A case for exploiting complex arithmetic circuits towards performance yield enhancement

Process variation has an increasingly dramatic effect on delay and power as process geometries shrink. Even if the amount of variation remains the same as in previous generations, it accounts for a greater percentage of process geometries as they get smaller. So an accurate prediction of path delay and power variability for real digital circuits in the current technologies is very important; however, its main drawback is the high runtime cost. In this paper, we present a new fast EDA tool which accelerates Monte Carlo based statistical static timing analysis (SSTA) for complex digital circuit. Parallel platforms like Message Passing Interface and POSIX <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">®</sup> Threads and also the GPU-based CUDA platform suggests a natural fit for this analysis. So using these platforms, Monte Carlo based SSTA for complex digital circuits at 32, 45 and 65 nm has been performed. and of the pin-to-output delay and power distributions for all basic gates are extracted using a memory lookup from Hspice and then the results are extended to the complex digital circuit in a hierarchal manner on the parallel platforms. Results show that the GPU-based platform has the highest performance (speedup of 19×). The correctness of the Monte Carlo based SSTA implemented on a GPU has been verified by comparing its results with a CPU based implementation.

An accurate prediction of circuit delay distribution is essential to verify the timing closure of digital circuits and to estimate the parametric yield. In the presence of process variations, the most accurate technique for circuit delay prediction is using Monte Carlo simulations. However, to get an accurate estimate, a large number of Monte Carlo simulations are required which is infeasible in the case of large circuits. Although statistical timing analysis techniques are widely used to predict the circuit delay distribution in a reasonable run time, the standard deviation of circuit delay is often inaccurate. In this work, an efficient technique for circuit delay prediction using limited number of Monte Carlo simulations is proposed. This is done using the concept of Bayesian inference with the results from statistical timing analysis as the prior information. The results indicate that combining statistical timing analysis along with limited number of Monte Carlo simulations increases the accuracy of prediction of circuit delay variance. The number of Monte Carlo simulations can be decided based on the accuracy requirements or run time constraints.

We propose a scalable efficient parameterized block-based statistical static timing analysis (SSTA) algorithm incorporating both Gaussian and non-Gaussian parameter distributions, capturing spatial correlations using a grid-based model. As a preprocessing step, we employ an independent component analysis to transform the set of correlated non-Gaussian parameters to a basis set of parameters that are statistically independent. Given the moments of the variational parameters, we use a Pade acute-approximation-based moment-matching scheme to generate the distributions of the random variables representing the signal arrival times and preserve correlation information by propagating arrival times in a canonical form. Our SSTA procedure is able to generate the circuit delay distributions with reasonably small prediction errors. For the ISCAS89 benchmark circuits, as compared to Monte Carlo simulations, we obtain average errors of 0.99%, 2.05%, 2.33%, and 2.36%, respectively, in the mean, standard deviation, and 5% and 95% quantile points of the circuit delay. Experimental results show that our procedure can handle as many as 256 correlated non-Gaussian variables in about 5 min of runtime. For a circuit with |G| gates and a layout with g spatial correlation grids, the complexity of our approach is O(g|G|) .

In this paper, we propose a calculation method of gate delay for SSTA (Statistical Static Timing Analysis) considering MIS (Multiple Input Switching). In SSTA, statistical maximum/minimum operation is necessary to calculate the latest/fastest arrival time of multiple input gate. Most SSTA approaches calculate the distribution in the latest/fastest arrival time under SIS (Single Input Switching assumption), resulting in ignoring the effect of MIS on the gate delay and the output transition time. MIS occurs when multiple inputs of a gate switch nearly simultaneously. Thus, ignoring MIS causes error in the statistical maximum/minimum operation in SSTA. We propose a statistical gate delay model considering MIS. We verify the proposed method by SPICE based Monte Carlo simulations. Experimental results show that the neglect of MIS effect leads to 80% error in worst case. The error of the proposed method is less than 20%.

Statistical static timing analysis (SSTA) methods, which model process variations statistically as probability distribution function rather than deterministically, have been thoroughly performed on traditional zero clock skew circuits. In the traditional zero clock skew circuits, the synchronizing clock signal is designed to arrive in phase with respect to each register. However, designers will often schedule the clock skew to different registers in order to decrease the minimum clock period of the entire circuit. Clock skew scheduling imparts very different timing constraints that are based, in part, on the topology of the circuit. In this paper, SSTA is applied to nonzero clock skew circuits in order to determine the accuracy improvement relative to their zero skew counterparts, and also to assess how the results of skew scheduling might be impacted with more accurate statistical modeling. For 99.7% timing yield (3σ variation), SSTA is observed to improve the accuracy, and therefore increase the timing margin, of nonzero clock skew circuits by up to 2.5×, and on average by 1.3×, the amount seen by zero skew circuits.

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.

Statistical static timing analysis (SSTA) tools have mostly addressed the process variations of devices and lumped interconnect RC effects. This paper provides an overview of interconnect process variations of capacitive coupling and its effect on crosstalk delay and noise. The correlations among parallel plate, lateral and total capacitance is shown. Correlations between resistance and capacitance are illustrated to enable development of a simple and efficient model for delay and noise analysis. Experimental results are shown to validate the assumptions on the linearity of sensitivity of delay and noise to process variations. A methodology to account for process variations in crosstalk delay and noise is proposed.

In this paper, a grid-based multiple supply voltage (MSV) assignment method is presented to statistically minimize the total power consumption of 3-D IC. This method consists of a statistical electro-thermal simulator to get the mean and variance of on-chip, a thermal-aware statistical static timing analysis (SSTA) to take into account the thermal effect on circuit timing, the statistical power delay sensitivity-slack product to be the optimization criterion, and an incremental update of statistical timing to save the runtime. The experimental results demonstrate the effectiveness of the developed methodology and indicate that the consideration of the thermal effect in the circuit simulation is imperative.

Side-Channel Attacks (SCAs), which are always considered a severe threat to the security of the cryptographic circuits, today can also be employed to extract IP secrets and neural network models. Hence, developing novel security solutions at different design levels is crucial. In this paper, we explore recent countermeasures at the circuit, algorithmic, and microarchitecture levels. First, we explain how Reconfigurable Field-Effect Transistor (RFET), as a beyond CMOS technology, enables us to provide both IP and data protection against SCAs at the circuit level. Second, we investigate an automated method for generating masked circuits as an algorithmic solution, and then we review machine learning-based SCA detection mechanisms at the microarchitecture level. Finally, we discuss emerging threats of SCAs from the industrial point of view.

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.

The advance in semiconductor technologies presents the serious problem of parameter variations. They affect threshold voltage of transistors and thus circuit delay also has variations. Recently, variable latency adders and long latency adders are proposed to manage the variation problem. Unfortunately, replacing a variation-affected adder with the long latency one has severe impact on processor performance. In order to maintain performance, the present paper proposes an instruction scheduling technique considering instruction criticality. By issuing and executing only uncritical instructions in the long latency ALU, we can maintain processor performance. From detailed simulations, we find that the proposed scheduling technique improves processor performance by 12.5% on average over the conventional scheduling and that performance degradation from a variation-free processor is only 4.0% on average, when 2 of 4 ALU's are affected by variations.

We present a new technique for statistical static timing analysis (SSTA) based on Markov chain Monte Carlo (MCMC), that allows fast and accurate estimation of the right-hand tail of the delay distribution. A ¿naive¿ MCMC approach is inadequate for SSTA. Several modifications and enhancements, presented in this paper, enable application of MCMC to SSTA. Moreover, such an approach overcomes inherent limitations of techniques such as importance sampling and Quasi-Monte Carlo. Our results on open source designs, with an independent delay variation model, demonstrate that our technique can obtain more than an order of magnitude improvement in computation time over simple Monte Carlo, given an estimation accuracy target at a point in the tail. Our approach works by providing a large number of samples in the region of interest. Open problems include extension of algorithm applicability to a broader class of synthesis conditions, and handling of correlated delay variations. In a broader context, this work aims to show that MCMC and associated techniques can be useful in rare event analyses related to circuits, particularly for high-dimensional problems.

This paper introduces a novel compact implicit model for a probabilistic set of waveforms (PSoW) which arise as representations for uncertain signal waveforms in Statistical Static Timing Analysis (SSTA). In traditional SSTA tools, signals are just represented as (distributions of) arrival time and slew. In our approach, to increase accuracy, PSoW's are used instead. However, to represent PSoW's explicitly, a very large amount of data is necessary, which can be problematic. To solve this problem, a compact implicit model is introduced, which can be characterized with just a handful of parameters. The results obtained show that the implicit model can generate real-life PSoW's with high accuracy.

In this paper, we explore the implementation of Monte Carlo based statistical static timing analysis (SSTA) on a Graphics Processing Unit (GPU). SSTA via Monte Carlo simulations is a computationally expensive, but important step required to achieve design timing closure. It provides an accurate estimate of delay variations and their impact on design yield. The large number of threads that can be computed in parallel on a GPU suggests a natural fit for the problem of Monte Carlo based SSTA to the GPU platform. Our implementation performs multiple delay simulations at a single gate in parallel. A parallel implementation of the Mersenne Twister pseudo-random number generator on the GPU, followed by Box-Muller transformations (also implemented on the GPU) is used for generating gate delay numbers from a normal distribution. The μ and σ of the pin-to-output delay distributions for all inputs and for every gate, are obtained using a memory lookup, which benefits from the large memory bandwidth of the GPU. Threads which execute in parallel have no data/control dependencies on each other. All threads compute identical instructions, but on different data, as required by the Single Instruction Multiple Data (SIMD) programming semantics of the GPU. Our approach is implemented on a NVIDIA GeForce GTX 8800 GPU card. Our results indicate that our approach can obtain an average speedup of about 260x as compared to a serial CPU implementation. With the recently announced quad 8800 GPU cards, we estimate that our approach would attain a speedup of over 785x. The correctness of the Monte Carlo based SSTA implemented on a GPU has been verified by comparing its results with a CPU based implementation.

Leakage power dissipation and parameter variations are main topics in current research. The problem of parameter variations leads to modified timing analysis. Traditionally, this is done with corner-case simulations, which are quite conservative and pessimistic approaches. This paper proposes a new statistical static timing analysis (SSTA) to improve performance predictions. Furthermore, the developed SSTA and a dual-V/sub th//dual-T/sub ox/ CMOS (DTTCMOS) design technique are combined to reduce the total leakage within designs. Compared to traditional corner-case timing analyses the proposed approach reduces leakage by an average of 70% for raw designs and by an average of 25% in pre-optimized DTTCMOS designs.