**4. Experimental results**

The characterization variables are delay and transition time. Cell delay is the delay from the 50%-point at the cell input to the 50%-point at the cell output. Cell transition time is also called the output slew time, and it is the time between the 20%-point and the 80%-point at cell output (20–80% for rising transition and 80–20% for falling transition). The goal is to find a model that best fits the relationship between the gate delay (output slew) and the explanatory parameters.

In our work, MARSP is implemented using a Matlab toolbox called ARESLab [25]. Some key settings for ARESLab are as follows: the maximum degree of interactions between explanatory parameters is 3; the maximum number of basis functions is 30; the threshold for the stopping criteria is set to 10−4. Please note that all the training data have been normalized.

A commercial library consisting of 247 standard cells was used, and every timing arc for every cell was characterized. The characterization results for some representative cells are shown in **Table 2**. The "*4\*N + 6*" column in the table means the number of parameters in the MARSP model, and the "Time(s)" column means characterization time. The "Error" column ("Mean" and "S.D.") means the average value and standard deviation of the errors between MARSP and golden reference (SPICE), respectively.

The interconnect characterization is similar to the gate although there are only five considered parameters in our work. The details of reduced-order model of interconnect transfer function is not covered in this chapter (Please refer to [8, 9] for more details). The interconnect results are also shown in the last row of **Table 2**. Interconnect variability (spacing, width) is not included in our experiments. It is also worth noting that our methodology can support a higher-order *H′(s)*-model which matches more moments of the original *H(s)* at the expense of adding more parameters to the MARSP models.

Multivariate Adaptive Regression Splines in Standard Cell Characterization for Nanometer… http://dx.doi.org/10.5772/intechopen.74854 57


**Table 2.** MARSP characterization results on representative standard cells.

#### **4.1. Validation using test paths**

**4. Experimental results**

56 Topics in Splines and Applications

and golden reference (SPICE), respectively.

adding more parameters to the MARSP models.

parameters.

The characterization variables are delay and transition time. Cell delay is the delay from the 50%-point at the cell input to the 50%-point at the cell output. Cell transition time is also called the output slew time, and it is the time between the 20%-point and the 80%-point at cell output (20–80% for rising transition and 80–20% for falling transition). The goal is to find a model that best fits the relationship between the gate delay (output slew) and the explanatory

**Figure 5.** A quadratic model is regressed from *h*(*X* 1, *X* 2) and it has poor accuracy compared to MARSP model.

In our work, MARSP is implemented using a Matlab toolbox called ARESLab [25]. Some key settings for ARESLab are as follows: the maximum degree of interactions between explanatory parameters is 3; the maximum number of basis functions is 30; the threshold for the stop-

A commercial library consisting of 247 standard cells was used, and every timing arc for every cell was characterized. The characterization results for some representative cells are shown in **Table 2**. The "*4\*N + 6*" column in the table means the number of parameters in the MARSP model, and the "Time(s)" column means characterization time. The "Error" column ("Mean" and "S.D.") means the average value and standard deviation of the errors between MARSP

The interconnect characterization is similar to the gate although there are only five considered parameters in our work. The details of reduced-order model of interconnect transfer function is not covered in this chapter (Please refer to [8, 9] for more details). The interconnect results are also shown in the last row of **Table 2**. Interconnect variability (spacing, width) is not included in our experiments. It is also worth noting that our methodology can support a higher-order *H′(s)*-model which matches more moments of the original *H(s)* at the expense of

ping criteria is set to 10−4. Please note that all the training data have been normalized.

Our framework was implemented with C++ and Perl, and the experiments were run on a Linux platform with a 2.27 GHz CPU and 1GB memory without using multi-threading.

Our experiments are based on ISCAS85 benchmark circuits where temperature and supply voltage are considered as global parameters, meaning that all the transistors across the circuit have the same values of temperature and voltage. However, it is worth noting that our methodology can support a temperature profile from a thermal simulator and a voltage profile from an IR-drop simulator. For process variation, as mentioned earlier, we have considered inter-die, intra-die, and intra-gate variations. For channel lengths, we have considered interdie and intra-die variation, and for threshold voltage, intra-gate variation is considered. This is because channel length is mostly impacted by lithography and etching which exhibit strong spatial correlations, while threshold voltage is strongly affected by random dopant fluctuations. Again, please note that our methodology can work with any inter- and intra-die variation model and with any distributions and any correlation profiles.

We have shown our MARSP models are perfectly accurate individually. Here we construct a framework to integrate our models and then verify its accuracy using test paths. We refer to our framework as GTSSTA hereafter. Two thousand Monte Carlo samples were run for 10 randomly selected test paths from ISCAS85 benchmark. As shown in the framework above, path delay is calculated for each sample. This obtained delay value is compared to the delay value from hSpice [26], using Eq. (6).

and delay is characterized or eaen sample. 1ms otoanueu denay value is conpiaren o teu aely value from h\$\\$p\$.\([26], using Eq. (6).\(\)\(\).\(\)
$$Error\_{\text{anh\\_s\\_\mu}} = \frac{DELAY\_{GSNR} - DEIAY\_{SIC}}{DELAY\_{S@\to 1}} \times 100\%\tag{6}$$

A quadratic delay model was also implemented and tested to give a comparison. The quadratic first generates a quadratic regression model as follows:

$$D = d\_0 + \sum a\_i X\_i + \sum b\_i X\_i^2 + \sum\_{\mu k} b\_{\downarrow k} X\_i X\_k \tag{7}$$

*D* denotes gate delay, *Xi* denotes the explanatory parameters, *d0* denotes the constant term, and *ai* and *bi* denote coefficients of first-order and second-order terms, respectively.

**Table 3** presents the results for our framework in comparison to hSpice using these 10 test paths. **Figure 6** gives the histogram comparison of one of the paths between hSpice and GTSSTA.Results in **Table 3** also show that quadratic model has limited accuracy for the 10 test paths.

#### **4.2. Runtime analysis**

Experimental results show our framework consumes only ~2% more runtime than quadratic delay model but achieves much better accuracy.

The quadratic delay model in Eq. (7) has a fixed number of operations, that is, 120 multiplications and 66 additions for a one-input gate and 224 multiplications and 120 additions for a two-input gate. The number of operations using MARSP models is not fixed, and it depends on which subspace the data sample falls into. Basically, calculating a MARSP model will have comparisons first and based on the comparison results, different equations (linear, quadratic etc.) are used for calculations. In average, the number of operations for the MARSP model is close to that of the quadratic delay model.

**5. Conclusion**

(2000 samples are run).

IR-drop models

**Author details**

Taizhi Liu

significant accuracy improvement.

Address all correspondence to: taizhiliu88@gatech.edu

Georgia Institute of Technology, Georgia, United States of America

This chapter talks about the technique called multivariate adaptive regression splines (MARSP). MARSP is a nonparametric regression without taking any pre-assumed form. Instead, it adaptively constructs the model according to the provided data. MARSP has been

**Figure 6.** Monte Carlo histogram comparison between GTSSTA and SPICE for test path (N85 to N724) in circuit c499

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http://dx.doi.org/10.5772/intechopen.74854

59

This chapter also gives an application of MARSP in semiconductor field, more specifically, in standard cell characterization. The objective of standard cell characterization is to create a set of high-quality models of a standard cell library that accurately and efficiently model cell behavior. In this work, the MARSP method is employed to characterize the gate delay as a function of many parameters including process-voltage-temperature parameters. Due to its ability of capturing essential nonlinearities and interactions, MARSP method helps to achieve

Some future work that is worth investigating includes extending the aging-aware MARSPbased timing analyzer to 3D integrated circuits (IC) to study the reliability of 3D ICs which tend to have reliability challenges due to the stronger heat issues. 3D ICs requires more sophisticated thermal models [27–29] and more complicated power-grid analysis [30]. As mentioned earlier, the methodology in this chapter is general to support other thermal and

widely used in high-dimension problems and particularly popular in data mining.


**Table 3.** Experimental results on 10 test paths for MARSP models and quadratic models (errors compared to golden SPICE results).

Multivariate Adaptive Regression Splines in Standard Cell Characterization for Nanometer… http://dx.doi.org/10.5772/intechopen.74854 59

**Figure 6.** Monte Carlo histogram comparison between GTSSTA and SPICE for test path (N85 to N724) in circuit c499 (2000 samples are run).
