformation module 200 is configured to perform hankerr transformation on the bezier integral according to an integrand formed by multiplying the integral kernel function by the bezier function; the integral kernel function is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from a source point to a field point, wherein the distance from the source point to the field point is the distance from the source point formed by a point current source with discrete continuous current on a copper-clad area of the integrated circuit to the field point acting on other layers;

the fixed sampling interval sequence screening module 300 is configured to calculate an error between a numerical solution of a bezier integral and an accurate solution of the bezier integral by using a sampling interval as a variable based on a hankel transform method of uniform sampling points, obtain a relation curve between the sampling interval and the calculated error of the bezier integral, and obtain a series of sampling intervals with locally minimum errors based on the relation curve to serve as a fixed sampling interval sequence;

the infinite filter expression module 400 is configured to discretize the hankerr transform by uniform sampling with a minimum fixed sampling interval in the fixed sampling interval sequence as a sampling interval, to obtain an expression of an infinite length filter;

the finite filter expression module 500 is configured to truncate the infinite length filter within a preset precision control range according to the layer thicknesses of different layers of the integrated circuit and the attenuation speed of the integrand determined by the characteristics of the interlayer medium, to obtain an expression of the finite length filter, and to obtain an equation set of the finite length filter according to the expression of the finite length filter;

the transformation pair construction module 600 is configured to construct a bezier integral-hankerr transformation pair by using a known bezier integral analysis expression, or construct a bezier integral closest to an actual kernel function by using typical parameters applicable to an integrated circuit for a bezier integral that cannot be actually analyzed, to form the bezier integral-hankerr transformation pair, and calculate the bezier integral in the bezier integral-hankerr transformation pair with high precision;

the matrix equation improvement module 700 is configured to substitute the constructed bezier integral-hankel transformation pair into the expression of the finite length filter to obtain a filter coefficient matrix equation corresponding to a filter equation set, and improve the filter coefficient matrix equation;

the filter coefficient calculation module 800 is configured to solve the improved filter coefficient matrix equation to obtain a filter coefficient;

the sampling point abandoning module 900 is configured to screen a sampling point which is preliminarily determined to be abandoned according to the contribution value of the sampling point for a sampling point obtained by uniform sampling with the minimum fixed sampling interval as a sampling interval, and determine whether the sampling point is abandoned according to the error influence of abandonment of the preliminarily determined abandoned sampling point on the calculation of the whole bessel integral in the bessel integral-hankerr transformation pair;

the bezier integral calculation module 1000 is configured to calculate a corresponding bezier integral based on the reserved sampling points and the corresponding filter coefficients and integral kernel functions;

the sampling point rejection module 900 includes a preliminary determination rejection sampling point screening unit 910, a bezier integral error influence calculation unit 920, and a rejection point update unit 930;

the preliminary determined abandoned sample point screening unit 910 is configured to determine, as a preliminary determined abandoned sample point, a sample point corresponding to a sample point whose contribution value is smaller than a reserved threshold value by calculating a contribution value of each sample point except for a first sample point and a last sample point; if the sampling points which are discarded in the preliminary determination do not exist, switching to the Bessel integral calculation module 1000;

the bezier integral error influence calculation unit 920 is configured to try to discard an ith sampling point of the preliminarily determined discarded sampling points and try to update the position of the retained sampling point based on a relationship between the current sampling interval and the sampling interval in the fixed sampling interval sequence to form a non-uniform sampling point, and calculate an error influence of discarding the ith sampling point of the preliminarily determined discarded sampling points on the calculation of the whole bezier integral in the bezier integral-hankel transform pair;

the point-cut updating unit 930 is configured to recover the attempt to discard an ith sampling point of the preliminarily determined discarded sampling points and recover the attempt to perform position updating on the remaining sampling points, let i = i +1, and shift to the bezier integral error influence calculation unit to obtain an error influence calculated on the entire bezier integral in the bezier integral-hankel transformation pair after all the preliminarily determined discarded sampling points are discarded separately, and find out an individually discarded sampling point with a minimum error influence; if the minimum error influence is less than or equal to the error threshold, discarding the sampling point corresponding to the minimum error influence, performing position update on the reserved sampling point, and turning to the preliminary determination discarded sampling point screening unit 910; if the minimum error influence result is greater than the error threshold, the Bessel integral calculation module 1000 is switched to.

As an implementable embodiment, the preliminary determination rejected sampling point screening unit 910 includes an error threshold setting subunit 911, a contribution value calculating subunit 912, and a preliminary determination rejected sampling point determining subunit 913;

the error threshold setting subunit 911 is configured to set the retention threshold based on the preliminary discarded sampling points of the contribution value and the error threshold based on the error discarded sampling points according to the layer thicknesses of different layers of the integrated circuit and the attenuation speed of the integrand determined by the characteristics of the interlayer medium;

the contribution value operator unit 912, configured to calculate a contribution value of each sampling point except the first sampling point and the last sampling point according to the filter coefficient and the sampling point of the filter;

the preliminary determined abandoned sampling point determining subunit 913 is configured to regard all the sampling points whose contribution values are smaller than the retention threshold of the sampling points as the preliminary determined abandoned sampling points, so as to form a set of preliminary determined abandoned sampling points; if the discarded sampling point set is determined to be an empty set preliminarily, switching to the Bessel integral calculation module 1000, otherwise, switching to a filter coefficient calculation subunit 921;

the bessel integral error influence calculation unit 920 includes a filter coefficient calculation subunit 921, a numerical solution calculation subunit 922, and an error influence calculation subunit 923;

the filter coefficient calculation subunit 921 is configured to try to discard an ith sampling point in the set of preliminarily determined discarded sampling points, update positions of sampling points around the ith sampling point, calculate filter coefficients of remaining sampling points based on the sampling points after the sampling point is tried to be discarded, and obtain the filter coefficients of the sampling points after the sampling point is tried to be discarded;

the numerical solution calculating subunit 922 is configured to calculate a bezier function integral based on the filter coefficient, and obtain a numerical solution of the bezier function integral in the bezier integral-hankerr transformation pair;

the error influence calculation subunit 923 is configured to calculate a relative error between the numerical solution and the precise solution based on a precise solution obtained by calculating the bezier function integral in the bezier integral-hankerr transformation pair by using an analytic expression or a high-precision calculation method, so as to obtain an error influence;

the cut-point updating unit 930 includes a relative error set summation operator unit 931, a minimum relative error calculation operator unit 932, and a cut-point updating subunit 933;

the relative error set calculating subunit 931, configured to restore the trying to discard an ith sampling point in the preliminarily determined discarded sampling point set and restore the trying to update positions of sampling points around the ith sampling point, set i = i +1, and shift to the filter coefficient calculating subunit 921, so as to obtain a relative error set after trying to discard each preliminarily determined discarded sampling point individually;

the minimum relative error calculation subunit 932 is configured to obtain a minimum relative error according to the error size of the relative error set;

the point dropping updating subunit 933 is configured to, if the minimum relative error is smaller than or equal to the preset error threshold, discard the sampling point and the filter coefficient corresponding to the minimum relative error, perform position updating on the sampling points around the sampling point corresponding to the minimum relative error, and shift the sampling points to the contribution value operator unit 912; and if the minimum relative error is larger than a preset error threshold, switching to the Bessel integral calculation module 1000.

after the ith sampling point is tried to be abandoned, the position of the sampling points around the ith sampling point is tried to be updated, so that the movement or non-movement of the (i-1) th sampling point and/or the (i + 1) th sampling point is met, and the deviation between the sampling interval between the (i-2) th sampling point and the (i-1) th sampling point, the (i-1) th sampling point and the (i + 2) th sampling point and one fixed sampling interval in the fixed sampling interval sequence is minimum; if all the deviations are smaller than a preset sampling interval deviation threshold value, the ith sampling point can be tried to be removed, and the sampling points around the ith sampling point are subjected to position updating according to the position updating attempt; if any deviation is larger than or equal to a preset sampling interval deviation threshold value, the ith sampling point cannot try to remove the deviation, and the position is not updated.

As an implementable mode, the attempting to discard an ith sampling point of the preliminary determination discarded sampling points and attempting to perform position update on sampling points around the ith sampling point based on the relation between the sampling interval and the fixed sampling interval sequence includes: adding corresponding deviation to each fixed sampling interval except the minimum fixed sampling interval in the fixed sampling interval sequence, and adjusting the deviation to be integral multiple of the minimum fixed sampling interval in the fixed sampling interval sequence; if the deviation added to the fixed sampling interval in the fixed sampling interval sequence is smaller than the preset fixed sampling interval rounding deviation, and the error value corresponding to the adjusted fixed sampling interval is smaller than the sampling interval error threshold, determining the fixed sampling interval corresponding to the deviation as a new fixed sampling interval sequence; if the deviation is greater than or equal to the preset fixed sampling interval rounding deviation, or the error value corresponding to the adjusted fixed sampling interval is greater than or equal to the sampling interval error threshold, removing the fixed sampling interval corresponding to the deviation;

Finally, it is noted that the above-mentioned embodiments illustrate rather than limit the invention, and that, while the invention has been described with reference to preferred embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the invention as defined by the appended claims.

Claims

1. A non-uniform sampling optimization method for a Hankel transform filter of an integrated circuit, comprising:

s100, acquiring all Bezier integrals contained in a dyadic Green function based on the dyadic Green function of a field generated by a current source at any point on a copper-clad region of any layer of the integrated circuit on other layers, and determining the order of the Bezier function used in the dyadic Green function; the Bessel integral is an infinite integral of an integrand containing a Bessel function;

s200, according to an integrand function formed by the product of the integral kernel function and the Bessel function, carrying out Hankel transformation on the Bessel integral; the integral kernel function is determined by electromagnetic parameters of materials of all layers of the integrated circuit, the thickness of all layers, the working frequency of the integrated circuit and the distance from a source point to a field point, wherein the distance from the source point to the field point is the distance from the source point formed by a continuous current discrete point current source on a copper-clad area of the integrated circuit to the field point acting on other layers;

s930, resuming the attempting to discard the ith sampling point of the preliminarily determined discarded sampling points and resuming the attempting to perform position updating on the remaining sampling points, letting i = i +1, and going to the step S920, obtaining an error influence calculated on the whole bezier integral in the bezier integral-hankerr transformation pair after all the preliminarily determined discarded sampling points are discarded individually, and finding out an individually discarded sampling point with a minimum error influence; if the minimum error influence is less than or equal to the preset error threshold, discarding the sampling point corresponding to the minimum error influence, updating the position of the reserved sampling point, and turning to step S910; if the minimum error influence result is larger than the preset error threshold, the step S1000 is switched to;

2. The method for non-uniform sampling optimization for a hankel transform filter of an integrated circuit according to claim 1, wherein S900 is specifically:

s913, identifying all the sampling points with the contribution values smaller than the retention threshold of the sampling points as the preliminarily determined abandoned sampling points to form a preliminarily determined abandoned sampling point set; if the discarded sampling point set is determined to be an empty set preliminarily, the step S1000 is carried out, otherwise, the step S921 is carried out;

s922, calculating a Bezier function integral based on the filter coefficient to obtain a numerical solution of the Bezier function integral in the Bezier integral-Hankel transformation pair;

3. The method for non-uniform sampling optimization of the hankel transform filter of the integrated circuit according to claim 2, wherein the attempting to discard an ith sample point of the preliminarily determined discarded sample points and attempting to perform position update on sample points around the ith sample point based on the relationship between the sampling interval and the fixed sampling interval sequence comprises:

after trying to abandon the ith sampling point, trying to update the positions of the sampling points around the ith sampling point to meet the requirement that the ith-1 sampling point and/or the (i + 1) th sampling point are moved or not moved, so that the deviation of the sampling interval between the (i-2) th sampling point and the (i-1) th sampling point, the (i-1) th sampling point and the (i + 2) th sampling point and one of the fixed sampling intervals in the fixed sampling interval sequence is minimum; if all the deviations are smaller than a preset sampling interval deviation threshold value, the ith sampling point can be tried to be removed, and the sampling points around the ith sampling point are subjected to position updating according to the position updating attempt; if any deviation is larger than or equal to a preset sampling interval deviation threshold value, the ith sampling point cannot try to remove the deviation, and the position is not updated.

4. The method for non-uniform sampling optimization of the hankel transform filter of the integrated circuit according to claim 2, wherein the attempting to discard an ith sample point of the preliminarily determined discarded sample points and attempting to perform position update on sample points around the ith sample point based on the relationship between the sampling interval and the fixed sampling interval sequence comprises: adding corresponding deviation to each fixed sampling interval except the minimum fixed sampling interval in the fixed sampling interval sequence, and adjusting the deviation to be integral multiple of the minimum fixed sampling interval in the fixed sampling interval sequence; if the added deviation of the fixed sampling interval in the fixed sampling interval sequence is smaller than the rounding deviation of the preset fixed sampling interval and the error value corresponding to the adjusted fixed sampling interval is smaller than the sampling interval error threshold, determining the fixed sampling interval corresponding to the deviation as a new fixed sampling interval sequence; if the deviation is greater than or equal to the preset fixed sampling interval rounding deviation, or the error value corresponding to the adjusted fixed sampling interval is greater than or equal to the sampling interval error threshold, removing the fixed sampling interval corresponding to the deviation;

5. The integrated circuit of claim 2, wherein the contribution is calculated by:

wherein, C _i Is the contribution of the ith sample point, r is the spatial distance of the Green function, h _i The filter coefficient of the ith sampling point is determined by the Bessel function order contained in the dyadic Green function of the field generated by the current source at any point on the copper-clad region of any layer of the integrated circuit on other layers, and lambda is _i For the ith sample point, the number of samples,g

is an input function determined by the electromagnetic parameters of the materials of the layers of the integrated circuit, the thickness of the layers, the operating frequency of the integrated circuit, and the distance from the source point to the field point,G

is an output function, 2L +1 is the filter length, 2L is the 2L sampling point, λ _m Is the sampling point corresponding to the mth subinterval of the Bessel integral, h _m And integrating the filter coefficient of the sampling point corresponding to the mth subinterval by Bessel.

6. An apparatus for non-uniform sampling optimization for a hankel transform filter in an integrated circuit, comprising:

the Bessel integral obtaining module is used for obtaining all Bessel integrals contained in the dyadic Green function based on the dyadic Green function of the field generated by a current source at any point on a copper-clad region of any layer of the integrated circuit on other layers, so that the order of the Bessel function used in the dyadic Green function is determined; the Bessel integral is an infinite integral of an integrand containing a Bessel function;

the fixed sampling interval sequence screening module is used for calculating errors of a numerical solution of the Bezier integral and an accurate solution of the Bezier integral by taking a sampling interval as a variable based on a Hankel transformation method of uniform sampling points, obtaining a relation curve of the sampling interval and the calculated errors of the Bezier integral, and obtaining a series of sampling intervals with the local minimum errors based on the relation curve to serve as a fixed sampling interval sequence;

the transformation pair construction module is used for constructing a Bessel integral-Hank transformation pair by adopting a known Bessel integral analysis expression, or constructing the Bessel integral closest to an actual kernel function by adopting typical parameters suitable for an integrated circuit aiming at the Bessel integral which cannot be analyzed actually to form the Bessel integral-Hank transformation pair, and calculating the Bessel integral in the Bessel integral-Hank transformation pair with high precision;

the matrix equation improvement module is used for substituting the constructed Bessel integral-Hankel transformation pair into the expression of the finite length filter to obtain a filter coefficient matrix equation corresponding to a filter equation set and improving the filter coefficient matrix equation;

the preliminary determination abandoned sampling point screening unit is used for calculating the contribution value of each sampling point except the first sampling point and the last sampling point, and determining the sampling point corresponding to the contribution value smaller than the reserved threshold value as the preliminary determination abandoned sampling point; if the sampling points which are determined to be abandoned preliminarily do not exist, switching to the Bessel integral calculation module;

the Bessel integral error influence calculation unit is used for trying to abandon the ith sampling point in the preliminarily determined abandoned sampling points and trying to update the positions of the reserved sampling points to form non-uniform sampling points based on the relation between the current sampling interval and the sampling intervals in the fixed sampling interval sequence, and calculating the error influence of the abandonment of the ith sampling point in the preliminarily determined abandoned sampling points on the calculation of the whole Bessel integral in the Bessel integral-Hankel transformation pair;

the system comprises a point-cut updating unit, a Bessel integral error influence calculating unit and a point-cut updating unit, wherein the point-cut updating unit is used for recovering the attempt to discard the ith sampling point in the preliminarily determined discarded sampling points and recovering the attempt to update the positions of the reserved sampling points, i = i +1 is converted into the Bessel integral error influence calculating unit, the error influence calculated on the whole Bessel integral in the Bessel integral-Hankel transformation pair after all the preliminarily determined discarded sampling points are discarded independently is obtained, and the independently discarded sampling points with the minimum error influence are found out; if the minimum error influence is less than or equal to a preset error threshold, discarding the sampling point corresponding to the minimum error influence, updating the position of the reserved sampling point, and switching to the primary determination discarded sampling point screening unit; and if the minimum error influence result is larger than a preset error threshold value, switching to the Bessel integral calculation module.

7. The integrated circuit of claim 6, wherein the integrated circuit further comprises a non-uniform sampling optimization device,

the preliminary determination abandoned sampling point screening unit comprises an error threshold value setting subunit, a contribution value calculating subunit and a preliminary determination abandoned sampling point determining subunit;

the preliminary determination abandoned sampling point determination subunit is used for determining all the sampling points with contribution values smaller than the retention threshold value of the sampling points as the preliminary determination abandoned sampling points to form a preliminary determination abandoned sampling point set; if the sampling point set which is preliminarily determined to be abandoned is an empty set, switching to the Bessel integral calculation module, and otherwise, switching to a filter coefficient calculation subunit;

the point-cut updating subunit is configured to, if the minimum relative error is smaller than or equal to the error threshold, discard the sampling point and the filter coefficient corresponding to the minimum relative error, update the positions of the sampling points around the sampling point corresponding to the minimum relative error, and switch to the contribution value operator unit; and if the minimum relative error is larger than the error threshold value, switching to the Bessel integral calculation module.

8. The apparatus for non-uniform sampling optimization for a hankel transform filter of an integrated circuit according to claim 7, wherein the attempting to discard an ith sample point of the preliminarily determined discarded sample points and attempting to perform position update on sample points around the ith sample point based on a relationship between a sampling interval and the sequence of fixed sampling intervals comprises:

9. The apparatus for non-uniform sampling optimization for a hankel transform filter of an integrated circuit according to claim 7, wherein the attempting to discard an ith sample point of the preliminarily determined discarded sample points and attempting to perform position update on sample points around the ith sample point based on a relationship between a sampling interval and the sequence of fixed sampling intervals comprises: adding corresponding deviation to each fixed sampling interval except the minimum fixed sampling interval in the fixed sampling interval sequence, and adjusting the deviation to be integral multiple of the minimum fixed sampling interval in the fixed sampling interval sequence; if the deviation added to the fixed sampling interval in the fixed sampling interval sequence is smaller than the preset fixed sampling interval rounding deviation, and the error value corresponding to the adjusted fixed sampling interval is smaller than the sampling interval error threshold, determining the fixed sampling interval corresponding to the deviation as a new fixed sampling interval sequence; if the deviation is greater than or equal to the preset fixed sampling interval rounding deviation, or the error value corresponding to the adjusted fixed sampling interval is greater than or equal to the sampling interval error threshold, removing the fixed sampling interval corresponding to the deviation;

10. The integrated circuit of claim 6, wherein the contribution value is calculated by:

wherein, C _i Is the contribution of the ith sample point, r is the spatial distance acted by the Green function, h _i The filter coefficient of the ith sampling point is determined by the Bessel function order contained in the dyadic Green function of the field generated by the current source at any point on the copper-clad region of any layer of the integrated circuit on other layers, and lambda is _i For the ith sample point, the number of samples,g