利用Hilbert变换提取信号瞬时特征的算法实现
研究了在工程中如何通过算法来实现利用 Hilbert 变换提取信号的瞬时特征。深入地分析了如何利用数值微分法提高提取瞬时频率特征的精度。最后,给出了一种可行的算法,并通过实验验证了这种方法可以在工程中有效地提取信号的瞬时频率特征。84微机发展第13卷①H(x)=y;H(x)=y;(i=0,1,…n)(j=0,1(11)②在每个小区间/x1,x1+17i=0,1,…,n-1)上由相关定理知:当划分的小区间的长度趋于零时H(x)是三次多项式。s(x)及其一至三阶导数分别一致收敛到f(x)及其一至可以写出分段三次 Hermite插值函数的分段表达式:三阶导数。所以用三次样条插值函数去近似表达用离散值(x)=(1+2x-x过+)2v;+表示的原函数,具有较高的可靠性。3)两种插值的比较挨尔米特 Hermite插值较三次样I-i,1+2条插值具有较好的稳定性与收敛性,但它只能休让各段曲线在连接点上的连续性,而不能保证整条曲线在这些点上y+1Ditl的光滑性。而有时不仅要求曲线连续,而且要求曲线的曲X/(i=0,18)率也连续即要求分段插值函数具有连续的一阶导数,埃H(x)的导数为尔米特 Hermite插值此时就不能满足上述要求6次样条插值较埃尔米特 Hermite插值具有较好的H(x光滑程度,用三次样条插值函数求数值导数比用埃尔米特+2(x-x2(xHermite插值可靠性大,但计算比较复杂,二者的区别见图h2yV+17, h(i-0,12)三次样条插值。已知函数y=f(x)在区间/a,b上的n+1个节点上的值y=f(x;)(i=0,1,…m),求插值函数s(x),使(i=0,1图4 Hermite插值与三次样条插值的比较图2在每个小区间x,x+1(=0.1.…n-1)上利用埃尔米特 Hermite插值得到的2FSK信号的瞬时s(x是三次多项式,记为s(x频率见图5,利用二次样条插值得到的该信号的瞬时频率③3(x)在la,b/上二阶连续可徵。见图6。数s(x)称为f(x)的三次样条插值函数可以利用节点处的二阶导数值为参数,也可以利用节点处的导数值为参数求三次样条插值涵数的表达式。若利用节点处的一阶导数值为参数,求得的三次样条插值函数的表达式为(x)=M-1x-x-)36 h6 hMihi5 DEMeN5a亩pai66hx∈[x;,x+17,b-x+1-x,S"(x)=M图5由 Hermite插值提取图6由三次样条插值提取(j=0,1的2FSK信号的瞬时频率的2FSK信号的瞬时频率对s(x)进行求导,利用S(x)在节点处一阶导数连从图5、图6可以看出利用三次样条插值得到的瞬时续的性质结合边界条件求解出参数M,把求得的参数代频率可以准确反映出信号具有的的摒时频率特征而利用入公式(10),即得三次样条插值函数的s(x)分段表示式。埃尔米特 Hermite插值得到的瞬时频率与信号具有的瞬s;(x)的导数为时频率特征不符。这是因为利用数值微分法求瞬时频率插值以后喫进行求导。三次样条插值函数具有连续的二阶M2 hiM; 2 hj导数,因而具有较好的光滑程度,符合求导条件,所以可以J+1-h(M2+1-M/)准确求出信号的瞬时频率;而埃尔米特 Hernite插值.不够光滑,虽能保证插值多项式收敛于原函数,但不能保证插x Elx,x;+1 h,=xi+I-x, S(xj )=M;值多顷式的导数收敛于原函数的导数,所以求得的值与信o1994-2010ChinaAcademicJournalElectronicPublishingHouse.Allrightsreservedhttp://www.cnki.net第6期刘慧婷等:利用 Hilbert变提取信号瞵时特征的算法实现号实际的瞬时频率值不符。实验结果和理论分析结果是(1) Hilbert变换只能近似应用于窄带信号,即形如纹的(t)=a(1)cosu+6(1)),其中>>B(B为信号带2.3.3结论宽)的信号。但实际应用中,存在许多非窄带信号, Hilbcrt利用数值微分法求瞬时频率ω(t)的步骤可以归纳变换对这些信号无能为力为:首先通过三次样条插值得到分段多项式p(1),(2)对于任意给定时刻,通过 Hilbert变换运算后的结pp(抄);然后分别对分段多项式p(t),Pp()关于变量t果只能存在一个频率值,即只能处理任何时刻为单一频率进行求导,得到pd(,ppd(t);最后求出每一时刻t所对的信号。这显然不合理,因为在实东中同一信号会含有多应的导数值,即求得t(t,u(t)。再把求得的值代入公种频率成分式(6)就完成了提取瞬时频率ω(1)的过程。求解结果见(3)对信号进行 Hilbert变换时,信号的两端会出现严图7重的端点效应。提取某些信号瞬时特征所得的瞬时频率在局部出现了负数,端点效应是造成负频率的一个原因而端点效应可以通过利用特征波对原有数据序列进行延拓的方法来解决,具体解决办法将在今后讨论。尽管目前出现了EMD担论4,其目的是将不满足Hibt变换的信号进行分解得到若干个IMF( intrinsic mode function),然后进行 Hilbert运算,达到提取信号瞬时特征的目的。该理论开辟了信号处理的新空间。但它还不够成熟还需喫进一步的完善和研究图7利用数值微分法提取信号的瞬时频率特征参考文献从图7可以看出,以三次烊条指值进行的数值微分可[]黄长蓉. Hilbert变换及其应用[J].成都气象学院学报以准确岀提取岀信号的瞬时频率特征。199,14(3):273-276.[2]杨小牛,楼A义,徐建良.软件无线电原理与应用[M].北3结束语京:电子工业出版社,2001在工程中, Hilbert变换使得我们对短信号和复杂信号[3]丁丽妤.数值计算方法[M].北京:北京理工大学出版社,的摒时特征的提取成为可能特别是对瞬时频率特征提1997取,在工程中具有十公重要的意义。文中讨论的利用三次[4] Huang N e. The empirical mode decomposition and the hilbert样条插值进行数值徵分以提取瞬时特征的方法是可行的,spectrum for nonlinear and nor stationary time series anal ysis但还存在着如下问题。[].Proc.R.soc.Lond.A,1998,454:903-995(上接第81页)218994。例22(x)=(1-2siny=223101075一般的(A算法计算了120代,求到的最大值为454176.219。154370083改进的α算法计算了34代,求到的最大值为1048575.875。改进后的αA算法收敛速度(指迭代次数)比一般GA算法几乎快了一个数量级,精度也提高了不少,特别是例2的最大值提高一倍多,速度提高这么快是未曾料到的y=74958参考文献+4X Axl Thla[1]陈国良.遗传算法及其应用[M]·北京:人民邮电出版社,图2函数2的图像1996一般GA算法计算了20代,求到的最大值为[2]袁亚湘,孙文瑜.最优化理沦与方法[M]北京:科学出版社,19991.218983[3]张铃,张钹·遗传算法杋理的硏究[J]·软件学报,改进(A算法计算了5代,求到的最大值为2000,11(7):945952o1994-2010ChinaacAdemicJournalElectronicPublishingHouse.Allrightsreservedhttp://www.cnki.net
- 2020-12-05下载
- 积分：1
Vehicle Dynamics Theory and Application
不错的汽车动力学教材，是参与汽车底盘电子开发的动力学基础。Reza n. jazarVehicle DynamicsTheory and ApplicationsSpringerReza n. jazarDept of Mechanical EngineeringManhattan collegeRiverdale. NY 10471ISBN:978-0-387-74243-4e-ISBN:978-0-387-74244-1Library of Congress Control Number: 200794219c 2008 Springer Science+ Business Media, LLCAll rights reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher(Springer Science+Business Media, LLC, 233 SpringStreet, New York, NY 10013, USA), except for brief excerpts in connection with reviews orscholarly analysis. Use in connection with any form of information storage and retrievalelectronic adaptation, computer software, or by similar or dissimilar methodology now knownor hereafter developed is forbidden. The use in this publication of trade names, trademarksservice marks and similar terms, even if they are not identified as such, is not to be taken as anexpression of opinion as to whether or not they are subject to proprietary rightsPrinted on acid-free paper987654321springer. comKavoshmy daughter, Vazan,and my wife, MojganHappiness is when you win a race against yourselfPrefaceThis text is for engineering students. It introduces the fundamental knowledge used in vehicle dymamics. This knowledge can be utilized to developcomputer programs for analyzing the ride, handling, and optimization ofroad vehiclesVehicle dynamics has been in the engineering curriculum for more thana hundred years. Books on the subject are available, but most of themare written for specialists and are not suitable for a classroom applicationA new student, engineer, or researcher would not know where and howto start learning vehicle dynamics. So, there is a need for a textbook forbeginners. This textbook presents the fundamentals with a perspective onfuture trendsThe study of classical vehicle dynamics has its roots in the work ofgreat scientists of the past four centuries and creative engineers in thepast century who established the methodology of dynamic systems. Thedevelopment of vehicle dynamics has moved toward modeling, analysisand optimization of multi-body dynamics supported by some compliantmembers. Therefore, merging dynamics with optimization theory was anexpected development. The fast-growing capability of accurate positioninsensing, and calculations, along with intelligent computer programming arethe other important developments in vehicle dynamics. So, a textbook helpthe reader to make a computer model of vehicles, which this book doesLevel of the bookThis book has evolved from nearly a decade of research in nonlineardynamic systems and teaching courses in vehicle dynamics. It is addressedprimarily to the last year of undergraduate study and the first year graduatestudent in engineering. Hence, it is an intermediate textbook. It providesboth fundamental and advanced topics. The whole book can be coveredin two successive courses, however, it is possible to jump over some sections and cover the book in one course. Students are required to know thefundamentals of kinematics and dynamics, as well as a basic knowledge ofnumerical methodsThe contents of the book have been kept at a fairly theoretical-practicallevel. Many concepts are deeply explained and their application empha-sized, and most of the related theories and formal proofs have been explained. The book places a strong emphasis on the physical meaning andapplications of the concepts. Topics that have been selected are of highinterest in the field. An attempt has been made to expose students to aPrefacebroad range of topics and approachese There are four special chapters that are indirectly related to vehicle dy-amics: Applied Kinematics, Applied Mechanisms, Applied dynamics, andApplied vibrations. These chapters provide the related background to understand vehicle dynamics and its subsystemsOrganization of the bookThe text is organized so it can be used for teaching or for self-studyChapter 1"Fundamentals, "contains general preliminaries about tire andrim with a brief review of road vehicle classificationsPart I"One Dimensional Vehicle Dynamics, " presents forward vehicledynamics, tire dynamics, and driveline dynamics. Forward dynamics refersto weight transfer, accelerating braking, engine performance, and gear ratiodesignPart II"Vehicle Kinematics, presents a detailed discussion of vehiclemechanical subsystems such as steering and suspensionsPart IIT"Vehicle Dynamics, employs Newton and Lagrange methodsto develop the maneuvering dynamics of vehiclesPart Iv "Vehicle Vibrations, presents a detailed discussion of vehi-cle vibrations. An attempt is made to review the basic approaches anddemonstrate how a vehicle can be modeled as a vibrating multiple degreeof-freedom system. The concepts of the Newton-Euler dynamics and La-grangian method are used equally for derivation of equations of motionThe RMS optimization technique for suspension design of vehicles is intro-duced and applied to vehicle suspensions. The outcome of the optimizationtechnique is the optimal stiffness and damping for a car or suspended equipmentMethod of presentationThis book uses a fact-reason-application"structure. The "fact"is themain subject we introduce in each section. Then the reason is given as a" proof. The application of the fact is examined in some examples. Theexamplesare a very important part of the book because they show howto implement the facts. They also cover some other facts that are neededto expand the subjectPrerequisitesSince the book is written for senior undergraduate and first-year graduatelevel students of engineering, the assumption is that users are familiar withmatrix algebra as well as basic dynamics. Prerequisites are the fundamentals of kinematics, dynamics, vector analysis, and matrix theory. Thesebasics are usually taught in the first three undergraduate yearsPrefaceUnit SystemThe system of units adopted in this book is, unless otherwise stated, theinternational system of units(SI). The units of degree(deg)or radian(rad)are utilized for variables representing angular quantitiesSymbolse Lowercase bold letters indicate a vector. Vectors may be expressed inan n dimensional Euclidian space. ExamplerCUppercase bold letters indicate a dynamic vector or a dynamic matrix, such as force and moment. ExampleFo Lowercase letters with a hat indicate a unit vector. Unit vectors arenot bolded. ExampleLowercase letters with a tilde indicate a 3 x 3 skew symmetric matrixassociated to a vector. Examplea3211An arrow above two uppercase letters indicates the start and endpoints of a position vector. ExampleON = a position vector from point o to point Ne The length of a vector is indicated by a non-bold lowercase letterExampleCapital letter B is utilized to denote a body coordinate frame. ExampleB(ocgB(Oxyz)B1(o1x19121)ⅹ11PrefaceCapital letter G is utilized to denote a global, inertial, or fixed coordinate frame. ExampleG(XYZG(OXYZRight subscript on a transformation matrix indicates the departureframes. ExampleRB= transformation matrix from frame B(oxyz)Left superscript on a transformation matrix indicates the destinationframe. ExampleRBtransformation matrix from frame B(o cgz)to frame G(OxYZ)Capital letter R indicates rotation or a transformation matrix, if itshows the beginning and destination coordinate frames. Example0BSIn a0Whenever there is no sub or superscript, the matrices are shown in abracket. ExampleCOS asin a osIn aCOs O0e Left superscript on a vector denotes the frame in which the vectoris expressed. That superscript indicates the frame that the vectorbelongs to; so the vector is expressed using the unit vectors of thatEr= position vector expressed in frame G(OXYZ)Right subscript on a vector denotes the tip point that the vector isreferred to. ExamplePsition vector ofexpressed in coordinate frame G(OXYZ)Right subscript on an angular velocity vector indicates the frame thatthe angular vector is referred to. ExampleB= angularof the body coordinate frame B(oxyz)
- 2020-12-12下载
- 积分：1
