Jaeyong Chung

Path Criticality Computation in Parameterized Statistical Timing Analysis Using a Novel Operator

This paper presents a method to compute criticality probabilities of paths in parameterized statistical static timing analysis (SSTA). We partition the set of all the paths into several groups and formulate the path criticality into a joint probability of inequalities. Before evaluating the joint probability directly, we simplify the inequalities through algebraic elimination, handling topological correlation. Our proposed method uses conditional probabilities to obtain the joint probability, and statistics of random variables representing process parameters are changed due to given conditions. To calculate the conditional statistics of the random variables, we derive analytic formulas by extending Clarks work. This allows us to obtain the conditional probability density function of a path delay, given the path is critical, as well as to compute criticality probabilities of paths. Our experimental results show that the proposed method provides 4.2X better accuracy on average in comparison to the state-of-art method.

Recursive Path Selection for Delay Fault Testing

This paper presents a new path selection algorithm for delay fault testing in a statistical timing framework. Existing algorithms which consider correlation between paths use an iterative process for each path or defect and require a Monte Carlo simulation for each iteration to calculate the conditional fault probability. The proposed algorithm does not require the iteration process and selects a requested number of paths simultaneously once it performs a statistical timing analysis at the beginning. If selection of k paths is required in a set of paths, it partitions the set into two path sets and determines how many paths should be selected in each path set out of the k paths. It recursively continues this process and ends up with k paths. The partitioning is easily performed during the recursive traversal of a circuit, which produces an imaginary path tree, where paths are already grouped based on their prefix. Experimental results show the proposed algorithm can effectively use structural correlation and spatial correlation to generate high quality path sets.

Post-Silicon Timing Validation Method Using Path Delay Measurements

In the nanometer era, the mismatch between the pre-silicon model and the post-silicon timing behavior is becoming severer. Therefore, it is necessary to validate timing with post-silicon data. We propose a method that estimates all the segment delays in the observed paths of a design from post-silicon path delay measurements. Our method is based on equality-constrained least squares methods, which enable us to find a unique and optimized solution of segment delays from underdetermined systems. Experimental results show that segment delays obtained using our method achieved correlation ranged from 0.848 to 0.992 to the sampled segment delays for different ISCAS-85 benchmark circuits.

A Built-In Repair Analyzer With Optimal Repair Rate for Word-Oriented Memories

This paper presents a built-in self repair analyzer with the optimal repair rate for memory arrays with redundancy. The proposed method requires only a single test, even in the worst case. By performing the must-repair analysis on the fly during the test, it selectively stores fault addresses, and the final analysis to find a solution is performed on the stored fault addresses. To enumerate all possible solutions, existing techniques use depth first search using a stack and a finite-state machine. Instead, we propose a new algorithm and its combinational circuit implementation. Since our formulation for the circuit allows us to use the parallel prefix algorithm, it can be configured in various ways to meet area and test time requirements. The total area of our infrastructure is dominated by the number of content addressable memory entries to store the fault addresses, and it only grows quadratically with respect to the number of repair elements. The infrastructure is also extended to support various types of word-oriented memories.

Testability driven statistical path selection

In the face of large-scale process variations, statistical timing methodology has advanced significantly over the last few years, and statistical path selection takes advantage of it in at-speed testing. In deterministic path selection, the separation of path selection and test generation is known to require time consuming iteration between the two processes. This paper shows that in statistical path selection, this is not only the case, but also the quality of results can be severely degraded even after the iteration. To deal with this issue, we consider testability in the first place by integrating a SAT solver, and this necessitates a new statistical path selection method. Our proposed method is based on a generalized path criticality metric which properties allow efficient pruning. Our experimental results show that the proposed method achieves 47% better quality of results on average, and up to 361x speedup compared to statistical path selection followed by test generation.

Reducing test time and area overhead of an embedded memory array built-in repair analyzer with optimal repair rate

This paper presents a built-in self repair analyzer with the optimal repair rate for embedded memory arrays. The proposed method requires only a single test, even in the worst case. By performing the must-repair analysis on the fly during the test, it selectively stores fault addresses, and the final analysis to find a solution is performed on the stored fault addresses. To enumerate all possible solutions, existing techniques use depth first search using a stack and a FSM. Instead, we propose a new algorithm and its combinational circuit implementation. Since our formulation for the circuit allows us to use the parallel prefix algorithm, it can be configured in various ways to meet area and test time requirements. The total area of our infrastructure is dominated by the number of CAM entries to store the fault addresses, and it only grows quadratically with respect to the number of repair elements.

LFSR-Based Performance Characterization of Nonlinear Analog and Mixed-Signal Circuits

This paper presents an efficient pseudorandom (PR) test method to characterize the performance of nonlinear analog and mixed-signal (AMS) circuits including those embedded in SoC devices. Previous applications of the PR test method to BIST have been limited to digital and linear analog circuits. In this paper, we extend the application of PR test to nonlinear AMS circuits. In doing so, we reduce the cost of testing nonlinear circuits, and increase the test coverage of embedded AMS circuits without incurring a large area overhead to accommodate a test stimulus generator. Our method maintains good test accuracy by using a Volterra series model to describe the behavior of the device under test (DUT). A PR sequence generated from a simple LFSR is used to excite the DUTs over a wide range of frequencies and estimate the parameters of the Volterra series, which are then used to predict the performance of DUTs. We present a method to reduce the test time by using a compressed cross-correlation method which reduces the complexity of the presented algorithm. The mathematical background and hardware measurement results are presented to validate our method.

Refactoring of Timing Graphs and Its Use in Capturing Topological Correlation in SSTA

Reconvergent paths in circuits have been a nuisance in various computer-aided design (CAD) algorithms, but no elegant solution to deal with them has been found yet. In statistical static timing analysis (SSTA), they cause difficulty in capturing topological correlation. This paper presents a technique that in arbitrary block-based SSTA reduces the error caused by ignoring topological correlation. We interpret a timing graph as an algebraic expression made up of addition and maximum operators. We define the division operation on the expression and propose algorithms that modify factors in the expression without expansion. As a result, the algorithms produce an expression to derive the latest arrival time with better accuracy in SSTA. Existing techniques handling reconvergent fanouts usually use dependency lists, requiring quadratic space complexity. Instead, the proposed technique has linear space complexity by using a new directed acyclic graph search algorithm. Our results show that it outperforms an existing technique in speed and memory usage with comparable accuracy. More important, the proposed technique is not limited to SSTA and is potentially applicable to various issues due to reconvergent paths in timing-related CAD algorithms.

Path criticality computation in parameterized statistical timing analysis

This paper presents a method to compute criticality probabilities of paths in parameterized statistical static timing analysis. We partition the set of all the paths into several groups and formulate the path criticality into a joint probability of inequalities. Before evaluating the joint probability directly, we simplify the inequalities through algebraic elimination, handling topological correlation. Our proposed method uses conditional probabilities to obtain the joint probability, and statistics of random variables representing process parameters are changed to take into account the conditions. To calculate the conditional statistics of the random variables, we derive analytic formulas by extending Clarks work. This allows us to obtain the conditional probability density function of a path delay, given the path is critical, as well as to compute criticality probabilities of paths. Our experimental results show that the proposed method provides 4.2X better accuracy on average in comparison to the state-of-art method.

A hierarchy of subgraphs underlying a timing graph and its use in capturing topological correlation in SSTA

This paper shows that a timing graph has a hierarchy of specially defined subgraphs, based on which we present a technique that captures topological correlation in arbitrary block-based statistical static timing analysis (SSTA). We interpret a timing graph as an algebraic expression made up of addition and maximum operators. We define the division operation on the expression and propose algorithms that modify factors in the expression without expansion. As a result, they produce an expression to derive the latest arrival time with better accuracy in SSTA. Existing techniques handling reconvergent fanouts usually use dependency lists, requiring quadratic space complexity. Instead, the proposed technique has linear space complexity by using a new directed acyclic graph search algorithm. Our results show that it outperforms an existing technique in speed and memory usage with comparable accuracy.