SAMPLED-DATA SIMULATION OF PERIODICALLY SWITCHED LINEAR CIRCUITS - Computer Methods for Analysis of Mixed-Mode Switching Circuits - page 121

these random parameters. Worst-case analysis determines the bound-
ary of the design objective using the so-called corner technique. Due
to the over-estimation nature of the method, the results obtained usu-
ally lead to expensive over-designs [52]. Monte Carlo analysis, available
in most commercial CAD tools for circuit design, yields a converging
results only if the number of samples, each obtained from one circuit
analysis, is large. The effectiveness of these methods is often quickly
offset by the excessive computation, especially for periodically switched
circuits where the cost of computation of each circuit analysis is high.
Most recently, interval analysis has been applied to statistical analysis
of linear analog circuits [53]. Its application to statistical analysis of
time-varying circuits, specifically periodically switched linear circuits,
however, is yet to be exploited. In practice, to quantify the effect of
random variation of circuit parameters, designers often rely on costly
multiple runs of SPICE-type simulators to determine the upper and
lower bounds of circuit parameters, resulting in exceedingly long design
cycles. In this section, we introduce an explicit and computationally
efficient non-Monte Carlo method for statistical analysis of periodically
switched linear circuits in the time domain. The method computes the
first and second moments of the time domain response of periodically
switched linear circuits with normally distributed circuit parameters at
time points of an equal interval.

5.2 First-Order Second-Moment Method
In analysis of the mean and variance of the response of circuits, it is
often assumed that the value of circuit elements is normally distributed
with the mean to be the nominal value of the elements. When the
coefficient of variance of the circuit parameters, defined as the ratio of
the standard deviation of the circuit parameters to the mean of the
parameters, is small, the response of the circuit at denoted by
where x is the vector containing all random variables of the
circuits, can be approximated by only considering the first and second-
order terms of its Taylor series expansion at the mean of

x