Abstract:
A characterized cell library for EDA tools includes one or more mathematical models for each cell, and one or more preconditioning functions (and/or inverse preconditioning functions) for each mathematical model. Each mathematical model represents a performance parameter (e.g., delay, power consumption, noise) or a preconditioned performance parameter of the cell. The preconditioning functions convert an operating parameter (e.g., input slew, output capacitance) associated with the performance parameter into a preconditioned input variable for the mathematical models. In doing so, the preconditioning functions allow for more accurate modeling of complex data relationships without increasing the complexity (e.g., order and number of coefficients) of the mathematical models. Also, because the cell library can be substantially similar to conventional polynomial-based cell libraries except for the inclusion of preconditioning functions, preconditioning does not significantly increase storage requirements and conventional EDA tools can be readily adapted to use the preconditioned cell library.

Description:
BACKGROUND OF THE INVENTION  
       [0001]     1. Field of the Invention  
         [0002]     The invention is in the field of electronic design automation (EDA), and more particularly, is related to cell libraries for efficient modeling of device properties.  
         [0003]     2. Related Art  
         [0004]     An electronic design automation (EDA) system is a computer software system used for designing integrated circuit (IC) devices. The EDA system typically receives one or more high level behavioral descriptions of an IC device (e.g., in HDL languages like VHDL, Verilog, etc.) and translates (“synthesizes”) this high level design language description into netlists of various levels of abstraction. A netlist describes the IC design and is composed of nodes (functional elements) and edges, e.g., connections between nodes. At a higher level of abstraction, a generic netlist is typically produced based on technology independent primitives.  
         [0005]     The generic netlist can be translated into a lower level technology-specific netlist based on a technology-specific (characterized) cell library that has gate-specific models for each cell (functional element). The models define performance parameters for the cells; e.g., parameters related to the operational behavior of the cells, such as power consumption, delay, transition time, and noise. The netlist and cell library are typically stored in computer readable media within the EDA system and are processed and verified using many well-known techniques.  
         [0006]     Before proceeding further with the description, it may be helpful to place these processes in context.  FIG. 1A  shows a simplified representation of an exemplary digital ASIC design flow. At a high level, the process starts with the product idea (step E 100 ) and is realized in an EDA software design process (step E 110 ). When the design is finalized, it can be taped-out (event E 140 ). After tape out, the fabrication process (step E 150 ) and packaging and assembly processes (step E 160 ) occur resulting, ultimately, in finished chips (result E 170 ).  
         [0007]     The EDA software design process (step E 110 ) is actually composed of a number of steps E 112 -E 130 , shown in linear fashion for simplicity. In an actual ASIC design process, the particular design might have to go back through steps until certain tests are passed. Similarly, in any actual design process, these steps may occur in different orders and combinations. This description is therefore provided by way of context and general explanation rather than as a specific, or recommended, design flow for a particular ASIC.  
         [0008]     A brief description of the components steps of the EDA software design process (step E 110 ) will now be provided:  
         [0009]     System design (step E 112 ): The designers describe the functionality that they want to implement and can perform what-if planning to refine functionality, check costs, etc. Hardware-software architecture partitioning can occur at this stage. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include Model Architect, Saber, System Studio, and DesignWare® products.  
         [0010]     Logic design and functional verification (step E 114 ): At this stage, the VHDL or Verilog code for modules in the system is written and the design is checked for functional accuracy. More specifically, the design is checked to ensure that it produces the correct outputs. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include VCS, VERA, DesignWare®, Magellan, Formality, ESP and LEDA products.  
         [0011]     Synthesis and design for test (step E 116 ): Here, the VHDL/Verilog is translated into a netlist. The netlist can be optimized for the target technology. Additionally, the design and implementation of tests to permit checking of the finished chip occurs. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include Design Compiler®, Physical Compiler, Test Compiler, Power Compiler, FPGA Compiler, Tetramax, and DesignWare® products.  
         [0012]     Design planning (step E 118 ): Here, an overall floorplan for the chip is constructed and analyzed for timing and top-level routing. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include Jupiter and Floorplan Compiler products.  
         [0013]     Netlist verification (step E 120 ): At this step, the netlist is checked for compliance with timing constraints and for correspondence with the VHDL/Verilog source code. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include VCS, VERA, Formality and PrimeTime products.  
         [0014]     Physical implementation (step E 122 ): The placement (positioning of circuit elements) and routing (connection of the same) occurs at this step. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include the Astro product.  
         [0015]     Analysis and extraction (step E 124 ): At this step, the circuit function is verified at a transistor level, this in turn permits what-if refinement. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include Star RC/XT, Raphael, and Aurora products.  
         [0016]     Physical verification (step E 126 ): At this step various checking functions are performed to ensure correctness for: manufacturing, electrical issues, lithographic issues, and circuitry. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include the Hercules product.  
         [0017]     Resolution enhancement (step E 128 ): This step involves geometric manipulations of the layout to improve manufacturability of the design. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include iN-Phase, Proteus, and AFGen products.  
         [0018]     Mask data preparation (step E 130 ): This step provides the “tape-out” data for production of masks for lithographic use to produce finished chips. Exemplary EDA software products from Synopsys, Inc. that can be used at this step include the CATS(R) family of products.  
         [0019]     In general, a characterized cell library can be used in the steps of synthesis, design planning, netlist verification, physical implementation, and analysis (as indicated by the bolded chevrons).  
         [0020]      FIG. 1B  illustrates an example cell  100 . Cell  100  represents an AND-OR-Invert (AOI) gate formed by AND gates  110  and  120  and an OR gate  130 , with the outputs of AND gates  110  and  120  being tied to the inputs of OR gate  130 . A characterized library entry associated with cell  100  would typically include performance parameter information for cell  100  across a range of operating conditions. For example, because each of AND gates  110  and  120  includes two inputs, cell  100  includes at least four paths (i.e., input  111  to output  133 , input  112  to output  133 , input  121  to output  133 , and input  122  to output  133 ) for which delay, power, noise, and other performance parameters can be specified.  
         [0021]     The performance parameter data associated with the cells in a characterized cell library is typically provided across a range of operating parameter values (e.g., values for input slew, output capacitance, voltage (e.g., cell operating voltage), and temperature), and can be provided by a foundry or can be calculated via a simulation program such as SPICE. The performance parameter data therefore may originally be compiled as a set of discrete data points.  
         [0022]     For example,  FIG. 1C , shows a sample performance parameter table  101  for one timing arc of the timing path from input  111  to output  133  of cell  100  (shown in  FIG. 1B ). For example, the timing arc could be the fall/rise delay arc or the fall/rise transition time arc. For the sake of simplicity, the timing arc is shown as a function of two operational parameters, although in other embodiments, the timing arc could be a function of any number of operational parameters (if more than two operational parameters are considered, a multi-dimensional table or multiple two-dimensional tables would be required).  
         [0023]     Table  101  could be an entry in a cell library, and includes performance parameter values PP 11 -PP 46  and two sets of operational parameter values X 1 -X 6  and Y 1 -Y 4 . Performance parameter values PP 11 -PP 46  can represent values for any single type of performance parameter (e.g., delay, noise, or power consumption), operational parameter values X 1 -X 6  can represent values for a first operational parameter (e.g., input slew, output capacitance, temperature, or voltage), while operational parameter values Y 1 -Y 4  can represent values for a second operational parameter.  
         [0024]     Each performance parameter value is referenced by a particular combination of operational parameter values (i.e., performance parameter value PP 11  is generated by the combination of operational parameter values X 1  and Y 1 ). Thus, for example, PP 11  could represent the delay from input  111  to output  133  for an output (load) capacitance X 1  and an input slew Y 1 . Note that interpolation or extrapolation may be required if the desired operational condition (i.e., the desired combination of operations parameter values) is not listed in the table. For example, if the operational parameter values are X 5  and Y 5 , with X 1 &lt;X 5 &lt;X 2  and Y 2 &lt;Y 5 &lt;Y 3 , then PP 21 , PP 22 , PP 31  and PP 32  may be used to calculate the performance parameter value at (X 5 , Y 5 ) using interpolation.  
         [0025]     Note that while performance parameter table  101  is relatively small for exemplary purposes, typical performance parameter tables will be much larger, as performance parameter values will be provided for many more combinations of operating parameter values in order to produce more accurate performance information and/or provide improved interpolation/extrapolation capabilities. Therefore, look-up tables can consume relatively large amounts of storage space and memory resources within an EDA system. For example, a single library can contain tens of thousands of look-up tables. In addition, each look-up tables may require a large number of data points if high interpolation and/or extrapolation accuracy is desired. The resulting large number of large lookup tables can be cumbersome for EDA tools (e.g., synthesis or analysis tools) and can significantly increase computational requirements and modeling time. Therefore, modern characterized cell libraries may choose to replace lookup tables with mathematical models (typically SPM formulas) of cell behavior.  
         [0026]     For example, some early mathematical models of functional element timing were based on fixed-form linear equations of input slew and output capacitance load. Later, these timing models (e.g., for generic CMOS) were based on a linear equation including both of these two variables (with fixed coefficients) and, similarly, the same linear equation form was used to model all of the gates of a given technology library. Although the linear equation form was the same for all gates, the coefficients of the linear equation could change from timing arc to timing arc within the same technology library. This allowed the timing calculations to be generalized across the entire technology library and thereby made the timing calculations easier to perform. However, the calculations were not entirely accurate because some library gates were not well modeled by the fixed-form linear equation.  
         [0027]     Recently, advanced cell libraries have begun to incorporate scalable polynomial-based models (“SPM models”) to specify cell performance parameters, as described in co-owned U.S. Pat. No. 6,732,341, issued May 4, 2004 to Chang et al., herein incorporated by reference. Replacing look-up table models with polynomial models significantly reduces memory usage, while increasing computation speed for tools that make use of the cell library (e.g., synthesis tools and static timing analysis tools).  
         [0028]     For example,  FIG. 1D  shows a sample cell library entry  102  generated from table  101  in  FIG. 1C . The performance parameter values associated with each different combination of operational parameter values (X 1 -X 4  and Y 1 -Y 4 ) have been replaced with a polynomial function FN 1  of operational parameters X and Y. Typically, a predetermined polynomial form is used for function FN 1 , so that only the coefficients of the polynomial function FN 1  are stored in the library. In doing so, the size of cell library entry  102  can be significantly reduced over table  101  shown in  FIG. 1C .  
         [0029]     The particular form of polynomial function FN that is used for modeling purposes is typically selected from a set of scalable polynomial systems (e.g., the decomposed Taylor Series and the Joint Taylor Series), and can have different orders (e.g., first, second, third, etc.) with respect to the input variables. The lower the orders used in the polynomial forms in a library, the less computationally expensive are the analyses performed using that library.  
         [0030]     For example, if the order of the polynomial using decomposed Taylor series is 2 (i.e., the highest order for each variable in the polynomial form is 2), and the function has 4 operation parameters, the total number of polynomial terms is (2+1) 4 =81. However, if the order of the polynomial is 3, the total number of polynomial terms jumps to (3+1) 4 =256. Therefore, it is extremely beneficial to limit the order of the library polynomial forms to minimize the number of polynomial terms, thereby minimizing polynomial model derivation and usage complexity.  
         [0031]     However, as device sizes continue to shrink, the performance parameters of those devices become more nonlinear, and hence more difficult to fit with conventional polynomial models. Although high-order polynomials or piecewise polynomials may be used to achieve better fitting accuracy, both have limitations. High-order polynomials require more terms and have the undesirable high-order oscillations that may cause large errors in certain fitting regions of performance parameter data. Piecewise polynomials can avoid such oscillations, but can require a large number of polynomials to provide an accurate fit.  
         [0032]     In either case, the large number of coefficients of both high-order polynomials and piecewise polynomials can make such approaches unattractive solutions to the problem of excessive library size and computational requirements. These shortcomings of high-order polynomials and piecewise polynomials only become more pronounced as device sizes continue to rapidly scale down and more operational parameters start to affect performance parameters.  
         [0033]     Furthermore, it is well known that a polynomial form has difficulty in accurately modeling/representing certain very nonlinear behavior (e.g., exponential functions). In such cases, a conventional scalable polynomial representation of the data is neither efficient nor robust.  
         [0034]     Accordingly, it is desirable to provide a method and system that can accurately model all types of performance parameter distributions without significantly increasing computational and storage requirements.  
       SUMMARY OF THE INVENTION  
       [0035]     To improve mathematical model accuracy without unduly increasing cell library size or analytical complexity, preconditioning (or mapping) of operational parameters and/or even performance parameters can be used. For example, in one embodiment, a characterized cell model can be generated by applying a preconditioning function to operating parameter values that reference a set of performance parameter values for a cell. The performance parameter for the cell is then modeled as a function of the preconditioned operational parameter, thereby providing greater modeling capabilities with a minimal increase in library storage requirements and computational complexity. For example, certain nonlinear patterns that are difficult to model using polynomial can be accurately modeled through the use of simple preconditioning functions (e.g., exponential functions, logarithmic functions, and rational functions). The use of preconditioning functions to “translate” from operational parameter values to preconditioned values in this manner can significantly reduce the difficult-to-model nonlinearity to be accommodated by the mathematical model (e.g. polynomial model), and thus provide dramatically improved modeling accuracy without increasing the order or complexity of the actual model.  
         [0036]     An entry in a characterized cell library can therefore include the mathematical model of the preconditioned operational parameters and the preconditioning functions. When the library is used by EDA tools, the preconditioning function(s) can first be applied to the operational parameters, after which the mathematical model can be applied to calculate the performance parameters using the preconditioned operational parameters. Performance parameter preconditioning functions may also be applied to “translate” performance parameters to preconditioned performance parameters that are then modeled by the mathematical models during the model characterization stage. In this case the library can also include the inverse preconditioning function(s) for those preconditioned performance parameters. Then, when the library is used by EDA tools, the mathematical model can be applied to calculate the preconditioned performance parameters, after which the inverse preconditioning function(s) can be applied to “translate” the preconditioned performance parameters back to the actual performance parameter values to be used by EDA tools.  
         [0037]     Because the characterized library can be implemented with minimal change to conventional polynomial-based libraries (i.e., by simply incorporating appropriate preconditioning functions), the benefits of preconditioning can be achieved using conventional EDA tools without requiring major modification. This “preconditioned” cell library can be stored on and accessed from any type of computer-readable medium (e.g., CDROM, DVD, hard drive, network server).  
         [0038]     According to various embodiments, the mathematical model for a cell can be a function of one or more preconditioned operational parameters, and any number of additional non-preconditioned operational parameters. The preconditioning function can be a linear function, a nonlinear function, or a function of two or more operational parameters (e.g., for merging multiple input variables into a single input variable for the mathematical model).  
         [0039]     In one embodiment, an EDA analysis tool can perform an analysis of an IC design by using a characterized cell library that includes one or more mathematical models for cell performance parameters that are functions of preconditioned operational parameters. The analysis could be performed by selecting a cell in the design, applying preconditioning functions to one or more operational parameter values for the cell, and then substituting the preconditioned operational parameter value into the mathematical model for a performance parameter of the cell. Note that if performance parameter values are preconditioned, then the output of the above mathematical model is a preconditioned performance parameter value, from which a final performance parameter value can be derived using the inverse preconditioning function.  
         [0040]     The invention will be more fully understood in view of the following description and drawings. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0041]      FIG. 1A  is a process flow diagram for a general EDA design flow.  
         [0042]      FIG. 1B  is a schematic diagram of a sample cell that can be included in a cell library.  
         [0043]      FIG. 1C  is a sample table of performance parameter data that could be associated with the cell of  FIG. 1B .  
         [0044]      FIG. 1D  is a conventional polynomial-based cell library entry that could be associated with the table of  FIG. 1C .  
         [0045]      FIGS. 2A and 2B  are flow diagrams of a characterized cell creation process.  
         [0046]      FIG. 3A  is a depiction of a translation from a lookup table of cell performance data into a library of mathematical models based on preconditioned operational parameters.  
         [0047]      FIG. 3B  is a sample graph of performance parameter values versus preconditioned operational parameter values, and a best fit polynomial curve.  
         [0048]      FIGS. 3C-3F  are sample cell libraries that include mathematical cell models having a variety of preconditioned and non-preconditioned input variables.  
         [0049]      FIG. 4  is a diagram of a computing system that includes logic for generating a characterized cell library using preconditioning functions.  
         [0050]      FIG. 5  is a flow diagram of a process for performing an EDA analysis using a cell library that includes preconditioning functions. 
     
    
     DETAILED DESCRIPTION  
       [0051]     Conventional characterized cell libraries are derived by modeling performance parameter data referenced by operational parameters. As noted above, providing a desired level of modeling accuracy using those conventional means can result in excessive computational and storage requirements.  FIG. 2A  shows a flow diagram for the generation of a characterized cell library generation process that overcomes these problems.  
         [0052]     In an “ACCESS ORIGINAL CELL LIBRARY” step  210 , a set of functional elements (cells) is accessed (typically by being read into the memory of a computer system). As is known in the art, this set can include any number and type of functional elements, including AND gates, OR gates, inverters, latches, multiplexers, half adders, and full adders, to name a few.  
         [0053]     In some embodiments, the original cell library can also include technology-specific performance parameter data (e.g., provided by a foundry or generated via simulation of the functional elements using a simulator such as SPICE). In other embodiments, the performance parameter data can then be associated with the functional elements in an optional “ADD PRECHARACTERIZATION DATA” step  220 . In any case, the performance parameter data provides a set of performance parameter values, each of which is referenced by particular values of one or more types of operational parameters.  
         [0054]     Then, in a “PRECONDITION DATA” step  230 , one or more preconditioning functions can be applied to the operating parameter values associated with each set of performance parameter data, and/or to the performance parameter data. For example, a set of delay data for a given timing arc might be associated with a set of operating parameter values that include a set of input slew values, a set of output capacitance values, a set of temperature values, and a set of voltage (e.g., cell voltage) values. Preconditioning functions can be applied to one or more of those operating parameter data sets and/or the performance parameters to generate a set of preconditioned values (as described in greater detail below with respect to  FIG. 3A ).  
         [0055]     Next, in a “DERIVE MATHEMATICAL MODEL(S)” step  240 , the preconditioned operational parameter values generated in step  230  can be used as input variables to derive mathematical models for the different cells in the library. In one embodiment, the mathematical models can comprise SPM models. Each mathematical model provides a model of the behavior of a particular performance parameter for a particular cell, as a function of one or more preconditioned operational parameters. For example, according to an embodiment of the invention, a mathematical model for delay can be a function of output capacitance preconditioned by a first function, and of input slew preconditioned by a second function.  
         [0056]     Note that step  240  can be performed in any manner that provides a desired level of accuracy in modeling the relationship between the performance parameter values and the preconditioned operational parameter values generated in step  230 . For example, according to an embodiment of the invention, the mathematical model derivation of step  240  can be performed via the method for translating look-up tables into scalable polynomial models described in U.S. Pat. No. 6,732,341, by substituting preconditioned operating parameter values and/or performance parameter values for the “raw” (i.e., non-preconditioned) operating parameter values and/or performance parameter values, respectively, used in U.S. Pat. No. 6,732,341.  
         [0057]     Note further that the process can loop back to step  230  if the desired accuracy is not achieved using the existing preconditioned operational parameter data values. A new preconditioning function(s) could then be applied to the raw operational parameter and/or performance parameter data in step  230 , and the new set(s) of preconditioned operating parameter data values and/or performance parameter data values, respectively, could then be used in the mathematical model derivation of step  240 . Once the desired model accuracy is achieved, the mathematical models can be compiled into a final characterized cell library in a “COMPILE CELL LIBRARY” step  250 . Note that while the original cell library accessed in step  210  will typically represent an uncharacterized cell library, in various other embodiments, the original cell library can represent a characterized cell library having a first data format (e.g., non-linear delay model (NLDM) format) that is translated into a characterized cell library (via steps  230  through  250 ) having a different mathematical model format (e.g., SPM format).  
         [0058]      FIG. 2B  shows a detailed embodiment of steps  230  and  240  (in  FIG. 2A ). Step  230  can begin with the selection of one or more preconditioning functions in a “SELECT PRECONDITIONING FUNCTION(S)” step  231 . The preconditioning functions can be selected in a variety of ways, including selection based on “known good” preconditioning functions (e.g., preconditioning functions that have provided desirable results in previous modeling efforts), or selection from a predefined set of preconditioning functions. Note also that while linear preconditioning functions may sometimes be useful to improve numerical stability during the subsequent model derivations, nonlinear preconditioning functions (e.g., exponential or logarithmic functions) will typically provide greater benefit in improving the accuracy of the final mathematical model.  
         [0059]     The one or more selected preconditioning functions can then be applied to one or more sets of operational parameter and/or performance parameter values in an “APPLY PRECONDITIONING FUNCTION(S) TO PRECHARACTERIZED DATA” step  232 . Note that according to various embodiments, preconditioning may be applied to some or all of the sets of operational parameter values and performance parameter values, and that different preconditioning functions may be applied to different sets of operational parameter or performance parameter values.  
         [0060]     Then, step  240  can begin with a “SELECT MATHEMATICAL FORM(S) step  241 , in which mathematical forms for the performance parameters of interest are selected. The mathematical forms can be functions of the preconditioned operational parameter values and/or can generate preconditioned performance parameter values. In one embodiment, polynomials or piecewise polynomials for an SPM model are chosen.  
         [0061]     Modeling of the precharacterized data is then performed using the mathematical form(s) from step  241  in a “GENERATE MODEL OUTPUT(S) USING PRECONDITIONED DATA” step  242 , and the fit provided by the mathematical model output is then evaluated in a “WITHIN TOLERANCE?” step  243 . Note that preconditioning functions can be applied to some or all of the operational parameter data and/or the performance parameter data. Note further that if a preconditioning function is applied to the performance parameter data, the output of the mathematical model is the preconditioned version of the performance parameter data, which can be converted back to the actual performance parameter data via the inverse function of the preconditioning function.  
         [0062]     The process then iterates, looping back to step  241  to adjust the mathematical forms (e.g., by changing coefficients or changing the order of a polynomial or changing the regions of piecewise polynomials) until a desired fit is achieved. The iterative process can also loop back to step  231 , and a different preconditioning function(s) can be applied to the sets of operational parameter values and/or the performance parameter values. Once the desired model accuracy is detected in step  243 , the mathematical models can be finalized in a “FINALIZE MATHEMATICAL MODEL(S)” step  244  to complete step  240 .  
         [0063]      FIG. 3A  shows an exemplary graphical representation of the characterized library generation process described with respect to  FIG. 2A . In  FIG. 3A , a performance parameter table  301  for the timing arc between input  111  and output  133  of cell  100  shown in  FIG. 1B  includes performance parameter values PP 11 -PP 46  and operational parameter data values X 1 -X 6  and Y 1 -Y 6 . Performance parameter values PP 11 -PP 46  can represent values for any single type of performance parameter (e.g., delay, noise, or power consumption), operational parameter values X 1 -X 6  can represent values for a first operational parameter (e.g., input slew, output capacitance, temperature, or voltage), while operational parameter values Y 1 -Y 4  can represent values for a second operational parameter. Note that while only two types of operational parameters (X and Y) are shown for exemplary purposes, the invention can be applied to performance parameter data associated with any number of different types of operational parameters.  
         [0064]     Each performance parameter value is referenced by a particular combination of operational parameter values (i.e., performance parameter value PP 11  is generated for the combination of operational parameter values X 1  and Y 1 ). Thus, for example, PP 11  could represent the delay between input  111  and output  133  for an output (load) capacitance X 1  and an input slew Y 1 . Note that while only two types of operational parameters (X and Y) are depicted in table  301  for clarity, the invention can be applied to performance parameter data that includes any number of different operational parameter types.  
         [0065]     Using the method described with respect to  FIG. 2A , the data in table  301  is translated (indicated by the dark arrow) into a mathematical model FM 1  in a cell library entry  302  of a cell library  390 . In general, a cell library entry in accordance with the invention includes a cell identifier (e.g., cell ID  302 A), a mathematical model for a performance parameter (or a preconditioned performance parameter) of the cell (e.g., model FM 1 ), and one or more preconditioning functions (e.g., functions FP 1  and FP 2 ) associated that mathematical model.  
         [0066]     Mathematical model FM 1  in  FIG. 3A  is a function of input parameters A and B, which are generated by applying preconditioning functions FP 1  and FP 2 , respectively, to operational parameters X and Y, respectively. In other words, operational parameters X and Y are mapped to input parameters A and B, respectively, using preconditioning functions FP 1  and FP 2 , respectively. A, B, and the performance parameter are then used to fit a SPM model FM 1 .  
         [0067]      FIG. 3B  shows an example of the fit that can be achieved through the use of data preconditioning. Performance parameter values PP 11 -PP 16  from  FIG. 3A  are plotted against input parameter A (i.e., the preconditioned operational parameter X) as points P 1 ′-P 6 ′, respectively, with input parameter B held constant at a value B_FIX for clarity. It can be seen that points P 1 ′-P 6 ′ exhibit a relatively linear relationship (indicated by idealized curve C_IDEAL) that can be readily modeled by mathematical model FM 1 .  
         [0068]     Note that while mathematical model FM 1  described with respect to  FIGS. 3A and 3B  is a function of two preconditioned operational parameters (input variables) for exemplary purposes, according to various other embodiments of the invention, a mathematical model for a cell can be a function of any number of preconditioned operational parameters.  
         [0069]     Note further that according to another embodiment of the invention, a mathematical model for a cell can be a function of both preconditioned and raw operational parameters. For example,  FIG. 3C  shows a cell library  392  that includes a cell library entry  304  in accordance with an embodiment of the invention. Like cell library entry  302  in  FIG. 3A , cell library entry  304  models the timing arc between input  111  and output  133  of cell  100  using a single function FM 6 . However, rather than applying a preconditioning function to all operational parameters, a preconditioning function FP 7  is applied only to operational parameter X, so that function FM 6  is a function of the preconditioned operational parameter X (i.e., variable G) and the raw operational parameter Y.  
         [0070]     Note further that according to another embodiment of the invention, a preconditioning function can even merge operational parameters. In other words, the preconditioning function itself can be a function of two or more operational parameters. For example,  FIG. 3D  shows a cell library  393  that includes a cell library entry  305  in accordance with an embodiment of the invention. Like cell library entry  302  in  FIG. 3A , cell library entry  305  models the timing arc between input  111  and output  133  of cell  100  using a single function FM 7 . However, rather than being a function of some combination of preconditioned or raw operational parameters X and Y, function FM 7  is a function of a single variable H. Variable H is generated by a preconditioning function FP 8  that is a function of both operational parameters X and Y. Preconditioning function FP 8  therefore merges two variables (i.e., operational parameters X and Y) into a single modeling variable H, which in turn can reduce the complexity of model function FM 7 .  
         [0071]     Note also that according to another embodiment of the invention, a preconditioning function can be applied to the performance parameter values during creation of the mathematical model. However, because the resulting mathematical model will then generate the preconditioned performance parameter, a cell library entry for such a model will include the “inverse” of the preconditioning function used on the performance parameter values during mathematical model generation. The inverse preconditioning function allows the output of the mathematical model (i.e., preconditioned performance parameter) to be converted back into the actual performance parameter.  
         [0072]     For example,  FIG. 3E  shows a cell library  394  that includes a cell library entry  306  in accordance with an embodiment of the invention. Like cell library entry  302  in  FIG. 3A , cell library entry  306  models the timing arc between input  111  and output  133  of cell  100 , but rather than modeling performance parameter PP directly, cell library entry  306  includes a mathematical model FM 8  that models a preconditioned performance parameter PPP as a function of operational parameters X and Y. Preconditioned performance parameter PPP represents a preconditioned function of performance parameter PP, and so cell library entry  306  also includes an inverse preconditioning function FV 1  that, when applied to preconditioned performance parameter PPP, generates the actual performance parameter PP.  
         [0073]     Note that according to another embodiment of the invention, a mathematical model for a preconditioned performance parameter can itself be a function of preconditioned operational parameters. For example,  FIG. 3F  shows a cell library  395  that includes a cell library entry  307  in accordance with an embodiment of the invention. Like cell library entry  306  in  FIG. 3E , cell library entry  307  models the timing arc between input  111  and output  133  of cell  100  in terms of a preconditioned performance parameter PPP. However, the mathematical model FM 9  that is used to generate preconditioned performance parameter PPP in cell library entry  307  is a function of input parameters J and K, which are generated by applying preconditioning functions FP 7  and FP 8 , respectively, to raw operational parameters X and Y, respectively. Cell library  307  also includes an inverse preconditioning function FV 2  that, when applied to preconditioned performance parameter PPP, generates the actual performance parameter PP.  
         [0074]      FIG. 4  shows a block diagram of a computer system  400  that includes a library generator  420 , in accordance with an embodiment of the invention, for translating an original cell library  410  (e.g., an uncharacterized cell library or a cell library having a format different from the desired one) into a new characterized cell library  460  (e.g., libraries  390 ,  392 , and  393  in  FIGS. 3A, 3C , and  3 D, respectively) having a desired format (e.g., SPM format). Library generator  420  performs this translation process as described with respect to  FIG. 2A . According to an embodiment of the invention, library generator  420  can comprise software stored within computer system  400 , software accessed remotely (e.g., run from a network server (not shown)), or software available on any other medium readable by computer system  400 .  
         [0075]     Library generator  420  includes an optional precharacterized data generator  430 , an optional preconditioner generator  440 , and a mathematical model generator  450 . If original cell library  410  does not include precharacterized data, precharacterized data generator  430  performs a simulation of the cells in library  410  (e.g., SPICE simulation). If library  410  already includes the precharacterized data, precharacterized data generator  430  can be bypassed. Note that according to another embodiment of the invention, precharacterized data generator  430  can be implemented outside of library generator  420  (e.g., in a separate software module or even a separate computer system).  
         [0076]     In either case, the precharacterized data provided to preconditioner generator  440  includes one or more sets of performance parameter values (e.g., PP 11 -PP 46  in  FIG. 3A ) associated with one or more sets of operational parameter values (e.g., X 1 -X 6  and Y 1 -Y 4  in  FIG. 3A ). Preconditioner generator  440  selects one of the sets of performance parameter values, and can apply one or more preconditioning functions (e.g., functions FP 1 -FP 2  in  FIG. 3A ) to some or all of the sets of operational parameter values that reference the performance parameter values and/or can apply a preconditioning function (e.g., the inverse of function FV 1  in  FIG. 3E ) to the performance parameter values. The preconditioned data is then passed to mathematical model generator  450 , which fits mathematical forms to the performance parameter data (or preconditioned performance parameter data) as referenced by the preconditioned (and any non-preconditioned) operational parameters, thereby generating the mathematical models (e.g., function FM 1  in  FIG. 3A  or function FM 6  in  FIG. 3E ) for the cells in the final characterized cell library  460 . Mathematical model generator then writes the characterized cell library  460  to some form of computer-readable medium, such as memory within computer system  400 , a removable storage medium (e.g., CDROM or DVD), or a network storage location. Note that according to another embodiment of the invention, writing the final characterized cell library to some form of medium can be performed outside of library generator  420  (e.g., in a separate software module or even a separate computer system).  
         [0077]     Note that, as indicated by the double-headed arrow between preconditioner generator  440  and mathematical model generator  450 , if the modeling performed by mathematical model generator  450  cannot provide a satisfactory approximation of the preconditioned data (i.e., the model fit to the data does not fall within a predetermined tolerance band), preconditioner generator  440  can apply a different set of preconditioning functions to the precharacterized data to generate a new set of preconditioned data. Mathematical model generator  450  can then use this new set of preconditioned data to derive more accurate mathematical models for the cells.  
         [0078]      FIG. 5  shows a flow diagram for an analysis process (e.g., synthesis or static timing analysis) using a characterized cell library employing preconditioning functions, in accordance with an embodiment of the invention. In a “SELECT CELL” step  510 , a first cell in an IC design is selected. Then, in a “DETERMINE OPERATIONAL PARAMETER VALUE(S)” step  520 , one or more operational parameter values (e.g., values for input slew and output capacitance) associated with a performance parameter type (e.g., delay) are specified for that cell. Next, in an “APPLY PRECONDITIONING” step  530 , the preconditioning function(s) associated with the cell are applied to the operational parameter value(s) to generate preconditioned input values.  
         [0079]     Those preconditioned input values are then substituted into the mathematical model for the performance parameter type being calculated for the cell in a “SUBSTITUTE IN TO MODEL” step  540 . The resulting output of the mathematical model is provided as the performance parameter value for the cell under the given conditions, in a “GENERATE PERFORMANCE PARAMETER VALUE” step  550  (note that if the mathematical model was derived using preconditioned performance parameter data, step  550  involves applying an inverse preconditioning function to the output of the mathematical model to generate the performance parameter value for the cell). Then, in an “ADDITIONAL CELLS?” step  555 , if more cells in the IC design remain to be analyzed, the process loops back to step  510 , where a new cell is selected. Otherwise, the process ends at “END” step  560 .  
         [0080]     In this manner, preconditioning functions associated with a cell library can be used to accurately and efficiently model the behavior of cell elements and therefore allows more efficient IC designs. Because the preconditioning can be implemented in a cell library that may only differ from a conventional polynomial-based library by a set of preconditioning functions, conventional EDA systems can be readily adapted to make use of a cell library in accordance with the invention.  
         [0081]     The various embodiments of the structures and methods of this invention that are described above are illustrative only of the principles of this invention and are not intended to limit the scope of the invention to the particular embodiments described. Thus, the invention is limited only by the following claims and their equivalents.