Abstract:
A system and method for estimating propagation noise that is induced by a non-zero noise glitch at the input of the driver circuit. Such propagation noise is a function of both the input noise glitch and the driver output effective capacitive load, which is typically part of the total wiring capacitance due to resistive shielding in deep sub-micron interconnects. The noise-driven effective capacitance solution provided herein also estimates the propagation noise induced by a non-zero noise glitch at the input of the driving gate. Gate propagation noise rules describing a relationship between the output noise properties and the input noise properties and the output loading capacitance are used within the noise-driven effective capacitance process to determine the linear Thevenin model of the driving gate. The linearized Thevenin driver model is then employed to analyze both the propagation noise and the combined coupling and propagation noise typically seen in global signal nets.

Description:
BACKGROUND OF THE INVENTION 
     The present invention relates to the design of integrated circuit chips, and more particuarly, to a method and a system for improving the accuracy of existing noise analysis tools in order to identify wires susceptible to noise problems such that other tools can resolve the noise problems in the design. 
     The continuous scaling of CMOS processes has lead to an increase of noise in digital integrated circuits (ICs). Noise glitches that propagate to a dynamic node or storage element (e.g., a latch) can alter the state of the circuit. This type of noise is often called functional noise. There are two types of noise glitches in a victim net. The first one is coupling noise, which refers to the noise due to the switching of the neighboring nets (referred hereinafter as aggressor nets) that are coupled to the victim net via coupling capacitances or coupling inductances. The second one is referred to propagation noise, which comes from an existing glitch at the input of the victim driver. 
     To gain a better understanding of the problem addressed by the invention, reference is made to  FIG. 1  that graphically illustrates the coupling noise and the propagation noise. The two sources of functional noise are shown occurring in victim net  102 . The first one is the aforementioned propagation noise, which is induced by an existing glitch at the input of the victim driver. The second, is the previously mentioned coupling noise, also known as crosstalk noise, which is caused by the simultaneous switching of aggressor net(s)  103  capacitively or inductively coupled to the victim net  102 . When the aggressor net  103  is not switching, only propagation noise is seen at the victim net  102 . When the arrivals of the aggressor switching signal  110  and the victim input noise pulse  114  are aligned with each other, the combined propagation and coupling noise, e.g.,  106 ,  107 ,  108 , will show in victim net  102 . Because of the non-linearity of the victim driver  100 , the combined propagation and coupling noise is typically much higher than a superposition of each individual noise. Therefore the victim driver modeling is of particular importance. 
     It is usually impractically slow to directly analyze the global signal net shown in  FIG. 1  using a non-linear circuit analyzer. In order to determine the worst-case noise peak at the input of each victim receiver (e.g.  104 ), the same circuit has to be re-analyzed with different alignments of arrival times of the aggressor switching signal  110  and the victim input noise glitch  114 , which is formidable. 
     Non-zero noise at both the input and output of the victim driver often makes the gate display a strong non-linear behavior. Therefore, the victim driver must be carefully modeled to ensure a correct functional noise analysis. 
     Two general approaches have been proposed for modeling the victim driver and the worst-case noise analysis. In the first approach, the victim driver is linearized while keeping the overall aggressor and victim circuit a linear system. The worst-case peak noise alignment between the victim and aggressors are obtained using superposition, which requires having the entire circuit analyzed only once. However, it is rather challenging to model the non-linear behavior of a gate using a simple linear model. By way of example, an extremely simple linear model can consist of only a load independent noise pulse and a driver quiet holding resistance. Such an approach typically introduces a considerable amount of error. It has been reported that for a 130 nm technology, the simplified flow underestimates the noise peak by as much as 70%. The second approach characterizes the driver using a simplified non-linear behavioral model. It consists of a non-linear voltage dependent DC current source and parasitic capacitors. To analyze such a model, a fast non-linear transient analysis engine with numerical integration techniques is employed. One limitation of such a behavioral model is that it works well only for single stage gates. Moreover, the introduction of the non-linear model in the circuit makes the worst-case peak alignment between the victim and aggressors costly. The alignment must be achieved iteratively, and in each iteration, a non-linear analysis of the behavioral model along with the entire interconnect circuit needs to be performed. 
     Another method that has been reported falls into the first category of computing a linear Thevenin model for the victim driver. To improve the accuracy, the computation of the linear victim driver parameters involves matching the linear driver current to the non-linear current through the interconnect driven by the non-linear behavioral model which, by nature, limits this approach to single-stage driving gates. Additionally, in this model, several Thevenin model parameters, such as the Thevenin resistance and Thevenin voltage pulse width are empirically chosen. However, this approach raises concerns in the computational cost of the model parameters. 
     Therefore, there is a need in industry for a better approach for determining a linear victim driver model whose model parameters are easy to compute, which is convenient for worst-case noise alignment and which is fully integratable with existing industry standard cell libraries. 
     OBJECTS AND SUMMARY OF THE INVENTION 
     Accordingly, it is an object of the invention to improve the accuracy of existing noise analysis tools in order to identify wires susceptible to noise problems such that other tools can solve the noise problems in the design. 
     It is another object of the invention to minimize the increase in run time of the noise analysis tool while executing a more accurate analysis. 
     It is yet another object of the invention to improve and accurately predict signal integrity issues in high performance digital integrated circuits. 
     These and other objects, aspects and advantages of the invention are achieved by a method for estimating propagation noise that is induced by a non-zero noise glitch at the input of the driver circuit. Such propagation noise is a function of both the input noise glitch and the driver output effective capacitive load, which is typically part of the total wiring capacitance due to resistive shielding in deep sub-micron interconnects. 
     The present invention provides further a noise-driven effective capacitance solution for estimating propagation noise induced by a non-zero noise glitch at the input of the driving gate. Gate propagation noise rules describing a relationship between the output noise properties and the input noise properties and the output loading capacitance are used within the noise-driven effective capacitance process to determine the linear Thevenin model of the driving gate. The noise rules are either pre-characterized as a look-up table using SPICE or analyzed using a simple non-linear behavioral model of the gate. The linearized Thevenin driver model is then employed to analyze both the propagation noise and the combined coupling and propagation noise typically seen in global signal nets. The present invention extends a conventional timing driven effective capacitance method into the noise domain. Similar to the effective capacitance method in timing analysis which is widely used in industry timing tools, this approach provides a successful mechanism for separating the non-linear driver analysis from the linear interconnect analysis. In addition, the linear driver model maintains the linear property of the overall circuit, upon which superposition is applied to ease the task of finding the peak alignment of all the propagation and coupling noise sources. 
     This present invention extends the conventional timing-driven effective capacitance method into the noise domain. Similar to timing, gate propagation noise rules describe the relationship existing between the output noise properties, the input noise properties and the output loading capacitance for use within the noise-driven effective capacitance process. The noise rules are either precharacterized as a look-up table using SPICE or analyzed using a simple non-linear behavioral model of the gate. Through an effective capacitance methodology, a linear Thevenin model is constructed for the non-linear driver. The linear Thevenin model consists of one parameter for a resistor and four parameters for a triangular voltage source. These parameters are then calculated from the propagation noise rules. 
     The propagation noise in an interconnect is computed by convolving the Thevenin voltage source with the transfer function of the linear interconnect circuit. The same Thevenin model is employed to analyze propagation noise in the presence of coupling noise caused by the simultaneous switching of aggressor nets capacitively or inductively coupled to the victim net. The coupling and propagation noise peak alignment is trivially achieved by superposition. This approach works both accurately and efficiently for estimating the propagation noise and the combined propagation and coupling noise. 
     Additionally, the present invention provides a method for minimizing noise problems in a chip design, the chip design including at least one driver gate driving a victim net and sinks attached to the victim net, the method including the steps of: a) identifying noise at the inputs of the driver gate and computing the noise at the output thereof; b) propagating the computed noise at the output to all the sinks attached to the victim net; c) identifying nets coupled to the victim net and signal transitions on thecoupled nets, and computing the coupled noise for all the sinks attached to thevictim net; d) adding the computed propagation noise to the computed coupled noise; e) iteratively repeating steps a) to d) until a latch is reached; and f) determining if the noise at the latch input overlaps a signal clock, the overlap being indicative of a chip failure. 
     The foregoing discussion has outlined rather broadly the features and technical advantages of the present invention in order for the detailed description of the invention that follows to be better understood. Additional features and advantages of the invention will be described hereinafter. 
    
    
     
       BRIEF DESCRIPTION OF THE FIGURES 
       For a more complete understanding of the present invention and the advantages thereof, reference is now made to the following description of the invention taken in conjunction with the accompanying figures, in which: 
         FIG. 1  is a prior art diagram illustrating coupling noise and propagation noise; 
         FIG. 2  is a prior art diagram illustrating a linear interconnect circuit for analyzing the combined coupling and propagation noise using linear drivers; 
         FIG. 3  is a prior art diagram illustrating a noisy waveform versus a switching waveform; 
         FIG. 4  is a detailed flow diagram of the method steps, expressed in generic form, according to the present invention; 
         FIG. 5  is a flow diagram detailing the same method steps of the present invention illustrated in  FIG. 4  expressed in mathematical form; 
         FIG. 6  is a diagram illustrating the conversion to a linear driver Thevenin model driving a load capacitance; 
         FIG. 7  is a diagram showing the rising transition time equation curve of the driver Thevenin voltage source; 
         FIG. 8  is a diagram illustrating the sensitivity-based Thevenin driver resistance R TH  of an inverter with respect to the input noise with various peaks and widths; 
         FIG. 9  is a diagram that illustrates the effective capacitance, π-model and higher reduced order model of the interconnect. The pi-model is used to represent the load as seen by the driver when it is connected to an RC wire; and 
         FIG. 10  shows a circuit typically found in a digital VLSI chip that illustrates various aspects of the present invention 
     
    
    
     DETAILED DESCRIPTION 
     In the following description, numerous specified details are set forth to provide a thorough understanding of the present invention. 
     Referring to  FIG. 2 , the noise-driven effective capacitance method will now be described to construct a linear victim driver (including  200  and  201 ). Theoretically, the state-of-art timing driven effective capacitance method is applied to the aggressor net to find the linear aggressor driver model (including  202  and  203 ). In order to focus on the modeling of the victim driver in the preferred embodiment of the present invention, it is assumed that the fastest possible switching signal  202  and the smallest possible aggressor driver resistance  203  are present. This assumption is often used in practice for worst-case coupling noise analysis. 
     It is common practice in the industry to specify the switching waveform by way of parameters such as delay and rise time, commonly referred to as timing rules. No such universally accepted noise rules exist for noise waveforms. Unlike a timing/switching waveform, the noise waveform does not perform a complete single 0-to-Vdd (or Vdd-to-0) transition. 
     Shown in  FIG. 3  are some of the parameters that specify the noise waveform that are required for the present invention. More specifically:  FIG. 3(   a ) shows a sample circuit being analyzed for noise rules generation.  FIG. 3(   b ) illustrates a simplified model for the circuit shown in  FIG. 3(   a ).  FIG. 3(   c ) depicts a typical timing waveform shown to distinguish between a noise waveform represented in  FIG. 3(   d ). Finally, in  FIG. 3(   d ) the noise waveform and various parameters of the noise waveform are specified in the noise rules.  FIGS. 3(   a )- 3 ( d ) will be explained hereinafter following a more detailed discussion of the method steps of the present invention. 
     The overall flow of the noise-driven effective capacitance algorithm of the present invention is illustrated in  FIG. 4 . 
     In step  1001 , a victim driver is shown driving a victim net, a list of sinks of the victim net to be analyzed based on the noise peak, and the width of the electrical signal as seen from the driver input which is provided. The victim driver is first analyzed using SPICE with a family of input noise pulse widths, peaks, output load capacitances and all the propagation noise rules listed in  1002  seen at the output of the driver, and which are stored in a table. The table facilitates calculating the propagated noise at the output of the gate as a function of the load capacitance connected to the gate output and the noise waveform at the gate input. Some of the specific parameters stored in the table are shown in the previously mentioned  FIG. 3(   d ). This table is generated by simulating the circuit shown in  FIG. 3(   b ) with various output load capacitances and different input noise waveforms. Once the table has been generated for the gate, it can be reused for all subsequent usages of this gate in the design. 
     The aforementioned rules are used in subsequent steps  1005 ,  1006 ,  1009 ,  1010  and  1011 . 
     In step  1003 , the total capacitance of the victim net is determined, and the effective capacitance at the driving point of the net to become the total capacitance is initialized. 
     In Steps  1005  and  1006 , two output noise widths from the noise rule table corresponding to the given input noise peak and width and two output capacitances with a small amount of difference are extracted. The driver linear Thevenin resistance is calculated in step  1007  by dividing the output noise width variance by the small variance in output capacitance. For the victim net, the reduced order driving point admittance Y(s) and the transfer function H(s) of each victim sink are determined by way of any known model order reduction (MOR) techniques (Step  1008 ). The use of MOR is prompted in order to speed the run time. In step  1012 , Y(s) computes the average current flowing into the victim net. In step  1015 , the voltage waveform at each victim sink is calculated. The noise rules are determined in step  1009  and used along with the Thevenin driver resistance obtained in step  1007 . The effective capacitance to form a non-linear equation is obtained and is solved to extract therefrom the driver Thevenin voltage source parameter of the rising transition time to peak (step  1010 ). 
     In step  1011 , another three Thevenin voltage source parameters are solved by way of three linear equations formed with the same noise rules of step  1009  including the Thevenin driver resistance and the present effective capacitance. An actual Thevenin voltage source is then obtained. The average current flowing from the voltage source through the Thevenin resistance into the victim net does not necessarily coincide with the average current flowing from the same voltage source through the same Thevenin resistance into the effective capacitance. Therefore, the effective capacitance in step  1012  is updated in order to match the two average currents. 
     In step  1012 , the convergence of the effective capacitance is checked. If the updated effective capacitance differs from the present effective capacitance, a second iteration is necessary. Thus, the actual capacitance becoming the updated effective capacitance is set (step  1014 ). The algorithm then branches back to step  1009 . Otherwise, convergence of the effective capacitance is achieved and the algorithm steps forward to step  1015  to find the victim sink voltage from the Thevenin voltage source and the transfer function H(s). The process then comes to a stop at step  1016 . 
     The aforementioned steps, which have been described generically with reference to  FIG. 4 , can also be expressed in mathematical form. Accordingly, detailed mathematical steps corresponding to the flow chart of  FIG. 4  are shown in  FIG. 5 . 
     The input to the algorithm consists of noise width W i    109  and peak P i    113  at the victim driver input, wand an output consisting of a linear Thevenin voltage source V TH    200 , Thevenin resistor R TH    201  and noise  210  at the receiver  104  inputs  206  (also known as sinks) of the net. The construction of the linear Thevenin model is based on an iterative effective capacitance process. 
     
       
         
               
               
               
             
           
               
                   
                   
               
             
             
               
                   
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     Steps mapping between  FIGS. 4 and 5  is summarized above in tabular form. In general, convergence of a variable that is calculated in an iterative manner implies that the variable differs from the previously calculated value by a known, small percentage. In the present embodiment, victim driver Thevenin voltage source parameters are obtained by achieving convergence of the effective capacitance. Experiments show that a typical convergence of C eff  is reached within a few iterations. More details on the algorithm shown in  FIG. 5  follow hereinafter. 
     For any particular effective capacitance C eff , the gate Thevenin model parameters ( 200  and  201 ) are constructed to match the gate propagation noise rules ( 309 ,  310 ,  311 ,  312  and  313 ) assuming a pure capacitive load of C L =C eff  ( 302 ). Next, for a fixed Thevenin driver, C eff    807  is updated so that the average current measured up to the noise peak arrival time is the same when driving either the pure effective capacitance C eff    807 , a π model  801 , or a reduced order model  802  of the interconnect. The pi-model is used to represent the load as seen by the driver when it is connected to an RC wire. Thus, the entire wire is reduced to a pi-model. However, for RLC wires or very long RC wires, the pi-model may not be sufficiently accurate. In such an instance, a higher order reduced model having more parameters for better accuracy is advantageously used. The reduced model is then converted to an “effective” capacitance using the inventive method. The reduced order model represents the original interconnect circuit using a reduced set of parameters. By way of example, four poles and residues are often sufficient to describe the response of an RC net consisting of a few thousand nodes. 
     The iterative process is sub-divided in two segments. In the first, the linear driver Thevenin model parameters are computed given a fixed effective capacitance C eff . In the second, given the fixed linear driver Thevenin model, C eff  is updated to match the average currents. Convergence of the algorithm will now be discussed. 
     Propagation noise rules are pre-characterized by analyzing each cell in the library (to be used as driving gates) by way of a family of input noise pulses  314  having a peak P i    308 , a width W i    307  and a set of output loading capacitances C L    302  obtaining by running SPICE, and a general purpose circuit simulation program. 
     Referring back to  FIG. 3(   a ), practitioners of the art will appreciate that for timing rule generation switching waveform  315  is to be applied to the gate input. In a similar manner, for noise rule generation, a noisy waveform  314  is applied to the input of the gate. Instead of timing properties such as delay  305  and slew  306 , the output noise properties, specifically, the gate intrinsic noise delay T 0    309 , noise peak P o    312 , noise transition time Δt  310 , post-peak noise integral A o    313  and output noise width W o    311 , are characterized. The resulting noise rules are stored in a look-up table or fitted to form the noise rule equations similar to the k-factor timing rule equations:
 
 T   0   =k   1 ( W   i   ,P   i   ,C   L )  (1)
 
Δ t=k   2 ( W   i   ,P   i   ,C   L )  (2)
 
 P   o   =k   3 ( W   i   ,P   i   ,C   L )  (3)
 
 A   o   =k   4 ( W   i   ,P   i   ,C   L )  (4)
 
 W   o   =k   5 ( W   i   ,P   i   ,C   L )  (5)
 
     It is worth noting that the complete input waveform is advantageously represented in three dimensions, i.e., its peak, rising transition time to peak, and falling transition time to peak. However, since the symmetry of the waveform is only a second order effect to the output noise, one may simplify the characterization by assuming symmetric input waveforms that are described by the width W i  and peak P i , where its rising transition time to peak equals its falling counterpart. Moreover, the noise rules described by Eqns. (1) to (5) include certain redundancies when representing the output noise waveform. For example, the smallest set of properties necessary for the overall computation may exclude Equations (1) and (4). A tradeoff must be made between the characterization time and the accuracy of the analysis. Alternatively, if a behavioral model of the gate is available, a fast non-linear analysis of the simple circuit is performed for calculating the output noise and, hence, the desired noise rules is determined on the fly. Noise rules generated in this fashion are no longer restricted to a symmetric input waveform. 
       FIG. 3(   b ) illustrates a behavioral model  303  consisting of a DC voltage dependent current source I o (V i ,V o )  316 , a miller capacitor C m    317  and an output capacitor C o    318 . The pre-characterization of this model often requires a two-dimensional DC current table I o (V i ,V o ), a one-dimensional DC voltage transfer table V o (V i ) and the characterization of the miller and output capacitors. 
     As previously described, the Thevenin voltage source is modeled using a time-varying triangular waveform having delay t o    500 , rising transition time to peak t r    501 , falling transition time from peak t f    502  and peak P k    503 . Given an input noise pulse width W i    109 , peak P i    113 , and a specified load capacitance C L =C eff    504 , and a specified Thevenin resistance R TH    505 , the gate Thevenin voltage source parameters  506  are selected to match the output waveform V c (t)  507  (shown in  FIG. 6 ) whose key characteristics are represented by the propagation noise rules, i.e., gate intrinsic noise delay T 0    508 , noise peak P o    509 , noise transition time to peak Δt  510 , post-peak noise integral A o    511 , and output noise width W o    512 . The delay t o    500  of the Thevenin voltage source is set to coincide with the gate intrinsic noise delay T 0    508 , i.e. t 0 =T 0 . 
     A theoretical derivation for calculating the linear Thevenin model parameters will now be discussed. To simplify the discussion, it is assumed that delay t 0    500  is zero. V TH (t)  506  is written as 
     
       
         
           
             
               
                 
                   
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                   8 
                   ) 
                 
               
             
           
         
       
     
     Three parameters t r &gt;0, t f &gt;0 and P k &gt;0 are selected to match the output noise rules as follows:
 
 V   c (Δ t )= P   o    (9)
 
 V   c (Δ t )=0   (10)
 
                     A   o     =       ∫     Δ   ⁢           ⁢   t     ∞     ⁢         V   c     ⁡     (   t   )       ⁢     ⅆ   t                 (   11   )               
wherein Eqn (9) matches the output noise peak  509 , Eqn (10) matches the output noise rising transition time to peak  510 , and Eqn (11) matches the post peak noise area  511 . Since the noise peak of V c (t) occurs between t r  and t r +t f , i.e., t r ≦Δt≦t r +t f , the partial derivative of the second portion of V c (t) in Eqn. (8) is equated to zero to obtain
 
     
       
         
           
             
               
                 
                   
                     
                       - 
                       
                         1 
                         
                           t 
                           f 
                         
                       
                     
                     - 
                     
                       
                         1 
                         
                           t 
                           r 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           - 
                           
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               t 
                             
                             RC 
                           
                         
                       
                     
                     + 
                     
                       
                         ( 
                         
                           
                             1 
                             
                               t 
                               r 
                             
                           
                           + 
                           
                             1 
                             
                               t 
                               f 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           - 
                           
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                               - 
                               
                                 t 
                                 r 
                               
                             
                             RC 
                           
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     The output noise peak P o    509  is evaluated at Δt  510  of the second portion of V c (t) 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           P 
                           o 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             P 
                             k 
                           
                           [ 
                           
                             1 
                             + 
                             
                               
                                 RC 
                                 + 
                                 
                                   t 
                                   r 
                                 
                               
                               
                                 t 
                                 f 
                               
                             
                             - 
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                               
                                 t 
                                 f 
                               
                             
                             + 
                             
                               
                                 RC 
                                 
                                   t 
                                   r 
                                 
                               
                               ⁢ 
                               
                                 ⅇ 
                                 
                                   - 
                                   
                                     
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       t 
                                     
                                     RC 
                                   
                                 
                               
                             
                             - 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             RC 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   1 
                                   
                                     t 
                                     r 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     t 
                                     f 
                                   
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             ⅇ 
                             
                               - 
                               
                                 
                                   
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   - 
                                   
                                     t 
                                     r 
                                   
                                 
                                 RC 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     To match the post-peak noise integral A o    511 , V c (t) is integrated from Δt  510  to ∞: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           A 
                           o 
                         
                         = 
                           
                         ⁢ 
                         
                           
                             ∫ 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               t 
                             
                             ∞ 
                           
                           ⁢ 
                           
                             
                               
                                 V 
                                 c 
                               
                               ⁡ 
                               
                                 ( 
                                 t 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ⅆ 
                               t 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               P 
                               k 
                             
                             ⁢ 
                             
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         t 
                                         r 
                                       
                                       + 
                                       
                                         t 
                                         f 
                                       
                                       - 
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         t 
                                         r 
                                       
                                       + 
                                       
                                         t 
                                         f 
                                       
                                       - 
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                       + 
                                       
                                         2 
                                         ⁢ 
                                         RC 
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   2 
                                   ⁢ 
                                   
                                     
                                       ( 
                                       RC 
                                       ) 
                                     
                                     2 
                                   
                                 
                               
                               
                                 2 
                                 ⁢ 
                                 
                                   t 
                                   f 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               P 
                               k 
                             
                             ⁢ 
                             
                               
                                 
                                   ( 
                                   RC 
                                   ) 
                                 
                                 2 
                               
                               
                                 t 
                                 r 
                               
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 - 
                                 
                                   
                                     
                                       - 
                                       Δ 
                                     
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     t 
                                   
                                   RC 
                                 
                               
                             
                           
                           - 
                           
                             
                               
                                 
                                   P 
                                   k 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   RC 
                                   ) 
                                 
                               
                               2 
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   1 
                                   
                                     t 
                                     r 
                                   
                                 
                                 + 
                                 
                                   1 
                                   
                                     t 
                                     f 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               ⅇ 
                               
                                 - 
                                 
                                   
                                     
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       t 
                                     
                                     - 
                                     
                                       t 
                                       r 
                                     
                                   
                                   RC 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     By combining (12) and (13), Eqn. (13) is simplified to  406   
     
       
         
           
             
               
                 
                   
                     P 
                     o 
                   
                   = 
                   
                     
                       P 
                       k 
                     
                     ⁢ 
                     
                       
                         
                           t 
                           r 
                         
                         + 
                         
                           t 
                           f 
                         
                         - 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                       
                         t 
                         f 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Similarly, by feeding (12) into (14), Eqn. (14) is simplified to 
     
       
         
           
             
               
                 
                   
                     A 
                     o 
                   
                   = 
                   
                     
                       
                         P 
                         k 
                       
                       2 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             
                               t 
                               r 
                             
                             - 
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               t 
                             
                           
                           
                             t 
                             f 
                           
                         
                         + 
                         1 
                       
                       ) 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           t 
                           r 
                         
                         + 
                         
                           t 
                           f 
                         
                         - 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                         + 
                         
                           2 
                           ⁢ 
                           RC 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Combining (15) and (16), leads to  403  and  405   
     
       
         
           
             
               
                 
                   
                     
                       t 
                       r 
                     
                     + 
                     
                       t 
                       f 
                     
                   
                   = 
                   
                     
                       
                         
                           2 
                           ⁢ 
                           
                             A 
                             o 
                           
                         
                         
                           P 
                           o 
                         
                       
                       + 
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                       - 
                       
                         2 
                         ⁢ 
                         RC 
                       
                     
                     = 
                     
                       T 
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     Eqn. (15) shows the relationship between the peak  503  of the Thevenin voltage and the peak  509  of the output noise waveform. Eqn. (17) reveals that the sum of t r    501  and t f    502  is constant, given the gate output waveform characteristics (A o    511 , P o    509  and Δt  510 ) as well as the values of R  505  and C  504 . 
     Applying Eqn. (15) and (17) into (12), one obtains the following non-linear equation of t r    404 : 
                     f   ⁡     (     t   r     )       =           (     1   -     ⅇ     -       Δ   ⁢           ⁢   t     RC           )     ⁢     t   r       +       T   c     ⁢     ⅇ       t   r     RC         -     T   c       =   0             (   18   )               
where 0&lt;t r &lt;T c . This equation is efficiently solved using Newton-Raphson, an iterative algorithm to solve a non-linear equation of f(x)=0, within a limited number of iterations. A typical portion of the f(t r ) curve with T c =0.063 ns, Δt=0.033 ns, R=36Ω and C=0.197 pF is shown in  FIG. 7 . The solution is found at t r =0.027 ns.
 
     Once t r    501  is computed, t f    502  and P k    503  is obtained from Eqn. (15) and Eqn. (17). Taking the partial derivative of Eqn. (17) with respect to C, one obtains 400 
     
       
         
           
             
               
                 
                   R 
                   = 
                   
                     
                       0.5 
                       ⁢ 
                       
                         
                           ∂ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     A 
                                     o 
                                   
                                   / 
                                   
                                     P 
                                     o 
                                   
                                 
                               
                               + 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                             ) 
                           
                         
                         
                           ∂ 
                           C 
                         
                       
                     
                     ≈ 
                     
                       
                         ∂ 
                         
                           W 
                           o 
                         
                       
                       
                         ∂ 
                         C 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Eqn. (19) shows that the driver Thevenin resistance  505  is approximated by the sensitivity of the output noise pulse width  512  with respect to the output capacitance  504 , which is obtained by querying the noise rule of W o    512  twice with two different capacitance values. Since a fixed value of R TH  can provide the efficiency for only one model order reduction of the entire interconnect circuit ( 111  or  212 ), at every C eff  iteration, one obtains R TH  by calculating the sensitivity value only at C eff   (0)  which is equal to the total wiring capacitance. The sensitivity based driver resistance as a function of input noise peak P i  at various input noise widths W i  for an inverter in a 0.13 μm technology, is shown in  FIG. 8 . Typically, the resistance value in the presence of an input noise pulse is higher than a quiet one. The coupling noise analysis using such a higher driver resistance often captures the non-linear effect of the gate. 
     It should be noted that the sensitivity based driver resistance in Eqn. (19) assumes a non-negligible output noise pulse with a finite pulse width. There are often cases when the non-zero input noise pulse is not sufficiently strong to cause a negligible output noise pulse. In such cases, the driver resistance is found from the DC load current table I o (V i ,V o ) and the DC voltage transfer table V o (V i ) by assuming an average input voltage level V i . 
     Thus far, it is observed that the equations to compute the linear Thevenin model from the propagation noise rules have been derived, given an input noise pulse width W i    109  and peak P i    113  for a specified load capacitance C L    302 . Described next, is how to update C eff  through matching the average current flowing into the effective capacitance with that flowing into the actual interconnect circuit. Given the closed form formula for V TH  (t) (Eqn. (6)) and V c (t) (Eqn. (8)), the current flowing into the effective capacitance  807  is written as 
                       I     C   eff       ⁡     (   t   )       =           V   TH     ⁡     (   t   )       -       V   c     ⁡     (   t   )           R   TH               (   20   )               
The average current is calculated as
 
                       I   avg     ⁡     (     C   eff     )       =       1     Δ   ⁢           ⁢   t       ⁢       ∫   0     Δ   ⁢           ⁢   t       ⁢         I     C   eff       ⁡     (   t   )       ⁢     ⅆ   t                   (   21   )               
By way of some mathematical manipulation, it is seen that
 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       avg 
                     
                     ⁡ 
                     
                       ( 
                       
                         C 
                         eff 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         P 
                         k 
                       
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             
                               
                                 t 
                                 r 
                               
                               + 
                               
                                 t 
                                 f 
                               
                               - 
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                             
                               t 
                               f 
                             
                           
                           ⁢ 
                           
                             C 
                             eff 
                           
                         
                         + 
                         
                           
                             C 
                             eff 
                             2 
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 
                                   R 
                                   TH 
                                 
                                 
                                   t 
                                   f 
                                 
                               
                               + 
                               
                                 
                                   
                                     R 
                                     TH 
                                   
                                   
                                     t 
                                     r 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     - 
                                     
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                       
                                         
                                           R 
                                           TH 
                                         
                                         ⁢ 
                                         
                                           C 
                                           eff 
                                         
                                       
                                     
                                   
                                 
                               
                               - 
                               
                                 
                                   
                                     R 
                                     TH 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           t 
                                           r 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           t 
                                           f 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   ⅇ 
                                   
                                     - 
                                     
                                       
                                         
                                           Δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           t 
                                         
                                         - 
                                         
                                           t 
                                           r 
                                         
                                       
                                       
                                         
                                           R 
                                           TH 
                                         
                                         ⁢ 
                                         
                                           C 
                                           eff 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     The average current flowing into the actual interconnect circuit ( 808  or  809 ) is calculated using any known reduced order modeling technique. For instance, the pi-model  808  is an order-two model. The poles and residues of the admittance Y(s)  802  including the Thevenin resistance R TH  (shown in  FIG. 9 ) are obtained from model order reduction techniques such as AWE, SyPVL and PRIMA, and the like. In the algorithm, since R TH  is fixed, only one model order reduction needs to be performed to get the poles and residues of Y(s): 
                     Y   ⁡     (   s   )       =       ∑     i   =   1     q     ⁢       r   i       s   -     p   i                   (   23   )               
where q is the order, and p i  and r i  are the ith pole and residue of Y(s), respectively. The current flowing into the actual interconnect  809  is given by
 
 I   actual ( t )= L   −1 ( I   actual ( s ))= L   −1 ( V   TH ( s ) Y ( s ))   (24)
 
where L −1  is the inverse Laplace Transform operator and V TH (s) is given in Eqn. (7). I actual (t) is
 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       actual 
                     
                     ⁡ 
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         P 
                         k 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         q 
                       
                       ⁢ 
                       
                         
                           
                             r 
                             i 
                           
                           
                             p 
                             i 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   1 
                                   - 
                                   
                                     ⅇ 
                                     
                                       
                                         p 
                                         i 
                                       
                                       ⁢ 
                                       t 
                                     
                                   
                                 
                                 
                                   t 
                                   r 
                                 
                               
                               ⁢ 
                               
                                 u 
                                 ⁡ 
                                 
                                   ( 
                                   t 
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     1 
                                     
                                       t 
                                       r 
                                     
                                   
                                   + 
                                   
                                     1 
                                     
                                       t 
                                       f 
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 ( 
                                 
                                   1 
                                   - 
                                   
                                     ⅇ 
                                     
                                       
                                         p 
                                         i 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             t 
                                             r 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 u 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       t 
                                       r 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               
                                 
                                   1 
                                   - 
                                   
                                     ⅇ 
                                     
                                       
                                         p 
                                         i 
                                       
                                       ⁡ 
                                       
                                         ( 
                                         
                                           t 
                                           - 
                                           
                                             t 
                                             r 
                                           
                                           - 
                                           
                                             t 
                                             f 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                 
                                 
                                   t 
                                   f 
                                 
                               
                               ⁢ 
                               
                                 u 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     t 
                                     - 
                                     
                                       t 
                                       r 
                                     
                                     - 
                                     
                                       t 
                                       f 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     The actual average current is calculated as 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       actual 
                     
                     _ 
                   
                   = 
                   
                     
                       1 
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     ⁢ 
                     
                       
                         ∫ 
                         0 
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                       ⁢ 
                       
                         
                           
                             I 
                             actual 
                           
                           ⁡ 
                           
                             ( 
                             t 
                             ) 
                           
                         
                         ⁢ 
                         
                           ⅆ 
                           t 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     Further derivation gives 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       actual 
                     
                     _ 
                   
                   = 
                   
                     
                       
                         P 
                         k 
                       
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           q 
                         
                         ⁢ 
                         
                           
                             
                               r 
                               i 
                             
                             
                               p 
                               i 
                             
                           
                           ⁢ 
                           
                               
                             
                               [ 
                               
                                 
                                   
                                     - 
                                     
                                       1 
                                       
                                         t 
                                         r 
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                       + 
                                       
                                         
                                           1 
                                           - 
                                           
                                             ⅇ 
                                             
                                               
                                                 p 
                                                 i 
                                               
                                               ⁢ 
                                               Δ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               t 
                                             
                                           
                                         
                                         
                                           p 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   
                                     ( 
                                     
                                       
                                         1 
                                         
                                           t 
                                           r 
                                         
                                       
                                       + 
                                       
                                         1 
                                         
                                           t 
                                           f 
                                         
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         t 
                                       
                                       - 
                                       
                                         t 
                                         r 
                                       
                                       + 
                                       
                                         
                                           1 
                                           - 
                                           
                                             ⅇ 
                                             
                                               
                                                 p 
                                                 i 
                                               
                                               ⁡ 
                                               
                                                 ( 
                                                 
                                                   
                                                     Δ 
                                                     ⁢ 
                                                     
                                                         
                                                     
                                                     ⁢ 
                                                     t 
                                                   
                                                   - 
                                                   
                                                     t 
                                                     r 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                         
                                         
                                           p 
                                           i 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                               ] 
                             
                             } 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     By equating (21) with (27), one obtains  408 
 
Ī( C   eff )=  I actual     (28)
 
     An iterative method such as Newton-Raphson converges within a few iterations to solve the above equation for C eff . 
     Given a noise pulse width W i    109  and peak P i    113  at the victim driver  110  input, the steps in the noise-driven effective capacitance algorithm ( FIG. 5 ) are summarized as follows:
         Initialize k as 0.   Set the initial effective capacitance C eff   (k)  to be the total victim net capacitance C total    212 .   Find the Thevenin resistance R TH    400  as the sensitivity of the output noise width  311  with respect to the total capacitance C total    212 . The output noise width is found from noise rule W o    311 , which is in the propagation noise rule look-up-table obtained from SPICE simulation of non-linear victim driver  301  or fast simulation of the simple behavioral model  303  of the driver.   Find the reduced-order poles and residues of driving point admittance Y(s)  401  (Eqn. (5)) and sink transfer function H(s)  402  (with similar form of Y(s) but different p i &#39;s and r i &#39;s) using known reduced order model reduction techniques such as AWE, SyPVL and PRIMA, etc.   Find additional noise rules T o   (k)    309 , Δt (k)    310 , P o   (k)    312  and A o   (k)    313  from look-up-table obtained from SPICE simulation of non-linear victim driver  301  or fast simulation of the simple behavioral model  303  of the driver.   Find Thevenin voltage source V TH   (k)  parameters t r   (k)    501 , t f   (k)    502 , P k   (k)    503  and t 0   (k)    500  by solving equations of  404 ,  405 ,  406  and  407 .   For the present Thevenin voltage source V TH   (k) , match the average current flowing into a new effective capacitance C eff   (k+1)  to the average current flowing into the reduced order model of the interconnect by solving equation  408 .   If C eff   (k+1)  differs from C eff   (k) , increment k by one and go to Step 5. Otherwise,   C eff  converges. Calculate the victim sink voltage V sink (t) by way of an inverse Laplace Transform of  410 .   Determine the output Thevenin resistance R TH , voltage source V TH =V TH   (k)  and victim sink voltage V sink (t).   Stop.       

     It should be noted that the convergence  409  of the effective capacitance procedure falls into the following three scenarios:
         Regular triangular Thevenin voltage source with t r &gt;0, t f &gt;0 and P k &gt;0 and regular effective capacitance C eff  where 0&lt;C eff &lt;C total .   Effective capacitance C eff =0 when  I actual    is close to zero. In this case propagation noise is negligible and therefore one sets P k =0.   Thevenin voltage source having a very sharp rising transition with t r  close to or equal to zero which corresponds to the case when Eqn. (18) has no non-zero solution for t r . In this case, one specially chooses the two parameters t f &gt;0 and P k &gt;0 to match the output noise rules as follows:
 
 V   c ( Δt )= P   o   (29)
       

                       ∫   0   ∞     ⁢         V   c     ⁡     (   t   )       ⁢           ⁢     ⅆ   t         =       A   total     ≈         P   o     ⁢   Δ   ⁢           ⁢     t   /   2       +     A   o                 (   30   )               
where Eqn (11) is to match the output noise peak, and Eqn (12) is to match the total output noise area. The analysis results using the noise-driven effective capacitance method follow hereinafter. The driving gates are precharacterized and the noise rules are stored in tables. The noise-driven effective capacitance algorithm is preferably implemented in C++.
 
     The method is advantageously tested with two global netlists extracted from a high performance microprocessor corresponding to a 0.13 μm technology and a supply voltage of 1.2V. One netlist has inverters driving two 1 mm -long capacitively coupled lines, while the other has inverters driving two 5 mm -long lines with both capacitive and inductive coupling, with one line being the aggressor and the other being the victim. All the experiments are preferably run on an AIX machine with 1 GB memory. 
     
       
         
               
               
               
               
             
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                   
                 TABLE 1 
               
             
             
               
                   
                   
               
               
                   
                 Input 
                 Sink noise 
                 Sink noise 
               
               
                   
                 noise 
                 (SPICE) 
                 (C eff ) 
               
             
          
           
               
                   
                 P i   W i   
                 P s   
                 A s   
                 P s   
                 Err % 
                 A s   
                 Err % 
               
               
                   
               
             
          
           
               
                   
                 0.86 0.025 
                 0.3615 
                 0.01133 
                 0.3817 
                 5.6% 
                 0.01226 
                 8.2% 
               
               
                   
                 0.86 0.100 
                 0.8454 
                 0.06973 
                 0.8368 
                 1.0% 
                 0.06661 
                 4.5% 
               
               
                   
                 0.86 0.200 
                 1.0162 
                 0.15006 
                 1.0025 
                 1.3% 
                 0.14210 
                 5.3% 
               
               
                   
                 0.63 0.025 
                 0.1509 
                 0.00444 
                 0.1590 
                 5.4% 
                 0.00463 
                 4.3% 
               
               
                   
                 0.63 0.100 
                 0.3531 
                 0.02415 
                 0.3672 
                 4.0% 
                 0.02490 
                 3.1% 
               
               
                 RC 
                 0.63 0.200 
                 0.4790 
                 0.05487 
                 0.4896 
                 2.2% 
                 0.05685 
                 3.6% 
               
               
                   
                 0.48 0.025 
                 0.0711 
                 0.00205 
                 0.0717 
                 0.8% 
                 0.00204 
                 0.5% 
               
               
                   
                 0.48 0.100 
                 0.1328 
                 0.00903 
                 0.1376 
                 3.6% 
                 0.00921 
                 2.0% 
               
               
                   
                 0.48 0.200 
                 0.1602 
                 0.01866 
                 0.1667 
                 4.1% 
                 0.01974 
                 5.8% 
               
               
                   
                 0.35 0.025 
                 0.0303 
                 0.00085 
                 0.0302 
                 0.3% 
                 0.00084 
                 1.2% 
               
               
                   
                 0.35 0.100 
                 0.0533 
                 0.00350 
                 0.0543 
                 1.9% 
                 0.00355 
                 1.4% 
               
               
                   
                 0.35 0.200 
                 0.0613 
                 0.00704 
                 0.0623 
                 1.6% 
                 0.00739 
                 5.0% 
               
               
                   
                 0.86 0.025 
                 0.2943 
                 0.01349 
                 0.2919 
                 0.8% 
                 0.01444 
                 7.0% 
               
               
                   
                 0.86 0.100 
                 0.6194 
                 0.06260 
                 0.6694 
                 8.1% 
                 0.06964 
                 11.2% 
               
               
                   
                 0.86 0.200 
                 0.9249 
                 0.14409 
                 0.9058 
                 2.1% 
                 0.14219 
                 1.3% 
               
               
                   
                 0.63 0.025 
                 0.1206 
                 0.00516 
                 0.1182 
                 2.0% 
                 0.00544 
                 5.4% 
               
               
                   
                 0.63 0.100 
                 0.2598 
                 0.02279 
                 0.2691 
                 3.6% 
                 0.02676 
                 17.4% 
               
               
                 RLC 
                 0.63 0.200 
                 0.3862 
                 0.05064 
                 0.4188 
                 8.4% 
                 0.05709 
                 12.7% 
               
               
                   
                 0.48 0.025 
                 0.0474 
                 0.00213 
                 0.0482 
                 1.7% 
                 0.00215 
                 0.9% 
               
               
                   
                 0.48 0.100 
                 0.0955 
                 0.00878 
                 0.0977 
                 2.3% 
                 0.00948 
                 8.0% 
               
               
                   
                 0.48 0.200 
                 0.1332 
                 0.01810 
                 0.1441 
                 8.2% 
                 0.01966 
                 8.6% 
               
               
                   
                 0.35 0.025 
                 0.0190 
                 0.00085 
                 0.0196 
                 3.2% 
                 0.00086 
                 1.2% 
               
               
                   
                 0.35 0.100 
                 0.0377 
                 0.00343 
                 0.0380 
                 0.8% 
                 0.00358 
                 4.4% 
               
               
                   
                 0.35 0.200 
                 0.0514 
                 0.00692 
                 0.0547 
                 6.4% 
                 0.00728 
                 5.2% 
               
               
                   
               
             
          
         
       
     
     Table 1 shows the sink propagation noise (without coupling) comparison between the C eff  method and SPICE. P i  and W i  are input noise peak (V) and width (ns). P s  and A s  are the sink noise peak (V) and area (V×ns). 
     
       
         
               
               
               
               
             
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                   
                 TABLE 2 
               
             
             
               
                   
                   
               
               
                   
                 Input 
                 Sink noise 
                 Sink noise 
               
               
                   
                 noise 
                 (SPICE) 
                 (C eff ) 
               
             
          
           
               
                   
                 P i  W i   
                 P s   
                 A s   
                 P s   
                 Err % 
                 A s   
                 Err % 
               
               
                   
               
             
          
           
               
                   
                 0.86 0.025 
                 0.5921 
                 0.02131 
                 0.534 
                 9.8% 
                 0.0186 
                 12.7% 
               
               
                   
                 0.86 0.100 
                 1.0551 
                 0.08338 
                 0.989 
                 6.3% 
                 0.0730 
                 12.4% 
               
               
                   
                 0.86 0.200 
                 1.2019 
                 0.16232 
                 1.154 
                 4.0% 
                 0.1485 
                 8.5% 
               
               
                   
                 0.63 0.025 
                 0.3569 
                 0.01263 
                 0.337 
                 5.6% 
                 0.0132 
                 4.5% 
               
               
                   
                 0.63 0.100 
                 0.5924 
                 0.03749 
                 0.545 
                 8.0% 
                 0.0335 
                 10.6% 
               
               
                 RC 
                 0.63 0.200 
                 0.7191 
                 0.07108 
                 0.668 
                 7.1% 
                 0.0655 
                 7.9% 
               
               
                   
                 0.48 0.025 
                 0.2441 
                 0.00900 
                 0.246 
                 0.8% 
                 0.0103 
                 14.4% 
               
               
                   
                 0.48 0.100 
                 0.3319 
                 0.01799 
                 0.312 
                 6.0% 
                 0.0174 
                 3.3% 
               
               
                   
                 0.48 0.200 
                 0.3698 
                 0.02920 
                 0.341 
                 7.8% 
                 0.0280 
                 4.1% 
               
               
                   
                 0.35 0.025 
                 0.1874 
                 0.00728 
                 0.183 
                 2.3% 
                 0.0073 
                 0.3% 
               
               
                   
                 0.35 0.100 
                 0.2164 
                 0.01038 
                 0.208 
                 3.9% 
                 0.0100 
                 3.7% 
               
               
                   
                 0.35 0.200 
                 0.2266 
                 0.01418 
                 0.216 
                 4.7% 
                 0.0138 
                 2.7% 
               
               
                   
                 0.12 0.025 
                 0.1514 
                 0.00622 
                 0.162 
                 7.0% 
                 0.0072 
                 15.8% 
               
               
                   
                 0.12 0.100 
                 0.1524 
                 0.00638 
                 0.165 
                 8.3% 
                 0.0074 
                 16.0% 
               
               
                   
                 0.12 0.200 
                 0.1527 
                 0.00656 
                 0.165 
                 8.1% 
                 0.0076 
                 15.9% 
               
               
                   
                 0.86 0.025 
                 0.6092 
                 0.03000 
                 0.6064 
                 0.5% 
                 0.03071 
                 2.4% 
               
               
                   
                 0.86 0.100 
                 0.9724 
                 0.08334 
                 0.9838 
                 1.2% 
                 0.08592 
                 3.1% 
               
               
                   
                 0.86 0.200 
                 1.2724 
                 0.16873 
                 1.2202 
                 4.1% 
                 0.15847 
                 6.1% 
               
               
                   
                 0.63 0.025 
                 0.4730 
                 0.02364 
                 0.4412 
                 6.7% 
                 0.02410 
                 1.9% 
               
               
                   
                 0.63 0.100 
                 0.5822 
                 0.04046 
                 0.5920 
                 1.7% 
                 0.04542 
                 12.3% 
               
               
                 RLC 
                 0.63 0.200 
                 0.7374 
                 0.07270 
                 0.7417 
                 0.6% 
                 0.07575 
                 4.2% 
               
               
                   
                 0.48 0.025 
                 0.3828 
                 0.01934 
                 0.3737 
                 2.4% 
                 0.02165 
                 11.9% 
               
               
                   
                 0.48 0.100 
                 0.4129 
                 0.02559 
                 0.4233 
                 2.5% 
                 0.02898 
                 13.2% 
               
               
                   
                 0.48 0.200 
                 0.4581 
                 0.03597 
                 0.4697 
                 2.5% 
                 0.03915 
                 8.8% 
               
               
                   
                 0.35 0.025 
                 0.3393 
                 0.01750 
                 0.3349 
                 1.3% 
                 0.01736 
                 0.8% 
               
               
                   
                 0.35 0.100 
                 0.3535 
                 0.02000 
                 0.3533 
                 0.1% 
                 0.02007 
                 0.4% 
               
               
                   
                 0.35 0.200 
                 0.3689 
                 0.02374 
                 0.3699 
                 0.3% 
                 0.02377 
                 0.1% 
               
               
                   
                 0.12 0.025 
                 0.3161 
                 0.01647 
                 0.3172 
                 0.3% 
                 0.01702 
                 3.3% 
               
               
                   
                 0.12 0.100 
                 0.3164 
                 0.01658 
                 0.3191 
                 0.9% 
                 0.01720 
                 3.7% 
               
               
                   
                 0.12 0.200 
                 0.3169 
                 0.01677 
                 0.3196 
                 0.9% 
                 0.01740 
                 3.8% 
               
               
                   
               
             
          
         
       
     
     Table 2 illustrates the sink total coupling and propagation noise comparison between the C eff  method and SPICE. P i  and W i  are input noise peak (V) and width (ns). P s  and A s  are the sink noise peak arrival and the aggressor switching signal. 
     
       
         
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                   
                 TABLE 3 
               
               
                   
                   
               
               
                   
                   
                 Input 
                 Input 
                 Total 
                 Effective 
                   
               
               
                   
                   
                 peak 
                 width 
                 cap 
                 cap 
                 Num 
               
               
                   
                   
                 (V) 
                 (ns) 
                 (pF) 
                 (pF) 
                 itr 
               
               
                   
                   
               
             
             
               
                   
               
             
          
           
               
                   
                   
                 0.86 
                 0.025 
                 0.20 
                 0.127 
                 5 
               
               
                   
                   
                 0.86 
                 0.100 
                 0.20 
                 0.171 
                 3 
               
               
                   
                   
                 0.86 
                 0.200 
                 0.20 
                 0.184 
                 3 
               
               
                   
                   
                 0.63 
                 0.025 
                 0.20 
                 0.121 
                 5 
               
               
                   
                   
                 0.63 
                 0.100 
                 0.20 
                 0.169 
                 3 
               
               
                   
                 RC 
                 0.63 
                 0.200 
                 0.20 
                 0.182 
                 3 
               
               
                   
                   
                 0.48 
                 0.025 
                 0.20 
                 0.117 
                 5 
               
               
                   
                   
                 0.48 
                 0.100 
                 0.20 
                 0.169 
                 3 
               
               
                   
                   
                 0.48 
                 0.200 
                 0.20 
                 0.184 
                 3 
               
               
                   
                   
                 0.35 
                 0.025 
                 0.20 
                 0.113 
                 5 
               
               
                   
                   
                 0.35 
                 0.100 
                 0.20 
                 0.169 
                 3 
               
               
                   
                   
                 0.35 
                 0.200 
                 0.20 
                 0.184 
                 3 
               
               
                   
                   
                 0.12 
                 0.025 
                 0.20 
                 0.000 
                 1 
               
               
                   
                   
                 0.12 
                 0.100 
                 0.20 
                 0.172 
                 3 
               
               
                   
                   
                 0.12 
                 0.200 
                 0.20 
                 0.186 
                 2 
               
               
                   
                   
                 0.86 
                 0.025 
                 0.89 
                 0.266 
                 6 
               
               
                   
                   
                 0.86 
                 0.100 
                 0.89 
                 0.480 
                 5 
               
               
                   
                   
                 0.86 
                 0.200 
                 0.89 
                 0.698 
                 4 
               
               
                   
                   
                 0.63 
                 0.025 
                 0.89 
                 0.245 
                 6 
               
               
                   
                   
                 0.63 
                 0.100 
                 0.89 
                 0.477 
                 5 
               
               
                   
                 RLC 
                 0.63 
                 0.200 
                 0.89 
                 0.678 
                 4 
               
               
                   
                   
                 0.48 
                 0.025 
                 0.89 
                 0.237 
                 5 
               
               
                   
                   
                 0.48 
                 0.100 
                 0.89 
                 0.488 
                 4 
               
               
                   
                   
                 0.48 
                 0.200 
                 0.89 
                 0.706 
                 3 
               
               
                   
                   
                 0.35 
                 0.025 
                 0.89 
                 0.222 
                 6 
               
               
                   
                   
                 0.35 
                 0.100 
                 0.89 
                 0.491 
                 4 
               
               
                   
                   
                 0.35 
                 0.200 
                 0.89 
                 0.731 
                 3 
               
               
                   
                   
                 0.12 
                 0.025 
                 0.89 
                 0.001 
                 1 
               
               
                   
                   
                 0.12 
                 0.100 
                 0.89 
                 0.532 
                 3 
               
               
                   
                   
                 0.12 
                 0.200 
                 0.89 
                 0.794 
                 3 
               
               
                   
                   
               
             
          
         
       
     
     Table 3 shows the total wiring capacitance versus the effective capacitance “seen” by the driving gate at different input noise widths and heights. As expected, all the effective capacitance remains between zero and the total wiring capacitance. For both the RC and RLC nets, the effective capacitance corresponding to the input pulses with P i =0.12 and W i =0.025 is zero or close to zero because the average current flowing out of the driving gate is zero or close to zero. In the experiments, convergence was seen to be achieved within 6 iterations. 
     EXAMPLE 
     Referring now to  FIG. 10 , a circuit typicaly found in a digital VLSI chip is shown and will be used to illustrate various aspects of the invention. 
     L 1  and L 2  are latches and G 1 , G 2 , and G 3  are combinational logic gates. RLC wires N 1 , N 2 , N 3 , and N 4  interconnect the latches and the gates with each other. L 1  is clocked by clock C 1 , and L 2  by clock C 2 . The clocking scheme is shown to the right of the figure. Data is launched from latch L 1  at the edge E 1  of C 1 , and after passing through gate G 1 , it is captured by latch L 2  at clock edge E 3  of C 2 . Simultaneously, the data after passing through G 2  and G 3  is captured by latch L 1  at the edge E 2  of C 1 . Also shown, are wires that run adjacent to N 1 , N 2  and N 4  denoted by P 1 , P 2 , P 3  and P 4 , respectively. These adjacent wires have both capacitive and inductive coupling to N 1 , N 2  and N 4 . Therefore, any signal transitions on these adjacent wires induce noise which interferes with the data launched by L 1 , forcing incorrect data to be captured at L 1  and L 2 , which causes the chip to fail. The noise induced on the wire due to transitions on an adjacent wire are the aforementioned coupled noise and the noise that propagates to the output of the gate (i.e., the propagated noise). The invention which is preferably integral to the noise analysis tool determines the occurrence of such noise events which are accurately predicted, allowing corrective action be taken when necessary. 
     The method begins at the wire connected at the output of latch L 1 , i.e., N 1 .
     1. The noise induced on N 1  by adjacent wires (in this case P 1  and P 2 ) is computed at all the fan-out points of the wire. In the example, there are two fan-out points: the inputs to gate G 1  and G 2 . Therefore, the coupled noise at the input to G 1  and G 2  are computed.   2. From the coupled noise computed at the input of G 1  from step  1 , the propagated noise is then computed at the output of G 1 .   3. From the noise at the output of G 1 , noise is then computed at the input of latch L 2 .   4. The coupled noise due to a signal transition at P 3  is computed at the input of latch L 2 .   5. The coupled noise computed in step 4 is added to the propagated noise calculated in step 3.   6. If the total noise computed in step 5 is greater than a predetermined threshold (part of the design specification), and if the noise occurs within a certain window (part of the design specification) of the capture edge E 3  of C 2 , a failure is flagged for a subsequent repair. Otherwise, no failure is flagged.   7. Similarly, the noise is propagated to the output of gate G 2 .   8. The propagated noise in step 7 is then used to compute the noise at the input of gate G 3 .   9. The noise computed in step 8 is propagated to the output of gate G 3 .   10. From the noise calculated in step 9, noise is computed at the input of L 1 .   11. The coupled noise at adjacent wire P 4  due to a transition is computed at the input of L 1 .   12. The coupled noise from step 11 is added to the propagated noise computed at step 10.   13. If the total noise computed in step 12 is greater than a threshold (part of the design specification) and if the noise occurs within a certain window (part of the design specification) of the capture edge E 2  of C 1 , a failure is flagged for subsequent repair. Otherwise, no failure is flagged.   

     Thus, the proposed invention identifies which latches fail due to noise before the chip is sent for fabrication. If any of the latches fail, then using the noise information from the tool, an appropriate repair strategy (not part of the invention) is implemented, such as rerouting the signal wires so that they are further apart. 
     While the present invention has been particularly described, in conjunction with a specific preferred embodiment, it is evident that many alternatives, modifications and variations will be apparent to those skilled in the art in light of the present description. It is therefore contemplated that the appended claims will embrace any such alternatives, modifications and variations as falling within the true scope and spirit of the present invention.