Wavelet-Based Data Reduction of B-Spline Curves

第 21 卷 第 8 期2023 年 8 月Vol.21，No.8Aug.，2023太赫兹科学与电子信息学报Journal of Terahertz Science and Electronic Information Technology小波包分析结合VMD的气体泄漏信号降噪朱寅非1,2a，常思婕2b，李鹏*2a，2b(1.南京城市职业学院智能工程学院，江苏南京211200；2.南京信息工程大学 a.江苏省气象探测与信息处理重点实验室；b.江苏省气象传感网技术工程中心，江苏南京210044)摘要：针对气体泄漏声波信号降噪的问题，提出一种集合小波包分析(WPA)与变分模态分解(VMD)相结合的降噪方法。

通过小波包变换对信号的噪声进行预处理；利用VMD对去除噪声的信号进行分解，得到所有的本征模函数(IMF)分量，并根据相关系数准则判断有效IMF；最后提取有效成分并进行信号重构。

对本文方法进行验证，结果表明，本文方法能够有效剔除气体泄漏信号中包含的各种噪声，降噪后信噪比为15.485 1，均方根误差为0.028，为后续信号分析减少了干扰，也为气体泄漏声波信号的特征提取与分析提供了新的思路。

关键词：降噪；气体泄漏；小波包分析；变分模态分解；预处理中图分类号：TN41 文献标志码：A doi：10.11805/TKYDA2021171WPA combined with VMD for noise reduction of gas leakage signalZHU Yinfei1,2a，CHANG Sijie2b，LI Peng*2a,2b(1.School of Intelligent Engineering，Nanjing City Vocational College，Nanjing Jiangsu 211200，China；2a.Jiangsu Key Laboratory ofMeteorological Observation and Information Processing；2b.Jiangsu Meteorological Sensor Network Technology Engineering Center，Nanjing University of Information Science and Technology，Nanjing Jiangsu 210044，China)AbstractAbstract：：Aiming at the problem of noise reduction of gas leakage acoustic signals, a method combining ensemble Wavelet Packet Analysis(WPA) and Variational Mode Decomposition(VMD) wasproposed to de-noise the collected gas leakage acoustic signals. Firstly, the wavelet packet transform isemployed to preprocess the noise of the signal. Then, the de-noised signal was decomposed by VMD toobtain all the Intrinsic Mode Function(IMF) components, and the effective IMF was judged according tothe correlation coefficient criterion. Finally, the active components were extracted and the signal wasreconstructed. The experimental results show that the above method can effectively eliminate all kinds ofnoises contained in the gas leakage signal. After noise reduction, the SNR is 15.485 1, and the root meansquare error is 0.028, which reduces the interference for the subsequent signal analysis. The abovemethod provides a new idea for the feature extraction and analysis of gas leakage acoustic signal.KeywordsKeywords：：noise reduction；gas leakage；wavelet packet analysis；Variational Mode Decomposition；pretreatment以压力容器为载体的化工产品在人类日常生活中得到广泛应用，但由于生产、存储和运输过程中的不当操作，极有可能泄漏并引发安全事故，不仅会给化工单位和国家带来巨大的经济损失，还会破坏泄漏地的生态环境，严重的甚至会威胁到周边人民的生命安全。

基于小波神经网络的光纤陀螺误差补偿方法骞微著;杨立保【摘要】为了提高光纤陀螺的测量精度,提出了一种基于小波神经网络的误差补偿方法.首先使用小波分析中的Mallat分解算法提取出陀螺信号中的主趋势项,对其误差余项进行重构.然后将重构信号作为小波神经网络的目标输出,将原始陀螺信号作为训练样本.为了提高小波神经网络的训练速度同时防止其陷入局部极小值,采用增加动量因子和自适应调整学习速率的方法来改进训练方法.训练后建立的神经网络模型对光纤陀螺误差具有良好的估计能力.结果表明,经过小波神经网络方法补偿后,光纤陀螺的输出精度达到了0.0194°/s,光纤陀螺的测量性能得到了提高.【期刊名称】《中国光学》【年(卷),期】2018(011)006【总页数】8页(P1024-1031)【关键词】光纤陀螺;小波神经网络;小波分析;误差补偿;趋势项提取【作者】骞微著;杨立保【作者单位】中国科学院长春光学精密机械与物理研究所,吉林长春130033;中国科学院大学,北京100039;中国科学院长春光学精密机械与物理研究所,吉林长春130033;长春理工大学机电工程学院,吉林长春130022【正文语种】中文【中图分类】TH8241 引言光纤陀螺利用光纤的Sagnac 效应来测量惯性空间的角速率，具有测量精度高、稳定性好、动态范围大等优点，目前已被广泛用于稳定平台和导航系统中。

在光纤陀螺受到外界条件影响后会产生漂移误差，影响其测量精度。

因此对于光纤陀螺误差补偿的研究近年来已经成为热点。

目前普遍采用的误差补偿方法有ARMA[1]模型和Kalman算法[2-3]。

其中，ARMA是一种线型模型，其假设误差为零均值、平稳的正态时间序列。

而Kalman算法要求获得准确的噪声先验统计信息[4]。

这些都影响了这些方法对误差的建模精度和补偿效果。

不同于ARMA模型和Kalman 算法，神经网络是一种基于非参数辨识的建模方法。

神经网络在获得足够多的信号样本后，便可以获得光纤陀螺的误差在频域上的特性。

第 54 卷第 4 期2023 年 4 月中南大学学报(自然科学版)Journal of Central South University (Science and Technology)V ol.54 No.4Apr. 2023基于随机森林与粒子群算法的隧道掘进机操作参数地质类型自适应决策刘明阳，陶建峰，覃程锦，余宏淦，刘成良(上海交通大学 机械与动力工程学院，上海，200240)摘要：考虑到隧道掘进机的性能对地质条件比较敏感且其操作依赖于司机经验，提出基于随机森林和粒子群算法的隧道掘进机操作参数地质条件自适应决策方法。

利用随机森林(RF)分别建立地质类型、操作参数与推进速度、刀盘转矩的映射关系模型；结合映射关系模型，构建以盾构机推进速度最大为目标，以刀盘转速、螺旋输送机转速、总推力、土仓压力4个操作参数为控制变量的优化方程；利用粒子群算法(PSO)求解各地质类型地层中的最优操作参数决策结果。

通过新加坡某地铁工程施工数据验证所提方法的有效性和优越性。

研究结果表明：建立的随机森林模型中推进速度和刀盘转矩预测的决定系数R 2分别达到0.936和0.961，均大于adaboost 、多元线性回归、岭回归、支持向量回归和深度神经网络模型中相应的R 2；基于粒子群算法的操作参数决策方法能够准确求解操作参数最优解，寻优用时均比遗传算法、蚁群算法和穷举法的短。

本文所提决策方法使隧道掘进机在该施工段的福康宁卵石地层、句容地层IV 、句容地层V 、海洋黏土地层中的推进速度分别提升了67.2%、41.8%、53.6%和15.0%。

关键词：隧道掘进机；操作参数决策；随机森林；粒子群优化中图分类号：TH17；TU62 文献标志码：A 文章编号：1672-7207（2023）04-1311-14Geological adaptive TBM operation parameter decision based onrandom forest and particle swarm optimizationLIU Mingyang, TAO Jianfeng, QIN Chengjin, YU Honggan, LIU Chengliang(School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200240, China)Abstract: Considering that the performance of TBM is affected by geological condition and driver experience, a geological adaptive TBM operation parameter decision based on random forest(RF) and particle swarm optimization algorithm(PSO) was proposed. RF was used to establish the mapping relation model between geological types, operating parameters and thrust speed, cutter head torque. An optimization equation was established using the mapping relationship model in which the maximum TBM thrust speed was taken as the target, and cutterhead speed, screw conveyor speed, total thrust and earth pressure were taken as control variables.收稿日期： 2022 −06 −19； 修回日期： 2022 −08 −21基金项目(Foundation item)：国家重点研发计划项目(2018YFB1702503) (Project(2018YFB1702503) supported by the National KeyR&D Program of China)通信作者：陶建峰，博士，教授，从事机械电子工程研究；E-mail ：**************.cnDOI: 10.11817/j.issn.1672-7207.2023.04.010引用格式： 刘明阳, 陶建峰, 覃程锦, 等. 基于随机森林与粒子群算法的隧道掘进机操作参数地质类型自适应决策[J]. 中南大学学报(自然科学版), 2023, 54(4): 1311−1324.Citation: LIU Mingyang, TAO Jianfeng, QIN Chengjin, et al. Geological adaptive TBM operation parameter decision based on random forest and particle swarm optimization[J]. Journal of Central South University(Science and Technology), 2023, 54(4): 1311−1324.第 54 卷中南大学学报(自然科学版)PSO was used to solve the optimal combination of operating parameters for each geological type. The validity and superiority of the proposed method were verified by the construction data of a subway project in Singapore. The results show that the R2 of the driving speed and cutter head torque predicted by random forest model reaches 0.936 and 0.961, which are greater than those of adaboost, multiple linear regression, ridge regression, SVR and DNN. PSO can accurately solve the optimal solution of operating parameters, and the time consumption is shorter than that of genetic algorithm, ant colony algorithm and exhaustive algorithm. By using the proposed method, the TBM thrust speed increases by 67.2%, 41.8%, 53.6%, 15.0% in the strata of Fokonnen Pebble Formation, Jurong Formation IV, Jurong Formation V and Marine Clay Formation in this construction section, respectively.Key words: tunnel boring machine; operating parameter decision; random forest; particle swarm optimization隧道掘进机是一种大型隧道掘进装备，具有开挖速度快、自动化程度高、施工质量好的优点，广泛地被应用于地铁、铁路、公路等隧道工程中[1]。

东南大学学报（自然科学版-Vol. 50 No. 6Nov . 2022第 5 卷第 6 期2020年4月JOURNAL OF SOUTHEAST UNIVERSITY (Natural SU/ce Edition )DOI 96. 3966/(. issn. 1601 -0507 2022. 06.01基于小波变换的内河航道 船行波时频特性分析毛礼磊陈一梅李鑫（东南大学交通学院，南京21149）摘要：为了准确分析内河航道中船行波的时频特性,将小波变换理论应用于内河航道船行波频谱分析中.通过建立水槽试验获得船舶以不同条件航行产生的水位波动数据，对9种工况下船舶 航行产生的船行波，从时间尺度解析船行波波列结构特征，从频率尺度探究船行波能量分布特征及船舶航行条件的影响•结果表明，船行波小波能谱呈现局部突出的特点，小波谱能量主要集中在船行波低频主波段，对应的频率范围为0~0. 35 Hz,与时间尺度上的水位波动过程相对应.当船舶航速、吃水深度增大时，船行波全局小波能量峰值显著增大，且同一位置处受吃水深度影响更大；当船舶航速、吃水深度相同时，同一位置处船行波全局小波能量峰值随离岸距离增大而减小.该研究可为内河航道中船行波频谱特征分析提供新途径.关键词：小波变换；内河航道;船行波；小波能谱;全局小波能谱中图分类号：U64； TV139. 2文献标志码：A 文章编号：101 -0505（2020）06045-08TimeSrequeecy analysis on ship waves O inland waterways using wavelet trcnsformMao LilelChen Yimel Li Xin(School of TraospoVation , SonUeast Universbp , Naiging 011139 , China)Abstryci : To accurately analyze the UmeOuquence choacteristies of shiq waves in inland water ­ways ,a wavelet transform Ueorg was applieP to Ue fuquexey spectrum analysis of shiq waves. The tume tests were esWOUshed to obtain water-level Uuctu0iop data caused by shiq sailing under differ ­ent conditions. For the waves eeueuWd by the shiq under 9 working conditions , the sWuctcai cho- acterishes of the shiq wave train were analyzed in the time domain , and Ue exergy distridutiop of shiq waves and the ^—/^ of shiq sailing conditions on U were explored in the fuquexey domain. The results show that the wavelet exergy spectrum of shiq waves is Uuk y prominent ； and the exergy of the wavelet spectrum is mainly concentrated in the main band of the water-level drop and the Uw fuquexey , and the corresponding fuquexey range is O-O. 39 He corresponding to Ue proces s of wa ­ter-level 1-0/3X00 on time scale. When the shiq speed and the draft increase ； the yloPci wavelet exergy peak of shiq waves increases significeo/y , and Ue inUueuce on the draft at the same position is greater. When the shiq speed and the draft oe Ue same , the yloPci waveUt coeTgy peak of shiq waves at the same position decreases with the increase of the offshore disWocc. The study provides a new way to analyze the frequency spectrum choacteristies of shiq waves in inland waterways.Key worCs : wavelet transform ； inUod waterways ； shiq waves ； wavelet exergy spectrum ； yloPciwavelet exergy spectrum收稿日期：2626O 4O 9.作者简介：毛礼磊（1992—-男，博士生；陈一梅（联系人-女，博士 ,教授，博士生导师,chevyimei @ sen . edb . c n .基金项目：国家自然科学基金资助项目（31074935 -、东南大学优秀博士学位论文培育基金资助项目（YBPY1385 -.引用本文：毛礼磊，陈一梅，李鑫•基于小波变换的内河航道船行波时频特性分析J ］东南大学学报（自然科学版-2222—2（2）：44-1104 • DOT ：62.8999/（. Usn. 8021 -2505. 0222.02.213.1116东南大学学报（自然科学版）第50卷内河水运具有运能大、占地少、能耗低等优势,是现代综合运输体系的重要组成部分，也是实现经济社会可持续发展的重要战略资源•随着内河水运需求的增长，船舶大型化、高速化趋势明显，由船舶航行产生的船行波给内河水域环境带来了显著的影响，主要包括:①对航道岸坡冲刷作用,破坏两侧护岸结构743］:②改变床底泥沙输运，造成河床冲淤变化7③增加水体浑浊度，降低透光率，影响水生生物生存J68;④影响其他航行或靠泊船只，破坏航道中固定或浮动的水工结构物［4］•船行波问题不仅为造船工程师所关注，也成为水利、环境、生态工程中关注的热点问题•船行波指船舶在水中航行时船体对水体的作用而产生的压力变化所引起的表面波动［1］，其特征不仅取决于船舶本身的参数,如航速、船型、船舶尺度、排水量等,也受到船舶航行环境的影响，如航道尺度、地形、边界条件等•因此，船舶在宽阔无限制水域和受限水域中产生的船行波具有明显不同的特征•在深水中,船行波波态为典型的Kelvin波态，由横波和散波组成，其理论解答可应用线性波理论71］•而在受限水域中，船行波波态则由水动力场和航道地形、边界条件共同决定J0］，会出现一系列非线性波734］，导致船行波及其引起的动力问题呈现出非线性、非平稳状态•通常，船行波波列可以在频率尺度上分解为7个主要组成部分［5］，一是具有长周期的主波，二是具有较短周期的次波.已有的船行波特征研究往往从时域出发，分析船行波的波要素特征.对于船行波频域特性的研究开始于船行波和风浪的区分756］，通过不同的频率分布范围量化7种波的重要程度;还用来分离组成船行波的主波和次波7214］，分别分析其动力特征.此外,少数学者也通过傅里叶变换方法来研究船行波在频域上的能量分布J O41］，分析船行波的频谱特征.然而，传统的分析方法是基于平稳信号假设的基础上进行的，对于非线性和非平稳过程的应用会造成局部瞬态信息的缺失，因此，传统的傅里叶变换对于内河航道中船行波的频域特性分析存在明显的缺陷作为傅里叶变换方法的拓展和延伸，小波变换继承和发展了短时傅里叶变换局部化的思想，也克服了窗口大小不随频率变化等缺点,能够提供一个随频率改变的“时间-频率”窗口，可用来分析各种不平稳信号［2］•小波变换是时间-频率上的局部化分析,通过伸缩平移运算对信号逐步进行多尺度细化,达到高频处时间细分、低频处频率细分的目的, ht/：//jonoa-.sen.e/u.cu自动适应时频信号分析的要求，解决了傅里叶变换的困难之处,也被称为“数学显微镜”.因此，本文通过建立室内水槽试验，复演内河航道中船行波的产生和传播过程,采集船舶以不同条件航行时引起的水位波动数据，解析船行波波列结构特征，并基于小波变换理论探究船行波的能量分布特征,为内河航道中船行波频谱特征分析提供新途径.1水槽试验l.1试验布置试验在南京水利科学研究院泥沙基本试验厅的水槽中进行，试验水槽尺寸为50m x4.0m x O8m,水槽两端设有进水和出水口.试验采用自航船模产生船行波，该船模依据限制性航道中50）级货船代表船型采用1•20模型比尺进行制作,原型尺寸为46m x8.7m,设计吃水2.45m,船模外形尺寸误差控制在土1mm之内.船模采用遥控操纵，最大航行速度可达2.50m?•试验段设置于水槽中间,长度为6.0m,水槽两端为试验船模加速和减速区域•在长三角地区,运河两岸受城镇、堤防等限制，其面宽往往受限7亢道整治断面常采用梯形断面•在水槽试验中，试验段采用典型梯形断面，底宽为2.25m,口宽为3.50 m,斜坡段坡比为1:3,如图1所示.割#-1對—通道电容式浪高测量系统和VM-301HA型电磁流速仪,测量仪器和设备在试验前进行标定,能够满足稳定性和灵敏度的要求•试验中使用的测量仪器布设于水槽一侧，波高仪和流速仪沿梯形断面布置情况如下：）试验共布设14个波高测点，4支波高仪分7排布置，每排5支,7排波高仪之间的距离为第6期毛礼磊，等:基于小波变换的内河航道船行波时频特性分析111210cm,其中波高仪P1设置于梯形断面的斜坡顶角处,距离航道岸坡9.175m,波高仪P、P3、P4、P5与P1的水平距离分别为9.S00、9.175、9.225和0.309m.其余5支波高仪（P6、P、P8、P和P19）布置位置与P1-P5沿试验段中间位置对称，如图1所示.各波高仪的采样频率为50Ha.2）试验共布设0个流速测点，3支流速仪EM1、EM2和EM0与波高仪Pl~P5布置在同一个横断面上,EM1、EM2和EM0与梯形横断面斜坡顶角的距离分别为9.050、9.220和9.825m,如图1所示3支流速仪测量靠近河床底部的水流流速，采样频率为22Ha .1.2试验工况在水深O=9.S2m的条件下开展一系列试验.通过遥控船模自动航行，控制船模的航线和速度，船模自水槽一端开始缓慢加速到一定速度并稳定后，匀速驶入试验段•此时,利用各测量仪器采集数据，同时记录船模通过试验段的时间，用来验证船模遥控系统指示的速度.当船模驶出试验段后逐步减速直至停止.根据船行波的影响因素,分别设置船模以不同航行速度（V s）、吃水深度（d）和离岸距离（y）通过试验段，开展船行波观测试验•试验共设置192组工况,船模航行速度范围为9.50~1S2ms；根据船模载重条件分为空载和重载2种工况,d分别为0.06和9.19m；船模按边线和中线2种离岸距离航线6分别为9.99和133m.采用深度傅汝德数F a表征船行波波态J0],上述试验工况中74组船速位于亚临界速度区（9.97W F W9.94）,26组船速位于跨临界速度区（9.95W F W9.82）.2小波变换原理对于离散时间序列X”来说，其连续小波变换W-i）定义为x”与母小波函数09（n）在缩放、平移后的卷积形式[20],即N-1Wc（i）=1严鋼（1式中,o、o,为时间序列编号;N为时间序列的点个数;n为无量纲时间参数;；为小波尺度M为母小波09无量纲化的结果；*表示复共轭;△[为时间步长•从本质上来说，小波变换是将函数空间内的函数表示成其在具有不同伸缩因子和平移因子的小波函数之上的投影的叠加•小波变换将一维时域函数映射到二维时间-尺度域上•在小波变换的实际应用中，母小波函数的选取对分析结果至关重要•目前在海浪分析中，Morlct小波应用最为广泛J5],它是一个由高斯包络调制的复平面波，在时域和频域都具有很好的局部性,其表达式为1.t?809（n）二^化02%-5（2）式中,s）为小波中心圆频率.根据小波变换的结果，小波变换系数可分解为实部和虚部，或振幅和相位，可定义振幅的平方W”（e）I2为小波变换的能量谱•同时，如果沿某一频率尺度切开小波图，在整个时间内进行平均,可得在整个时间范围内的全局小波能谱，即1N-1珡2（1）=辽Wc（i）2（2）利c=9全局小波能谱可以给出占优势的周期分量的强度信息•在本文中,连续小波变换W c（）定义为由船舶航行引起的水位波动时间序列与Morlct母小波函数在缩放、平移后的卷积3船行波时频特性分析32船行波波列结构为了解析船舶以不同条件航行时产生的船行波波列结构，本节根据水槽试验工况设置情况选取了包含船模空载、重载、边线、中线、亚临界航速和跨临界航速航行条件的5种工况进行分析6种工况下船模的航行条件如表1所示.8种工况既包含船模空载（工况12、3、4）和重载（工况5、6、7、8）的情况，也包含船模沿边线（工况16、5、6）和中线（工况3、4、7、8）航行的情况.此外，工况13、5、7中船模航行速度位于亚临界速度区，工况、、4、6、8中船模航行速度位于跨临界速度区表18种工况中船模航行条件工况h/m O/m y/m V s/（m・h、）F/19.12.962.992.8769.7749.12.962.992.9972.539.12.961852.8442.6749.12.961852.94（2.7559.19.S62.992.83686769.19.S62.99194679.19.S61852.85 2.7189.19.S6185 2.9999.795种工况下船模航行时在Pl~P5处产生的水位波动过程线，如图2所示.对于同一种工况，波高仪Pl-P5实测水位波动情况存在差别，这主要是因为船行波在向岸传播过程中逐渐衰减.当船模航行通过试验段时，对于5种不同工况下的实测水位过程线，其波动历经相似的过程：小http：//jomnaO .co1119东南大学学报(自然科学版)第50卷幅的水位上升一历时较长且较大的水位下降一剧烈的水位波动，这分别对应船首波、船行波中的低频主波和船行波中的高频次波•对比工况1与2、3与4、5与6、4与9可知，当船模吃水和航线相同，船模航速位于跨临界速度区时，即航行速度更快时，产生更明显的船首波，对应于水位波动过程线中水位下降前更明显的水位上升.由工况1、2与3、4及工况5、6与4、9对比可知,当船模航线距离岸坡较远时，所测得的水位下降值要小得多，这主要是因为船行波在传播过程中随着距离逐渐衰减•对于船模吃水相同的工况，由工况-与2、3与4、5与6、7与9对比可知，当船模沿相同航线航行时,耳较大即船模航行速度较快时，水位下降值相差并不大，已有的研究也表明,该最大水位下降值主要与船型及其吃水有关⑹.由不同船模吃水的工况对比可知，工况4和9中船模吃水较大，即使船模离岸距离较远，在波高测点处引起的最大水位下降值仍然较大•0510********时间/s(0)工况4051015202530时间/s(e)工况512345D一D一D一D一D一2£肩¥工况-2r)迫«0510********时间/s(3)工况60510********时间/s(g)工况4-P4-P50510********时间/s(h)工况8图28种工况下船模航行时在P4-P5处产生的水位波动过程线根据船行波主波和次波的不同频率于分布范围，可对实测水位波动时间序列进行分离•对于上述9种工况，由实测水位波动时间序列判定出船行波主波和次波的周期，确定出主波和次波的频率分布范围分别为/<0.35He和0.35He今<1.00 He.本文采用快速傅里叶变换(FFT)方法将水位波动时间序列从时域转换到频域，基于Matlad构建FFT的低通滤波器(2<0.35He)对船行波主波进行分离、带通滤波器(2.35He<<1.02He)对htU：//joprnc-.sen.edb.en 船行波次波进行分离•图3为9种工况下P1~P5实测水位波动时间序列经分离后所得的船行波主波和次波•对于每种工况,P1〜P5实测水位波动由于船行波衰减略有不同，分离后所得的主波和次波也具有同样的特征.9种工况下船行波结构在频率尺度上具有相似的特征，船行波主波具有较大的周期和波长，船行波次波波动周期较小，波动频率较高•因此，当船行波传播至近岸浅水区域时，主波往往可以产生较大的底部流速,改变自然情况下河床第6期毛礼磊，等:基于小波变换的内河航道船行波时频特性分析1 1 1 951015 202530时间/s2 0-?51015 20 25 30时间/s (a )工况151015 202530时间/s一 P1； —- P2； P3； 一 P4； - - - P5I IIII5 1015 20 25 30时间/s (b )工况7-——Pl ； —-P2；……P3；——P4； ―-P5 ______ ) 5 10 15 20 25 30时间/s----P1 ； -----P2；.......P3 ； — P4 ； — P551015 20 25 30时间/s(c )工况8--------P 1 ； P2 ； P3 ； — P4 ； — P5)5 W 1520 25 30时间/s——P1； —- P2； P3；——P4；——P55 1015 20 25 30时间/s(d )工况42O-2-4O 2 0-9OP2P3P45P 510S5C. 1州2052302 O-2-4P1;vp2P3P4P5105旬1州202530P1亠『P3P45P O205230€o2-2冬P15vr p?10翠亠S5C.1叭P4•75P 2052305 1015 20 25 30时间/s5 W 1520 25 30时间/s(g )工况9——P1； —- P2；…P3；——P4； — P55 W 1520 25 30时间/s (h )工况8图3 8种工况下P1~P5实测船行波主波和次波上的泥沙状态,而船行波次波波高较大时会对航道 岸坡产生较强的淘刷作用• 8种工况下，船行波传 播至P5处水位过程线最低水位值分别为2.77、2. 46、1.30、、. 62、4. 22、4. 62、2. 91 和 2. 74 cm,水位下降段的历时分别为2. 60、2.64、3. 40、2. 36、2. 76、3.1 6、3.94和2.62 /以船模吃水深度进行 分类，工况5、6、7、8吃水深度较大时，总体上看船行波主波最大水位下降值也更大•以船模航行离岸ht/：//jono/. sen . edu .cu112东南大学学报(自然科学版)第52卷距离来看，工况5和6中船行波次波比工况7和8 更为显著•以船模航行速度来看，当船模吃水深度 和航线离岸距离相同时，由工况(与2、3与4、5与 6、7与8对比可知，船行波主波和次波在周期和波幅上相差不大，这主要受船模航行速度的限制,船 模航行速度比较接近•32船行波频谱特征以上述8种工况中P5实测水位波动时间序列 为例,采用式(-)52)和(3)进行连续小波变换后得到各自的小波能谱和全局小波能谱，进而分析船行波的频谱特征•在利用小波能谱图对船行波频谱特征进行分析时,主要指标包括波浪能量峰值及其 出现的时间和频率位置•图、所示为8种工况下船行波小波能谱图及其对应的全局小波能谱图.从船行波小波能谱图来看6种工况下船行波小波能谱 呈现局部突出的特点:在时间尺度上,船行波能量 主要集中在水位下降段，即船行波主波段，工况-和2主要分布在5〜1 s 时间段内，工况3和4主要能量/10-3 m 2：能量/10-3 m 2：24681012I时间/s51015 20时间/s(?)工况2航行波小波能谱图(b )工况(全局小波能谱图(a )工况(航行波小波能谱图(0工况3航行波小波能谱图能量/10-3 m 2：12 3 4 5 6时间/s(g )工况、航行波小波能谱图能量/10-3 m 2(h )工况、全局小波能谱图⑴工况3全局小波能谱图能量/10-3 m 2：510 15 20 25能量/10-3 m 2：5 10 15 20 25 30 35 3.0,0 5 0 5 03 2 2 1 130300.5051015 20 25时间/s(-)工况5航行波小波能谱图 3.0°0 2 4 6 8能量/IO-? m 25 1015 20 25时间/s302.5S ：:能量/10-3 m 2：51015 20时间/s(m )工况7航行波小波能谱图(j)工况5全局小波能谱图(k )工况2航行波小波能谱图(-)°0 5 10能量/IO-? m 2工况2全局小波能谱图Z H、<^能量/IO 』m 2：5 10 15 20 25(?)工况7全局小波能谱图3.02.52.01.51.00.5°05 10 15 20 25时间/s(o )工况8航行波小波能谱图303.0r 2.5-n 2.0 -書1彳 於 1.0-能量/10-3 m 2to(p )工况8全局小波能谱图图4 8种工况船行波小波能谱图及其对应的全局小波能谱图hth ：//jomnai. sen . cdo .co第6期毛礼磊，等:基于小波变换的内河航道船行波时频特性分析1121分布在10-20)时间段内，工况5、6、4、3主要分布 在5〜20 s 时间段内；在频率尺度上，船行波能量主要集中在低频段，工况-〜9均主要分布在0〜0. 35 Hz 范围内.从全局小波能谱来看，上述9种 工况下船行波全局小波能谱峰值(m 2 -及其对应的 频率值(Hz)分别为9. 994 x 1-3和 0. 1 46、3. 783x1-3 和 0. 15、1.992 x 1-和 0. 073、1.565 x1-和 0.073、5.636 x 1-和 0.099、9.974 x 1- 和 0. 099、6. 7 1 x 1-和 0. 073、6. 345 x 1_3 和0. 099.可见，对于上述9种工况，船行波全局小波 能谱峰值全部出现在频率较小值处在分析船舶不同航行条件对船行波全局小波能谱的影响时，以船模吃水深度2离岸距离和航速对上述9种工况进行分类•图5所示为9种工况下 船行波全局小波能谱对比情况 当船模吃水深度和航线离岸距离相同时，可比较船模航速对船行波全 局小波能谱的影响•由工况1与2、5与6、4与9对 比可知，当船模航速增大时，产生的船行波全局小波能量峰值更大;而对于工况3和4,当船模航速增大时，产生的船行波全局小波能量峰值并未增 大,这是由于工况3和4中全局小波能量均出现了2 个峰值，分别为 1. 982 x 10-、1. 264 x 10-3 和1.525 x1-、1.399 xl0-.当船速从0.976 m/s 增加到0 . 564 Is 时，工况2全局小波能量峰值增大为工况1的1.59倍;当船速从0. 336 Is 增加到1.046 ms 时，工况5全局小波能量峰值增大为工况6的1.55倍;当船速从0. 382 Is 增加到0. 990Is 时，工况4全局小波能量峰值增大为工况8的-• 02倍•对比工况1和5,船模航线相同，工况5中 船模吃水深度大于工况1,尽管船模航行速度为0. 336 Is,小于工况1中船模的速度0. 376 Is, 此时工况5全局小波能量峰值为工况1的—97倍;对比工况2与5、工况4与4 ,结论亦相同，此时 工况5全局小波能量峰值为工况2的1.49倍，工8O.二二12345678况况况况况况况况工工工工工工工工图5 8种工况下船行波全局小波能量对比况4全局小波能量峰值为工况4的3. 54倍.由此可得，当船舶航线相同，在同一测点处船行波全局小波能量峰值受船舶吃水深度影响较大 对比工况-和4,当船模吃水深度相同时，工况4中船模航行 离岸距离较大，尽管船模航行速度为0 • 941 ms , 大于工况1中船模的航行速度0. 876 ms,此时工况1全局小波能量峰值为工况4的1. 21倍，说明 在同一测点处船行波全局小波能量峰值随离岸距 离增大而减小•4结论-通过概化的内河航道和通航船舶,建立室内水槽试验,复演了内河航道中船舶以不同条件航 行时船行波的产生及传播过程，获得-03组实测的水位波动时间序列数据，补充了内河航道船行波试 验资料2)从时间尺度上解析了内河航道中船行波波列结构，船行波传播至某处的水位过程线历经相似的过程:小幅的水位上升一历时较长且较大的水位下降一剧烈的水位波动，对应船首波、船行波中的低频主波和船行波中的高频次波3-从频率尺度分析了内河航道中船行波频谱特征,船行波小波能谱呈现局部突出的特点,船行波能量主要集中在水位下降段，即低频主波段，对应的频率范围为0〜0.35 He.当吃水深度和航线 相同，船舶航速更大时，产生的船行波全局小波能 量峰值更大;当航线相同时，相比与航速，同一位置处船行波全局小波能量峰值受吃水深度影响较大； 当吃水深度和航速相同，同一位置处船行波全局小波能量峰值随离岸距离增大而减小参考文献(Refeences)1]李志松，吴卫，陈虹，等.内河航道中船行波在岸坡爬高的数值模拟[J ]・水动力学研究与进展(A 辑)，201,31(5) 956 -566. 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Boca Ratou : CRC Puss, 201： 1 309-1 3 1 6.http ：//joorycd sen. edb.cn112东南大学学报（自然科学版）第50卷[3]de Roo S,Troch P•Eveluation of the effectiveness of aliving shorehne in a confinO,non-tiXai waterway sub­ject to heavy shipping traffi?[J].ReseercS and Applicadoos,208,31(8)：123—139.DOI：1.102/4(.2790.[4]Bauer B O,Loravg M S,Sherman D J.Estimatingboat-wake-induced levee erosion using seOiment shsyen-sion measurements[J]•Jocroai op Wdema/,Port, Coastai,and Oceo Engineering,2002,128(4):159 -162.DOI:10.1061/jasce)0733-050p(2002)128：4(52).[5]Houser C.SeOiment resuspension by vessO-genfawnwaves along the Savennah River,Georgia[J].Jocrnai oC Watema/,Port,and Oceeo Engioeering,2011,157(5):246—257.DOI：10.1061/(asce)ww.1943-2460.0000088.[6]Gdransson G,Lbson M,Althagc J.Ship-goeuwnwaves and induced Uirbidity in the Gote Alv river in SwedoJ].Jocroai oC Watema/,Port,andOceeo Engineering,2010,150(3):04410404.DOI：10.1061/(asce)ww.1943-2460.0000224.[7]Wolter C,Arlinghaos R.Navigation impacts on fresh­water fish assembWges:The ecological relevence of swimming performance[J]•Reviees in Fish占匚。

计算技术与自动化Computing Technology and Automation第40卷第1期2 0 2 1年3月Vol. 40,No. 1Mar. 2 02 1文章编号：1003-6199( 2021 )01-0101 — 03DOI ： 10. 16339/j. cnki. jsjsyzdh. 202101019基于3次B 样条小波变换的改进自适应阈值边缘检测算法王 煜J 谢 政，朱淳钊，夏建高(湖北工程职业学院建筑与环境艺术学院，湖北黄石435005)摘要：针对含噪声图像边缘提取问题，提出了一种改进NormalShrink 自适应阈值去噪算法。

该算法首先通过小波变换和局部模极大值法提取出可能包含图像边缘特征的小波系数，利用边缘像素之间特殊的空间关系以及噪声在各级小波分解尺度下的不同效应，构建适合各个尺度级的改进NormalShrink 自适应阈值，并依此对提取出的小波系数进行筛选。

实验结果表明，与改进的Candy 算子和传统的NormalShrink 自 适应阈值相比，本方法提取出的图像边缘较为完整清晰，峰值信噪比提升约6 db o关键词：边缘提取；小波变换；自适应阈值;峰值信噪比中图分类号:TP312文献标识码：AAn Improved Adaptive Threshold Edge Detection AlgorithmBased on Cubic B-spline Wavelet TransformWANG Yu f , XIE Zheng,ZHU Chun-zhao ,XIA Jian-gao(School of Architecture and Environmental Art, Hubei Engineering Institute, Huangshi, Hubei 435005, China)Abstract : In order to solve the problem of noisy image edge detection, an improved NormalShrink adaptive waveletthreshold is put forward on the foundation of combining edge detection and denoising . According to the different characteris ­tics of noise at different wavelet scales and the special spatial relationship between the edge pixels , the algorithm first extract wavelet coefficients which may contain image edge feature by using wavelet transform and local maximum mode, and thenconstruct an improved NormalShrink adaptive threshold of each scale level which is used to select the extracted wavelet coef ­ficients. Experimental results show that this method can keep imagers edges clear and increase PSNR about 6 db.Key words :edge detection ； wavelet transform ； adaptive threshold ； PSNR图像边缘信息的识别和提取在图像分割、图像 识别等领域有着重要的应用，提取出清晰有效的边缘是一个热点研究方向。

INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONSInt.J.Circ.Theor.Appl.2006;34:559–582Published online in Wiley InterScience().DOI:10.1002/cta.375A wavelet-based piecewise approach for steady-state analysisof power electronics circuitsK.C.Tam,S.C.Wong∗,†and C.K.TseDepartment of Electronic and Information Engineering,Hong Kong Polytechnic University,Hong KongSUMMARYSimulation of steady-state waveforms is important to the design of power electronics circuits,as it reveals the maximum voltage and current stresses being imposed upon specific devices and components.This paper proposes an improved approach tofinding steady-state waveforms of power electronics circuits based on wavelet approximation.The proposed method exploits the time-domain piecewise property of power electronics circuits in order to improve the accuracy and computational efficiency.Instead of applying one wavelet approximation to the whole period,several wavelet approximations are applied in a piecewise manner tofit the entire waveform.This wavelet-based piecewise approximation approach can provide very accurate and efficient solution,with much less number of wavelet terms,for approximating steady-state waveforms of power electronics circuits.Copyright2006John Wiley&Sons,Ltd.Received26July2005;Revised26February2006KEY WORDS:power electronics;switching circuits;wavelet approximation;steady-state waveform1.INTRODUCTIONIn the design of power electronics systems,knowledge of the detailed steady-state waveforms is often indispensable as it provides important information about the likely maximum voltage and current stresses that are imposed upon certain semiconductor devices and passive compo-nents[1–3],even though such high stresses may occur for only a brief portion of the switching period.Conventional methods,such as brute-force transient simulation,for obtaining the steady-state waveforms are usually time consuming and may suffer from numerical instabilities, especially for power electronics circuits consisting of slow and fast variations in different parts of the same waveform.Recently,wavelets have been shown to be highly suitable for describingCorrespondence to:S.C.Wong,Department of Electronic and Information Engineering,Hong Kong Polytechnic University,Hunghom,Hong Kong.†E-mail:enscwong@.hkContract/sponsor:Hong Kong Research Grants Council;contract/grant number:PolyU5237/04ECopyright2006John Wiley&Sons,Ltd.560K.C.TAM,S.C.WONG AND C.K.TSEwaveforms with fast changing edges embedded in slowly varying backgrounds[4,5].Liu et al.[6] demonstrated a systematic algorithm for approximating steady-state waveforms arising from power electronics circuits using Chebyshev-polynomial wavelets.Moreover,power electronics circuits are piecewise varying in the time domain.Thus,approx-imating a waveform with one wavelet approximation(ing one set of wavelet functions and hence one set of wavelet coefficients)is rather inefficient as it may require an unnecessarily large wavelet set.In this paper,we propose a piecewise approach to solving the problem,using as many wavelet approximations as the number of switch states.The method yields an accurate steady-state waveform descriptions with much less number of wavelet terms.The paper is organized as follows.Section2reviews the systematic(standard)algorithm for approximating steady-state waveforms using polynomial wavelets,which was proposed by Liu et al.[6].Section3describes the procedure and formulation for approximating steady-state waveforms of piecewise switched systems.In Section4,application examples are presented to evaluate and compare the effectiveness of the proposed piecewise wavelet approximation with that of the standard wavelet approximation.Finally,we give the conclusion in Section5.2.REVIEW OF WA VELET APPROXIMATIONIt has been shown that wavelet approximation is effective for approximating steady-state waveforms of power electronics circuits as it takes advantage of the inherent nature of wavelets in describing fast edges which have been embedded in slowly moving backgrounds[6].Typically,power electronics circuits can be represented by a time-varying state-space equation˙x=A(t)x+U(t)(1) where x is the m-dim state vector,A(t)is an m×m time-varying matrix,and U is the inputfunction.Specifically,we writeA(t)=⎡⎢⎢⎢⎣a11(t)a12(t)···a1m(t)............a m1(t)a m2(t)···a mm(t)⎤⎥⎥⎥⎦(2)andU(t)=⎡⎢⎢⎢⎣u1(t)...u m(t)⎤⎥⎥⎥⎦(3)In the steady state,the solution satisfiesx(t)=x(t+T)for0 t T(4) where T is the period.For an appropriate translation and scaling,the boundary condition can be mapped to the closed interval[−1,1]x(+1)=x(−1)(5) Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582A WA VELET-BASED PIECEWISE APPROACH FOR STEADY-STATE ANALYSIS561 Assume that the basic time-invariant approximation equation isx i(t)=K T i W(t)for−1 t 1and i=1,2,...,m(6) where W(t)is any wavelet basis of size2n+1+1(n being the wavelet level),K T i=[k i,0,...,k i,2n+1] is a coefficient vector of dimension2n+1+1,which is to be found.‡The wavelet transformedequation of(1)isKD W=A(t)K W+U(t)(7)whereK=⎡⎢⎢⎢⎢⎢⎢⎢⎣k1,0k1,1···k1,2n+1k2,0k2,1···k2,2n+1............k m,0k m,1···k m,2n+1⎤⎥⎥⎥⎥⎥⎥⎥⎦(8)Thus,(7)can be written generally asF(t)K=−U(t)(9) where F(t)is a m×(2n+1+1)m matrix and K is a(2n+1+1)m-dim vector,given byF(t)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣a11(t)W T(t)−W T(t)D T···a1i(t)W T(t)···a1m W T(t)...............a i1(t)W T(t)···a ii(t)W T(t)−W T(t)D T···a im W T(t)...............a m1(t)W T(t)···a mi(t)W T(t)···a mm W T(t)−W T(t)D T⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(10)K=[K T1···K T m]T(11)Note that since the unknown K is of dimension(2n+1+1)m,we need(2n+1+1)m equations. Now,the boundary condition(5)provides m equations,i.e.[W(+1)−W(−1)]T K i=0for i=1,...,m(12) This equation can be easily solved by applying an appropriate interpolation technique or via direct numerical convolution[11].Liu et al.[6]suggested that the remaining2n+1m equations‡The construction of wavelet basis has been discussed in detail in Reference[6]and more formally in Reference[7].For more details on polynomial wavelets,see References[8–10].Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582562K.C.TAM,S.C.WONG AND C.K.TSEare obtained by interpolating at2n+1distinct points, i,in the closed interval[−1,1],and the interpolation points can be chosen arbitrarily.Then,the approximation equation can be written as˜FK=˜U(13)where˜F= ˜F1˜F2and˜U=˜U1˜U2(14)with˜F1,˜F2,˜U1and˜U2given by˜F1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣[W(+1)−W(−1)]T(00···0)···(00···0)(00···0)[W(+1)−W(−1)]T···(00···0)............(00···0)2n+1+1columns(00···0)···[W(+1)−W(−1)]T(2n+1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭m rows(15)˜F2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣F( 1)F( 2)...F( 2n+1)(2n+1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭2n+1m rows(16)˜U1=⎡⎢⎢⎢⎣...⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭m elements(17)˜U2=⎡⎢⎢⎢⎢⎢⎣−U( 1)−U( 2)...−U( 2n+1)⎤⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭2n+1m elements(18)Finally,by solving(13),we obtain all the coefficients necessary for generating an approximate solution for the steady-state system.Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582A WA VELET-BASED PIECEWISE APPROACH FOR STEADY-STATE ANALYSIS5633.WA VELET-BASED PIECEWISE APPROXIMATION METHODAlthough the above standard algorithm,given in Reference[6],provides a well approximated steady-state solution,it does not exploit the piecewise switched nature of power electronics circuits to ease computation and to improve accuracy.Power electronics circuits are defined by a set of linear differential equations governing the dynamics for different intervals of time corresponding to different switch states.In the following,we propose a wavelet approximation algorithm specifically for treating power electronics circuits.For each interval(switch state),we canfind a wavelet representation.Then,a set of wavelet representations for all switch states can be‘glued’together to give a complete steady-state waveform.Formally,consider a p-switch-state converter.We can write the describing differential equation, for switch state j,as˙x j=A j x+U j for j=1,2,...,p(19) where A j is a time invariant matrix at state j.Equation(19)is the piecewise state equation of the system.In the steady state,the solution satisfies the following boundary conditions:x j−1(T j−1)=x j(0)for j=2,3,...,p(20) andx1(0)=x p(T p)(21)where T j is the time duration of state j and pj=1T j=T.Thus,mapping all switch states to the close interval[−1,1]in the wavelet space,the basic approximate equation becomesx j,i(t)=K T j,i W(t)for−1 t 1(22) with j=1,2,...,p and i=1,2,...,m,where K T j,i=[k1,i,0···k1,i,2n+1,k2,i,0···k2,i,2n+1,k j,i,0···k j,i,2n+1]is a coefficient vector of dimension(2n+1+1)×p,which is to be found.Asmentioned previously,the state equation is transformed to the wavelet space and then solved by using interpolation.The approximation equation is˜F(t)K=−˜U(t)(23) where˜F=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣˜F˜F1˜F2...˜Fp⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦and˜U=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣˜U˜U1˜U2...˜Up⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(24)Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582564K.C.TAM,S.C.WONG AND C.K.TSEwith ˜F0,˜F 1,˜F 2,˜F p ,˜U 0,˜U 1,˜U 2and ˜U p given by ˜F 0=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣F a 00···F b F b F a 0···00F b F a ···0...............00···F b F a (2n +1+1)×m ×p columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭m ×p rows (F a and F b are given in (33)and (34))(25)˜F 1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣F ( 1)0 0F ( 2)0 0............F ( 2n +1) (2n +1+1)m columns 0(2n +1+1)m columns···0 (2n +1+1)m columns(2n +1+1)×m ×p columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭2n +1m rows(26)˜F 2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣0F ( 1)···00F ( 2)···0............0(2n +1+1)m columnsF ( 2n +1)(2n +1+1)m columns···(2n +1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(27)˜F p =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣0···0F ( 1)0···0F ( 2)...... 0(2n +1+1)m columns···(2n +1+1)m columnsF ( 2n +1)(2n +1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(28)˜U0=⎡⎢⎢⎢⎣0 0⎤⎥⎥⎥⎦⎫⎪⎪⎪⎬⎪⎪⎪⎭m ×p elements(29)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582A WA VELET-BASED PIECEWISE APPROACH FOR STEADY-STATE ANALYSIS565˜U1=⎡⎢⎢⎢⎢⎢⎣−U( 1)−U( 2)...−U( 2n+1)⎤⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎭2n+1m elements(30)˜U2=⎡⎢⎢⎢⎢⎣−U( 1)−U( 2)...−U( 2n+1)⎤⎥⎥⎥⎥⎦(31)˜Up=⎡⎢⎢⎢⎢⎢⎣−U( 1)−U( 2)...−U( 2n+1)⎤⎥⎥⎥⎥⎥⎦(32)F a=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣[W(−1)]T0 00[W(−1)]T 0............00···[W(−1)]T(2n+1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭m rows(33)F b=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣[−W(+1)]T0 00[−W(+1)]T 0............00···[−W(+1)]T(2n+1+1)m columns⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭m rows(34)Similar to the standard approach outlined in Section2,all the coefficients necessary for gener-ating approximate solutions for each switch state for the steady-state system can be obtained by solving(23).It should be noted that the wavelet-based piecewise method can be further enhanced for approx-imating steady-state solution using different wavelet levels for different switch states.Essentially, Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582566K.C.TAM,S.C.WONG AND C.K.TSEwavelets of high levels should only be needed to represent waveforms in switch states where high-frequency details are present.By using different choices of wavelet levels for different switch states,solutions can be obtained more quickly.Such an application of varying wavelet levels for different switch intervals can be easily incorporated in the afore-described algorithm.4.APPLICATION EXAMPLESIn this section,we present four examples to demonstrate the effectiveness of our proposed wavelet-based piecewise method for steady-state analysis of switching circuits.The results will be evaluated using the mean relative error (MRE)and mean absolute error (MAE),which are defined byMRE =12 1−1ˆx (t )−x (t )x (t )d t (35)MAE =12 1−1|ˆx (t )−x (t )|d t (36)where ˆx (t )is the wavelet-approximated value and x (t )is the SPICE simulated result.The SPICE result,being generated from exact time-domain simulation of the actual circuit at device level,can be used for comparison and evaluation.In discrete forms,MAE and MRE are simply given byMRE =1N Ni =1ˆx i −x i x i(37)MAE =1N Ni =1|ˆx i −x i |(38)where N is the total number of points sampled along the interval [−1,1]for error calculation.In the following,we use uniform sampling (i.e.equal spacing)with N =1001,including boundary points.4.1.Example 1:a single pulse waveformConsider the single pulse waveform shown in Figure 1.This is an example of a waveform that cannot be efficiently approximated by the standard wavelet algorithm.The waveform consists of five segments corresponding to five switch states (S1–S5),and the corresponding state equations are given by (19),where A j and U j are given specifically asA j =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩0if 0 t <t 10if t 1 t <t 21if t 2 t <t 30if t 3 t <t 40if t 4 t T(39)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582A WA VELET-BASED PIECEWISE APPROACH FOR STEADY-STATE ANALYSIS567S1S2S3S4S50t1t2t3t4THFigure 1.A single pulse waveform consisting of 5switch states.andU j =⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩0if 0 t <t 1H /(t 2−t 1)if t 1 t <t 2−Hif t 2 t <t 3−H /(t 4−t 3)if t 3 t <t 40if t 4 t T(40)where H is the amplitude (see Figure 1).Switch states 2(S2)and 4(S4)correspond to the rising edge and falling edge,respectively.Obviously,when the widths of rising and falling edges are small (relative to the whole switching period),the standard wavelet method cannot provide a satisfactory approximation for this waveform unless very high wavelet levels are used.Theoretically,the entire pulse-like waveform can be very accurately approximated by a very large number of wavelet terms,but the computational efforts required are excessive.As mentioned before,since the piecewise approach describes each switch interval separately,it yields an accurate steady-state waveform description for each switch interval with much less number of wavelet terms.Figures 2(a)and (b)compare the approximated pulse waveforms using the proposed wavelet-based piecewise method and the standard wavelet method for two different choices of wavelet levels with different widths of rising and falling edges.This example clearly shows the benefits of the wavelet-based piecewise approximation using separate sets of wavelet coefficients for the different switch states.Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582568K.C.TAM,S.C.WONG AND C.K.TSE0−0.2−0.4−0.6−0.8−1−20−15−10−50.20.40.60.81−0.2−0.4−0.6−0.8−10.20.40.60.81(a)051015(b)Figure 2.Approximated pulse waveforms with amplitude 10.Dotted line is the standard wavelet approx-imated waveforms using wavelets of levels from −1to 5.Solid lines are the actual waveforms and the wavelet-based piecewise approximated waveforms using wavelets of levels from −1to 1:(a)switch states 2and 4with rising and falling times both equal to 5per cent of the period;and (b)switch states 2and 4with rising and falling times both equal to 1per cent of the period.4.2.Example 2:simple buck converterThe second example is the simple buck converter shown in Figure 3.Suppose the switch has a resistance of R s when it is turned on,and is practically open-circuit when it is turned off.The diode has a forward voltage drop of V f and an on-resistance of R d .The on-time and off-time equivalent circuits are shown in Figure 4.The basic system equation can be readily found as˙x=A (t )x +U (t )(41)where x =[i L v o ]T ,and A (t )and U (t )are given byA (t )=⎡⎢⎣−R d s (t )L −1L 1C −1RC⎤⎥⎦(42)U (t )=⎡⎣E (1−s (t ))+V f s (t )L⎤⎦(43)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582Figure3.Simple buck convertercircuit.Figure4.Equivalent linear circuits of the buck converter:(a)during on time;and(b)during off time.Table ponent and parameter values for simulationof the simple buck converter.Component/parameter ValueMain inductance,L0.5mHCapacitance,C0.1mFLoad resistance,R10Input voltage,E100VDiode forward drop,V f0.8VSwitching period,T100 sOn-time,T D40 sSwitch on-resistance,R s0.001Diode on-resistance,R d0.001with s(t)defined bys(t)=⎧⎪⎨⎪⎩0for0 t T D1for T D t Ts(t−T)for all t>T(44)We have performed waveform approximations using the standard wavelet method and the proposed wavelet-based piecewise method.The circuit parameters are shown in Table I.We also generate waveforms from SPICE simulations which are used as references for comparison. The approximated inductor current is shown in Figure5.Simple visual inspection reveals that the wavelet-based piecewise approach always gives more accurate waveforms than the standard method.Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582−0.5−10.51−0.5−10.51012345670123456712345671234567(a)(b)(c)(d)Figure 5.Inductor current waveforms of the buck converter.Solid line is waveform from piecewise wavelet approximation,dotted line is waveform from SPICE simulation and dot-dashed line is waveform using standard wavelet approximation.Note that the solid lines are nearly overlapping with the dotted lines:(a)using wavelets of levels from −1to 0;(b)using wavelets of levels from −1to 1;(c)using wavelets oflevels from −1to 4;and (d)using wavelets of levels from −1to 5.Table parison of MREs for approximating waveforms for the simple buck converter.Wavelet Number of MRE for i L MRE for v C CPU time (s)MRE for i L MRE for v C CPU time (s)levels wavelets (standard)(standard)(standard)(piecewise)(piecewise)(piecewise)−1to 030.9773300.9802850.0150.0041640.0033580.016−1to 150.2501360.1651870.0160.0030220.0024000.016−1to 290.0266670.0208900.0320.0030220.0024000.046−1to 3170.1281940.1180920.1090.0030220.0024000.110−1to 4330.0593070.0538670.3750.0030220.0024000.407−1to 5650.0280970.025478 1.4380.0030220.002400 1.735−1to 61290.0122120.011025 6.1880.0030220.0024009.344−1to 72570.0043420.00373328.6410.0030220.00240050.453In order to compare the results quantitatively,MREs are computed,as reported in Table II and plotted in Figure 6.Finally we note that the inductor current waveform has been very well approximated by using only 5wavelets of levels up to 1in the piecewise method with extremelyCopyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582123456700.10.20.30.40.50.60.70.80.91M R E (m e a n r e l a t i v e e r r o r )Wavelet Levelsinductor current : standard method inductor current : piecewise methodFigure parison of MREs for approximating inductor current for the simple buck converter.small MREs.Furthermore,as shown in Table II,the CPU time required by the standard method to achieve an MRE of about 0.0043for i L is 28.64s,while it is less than 0.016s with the proposed piecewise approach.Thus,we see that the piecewise method is significantly faster than the standard method.4.3.Example 3:boost converter with parasitic ringingsNext,we consider the boost converter shown in Figure 7.The equivalent on-time and off-time circuits are shown in Figure 8.Note that the parasitic capacitance across the switch and the leakage inductance are deliberately included to reveal waveform ringings which are realistic phenomena requiring rather long simulation time if a brute-force time-domain simulation method is used.The state equation of this converter is given by˙x=A (t )x +U (t )(45)where x =[i m i l v s v o ]T ,and A (t )and U (t )are given byA (t )=A 1(1−s (t ))+A 2s (t )(46)U (t )=U 1(1−s (t ))+U 2s (t )(47)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582Figure7.Simple boost convertercircuit.Figure8.Equivalent linear circuits of the boost converter including parasitic components:(a)for on time;and(b)for off time.with s(t)defined earlier in(44)andA1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−R mL mR mL m00R mL l−R l+R mL l−1L l1C s−1R s C s000−1RC⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(48)A2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−R mR dL mR m R dL m0−R mL m d mR m R dL l−R mR d+R lL l−1L lR mL l d m1C s00R mC(R d+R m)−R mC(R d+R m)0−R+R m+R dC R(R d+R m)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(49)Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582U1=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣EL m⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(50)U2=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣EL m−R m V fL m d mR m V fL l(R d+R m)−V f R mC(R d m⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(51)Again we compare the approximated waveforms of the leakage inductor current using the proposed piecewise method and the standard wavelet method.The circuit parameters are listed in Table III.Figures9(a)and(b)show the approximated waveforms using the piecewise and standard wavelet methods for two different choices of wavelet levels.As expected,the piecewise method gives more accurate results with wavelets of relatively low levels.Since the waveform contains a substantial portion where the value is near zero,we use the mean absolute error(MAE)forTable ponent and parameter values for simulation ofthe boost converter.Component/parameter ValueMain inductance,L m200 HLeakage inductance,L l1 HParasitic resistance,R m1MOutput capacitance,C200 FLoad resistance,R10Input voltage,E10VDiode forward drop,V f0.8VSwitching period,T100 sOn-time,T D40 sParasitic lead resistance,R l0.5Switch on-resistance,R s0.001Switch capacitance,C s200nFDiode on-resistance,R d0.001Copyright2006John Wiley&Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–5820−0.2−0.4−0.6−0.8−1−50.20.40.60.815100(a)(b)−50.20.40.60.81510Figure 9.Leakage inductor waveforms of the boost converter.Solid line is waveform from wavelet-based piecewise approximation,dotted line is waveform from SPICE simulation and dot-dashed line is waveform using standard wavelet approximation:(a)using wavelets oflevels from −1to 4;and (b)using wavelets of levels from −1to 5.Table IV .Comparison of MAEs for approximating the leakage inductor currentfor the boost converter.Wavelet Number MAE for i l CPU time (s)MAE for i l CPU time (s)levels of wavelets(standard)(standard)(piecewise)(piecewise)−1to 3170.4501710.1250.2401820.156−1to 4330.3263290.4060.1448180.625−1to 5650.269990 1.6410.067127 3.500−1to 61290.2118157.7970.06399521.656−1to 72570.13254340.6250.063175171.563evaluation.From Table IV and Figure 10,the result clearly verifies the advantage of using the proposed wavelet-based piecewise method.Furthermore,inspecting the two switch states of the boost converter,it is obvious that switch state 2(off-time)is richer in high-frequency details,and therefore should be approximated with wavelets of higher levels.A more educated choice of wavelet levels can shorten the simulationCopyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582345670.050.10.150.20.250.30.350.40.450.5M A E (m e a n a b s o l u t e e r r o r )Wavelet Levelsleakage inductor current : standard method leakage inductor current : piecewise methodFigure parison of MAEs for approximating the leakage inductor current for the boost converter.time.Figure 11shows the approximated waveforms with different (more appropriate)choices of wavelet levels for switch states 1(on-time)and 2(off-time).Here,we note that smaller MAEs can generally be achieved with a less total number of wavelets,compared to the case where the same wavelet levels are employed for both switch states.Also,from Table IV,we see that the CPU time required for the standard method to achieve an MAE of about 0.13for i l is 40.625s,while it takes only slightly more than 0.6s with the piecewise method.Thus,the gain in computational speed is significant with the piecewise approach.4.4.Example 4:flyback converter with parasitic ringingsThe final example is a flyback converter,which is shown in Figure 12.The equivalent on-time and off-time circuits are shown in Figure 13.The parasitic capacitance across the switch and the transformer leakage inductance are included to reveal realistic waveform ringings.The state equation of this converter is given by˙x=A (t )x +U (t )(52)where x =[i m i l v s v o ]T ,and A (t )and U (t )are given byA (t )=A 1(1−s (t ))+A 2s (t )(53)U (t )=U 1(1−s (t ))+U 2s (t )(54)Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–5820−0.2−0.4−0.6−0.8−1−6−4−20.20.40.60.81024680−0.2−0.4−0.6−0.8−1−6−4−20.20.40.60.81024680−0.2−0.4−0.6−0.8−1−6−4−20.20.40.60.81024680−0.2−0.4−0.6−0.8−1−6−4−20.20.40.60.8102468il(A)il(A)il(A)il(A)(a)(b)(c)(d)Figure 11.Leakage inductor waveforms of the boost converter with different choice of wavelet levels for the two switch states.Dotted line is waveform from SPICE simulation.Solid line is waveform using wavelet-based piecewise approximation.Two different wavelet levels,shown in brackets,are used for approximating switch states 1and 2,respectively:(a)(3,4)with MAE =0.154674;(b)(3,5)withMAE =0.082159;(c)(4,5)with MAE =0.071915;and (d)(5,6)with MAE =0.066218.Copyright 2006John Wiley &Sons,Ltd.Int.J.Circ.Theor.Appl.2006;34:559–582。