Variable line width differential inductor multi-objective optimization algorithm
Through the variable line width differential inductor multi-objective optimization algorithm, the gradient boosting tree and K-nearest neighbor algorithm are used to fit the simulation data, and the differential evolution algorithm is combined to optimize the inductor geometric parameters. The complexity problem of inverse design of RF integrated circuit passive components is solved, and efficient and accurate inductor design is achieved.
Patent Information
- Application Number
- CN202510771325.8
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-06-10
- Publication Date
- 2025-09-30
AI Technical Summary
In the design of passive integrated devices for radio frequency integrated circuits, inverse design problems, especially the structural parameter optimization of variable differential inductors, are difficult to meet high-frequency performance and area requirements. Existing technologies have problems of design complexity and low efficiency.
A multi-objective optimization algorithm for variable linewidth differential inductors is adopted. The simulation data is fitted by gradient boosting tree and K-nearest neighbor algorithm. Combined with the differential evolution algorithm, a weighted objective function is constructed to optimize the geometric parameters of the inductor to achieve multi-objective optimization.
It simplifies the design process of RF integrated circuit passive integrated devices, improves design efficiency and accuracy, achieves area minimization and performance optimization, and expands the design flexibility and scope of application.
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Figure CN120724965A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of radio frequency integrated circuits, and in particular to a multi-objective optimization algorithm for variable line width differential inductors. Background Art
[0002] Radiofrequency integrated circuits (RFICs) play a vital role in modern wireless communication systems. Passive integrated devices, such as inductors, capacitors, and resistors, operate without external power and are essential components of integrated circuits. These devices play a key role in signal tuning, matching, filtering, and energy storage. However, their integration presents numerous challenges during design, particularly during the layout stage. First, to achieve good performance at high frequencies, these devices require precise physical dimensions and shapes. This is particularly challenging in the compact layout of integrated circuits, where a large number of components must be placed on a limited silicon die while meeting design rules and performance requirements. Second, during layout design, the parasitic parameters of each passive integrated device, such as parasitic capacitance and parasitic inductance, must be carefully considered. These parasitic effects can significantly affect the device's operating frequency and performance, leading to discrepancies between the actual circuit and the intended design. Engineers often need to repeatedly adjust and verify the design to ensure that the device performs as expected.
[0003] In summary, in practical applications, the layout design of RFIC passive integrated devices is a complex and challenging task, requiring deep expertise and extensive experience. It involves repeated comparisons and adjustments to structural parameters such as the size and position of capacitors and inductors, consuming significant time through numerous simulation iterations. Through deep learning optimization, we can overcome these difficulties and design high-performance, high-reliability RFIC passive integrated devices, providing a solid foundation for the development of modern wireless communication technology.
[0004] Previous research on machine learning-based passive component optimization methods has primarily focused on forward design, deriving the performance parameters and metrics of an inductor from given structural parameters. However, the inverse design problem of inductors—deriving the structural parameters of a variable differential inductor from specified performance metrics—faces greater challenges and difficulties. For example: 1) The complex nonlinear relationship between the performance parameters and structural parameters of passive integrated devices makes the inverse design process more difficult. 2) The structure of a passive integrated device is often composed of multiple parameters, and inverse design requires optimizing these parameters simultaneously to meet the given performance metrics, increasing the complexity of the design. Summary of the Invention
[0005] The purpose of the present invention is to provide a multi-objective optimization algorithm for variable linewidth differential inductors.
[0006] The technical solution to achieve the purpose of the present invention is: a multi-objective optimization algorithm for variable line width differential inductance, comprising the following steps:
[0007] S1: Confirm the performance indicators and geometric parameter ranges of the on-chip passive device inductor, select the parameters to be optimized under the premise of process constraints, use simulation software to simulate, collect data and construct a small sample data set, and store and compress the data;
[0008] S2, the collected data are fitted using the gradient boosting tree and K-nearest neighbor algorithm respectively, and the results of the two are weighted and fused to obtain the alternative model;
[0009] S3 sets the initialization conditions of the differential evolution optimization algorithm, determines the boundaries of the geometric parameters, calls the alternative model to calculate the inductance value, quality factor and area value, and uses the weighted objective function to simultaneously optimize multiple objectives to obtain the geometric parameters that meet the design requirements of differential inductors with variable line width and scalability, and achieve area minimization.
[0010] Furthermore, in S1, the geometric parameters of the variable line width and scalable differential inductor include the maximum line width, line spacing, minimum line width, number of turns, inductor inner diameter and scalability ratio; the performance indicators of the variable line width and scalable differential inductor are composed of the operating frequency, target inductance value Ltarget and target quality factor Qtarget.
[0011] Furthermore, in S1, data is acquired through simulation software to establish a small sample data set, which specifically includes the following steps: after determining the range of geometric parameters, the Latin hypercube sampling method is used to generate a representative set of initial sample points to cover the design space, and each set of geometric parameters is simulated using simulation software to collect the corresponding inductance value L, quality factor Q, and inductance area value. The geometric parameters and performance indicators are combined into complete sample pairs to form a correspondence between input features and output targets. The data is then preprocessed and cleaned: the integrity of the data is checked, and outliers or samples that do not meet the process constraints are removed; the input geometric parameters and output performance indicators are normalized to eliminate the influence of dimensional differences on model fitting.
[0012] Furthermore, in S1, the simulated data is compressed, and all sample data matrices are represented as X, with dimensions of m×n, where m is the number of samples and n is the number of features. Singular value decomposition is performed on X, and it is decomposed into X=UΣV T , where U is an m×m left singular vector matrix, Σ is an m×n diagonal matrix, the diagonal elements are singular values, representing the energy contribution in different feature directions, V T is an n×n right singular vector matrix; the dominant singular values in Σ are retained, and the characteristic components corresponding to the singular values less than the set threshold are removed to achieve dimensionality reduction and compression. The compressed data is used to build the proxy model.
[0013] Furthermore, in S2, the gradient boosting tree gradually builds a tree and optimizes the model error, and the K-nearest neighbor algorithm selects the average of k neighbors in the data as the prediction result; the results of the two models are weighted fused to finally construct a complete proxy model, and the weights are dynamically adjusted according to the model performance; through the above steps, a complete proxy model that can simultaneously capture global nonlinear relationships and local sample distributions is obtained.
[0014] The gradient boosting tree training process is as follows: initially, a decision tree is constructed to predict the output value of the current sample data set; based on the prediction error of the current model, the negative gradient of the loss function is calculated, and the gradient is used to generate a new decision tree to gradually reduce the error; iterative training is performed until the loss function converges or the preset number of iterations is reached; and a weighted combination of multiple decision trees is output to form the final gradient boosting tree model.
[0015] Furthermore, in S3, the optimization process of the differential evolution algorithm includes four steps: initialization, mutation, crossover, and selection. During the crossover process, the test vector is evaluated using the substitution model to obtain its corresponding objective function value; the next generation of population is selected based on the quality of the objective function value, and the objective function value of each test vector is predicted using the substitution model to decide whether to replace the current individual. Through the above process, the differential evolution algorithm uses the substitution model to continuously evaluate and optimize the geometric parameters of the population individuals, thereby achieving the purpose of multi-objective optimization.
[0016] Furthermore, in S3, a weighted objective function is set to balance the importance of different objectives, and finally convergence is achieved to meet the cutoff conditions to obtain the differential variable linewidth inductor geometric parameter design variables that minimize the area and meet the performance indicators.
[0017] A computer device comprises a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the steps of the above method are implemented when the processor executes the program.
[0018] A computer-readable storage medium stores a computer program, which implements the steps of the above method when executed by a processor.
[0019] Compared with the existing technology, the significant advantages of the present invention are: through the differential evolution design method, the design process of radio frequency integrated circuit passive integrated devices is simplified, the layout design efficiency is significantly improved, resources and time are saved, and fast and efficient design is achieved; the cost function is used to optimize the selection of geometric parameters to improve the reliability and availability of generated data; it also expands the flexibility, diversity and scope of application of the design, allowing relevant circuit designers to easily obtain radio frequency on-chip passive inductor devices with a variety of different performance requirements to better meet specific application requirements; and while meeting specific performance requirements, it helps to minimize the area of radio frequency passive integrated devices, thereby enhancing the practicality, performance and application value of on-chip passive inductor devices. BRIEF DESCRIPTION OF THE DRAWINGS
[0020] Figure 1 This is a flow chart of a multi-objective optimization algorithm for a variable linewidth differential inductor according to an embodiment of the present invention.
[0021] Figure 2 FIG. 3 is a 3D structural diagram of a variable line width differential inductor according to an embodiment of the present invention.
[0022] Figure 3 FIG. 2 is a two-dimensional plan view of an on-chip passive inductor device according to an embodiment of the present invention.
[0023] Figure 4 This is a diagram showing the fitting effect of the alternative model according to an embodiment of the present invention.
[0024] Figure 5 This is a diagram showing the convergence effect of the optimization method according to an embodiment of the present invention. DETAILED DESCRIPTION
[0025] The present invention proposes a multi-objective optimization algorithm for variable linewidth differential inductors. In the existing technology, it is difficult to achieve efficient optimization while ensuring accuracy in the face of high-frequency working conditions and multi-objective requirements (such as inductance performance, frequency characteristics and area optimization) in inductor design. The present invention proposes an efficient differential inductor design optimization scheme by constructing a substitution model based on simulation data and combining it with a differential evolution algorithm. The algorithm uses a gradient boosting tree (GBDT) and a K-nearest neighbor algorithm (KNN) to fit and weightedly fuse the simulation data to construct an accurate and efficient substitution model. Combined with the differential evolution algorithm, multi-objective optimization is achieved through a weighted objective function, and key performance indicators such as inductance value, quality factor, self-resonant frequency and area are comprehensively balanced. At the same time, Latin hypercube sampling and singular value decomposition technology are introduced to improve optimization efficiency. The present invention is suitable for multi-objective optimization needs in the design of high-frequency electronic components, especially millimeter-wave inductors, significantly improves design efficiency and accuracy, and has broad engineering application prospects.
[0026] A variable-width differential inductor (VTI) is an inductor whose inductance and quality factor are controlled by adjusting the line width or other geometric parameters. This design allows precise control of the inductance without changing the overall inductor structure, making it suitable for high-frequency applications requiring high inductance precision. By optimizing the line width, parasitic effects can be effectively reduced, the inductor's operating efficiency can be improved, and the inductor's inductance can be maintained relatively stable over a wide frequency band.
[0027] The present invention provides a multi-objective optimization algorithm for variable line width differential inductance, such as Figure 1 As shown in the figure, the present invention proposes a variable line width differential inductor multi-objective optimization algorithm and alternative model design method, which includes three core steps: constructing a sample set and preprocessing it; using GBDT and KNN to establish a weighted fusion proxy model; and performing multi-objective optimization based on a differential evolution algorithm. Figure 2 As shown in FIG, a three-dimensional structure diagram of a variable line width differential inductor in an embodiment of the present invention is shown. The figure shows the multi-turn line width adjustable characteristics of the inductor, including geometric structural parameters such as maximum line width, minimum line width, line spacing and inner diameter. Figure 3 The figure shows a two-dimensional planar structure diagram of a variable line width differential inductor according to an embodiment of the present invention. The figure shows the layout structure of the inductor on the chip from a layout perspective, and marks the actual layout form of key geometric parameters such as line width, number of turns and line spacing, providing an intuitive basis for subsequent geometric parameter optimization. The geometric parameters of the variable line width scalable differential inductor include maximum line width, line spacing, minimum line width, number of turns, inductor inner diameter, and expansion ratio; the performance indicators of the variable line width scalable differential inductor are determined by the operating frequency, target inductance value L target , target quality factor Q target constitute.
[0028] Acquire data through simulation software and build a small sample data set, which includes the following steps:
[0029] After determining the range of geometric parameters, the Latin Hypercube Sampling (LHS) method is used to generate a representative set of initial sample points to cover the design space. Simulation software is used to simulate each set of geometric parameters, and the corresponding inductance value L, quality factor Q, and inductor area value are collected. The geometric parameters and performance indicators are combined into complete sample pairs to form a correspondence between input features (geometric parameters) and output targets (performance indicators).
[0030] Preprocess and clean the data: Check the data integrity and remove outliers or samples that do not meet process constraints. Normalize the input geometric parameters and output performance indicators to eliminate the impact of dimensional differences on model fitting.
[0031] The data obtained by simulation is compressed, and all sample data matrices are represented as A (dimension is m×n, where m is the number of samples and n is the number of features), and the singular value decomposition (SVD) is applied to decompose A m×n =UΣV T , where U is the left singular vector matrix of m×m, Σ is the diagonal matrix of m×n, and the diagonal elements are singular values, representing the energy contribution in different feature directions, V T is an n×n right singular vector matrix. The dominant singular values in Σ are retained, and the eigenvalues corresponding to singular values below a set threshold are removed. This achieves dimensionality reduction and compression of the feature space, reducing computational complexity. The compressed data is used to construct a surrogate model, improving model training efficiency and prediction accuracy.
[0032] The gradient boosting tree is suitable for fitting complex nonlinear relationships by gradually building a tree and optimizing the model's error. The K-nearest neighbor algorithm uses the average of k neighbors in the data as the prediction result. The results of these two models are weighted and fused to ultimately construct a complete proxy model. The weights are dynamically adjusted based on model performance, so that higher-performing models are automatically assigned higher weights. Through these steps, a complete proxy model is obtained that simultaneously captures global nonlinear relationships and local sample distributions, providing efficient performance prediction capabilities for optimization.
[0033] Construction of gradient boosting tree: Gradient boosting tree is an ensemble learning method based on decision tree, which reduces the overall prediction error by gradually fitting the error of each round.
[0034] The training process is as follows: initially construct a weak learner (decision tree) to predict the output value of the current sample data set; based on the prediction error of the current model, calculate the negative gradient of the loss function, and use this gradient to generate a new decision tree to gradually reduce the error; iterative training is performed until the loss function converges or the preset number of iterations is reached; output a weighted combination of multiple decision trees to form the final gradient boosting tree model.
[0035] Construction of K-nearest neighbor algorithm: K-nearest neighbor algorithm is an instance-based learning method. Its prediction process is as follows: In the geometric parameter space, calculate the Euclidean distance between the target point and other training samples, select the nearest y1, y2, ..., y k ,k samples as neighbors; calculate the average value of the performance indicators (inductance value, quality factor, etc.) of these neighbor samples as the predicted value of the target point
[0036]
[0037] Weighted fusion of results: To build a complete proxy model, the prediction results of the gradient boosting tree and the K-nearest neighbor algorithm are weighted and fused. The weights are dynamically adjusted based on the model performance, and the performance indicator of the model is defined as the mean square error (MSE).
[0038]
[0039] where y i is the simulation value, is the predicted value. The performance of the two models on the validation set is calculated, and the model with better performance has a higher weight. The weighting formula is as follows:
[0040]
[0041] in, is the final predicted value, and are the predicted values of the gradient boosting tree and the K-nearest neighbor algorithm, w1 and w2 are weights, satisfying w1+w2=1. The weights are dynamically adjusted and determined dynamically according to the model performance:
[0042]
[0043] Among them, MSE GBT is the mean square error of the gradient boosting tree, MSE KNN is the mean square error of the K-nearest neighbor algorithm. In this way, the model with better performance will automatically be assigned a higher weight. Through the above steps, a complete proxy model that can capture both global nonlinear relationships and local sample distributions is obtained, providing efficient performance prediction capabilities for optimization. Figure 4 Figure 2 shows the performance of the proposed proxy model in the inductance prediction task. The horizontal axis represents the number of optimization iterations, and the vertical axis represents the target inductance value, L, in nH. Triangular symbols represent the true simulated values, and the solid line represents the proxy model's predicted values. The predicted values are highly consistent with the true values, with a consistent overall trend. This demonstrates that the constructed GBDT-KNN weighted fusion model can effectively fit the nonlinear distribution of inductance values and exhibits good generalization capabilities, making it suitable for rapid performance evaluation in lieu of computationally expensive electromagnetic simulations.
[0044] The optimization process of the differential evolution algorithm includes four steps: initialization, mutation, crossover and selection.
[0045] Initialization: During the initialization phase of the differential evolution algorithm, Latin Hypercube Sampling (LHS) is used to generate an initial population to ensure uniform coverage of the geometric parameter space. Each individual represents a candidate solution, containing multiple geometric parameters such as maximum line width, minimum line width, line spacing, number of turns, inductor inner diameter, and expansion ratio.
[0046] First, each member of the initial population is evaluated through the surrogate model to calculate its corresponding objective function value, such as inductance L, quality factor Q and area.
[0047] In the mutation phase, three different individuals X are randomly selected from the current population. r1 , X r2 , X r3 , and calculate the mutation vector V i :
[0048] V i =X r1 +F·(X r2 -X r3 )
[0049] Where F is the scaling factor, which controls the magnitude of the mutation and affects the intensity of the mutation. Each time a mutation occurs, the performance of the newly generated individual is predicted by the surrogate model, and the objective function value of the mutation vector is calculated;
[0050] The crossover phase will mutate the vector V i With the current target vector X i Perform crossover to generate the test vector U i :
[0051]
[0052] where u ij is the test vector U i The jth component of ij is the mutation vector V i The jth component of x ij is the target vector X i The jth component of rand is a random index, ensuring that each test vector inherits at least one gene C in the mutation vector R is the crossover probability, which determines whether each component comes from the mutation vector V i ,During the crossover process, the trial vector is evaluated using the surrogate model to obtain its corresponding objective function value;
[0053] In the selection phase, the trial vector U i Target vector X i Calculate the objective function value and compare it. If the objective function value of the test vector is better, replace the target vector with the test vector, otherwise keep the current target vector X. i :
[0054]
[0055] The next generation of population is selected based on the quality of the objective function value, and the objective function value of each test vector is predicted using the substitution model to decide whether to replace the current individual. Through the above process, the differential evolution algorithm uses the substitution model to continuously evaluate and optimize the geometric parameters of the population individuals, thereby achieving the purpose of multi-objective optimization.
[0056] The differential evolution algorithm uses the substitution model to continuously evaluate and optimize the geometric parameters of the individuals in the population, thereby achieving the purpose of multi-objective optimization.
[0057] The weighted objective function is:
[0058] f(x)=a1·P L +a2·P Q +a3·P area
[0059] Among them, P L , P Q , P A is the penalty term for inductance value L, quality factor Q and area, a1, a2, a3 are the weights of each penalty term, which are used to balance the importance of different goals. The penalty term for inductance value L is defined as the inductance value L deviates from the target inductance value L target Degree:
[0060] P L =|LL target |
[0061] The penalty for the quality factor Q is defined as the penalty for the quality factor Q being lower than the target quality factor Q. target Punishment is given:
[0062] P Q =max(0,Q target -Q)
[0063] The penalty term of the area directly uses the true value of the area, that is:
[0064] P A =Area
[0065] By adjusting the weight of each penalty term, the device area can be compressed as much as possible while meeting the inductance and quality factor indicators, achieving an optimal balance between performance and resource usage. Figure 5 Figure 2 shows the convergence curve of the weighted objective function value during the multi-objective optimization process in an embodiment of the present invention. The figure demonstrates the joint convergence trend of the algorithm for the inductance performance index and area objective during the iterative process, demonstrating the good stability and effectiveness of the optimization process.
[0066] This application combines an electrical parameter prediction model with differential evolution optimization methods to adjust parameters such as line width, line spacing, and inner diameter to meet specific inductance values and quality factors. This method can effectively overcome the complexity of the inverse design problem, improve the efficiency and accuracy of RFIC passive integrated device design, and has great application prospects and practical value.
[0067] The above embodiments are illustrations of specific implementation methods of the present invention, rather than limitations of the present invention. Technicians in the relevant technical fields can make various changes and modifications to obtain corresponding equivalent technical solutions without departing from the spirit and scope of the present invention. Therefore, all equivalent technical solutions should be included in the patent protection scope of the present invention.
Claims
1. A multi-objective optimization algorithm for variable linewidth differential inductors, characterized in that: The steps include: S1: Confirm the performance indicators and geometric parameter ranges of the on-chip passive device inductor, select the parameters to be optimized under the premise of process constraints, use simulation software to simulate, collect data and construct a small sample data set, and store and compress the data; S2, the collected data are fitted using the gradient boosting tree and K-nearest neighbor algorithm respectively, and the results of the two are weighted and fused to obtain the alternative model; S3 sets the initialization conditions of the differential evolution optimization algorithm, determines the boundaries of the geometric parameters, calls the alternative model to calculate the inductance value, quality factor and area value, and uses the weighted objective function to simultaneously optimize multiple objectives to obtain the geometric parameters that meet the design requirements of differential inductors with variable line width and scalability, and achieve area minimization.
2. The variable linewidth differential inductor multi-objective optimization algorithm according to claim 1, characterized in that: In S1, the geometric parameters of the variable-linewidth scalable differential inductor include the maximum line width, line spacing, minimum line width, number of turns, inductor inner diameter, and scalability ratio; the performance indicators of the variable-linewidth scalable differential inductor are composed of the operating frequency, target inductance value Ltarget, and target quality factor Qtarget.
3. The multi-objective optimization algorithm for variable linewidth differential inductance according to claim 1, characterized in that: In S1, data is acquired through simulation software to establish a small sample data set, which specifically includes the following steps: after determining the range of geometric parameters, the Latin hypercube sampling method is used to generate a representative set of initial sample points to cover the design space, and each set of geometric parameters is simulated using simulation software to collect the corresponding inductance value L, quality factor Q, and inductance area value. The geometric parameters and performance indicators are combined into complete sample pairs to form a correspondence between input features and output targets. The data is then preprocessed and cleaned: the integrity of the data is checked, and outliers or samples that do not meet process constraints are removed; the input geometric parameters and output performance indicators are normalized to eliminate the impact of dimensional differences on model fitting.
4. The multi-objective optimization algorithm for variable linewidth differential inductance according to claim 3, characterized in that: In S1, the simulated data is compressed, and all sample data matrices are represented as X, with dimensions of m×n, where m is the number of samples and n is the number of features. Singular value decomposition is performed on X, and it is decomposed into X=UΣV T , where U is an m×m left singular vector matrix, Σ is an m×n diagonal matrix whose diagonal elements are singular values, representing the energy contribution in different feature directions, and VT is an n×n right singular vector matrix. The dominant singular values in Σ are retained, and the feature components corresponding to the singular values less than the set threshold are removed to achieve dimensionality reduction and compression. The compressed data is used to build the proxy model.
5. The multi-objective optimization algorithm for variable linewidth differential inductor according to claim 1, characterized in that: In S2, the gradient boosting tree gradually builds a tree and optimizes the model error, and the K-nearest neighbor algorithm selects the average of k neighbors in the data as the prediction result; the results of the two models are weighted fused to finally build a complete proxy model, and the weights are dynamically adjusted according to the model performance; through the above steps, a complete proxy model that can simultaneously capture global nonlinear relationships and local sample distributions is obtained.
6. The multi-objective optimization algorithm for variable linewidth differential inductance according to claim 5, characterized in that: The gradient boosting tree training process is as follows: initially, a decision tree is constructed to predict the output value of the current sample data set; based on the prediction error of the current model, the negative gradient of the loss function is calculated, and the gradient is used to generate a new decision tree to gradually reduce the error; iterative training is performed until the loss function converges or the preset number of iterations is reached; and a weighted combination of multiple decision trees is output to form the final gradient boosting tree model.
7. The multi-objective optimization algorithm for variable linewidth differential inductor according to claim 1, characterized in that: In S3, the optimization process of the differential evolution algorithm includes four steps: initialization, mutation, crossover, and selection. During the crossover process, the test vector is evaluated using the substitution model to obtain its corresponding objective function value; the next generation of population is selected based on the quality of the objective function value, and the objective function value of each test vector is predicted using the substitution model to determine whether to replace the current individual. Through the above process, the differential evolution algorithm uses the substitution model to continuously evaluate and optimize the geometric parameters of the population individuals, thereby achieving the purpose of multi-objective optimization.
8. The multi-objective optimization algorithm for variable linewidth differential inductor according to claim 3, characterized in that: In S3, a weighted objective function is set to balance the importance of different objectives, and finally convergence is achieved to meet the cutoff conditions to obtain the differential variable linewidth inductor geometric parameter design variables that minimize the area and meet the performance indicators.
9. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein: When the processor executes the program, the steps of the method according to any one of claims 1 to 8 are implemented.
10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the steps of the method according to any one of claims 1 to 8 are implemented.