IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 40, NO. 7, JULY 2021 1437 Efficient and Accuracy-Ensured Waveform Compression for Transient Circuit Simulation Lingjie Li and Wenjian Yu , Senior Member, IEEE Abstract—Efficient and accurate waveform compression is to large data storage for the waveforms. For accurate transient essentially important for the application of analog circuit tran- simulation, the number of time points ranges from several hun- sition simulation nowadays. In this article, an analog waveform dred thousands to even a billion, and voltage/current values compression scheme is proposed which includes two compressed formats for representing small and large signal values, respec- are stored as floating-point numbers. Therefore, the simulator tively. The compressed formats and the corresponding compres- may output a large hard-disk file occupying multiple GBs of sion/decompression approaches ensure that both the absolute and disk space. This further leads to considerable time for writ- relative errors of each signal value restored from the compres- ing the waveforms into a file and for loading the file into a sion are within specified criteria. The formats are integrated in viewer, and largely worsens the performance of circuit simu- a block-by-block compressing procedure which facilitates a sec- ondary lossless compression and a three-stage pipeline scheme lator and the customer experience of waveform viewer. Hence, for fast conversion from the simulator’s output to the com- it is greatly demanded to have an efficient waveform compres- pressed hard-disk file. Theoretical analysis is presented to prove sion approach which reduces the data storage of waveforms the accuracy-ensured property of our approach. Two schemes and ensures an acceptable accuracy as well [2]. are also proposed to incorporate the prediction method. They Some work on waveform compression or related problems achieve larger compression ratio for some cases while preserving the accuracy and runtime efficiency. Experiments are carried out have been reported in the literature. In [3], a lossless adap- with voltage waveforms from industrial circuits. The results val- tive prediction approach using a recursive least squares lattice idate the accuracy and the efficiency of the proposed techniques. with arithmetic coding was proposed. However, the compres- The obtained compression ratio is 2.6X larger than existing work sion ratio is unsatisfactory. In [4], an approach was proposed without overhead, even though the latter induces much larger for digital waveform compression, instead of the analog tran- error. Compared with the original double-precision floating num- ber format, the proposed approach achieves the compression ratio sient waveforms considered in this work. Hatami et al. [5], [6] of 26 averagely and up to 70, while keeping the relative error proposed to compress the waveforms by performing principal less than 10−3 and absolute error less than 10−5.Andwith component analysis (PCA) on signal values. It however means the pipelined computation, the proposed compression approach a lot of computation and large memory usage [7]. Later on, hardly increases time cost to the transient simulation. Liu et al. [8] proposed a stream compression method with Index Terms—Data compression, floating-point number, quan- waveform prediction and quantization. It reduces the storage tization, the prediction method, transient simulation waveform. of original waveforms in double-precision floating-point num- bers by 10×, with about 0.1% error. Nonetheless, it cannot guarantee any accuracy criterion. In [9] and [10], an effective I. INTRODUCTION lossy compression technique for high-performance computing RANSIENT circuit simulation has become one of the (HPC) data is proposed based on multilayer prediction and T most important steps in the design of analog or mixed- error-bounded quantization. Yet, in HPC applications, such signal integrated circuits. Its output is voltage waveforms for as climate simulation and hurricane simulation, the data for nodes and/or current waveforms along branches in the cir- compression exhibit strong correlation in local regions of a cuit. Each waveform consists of voltage/current signal values snapshot. The effectiveness of the compressor highly relies on at discrete time points [1]. The waveforms are subsequently the prediction based on the correlation, which is however not displayed or used for validating performance metrics of the suitable for transient circuit simulation. Moreover, the error design. With the increase of circuit size, the number of wave- control in [9] and [10] does not consider the relative error. forms in the result of transient simulation increases, leading Recently, Saurabh and Mittal [11] used a recursive polyno- mial representation to compress technology libraries. Notice Manuscript received May 20, 2020; revised July 28, 2020; accepted August that the problem in [11] allows as large as 5% relative error, 24, 2020. Date of publication August 31, 2020; date of current version June 18, 2021. This work was supported in part by the National Natural which is quite different from the problem considered in this Science Foundation of China under Grant 61872206; in part by the Beijing work. Besides, their approach is rather simple which could not National Research Center for Information Science and Technology under produce large compression ratio. Grant BNR2019ZS01001; and in part by the Tsinghua University Initiative Scientific Research Program. This article was recommended by Associate There are commercial tools that support dumping com- Editor N. Wong. (Corresponding author: Wenjian Yu.) pressed waveform formats, like FSDB [12]. Unfortunately, The authors are with the Department of Computer Science and their formats and algorithms are not disclosed. Some common Technology, BNRist, Tsinghua University, Beijing 100084, China (e-mail: [email protected]; [email protected]). lossless compression algorithms, such as Huffman coding, Digital Object Identifier 10.1109/TCAD.2020.3020496 Deflate [13], and LZW [14], and some lossless floating-point 0278-0070 c 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: Tsinghua University. Downloaded on June 19,2021 at 01:14:33 UTC from IEEE Xplore. Restrictions apply. 1438 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 40, NO. 7, JULY 2021 compressor [15], [16] may also be used to reduce the data latter shows much larger error. The proposed compres- storage of waveforms. However, these algorithms do not con- sion/decompression algorithm consumes equal or less sider the correlation of waveform data and thus cannot provide time than existing methods. And with the pipelined sufficient compression. parallel computing, our approach hardly adds time In this work, we develop an efficient and accuracy-ensured cost to the transient simulation. Finally, combined with waveform compression scheme as a companion of transient prediction our approaches can achieve the compression circuit simulator. It includes two compressed formats for rep- ratio up to 70.4 with the secondary lossless compression resenting small and large signal values, respectively, and the and 38.2 without the lossless compression. approaches for compressing/decompressing the waveforms. By The remainder of this article is organized as follows. distinguishing between the small value and large value with The problem formulation and related work are presented in a calculated threshold, we can ensure that both the abso- Section II. Section III includes the proposed storage formats lute and relative errors of each compressed signal value are and the compression/decompression algorithms. The combi- within specified criteria. The proposed approach exploits the nation with the prediction method is discussed in Section IV, fact that the signal values at successive time points vary less, and the experimental results are demonstrated in Section V. and incorporates the lossless compression technique, so as to Finally, conclusions are drawn in Section VI. deliver large compression ratio. It also enables a three-stage pipeline scheme for fast conversion from the simulator’s out- II. BACKGROUND put to the compressed file on the hard disk. Besides, we can combine the proposed scheme with the prediction method, A. Problem Formulation leading to larger compression ratio for some cases. Two com- The waveforms produced during the transient circuit simu- bination schemes are proposed, which perform differently for lation can be represented as a vector-valued function different cases. Experimental results with voltage waveforms ( ) = v (t) v (t) ··· v (t) T from analog circuit simulation have validated the accuracy and v t 1 2 Ns (1) the efficiency of the proposed techniques. They demonstrate where t is the simulation time, Ns is the number of simulated remarkable advantages over the existing methods, like [8]. signals, and vi(t) means the value of the ith signal at time t.We The impact of error tolerance on the compression ratio and do not store the waveform continuously in practice. Instead, the convenience of decompressing partial waveforms from the a set of Nt discrete time-points t = t1 t2 ··· tN compressed file are also presented. t is stored along with their corresponding signal values V = The main contributions