Efficient timing analysis for CMOS circuit considering data dependent delays

Circuit delay calculation considering data dependent delays

We describe a procedure based on an existing static timing analysis tool for selecting path delay faults to target during test generation. The use of an existing static timing analysis tool ensures that a state-of-the-art process can be used for estimating path delays. However, static timing analysis, by itself, can be inaccurate as it does not take into consideration conditions that are necessary for detecting path delay faults. In the proposed method, these conditions are captured as what are called input necessary assignments, which static timing analysis tools are able to use. By providing the static timing analysis process with the input necessary assignments for a selected path, the static timing analysis process can estimate the delay of the path more accurately. It can also identify additional paths whose delays are at least as high as those of the selected paths. Thus, feeding back the input necessary assignments to the static timing analysis process enhances the correlation between static timing analysis and actual timing of tests on silicon. The result is a set of potentially detectable path delay faults associated with critical paths based on more accurate estimates of the path delays that can be exhibited by a test set, compared with the set that would be obtained by static timing analysis alone.

This paper addresses the problem of true delay estimation during high level design. The true delay is the delay of the longest sensitizable path in the resulting circuit, as opposed to the topological delay which is the delay of the longest path in the circuit. The existing delay estimation techniques either estimate the topological delay, which may be pessimistic if the longest path is unsensitizable or false, or estimate the true delay using gate-level timing analysis which may be prohibitively expensive. Resource sharing in high level synthesis can create false paths in the circuit implementation. Hence, determining the clock period using topological delay can be unduly conservative, resulting in excessive hardware to meet tight timing specifications. In this paper, we introduce an efficient technique to compute an estimate of the true delay. The proposed technique relies on partitioning the paths in the circuit and topological delay computation, and not on path sensitization. The paths in the implementation are partitioned into two sets given the high level information on scheduling and resource sharing: the complete determining path set (CDP/sub R/) and the nondetermining path set (NDP/sub R/). We prove that the delay of the longest path in CDP/sub R/ is lower bounded by the true delay and upper bounded by the topological delay of the circuit. Consequently, an estimate of the true delay of the resulting circuit can be computed by measuring the topological delay of the longest path in CDP/sub R/. We have developed a Functional delay ESTimation tool (FEST). Experimental results on a set of benchmarks reveal the following: approximately 50% of all paths are in NDP/sub R/ and can be ignored for true delay estimation, and the true delay estimates are on the average 15% less than the topological delay. The high level true delay estimates are accurate, as verified by comparing with the true delays obtained by gate-level timing analysis on actual implementations. Furthermore, results reveal that high level true delay estimation can be done very fast, even when gate-level true delay estimation becomes infeasible.

Existing statistical static timing analysis (SSTA) techniques suffer from limited modeling capability by using a linear delay model with Gaussian distribution, or have scalability problems due to expensive operations involved to handle non-Gaussian variation sources or non-linear delays. To overcome these limitations, we propose a novel SSTA technique to handle both nonlinear delay dependency and non- Gaussian variation sources simultaneously. We develop efficient algorithms to perform all statistical atomic operations (such as max and add) efficiently via either closed- form formulas or one-dimensional lookup tables. The resulting timing quantity provably preserves the correlation with variation sources to the third-order. We prove that the complexity of our algorithm is linear in both variation sources and circuit sizes, hence our algorithm scales well for large designs. Compared to Monte Carlo simulation for non-Gaussian variation sources and nonlinear delay models, our approach predicts all timing characteristics of circuit delay with less than 2% error.

Static timing analysis techniques are widely used to verify the timing behavior of large digital designs implemented predominantly in conventional static CMOS. These techniques, however, are not sufficient to completely verify the dynamic circuit families now finding favor in high-performance designs. In this paper, we describe an approach that extends static timing analysis to a high-performance dynamic CMOS logic family called self-resetting CMOS (SRCMOS). Due to the circuit structure employed in SRCMOS, designs naturally decompose into a hierarchy of gates and macros; timing analysis must address and preferably exploit this hierarchy. At the gate level, three categories of constraints on pulse timing arise from considering the effects of pulse width, overlap, and collisions. Timing analysis is performed at the macro level, by a) performing timing tests at macro boundaries and b) using macro-level delay models. We define various macro-level timing tests which ensure that fundamental gate-level timing constraints are satisfied. We extend the standard delay model to handle leading and trailing edges of signal pulses, across-chip variations, trading of signals, and slow and fast operating conditions. We have developed an SRCMOS timing analyzer based on this approach; the analyzer implemented as extensions to a standard static timing analysis program, thus facilitating its integration into an existing design system and methodology.

In recent years, there has been a lot of research into statistical static timing analysis (SSTA) to compute the critical path delay of a circuit under timing variation. In order to compute the true critical path delay, however, false paths that cannot be sensitized by any input vector must be identified first. Since SSTA is unable to capture the dynamic timing behavior of a circuit, it is completely blind to false paths, and thus it overestimates the circuit timing. In this paper, we propose a new concept of timing analysis approach called statistical dynamic timing analysis (SDTA), which is able to precisely express the statistical behavior of dynamic transitions at the output of gate into a compact form and to directly evaluate and propagate the expressions throughout the circuit, by which the false paths can be cleaned effectively. In addition, to be practical, we propose a couple of techniques that enable a fast computation of the SDTA. We tested the proposed approach on ISCAS benchmarks and carry skip adders under timing variation to show its accuracy in computing the distribution of the true critical path delay of a circuit. In summary, compared to the previous approach of false path-aware statistical timing analysis, our timing analysis technique is able to reduce the accuracy error in the mean and standard deviation of true critical path delay distribution from 9.8% to 1.9% and from 29.4% to -3.4%, respectively.

This paper proposes an accurate technique for computing critical delay of a circuit under a bounded delay model. The bounded delay model is better adapted to capture real time variations in the gate delays due to changes in operating conditions. But this flexibility comes at a price, since the uncertainty in gate delays increases the complexity of the timing analysis problem greatly. But we have shown in this paper that using fixed delay timing analysis with worst case delay values for gates can potentially underestimate the critical delay of a circuit. We propose a SAT based methodology for timing analysis in a bounded delay framework which utilises the phenomenal speed and efficiency of modern SAT solvers, and report encouraging results on the ISCAS benchmark circuits

We considered a post-layout transistor sizing problem in a static CMOS module layout. The transistor sizer proposed in this paper is different from previous approaches which work on a full-custom layout and optimizes a module layout on several rows of automatically generated leaf cells. The sizing is performed at two levels. At the module level, a leaf cell is chosen based on a height slack (usable area) and timing slack. At the cell level, the cell is sized based on a width constraint imposed from the module level. The object is to minimize the difference of the actual arrival time and the required arrival time. The problem of sizing a cell is formulated as a nonlinear program. A new objective function is defined so that not only the long delay is shortened but also the short delay is lengthened. We applied an extended empirical method to solve the nonlinear programming problem. A benchmarking process has been conducted at both cell level and module level. Experiments on a set of cells show that an average of 26% performance improvement was obtained by using 0.06% more area. Moreover, for a leaf cell with multiple outputs, the sizer can indeed simultaneously make the long delay paths shorter and short delay paths longer. The results of a module level experiment show that by using height slack, the maximum delay of the circuit can be reduced up to 17.7% without area penalty for the example shown.

As design size gets larger and becomes more complicated with feature integration, the runtime for Static Timing Analysis (STA) becomes more of a concern. Due to time-to-market pressure, the validation of the design is performed in parallel with the physical synthesis flow; therefore it is not uncommon to find last minute critical logic bugs during final design integration iterations. However, timing verification needs to be run on full-chip level to ensure that the check is comprehensive. This paper proposes a method to isolate only those logics which are affected by the Engineering Change Order (ECO) for STA. This simplification will allow faster ECO iteration to enable a more efficient timing convergence. The proposed method is also suitable to be used for multithreading in STA engines to speed-up timing verification due to ECO changes. Engineering Change Order (ECO), Static Timing Analysis (STA), Incremental Static Timing Analysis, VLSI design

The authors provide a method of finding all sensitizable paths in a non-scan synchronous sequential circuit. Path activation conditions of the circuit are mapped onto a single stuck type fault by adding a few modeling gates to the netlist. The path is considered to be sensitizable only if the corresponding stuck type fault is found detectable by a sequential circuit test generator. A depth-first analysis of circuit topology that determines all paths between primary inputs, primary outputs and flip-flops employs a partial path hierarchy. All paths with a common unsensitizable segment need not be examined separately. Results on benchmark circuits show that: (1) the number of sensitizable paths can be significantly smaller than that found by a static timing analyzer, and (2) the partial path analysis adds to efficiency when the number of sensitizable paths is less than 20%. >

This paper describes a combination transistor sizing/layout compaction tool used to synthesize high performance CMOS circuits. This optimization tool is part of a large integrated layout system, called Tailor. Given any CMOS circuit layout, Tailor's transistor size optimizer will simultaneously adjust transistor sizes and compact the layout so that the minimum required area (cell pitch) for a specified upper bound on circuit delay is achieved. All delay paths are considered by modeling circuit delay with a logic independent delay graph. Tailor's optimizer globally optimizes circuit area (in one dimension) and delay by use of compaction and nonlinear programming algorithms. The optimizer does not yet optimize in two dimensions simultaneously or optimize hierarchical circuits. Results for a few optimized CMOS circuits are presented.

Recently, an unconventional clock distribution scheme, called Branch-and-Combine (BaC) was proposed. The scheme is the first to guarantee constant skew upper bound irrespective of the clocked network's size. In BaC clocking, a set of interconnected nodes perform simple processing on clock signals such that the path from the source to any node is automatically and adaptively selected such that it is the shortest delay path. The graph underlying a BaC network is constrained by the requirement that each pair of adjacent nodes is in a cycle of length /spl les/k, where k is the feature cycle length. The graph representing such a network is called a BaC(k) graph. The feature cycle length (k) is an important parameter upon which skew bound and node function depend. We study the complexity of the general problem of designing a minimum cost BaC network for clocking a data processing network of arbitrary topology so that a certain feature cycle length is satisfied. We define two versions of the problem, differing in the way we are allowed to place edges in the graph representing the BaC network. We show that, in both cases, the general optimization problem is NP hard. We also provide efficient heuristic algorithms for both versions of the optimization problem. When k=2, the two versions of the optimization problem become the same and can be solved in polynomial time. For k=3, the complexity is still unknown.

Aging is an important determinant for the reliability of circuit designs and has been addressed by a number of protection techniques based on static timing analysis (STA). The timing reported by STA, however, is often too optimistic without considering the functional behavior of the circuit. Furthermore, signal probability has also been found to be a significant factor in the aging effect. As such, we present in this paper a timing-critical path analysis that takes function and aging into account as well as signal probability. Functional timing analysis (FTA) eliminates the false paths and generates more accurate timing. Furthermore, machine learning can be used to build models for predicting the timing of each cell for various aging lifetimes and signal probabilities. Experimental results indicate that there can be a difference of up to 6% on path delay between STA and FTA. The path ranks also differ for most of the benchmark circuits after considering aging with signal probability, resulting in the delay differences of up to 6.12 %. In conclusion, it is necessary to consider function, aging, and signal probability simultaneously when analyzing timing-critical paths in a circuit design.

Combinational circuits are often embedded in synchronous designs with memory elements at the input and output ports. A performance metric for a circuit is the cycle time of the clock signal. Correct circuit operation requires that all paths have a delay that lies between an upper bound and a lower bound. Traditional approaches in delay optimization for combinational circuits have dealt with methods to decrease the delay of the longest path. We address the issue of satisfying the lower bound constraints. Such a problem also arises in wave pipelining of circuits. We propose to handle short path constraints as a post processing step after traditional delay optimization techniques. There are two issues presented in this paper. We first discuss necessary and sufficient conditions for successful delay insertion without increasing delays of any long paths. In the second part, we present a naive approach to padding delays (greedy heuristic) and an algorithm based on linear programming. We describe an application of the theory to wave pipelining of circuits. Results are presented on a set of benchmark circuits, using two delay models.

One of the critical issues in MTCMOS design is how to estimate a circuit delay quickly. In this paper, we propose a delay modeling and static timing analysis (STA) methodology targeting at MTCMOS circuits. In the proposed method, we prepare a delay look-up table (LUT) consisting of the input slew, the output load capacitance, the virtual ground length, and a power-switch size. Using this LUT, we compute a circuit delay for each logic cell by applying the linear interpolation. Experimental results show that the proposed methodology enables to estimate the critical path delay in a good accuracy.