Abstract:
A method of data compression for continuous or piecewise linear curves in two variables is presented which can guarantee that any compression error is exclusively on one selected side of the curve. Limiting errors to one side is required when simulating integrated circuit performance to determine if a design will have speed-related problems. In such a simulation it is necessary to calculate both the minimum and maximum possible time delays for a logic chain of circuit elements. Data compression of the transistor or gate voltage versus time relationship is necessary to reduce the very large amount of data that is required for the simulation. Data compression may introduce errors into the data in either direction. If it is necessary to have any possible error confined to one side of the curve, the compressed data must be shifted toward the desired error side by at least the maximum possible data error. This shifting increases the total error between the compressed data and reality by more than is acceptable in current simulations. The present method approximates the data curve with piecewise linear segments all of which are on a selected side of the curve, for example the late side. This avoids increasing the error between the data curve and the stored approximation due to shifting of the stored data to ensure that all points are on the selected side.

Description:
BACKGROUND OF THE INVENTION 
     During the design of silicon integrated circuits (ICs), a method is needed to verify that the circuit design will work at the required clocking speed before the IC is manufactured. This is necessary because the time delay in verifying that a design is too slow for its intended application by waiting for actual ICs to be fabricated results in a product that may be many months late coming into the market. Arriving late into the electronic market place frequently results in lower prices for an electronic component, and in lower market share, both typically causing reduced earnings. 
     One known method of verifying the functionality at operating speed of an IC design is to use what is known as a min max static timing verifier. Examples of commercial timing verifiers include Path Mill by Synopsis, Inc., Pearl by Cadence, Inc., and Veritime by Cadence Inc. Typical timing verifiers work by calculating the minimum possible time delay and the maximum possible time delay for a logic signal traversing a circuit path comprised of transistors or logic gate, and labeling any path that does not meet the required timing limits or constraints. 
     Within any of the known timing verifiers there are many different techniques for calculating the logic signal propagation delay along a circuit path of interest. Typically the circuit path is broken down into smaller and more easily calculated sections, for example, individual logic gates such as NAND gates, NOR gates, inverters, and channel connected regions (CCR). The logic delay for the circuit path is then simply the sum of all of the delays for the individual parts. The delays used by the timing verification tools for the NAND, NOR, inverters and CCRs are typically computed from characterization for delay models. These models characterize the delay and voltage transitions for each output of each given gate or CCR in a circuit path. If the total computed path delay results in a design violation along the circuit path of interest, then the designer will employ more accurate delay estimation methods to verify the path performance before potentially redesigning the circuit path. Commonly designers will use a circuit simulation tool, such as SPICE, Timing Mill by Synopsis, Inc., Starsim by Avant!, Inc., and ASX by IBM, Inc., as the accurate delay estimator tool for the circuit logic element, such as a NAND gate, CCR, or for the entire circuit path. In certain instances the timing verifier may be able to directly invoke the circuit simulator tool. 
     The circuit simulation tool produces a behavior model for the circuit that is represented as a waveform. These waveforms must be stored in a memory device for later use. Since memory resources are expensive, it is beneficial to provide a method for compressing the waveform data. Since minimum delay and maximum delay estimations are required by timing verifiers, known as min/max sides, then two waveforms may have to be stored, one for the minimum delay or fast estimations (known as early signals) and one for the maximum delay or slow estimations (known as late signals). 
     The waveforms just discussed are an example of typical two variable curves, in this particular case voltage versus elapsed time. To reduce the amount of data that needs to be stored, the voltage versus time curve maybe expressed as a series of voltage values at a series of evenly spaced time intervals. The more time intervals recorded (i.e., the shorter the time interval), the more accurately the stored compressed data approaches the original. 
     Limiting errors to one side of a curve is required when simulating integrated circuit performance to determine if a design will have speed related problems. In such a simulation it is necessary to calculate both the minimum and maximum possible time delays for a logic chain of circuit elements. Data compression of the transistor or logic gate voltage versus time relationship is necessary to reduce the very large amount of data that is produced by the simulation. Data compression may introduce errors into the data in either direction, i.e., either to the fast operating side or to the late side. Since it is necessary to have all possible error confined to the slow (i.e., late) side of the curve when calculating the maximum delay, or to the fast (i.e., early) side of the curve when calculating the minimum delay for the logic chain, the compressed data must be shifted in the desired direction by the maximum possible data error in order to ensure that all of the compressed data points are either early, or late. This shifting increases the total error between the compressed data and the original waveform by more than is acceptable in current calculation tools. 
     SUMMARY OF THE INVENTION 
     Therefore, it would solve a problem in the art to provide a method or a tool for ensuring that all of the data compression error in a timing verifier tool is confined to a selected side of the curve, since with such a method there would be no need to shift the compressed data by an amount equal to the maximum error toward the desired region, and no consequent increase in error introduced by the compression. 
     A method of data compression for open curves in two variables is presented which can guarantee that any compression error is exclusively on one selected side of the curve. In the example of a circuit model waveform, the variables are voltage and time, or current and time, etc., and the area to the right of the curve (i.e., increasing time) is the slow (or late) side. The method requires inputs to define a beginning point, an ending point, an accuracy limit, and the side to which the error is to be confined. 
     The curve is divided into rising and falling segments, and into curve sections, each of which is either convex shaped or concave shaped toward, for example, the late side. On convex sections of the curve, the compression method draws a tangent to the curve at the beginning point and calculates the difference between the tangent line and the curve until the difference reaches a predetermined accuracy value. A new tangent line is drawn to the curve at the point where the deviation from the original tangent reaches the limit, and the intersection point of the two tangents is stored in memory. This process is repeated until the end point of the convex segment is reached, where another tangent is drawn to conclude the convex section approximation and data compression, and to begin the concave curve section approximation and data compression. The convex portion of the curve has been thus been approximated by a piecewise linear set of lines connecting the intersection points of the tangent lines drawn on the selected side of the curve, and none of the points on the piecewise set of lines is ever on the opposite side of the original curve. 
     The concave section approximation continues the tangent line drawn at the inflection point between the convex and concave sections of the curve, until the deviation in value exceeds the accuracy value. A new tangent line is constructed at the point on the curve where the deviation exceeded the set accuracy and the intersection point of the two tangent lines is found. The angle between the two tangent lines is calculated and bisected, and a perpendicular line to the bisecting line is constructed. The intersection point between the perpendicular line and the original curve is stored in memory. Thus the concave section of the curve has been approximated by a piecewise linear set of lines connecting the points of the original curve that are nearest to the intersection of the tangent lines. Since the points are all on the original curve and the segment is concave, then every point on the piecewise linear set of lines is on the opposite side of the original curve from the tangent lines, and all points are on the selected side of the curve. 
     The curve or waveform maybe divided into rising and falling segments by using the predefined beginning and ending points, and all locations on the curve having a first derivative equal to zero, i.e., being horizontal. A falling segment may be followed by either another falling segment or by a rising segment. One method of determining whether a segment is rising or falling is to measure the difference in the value of the curve at the beginning point of the segment and at the end point, where either beginning or ending points of a segment may be locations where the curve is horizontal. Another method of determining whether a segment is rising or falling is to examine the first derivative of the curve. A falling curve has negative slope and a negative derivative. 
     The inflection points that divide the rising and falling segments of the original curve into convex and concave sections may be found by calculating the second derivative of the curve function and setting it equal to zero. Between the beginning point of a curve section and the point where the second derivative equals zero, the curve section will be either convex toward the selected side or concave. The first derivative will be negative for a falling curve segment, and the change in the value of the first derivative while traversing the curve toward the right will be more negative for a convex right falling curve, and more positive for a concave right curve. The opposite is true for rising curve segments. Another way of looking at this is that the value of the second derivative will be negative for a convex right falling curve, and positive for a concave right falling curve. 
     The described method is useful for data compression of any open curve or waveform in two variables. This means that the curve may not have any loops that cause intersections between portions of the curve. The curve may be a continuous function or a piecewise linear curve. There may be numerous changes in direction of the curve, such as glitches or overshoots in voltage versus time waveforms. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The foregoing and other objects, features and advantages of the invention will be apparent from the following more particular description of preferred embodiments of the invention, as illustrated in the accompanying drawings in which like reference characters refer to the same parts throughout the different views. The drawings are not necessarily to scale, emphasis instead being placed upon illustrating the principles of the invention. 
     FIG. 1 is a graph of a curve in two variables; 
     FIG. 2 is a graph of the compression of a falling convex curve, in accordance with the invention; 
     FIG. 3 is a graph of the compression of a falling concave curve, in accordance with the invention; 
     FIG. 4 is a graph of the compression of a rising convex curve, in accordance with the invention; 
     FIG. 5 is a graph of the compression of a rising concave curve, in accordance with the invention; and 
     FIG. 6 is an overall view of the illustrative curve with the approximation points shown in accordance with the invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In FIG. 1, a curve  10  represents a functional relationship in two variables. Curves of the general type of curve  10  may represent any sort of harmonic, oscillatory, damped harmonic, growth or decay curves or waveforms. Such curves may be used to represent physical behavior such as voltage changes in electronic devices, vibrations in mechanical structures, or the movement of masses on springs. A curve or waveform such as  10  may be either a continuous function as shown, or a piecewise linear curve of connected straight line segments. The line segments comprising a piecewise linear curve may have any length, with shorter line lengths and greater numbers of line segments tending to more closely approximate the value of a continuous curve. A horizontal axis  12  records one of the two variables, in this illustrative embodiment, representing time. A vertical axis  14  records the other variable, in this illustrative embodiment representing voltage on a electronic device. The curve has a defined starting or beginning point  16 , and a defined ending point  18 . The curve  10  breaks the area of the chart into two regions, region  20  to the right of the curve  10 , and region  22  to the left. Using the two axes of the illustrative embodiment, the region  20  represents action of the electronic device that is later (i.e., slower) than that represented by the curve  10  over the first part of the curve, and the region  22  represents earlier (i.e., faster) electronic device action. This is true only of the falling part of the curve that runs from the starting point  16  to the first point  24  where the curve  10  becomes horizontal, or where the slope of curve  10  becomes zero. Point  24  may be found by setting the first derivative of the function of curve  10  equal to zero, or other means and methods as known in the art. 
     For the rising part of curve  10 , from point  24  to point  26 , where the first derivative is again zero, the roles of regions  20  and  22  reverse, and region  20  becomes the earlier (i.e., faster) operating region and region  22  becomes the later (i.e., slower) operating region for the rising curve. For the falling curve region between point  26  and the ending point  18 , regions  20  and  22  revert back to their initial representations. Essentially, points to the right of any curve segment are slower, or later. Thus any open curve which is single valued in one of it variables (in this illustrative example the variable of time) may be simply partitioned into falling segments and rising segments. 
     Within each of the falling or rising segments of the curve  10 , there may be more than one direction of curvature. The illustrative embodiment shows two curvatures for each of the segments of curve  10 . The first falling segment, from  16  to  24 , begins as a convex right curve, i.e., with the convex portion pointing toward the right at region  20 , which in this example represents later electronic action. The convex section ends at inflection point  28 , and the curve continues as a concave right curve, which may also be called convex left, to point  24 . The inflection point  28  may be found by setting the second derivative of the function of curve  10  to zero, or by other means and methods. 
     The second segment of curve  10 , running from point  24  to point  26 , is a rising curve, again having a concave portion and a convex portion. The inflection point  30  defines the boundary of the convex right section of the curve from the concave right section of the rising curve segment of curve  10 . 
     The second falling curve segment of curve  10  runs from point  26  to the ending point  18 , and again has a convex right portion extending to inflection point  32 , and a concave right portion extending to the end point  18 . In all cases the inflection points  28 ,  30  and  32  may be found by setting the second derivative equal to zero, or other methods of examining changes in slope. 
     The above described method of breaking up complex curves into rising and falling segments, and into concave and convex sections, may be applied to any single valued curve having two variables, and serves to simplify the analysis of the curve and the compression of the curve data into a more easily stored form. A typical method of representing either a continuous curve function or a very fine piecewise curve numerically is to approximate the curve by drawing a series of straight line segments between points spaced on the curve, and storing the location of the points in memory. While a waveform such as curve  10  may require several hundred sets of points to store the curve data accurately, thus resulting in a high requirement for storage memory, a piecewise approximation of the curve  10  may have only 20 sets of points to store and to use in calculations, thus resulting in improved calculation speed, but at the expense of reduced accuracy. In general, the more individual points of the curve that are stored, the better the accuracy, but the slower the calculation. Clearly the placement of the points is very important. For example, a 20 point approximation of a waveform may have more error than a 10 point approximation if the 20 points are not optimally placed. 
     For example, imagine using the numbered points on curve  10  as the end points of line segments intended to approximate the value of curve  10 . The amount of storage has clearly been reduced since only 7 points need to be stored, but the deviation between the value of curve  10  at some time point  34  on horizontal axis  12 , and the value of the line segment at that same time point may be large, leading to inaccuracy in calculations using the compressed data. The dotted line  36 , between point  28  and point  24  on curve  10 , has a value  38  at time  34 , while the curve  10  has a very different value  40  at the same time. Therefore, calculations using the approximate line  36  to represent the value of curve  10  will introduce unacceptable errors. In this example the error introduced was in region  20 , and thus represents a later operation of an electronic device than the correct value as represented by curve  10 . However, the direction or the sign of the introduced error is not always the same, as can be seen by drawing a straight line between point  16  and point  28 . In that region the approximation introduces an error toward the faster or earlier region  22 . Clearly the more points that are used to draw the straight line approximations the better the accuracy, assuming proper point placement, and the more storage memory is required. Each application will have its own balance between speed and accuracy. 
     There are certain applications where it is necessary that all of the error introduced by the approximation and compression of the data of curve  10  be either on the late side, or on the early side. As mentioned previously, region  20  is not always the late side. An example of such an application is calculating the maximum time that a signal will take to traverse a series of electronic devices, each one having a known voltage versus time behavior represented by a function having a curve such as curve  10 . It is necessary to ensure that all points of the first falling section (between points  16  and  24 ) of the compressed data of curve  10  be located in region  20 , i.e., the late operation region to the right, in order to ensure that the overall calculation results in a true maximum time for signal propagation. Other examples of max/min strengths, or structural flexion, exist that have the same requirement that all of the compression error to be on one side or the other of the original curve. What is typically done in the art to ensure that all points are on one side, is to shift the entire compressed (i.e., the piecewise linear line segments) curve over toward the desired side by at least the maximum possible error, thus resulting in a true maximum, or minimum, calculation. 
     In FIG. 2, a falling curve section  110 , having a horizontal variable axis  112 , in this illustrative embodiment representing time, and a vertical variable axis  114 , representing electronic device voltage levels, is convex right. The curve  110  has a beginning point  116  and an ending point  128 , which corresponds to inflection point  28  of FIG. 1 where the second derivative of curve  10  was equal to zero. Assuming for the present example that the desired direction in which to confine the compression error is toward the right, in this example the late side, a tangent  142  line (in this illustrative embodiment happening to be horizontal) is drawn from point  116 . Following the curve  110  in the direction of increasing time, at a time t 1  a point  144  is found where the difference in value between the curve  110  and tangent  142  is equal to a previously defined accuracy limit,  146 . A new tangent line  148  is drawn to the curve  110  at point  144 , and the intersection  150  of the two tangent lines is calculated and stored in a memory location in an ordered list along with the initial location  116 . These two stored points,  116  and  150 , are the end points of one of the linear line segments that will approximate the value of curve  110 . Note that the maximum difference between the two lines of the stored approximation and the curve  110  is the difference between point  150  and the curve  110 , which will be less than the defined accuracy limit  146 . 
     Now starting from point  144  on the curve  110 , and again proceeding toward increasing time in this illustrative embodiment, at a second time t 2 , point  152  on the curve  110  is reached, where the difference in value between tangent  148  and curve  110  again reaches the predetermined accuracy limit  154 . Typically all of the predetermined accuracy limits will be equal to a single value for a given curve  110 . In a preferred embodiment of the invention the accuracy limit is set at 150 millivolts. A new tangent  156  is drawn at point  152  and the intersection point  158  between tangents  152  and  148  is calculated and stored in the proper place on the ordered list. The straight line from  150  to  158  is the second line segment in the piecewise linear approximation and data compression of curve  110 . 
     Following the procedure again results in point  159  where the value of the tangent  156  differs from curve  110  by a predetermined accuracy limit  160  and a new tangent line  162  drawn at point  159 . The intersection point  164  between tangents  162  and  156  is again stored in the ordered list of end points of linear segments. 
     Finally, a tangent  166  is drawn through inflection or ending point  128 . The intersection point  168  and ending or inflection point  128  are recorded in the ordered list. 
     Thus the curve  110  is data compressed and approximated by the six points  116 ,  150 ,  158 ,  164 ,  168  and  128 , which represent the end points of line segments, all of whose points are within a preset accuracy value of the curve  110 , and all of whose points are on or to the right side of curve  110 . 
     In FIG. 3, a falling concave right curve  210  has a starting point  228 , which may also be an inflection point from a previous curve section, and a tangent line  212  constructed at point  228 . The tangent  212  may also be the tangent line  166  from the previous section of the curve as shown in FIG.  2 . Again in this example, it is assumed that the error is to be confined to the right, or later, side of the curve. At a point  230  on curve  210  at a time t 3 , the difference in value between the curve  210  and the tangent  212  reaches the accuracy limit at point  231 , and a new tangent line  232  to curve  210  is constructed at point  230 . The intersection point  234  between tangents  232  and  212  is determined, and the point on curve  210  closest to intersection  234  is calculated and recorded, in this example point  240 . In a preferred embodiment of the invention the point  240  is calculated by bisecting the angle between the tangents  232  and  212 , resulting in dotted line  236 . A line  238  is calculated perpendicular to bisection line  236 . The intersection (i.e., point  240 ) of perpendicular line  236  and the curve  210  is calculated and recorded along with starting point  228  in an ordered list. Note that there are other possible methods for finding the closest point on the curve  210  to the intersection point  234 . Points  228  and  240  are the end points of a straight line that approximates the curve  210  to within a preset accuracy value, with all of the values of the approximation on the right, i.e., the later or slower side of curve  210 . 
     Continuing to follow curve  210  in the direction of increasing time value, a point  242  at a time value t 4  is reached where the value of the curve  210  differs from the value of tangent  232  by the predetermined accuracy value. In the preferred embodiment of the invention all of the predetermined accuracy values will be the same. New tangent  244  is constructed at point  242 , and the intersection  246  of tangents  232  and  244  is calculated. The angle between the tangents is calculated and bisecting dotted line  248  is used to calculate perpendicular dotted line  250 . The intersection  252  of perpendicular line  250  and curve  210  is calculated and stored in the ordered list. The points  240  and  252  on the curve  210  are the end points of approximation line segment  254 . Note again that all of the points on line  254  are either on curve  210 , or are on the selected side of curve  210  to confine any compression errors and are within the preselected accuracy value  231 . 
     The above described process is continued until the predefined end point or inflection point  224  is reached, in this example the point where the descending curve becomes horizontal, and begins to ascend. A tangent line  256  is constructed to curve  210  at the end point  224  and the intersection with tangent line  244  is calculated as before. The nearest point on curve  210  is point  258 , and this point is recorded in the ordered list. If the point  224  is the endpoint of the entire curve to be data compressed, then the point  224  is recorded as the last entry in the ordered list. 
     The process of approximating and data compressing the value of the curves  110  and  210  as described above with reference to FIGS. 2 and 3, were discussed with the selected side upon which the compression error is to be confined being the right, or late side, region  20 . The process is exactly the same for the case of confining the error to the left, or early side region  22 , except that the curve section that was convex right becomes concave left, and thus the concave process of FIG. 3 would be performed first, and then followed by the second section of the curve, which is now convex left, performed as described in the FIG. 2 discussion. 
     In FIG. 4, a rising curve section  310  has a starting point  324 , which may also be the zero slope point of a previous curve segment, and an ending point  330 , which may also be the inflection point for a subsequent curve section. The process as previously is used for approximating and data compressing the curve  310  with the error confined to the right, i.e., the later side, region  22 . Note that the late side of the rising curve segment is now region  22  as opposed to region  20  as discussed above with reference to the falling curve segment. The rising curve  310  is a convex right curve section, and thus the process will be seen to be similar to that for the convex right falling curve of the FIG. 2 description. 
     Tangent  332  is constructed from point  324  and the point on curve  310  where the deviation reaches the predefined accuracy limit is  334 . Tangent  336  is constructed at point  334  and the intersection  338  of tangents  332  and  336  is recorded in an ordered list as previously described. At point  340  on curve  310  the deviation from tangent line  336  exceeds the accuracy limit, tangent line  342  is constructed, and intersection point  344  is recorded in the ordered list. The predefined accuracy limit is not reached again before the ending point  330  of the curve section  310 , and thus the last tangent line  346  is constructed at point  330 , and the intersection point  348  and ending point  330  are recorded on the ordered list. 
     Again, note that the illustrative example is confining the compression error to the slow side of the curve  310 , in region  22 , and that all points on the line segments  324 ,  338 ,  344 ,  348  and  330  are either on curve  310 , or on the selected side to confine error, and are within the predetermined accuracy value. 
     In FIG. 5, a concave right rising curve section  410  has a beginning point  430  and an ending point  426 , either of which may be inflection points or zero slope points of preceding or subsequent curve sections. A tangent  432  is constructed at starting point  430 . A new tangent line  436  is constructed at the point  434  on curve  410  where the deviation between the tangent  432  and the curve  410  reaches the accuracy limit, and the intersection  438  of the tangent line is found. The angle between the tangent lines is bisected with dotted line  440  and the perpendicular dotted line  442  is constructed. The intersection  444  of the line  442  and the curve  410  is recorded on the ordered list as one of the end points of the straight lines that approximate the curve  410 . 
     At point  450  on curve  410  the accuracy limit is reached and new tangent  452  is constructed. The intersection  453  of tangents  452  and  436  is found, the angle bisected and the perpendicular line is found, and the point  454  on curve  410  is recorded in the ordered list. The process is repeated to find point  460 , draw tangent  462 , find intersection  464 , and perpendicular point  466  on curve  410  is found and recorded. 
     The accuracy is not exceeded prior to reaching the end point  426  of the curve section  410 , then a tangent  470  is constructed at end point  426 , the intersection with tangent  462  is found at point  472 , the nearest point on curve  410  is found at point  474  and is recorded in the ordered list along with the end point  426  and the process concludes. The stored points  430 ,  444 ,  454 ,  466 ,  474  and  426  are the end points of straight lines that approximate the curve  410  with only six sets of points, all within the defined accuracy limit and with all data points on the late side of the curve  410 , in region  22 . An exemplary straight line segment  480  is shown between stored points  454  and  466 , and demonstrates that all points on the compressed data line segments will again be either on the curve  410 , or on the desired side to confine compression errors. 
     In FIG. 6, the entire curve is shown with the approximation line segments discussed and shown above with reference to FIGS. 1-5. Note that the side to which the error is confined is always to the right side of the curve  510  for the illustrative example of late electronic response. Also notice that the late approximation line segments cross over the curve  510  at points  520  and  530 . The curve tangents are shown as dotted lines and the approximation line segment end points are shown as bold dots. The compression method records the endpoint values as an accurate representation of the entire curve, with a consequent compression ratio of greater than 300%. 
     Using the above-described process of dividing a curve into rising and falling segments, and the segments into convex and concave sections, an automatic method of curve data compression with predetermined accuracy and with all of the compression error confined to a chosen side of the curve may be obtained. The method requires that a starting point, an ending point, at least one accuracy limit, and a chosen side upon which to confine any data compression error be provided. The method also requires that the curve to be data compressed be single valued in at least one of the two variables, i.e., it must have no locations where the curves loops or contacts itself. A specific example is the voltage versus time relationship of an electronic device such as a logic gate. The logic gate will not have two different voltage values at any single time point, and it thus single valued with respect to time. 
     While this invention has been particularly shown and described with references 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.