An Effective Method to Extract Probability Distribution of Circuit Performance Modeled by Quadratic Response Surface Model

The significantly increased relative variations in integrated circuit process necessitate using quadratic response surface model to capture their nonlinear effect on circuit performance. However, the exact probability distribution of the circuit performance modeled by quadratic response surface model is unknown. This paper proposes a method to obtain the approximated probability distribution. We first develop the exact expression of the Continuous Time Fourier Transform of the unknown probability density function, and then apply frequency-domain sampling and Inverse Discrete Fourier Transform to obtain the approximated points of the unknown probability density function. Finally, spline functions are built to approximate the unknown probability density function and cumulative distribution function. An example of bandgap voltage reference circuit demonstrates that this method is more accurate and efficient than Monte Carlo simulation with 10000 samples. The algorithm can be incorporated into integrated circuit Computer Aided Design software for yield analysis and optimization.