Efficient kernel functions for support vector machine regression model for analog circuits’ performance evaluation Dharmendar Boolchandani • Abrar Ahmed • Vineet Sahula Received: 17 September 2009 / Revised: 1 April 2010 / Accepted: 5 April 2010 / Published online: 28 April 2010 Ó Springer Science+Business Media, LLC 2010 Abstract Support vector machines (SVMs) have been widely used for creating fast and efficient performance macro-models for quickly predicting the performance parameters of analog circuits. These models have proved to be not only effective and fast but accurate also while pre- dicting the performance. A kernel function is an integral part of SVM to obtain an optimized and accurate model. There is no formal way to decide, which kernel function is suited to a class of regression problem. While most com- monly used kernels are radial basis function, polynomial, spline, multilayer perceptron; we have explored many other un-conventional kernel functions and report their efficacy and computational efficiency in this paper. These kernel functions are used with SVM regression models and these macromodels are tested on different analog circuits to check for their robustness and performance. We have used HSPICE for generating the set of learning data. Least Square SVM toolbox along with MATLAB was used for regression. The models which contained modified compo- sitions of kernels were found to be more accurate and thus have lower root mean square error than those containing standard kernels. We have used different CMOS circuits varying in size and complexity as test vehicles—two-stage op amp, cascode op amp, comparator, differential op amp and voltage controlled oscillator. Keywords Analog synthesis  Macromodels  Support vector machine  Kernel  Regression modeling 1 Introduction In order to characterize an analog system, a set of perfor- mance parameters are used to quantify the properties of the circuit. During analog synthesis, macromodel of an analog circuit helps in efficient design space exploration to obtain optimally sized circuit. Given a fixed topology, circuit sizing is the process of determining numerical values for all components in the circuit such that the circuit conforms to a set of performance constraints. Performance parameters of various design instances need to be evaluated to reach a suitable solution. Generally, SPICE is used to obtain per- formance parameter from circuit simulation, however it is computationally very intensive. An efficient and faster way is to use macromodel, which approximates the relationship between the device sizes and performance parameters. In past, many techniques have been proposed in literature [1–9] for analog circuit macromodeling such as knowl- edge-based approaches, response-surface-methods based on posynomial templates, neural network based approa- ches, symbolic modeling technique, and support vector machines (SVMs) based regression technique. SVM regression offers solution for such performance macromodeling. SVMs are class of machine learning approaches. An SVM model can be trained using data gen- erated directly from SPICE. These SVM models are build around suitable kernel functions as regression functions and are able to provide SPICE like accuracy. Extraction of data for use with SVM is although expensive yet affordable, as it is a one time cost per topology, for the chosen technology. Once the models are developed, execution times for per- formance evaluation are very small, leading to a consider- able reduction in synthesis time. While directly employing SPICE during synthesis, any topology can be readily han- dled, whereas SVMs require an extraction step which is D. Boolchandani (&)  A. Ahmed  V. Sahula Malaviya National Institute of Technology, Jaipur, India e-mail: dbool@ieee.org 123 Analog Integr Circ Sig Process (2011) 66:117–128 DOI 10.1007/s10470-010-9476-6