Defect Level Estimation for Pseudorandom Testing Using Stochastic Analysis

Pseudorandom testing has been widely used in built-in self-testing of VLSI circuits. Although the defect level estimation for pseudorandom testing has been performed using sequential statical analysis, no closed form can be accomplished as complex combinatorial enumerations are involved. In this work, a Markov model is employed to describe the pseudorandom test behaviors. For the first time, a closed form of the defect level equation is derived by solving the differential equation extracted from the Markov model. The defect level equation clearly describes the relationships among defect level, fabrication yield, the number of all input combinations, circuit detectability (in terms of the worst single stuck-at fault), and pseudorandom test length. The Markov model is then extended to consider all single stuck-at faults, instead of only the worst single stuck-at fault. Results demonstrate that the defect level analysis for pseudorandom testing by only dealing with the worst single stuck-at fault is not adequate (In fact, the worst single stuck-at fault analysis is just a special case). A closed form of the defect level equation is successfully derived to incorporate all single stuck-at faults into consideration. Although our discussions are primarily based on the single struck-at fault model, it is not difficult to extend the results to other fault types.

Limit distributions for coefficients of iterates of polynomials with applications to combinatorial enumerations

This paper studies coefficients yh, n of sequences of polynomialsdefined by non-linear recurrences. A typical example to which the results of this paperapply is that of the sequencewhich arises in the study of binary trees. For a wide class of similar sequences a general distribution law for the coefficients yh, n as functions of n with h fixed is established. It follows from this law that in many interesting cases the distribution is asymptotically Gaussian near the peak. The proof relies on the saddle point method applied in a region where the polynomials grow doubly exponentially as h → ∞. Applications of these results include enumerations of binary trees and 2–3 trees. Other structures of interest in computer science and combinatorics can also be studied by this method or its extensions.