From statistical model checking to statistical model inference: Characterizing the effect of process variations in analog circuits

Abstract

This paper studies the effect of parameter variation on the behavior of analog circuits at the transistor (netlist) level. It is well known that variation in key circuit parameters can often adversely impact the correctness and performance of analog circuits during fabrication. An important problem lies in characterizing a safe subset of the parameter space for which the circuit can be guaranteed to satisfy the design specification. Due to the sheer size and complexity of analog circuits, a formal approach to the problem remains out of reach, especially at the transistor level. Therefore, we present a statistical model inference approach that exploits recent advances in statistical verification techniques. Our approach uses extensive circuit simulations to infer polynomials that approximate the behavior of a circuit. A procedure inspired by statistical model checking is then introduced to produce "statistically sound" models that extend the polynomial approximation. The resulting model can be viewed as a statistically guaranteed over-approximation of the circuit behavior. The proposed technique is demonstrated with two case studies in which it identifies subsets of parameters that satisfy the design specifications.

Sparse statistical model inference for analog circuits under process variations

“…In this section, we review the statistical verification framework SMI with a running example [13]. Consider a circuit with process parameters p from a parameter space P with a joint distribution F. Let φ = f (p) be a output response of interest.…”

“…In the first phase, we construct a polynomial approximation q. A simple approach based on ordinary least squares (OLS) is used in Zhang et al [13], which does not scale well to circuits with many parameters. Therefore, a novel algorithm is presented in Section III to improve SMI by using sparse regression techniques.…”

“…The threshold K is chosen so that when it is achieved, the observed data provide strong statistical evidence to accept H 0 , indicating that the interval I is sufficiently large. The rationale of SMI including the choice of K is discussed by Zhang et al [13].…”

“…In this section, we experimentally evaluate the sparse-SMI approach proposed in this paper, comparing it with the original SMI proposed by Zhang et al [13] and a yield estimation using Monte-Carlo simulation. Our comparison involves two analog circuits: a three-stage ring oscillator and a two-stage operational amplifier (opamp) over a range of output response functions (properties) for these circuits.…”

“…Then the regression algorithm computes the coefficients for the remaining terms under L 1 regularization. This technique is used inside statistical model inference (SMI ), a recently proposed statistical verification framework for transistor-level circuits [13]. SMI addresses the problem of estimating a safe region of the parameter space that guarantees a given design specification of a circuit, and examines specifications of the form φ(p) ∈ [ , u] where φ is a performance metric, p a vector of process parameters, and and u are real-valued performance limit.…”