Finite-Dimensional Vector Spaces - P. Halmos (Springer, 1987)
在学习代数学之余，值得一看的代数学书籍。里面介绍了更为丰富的代数学概念和结论。PREFACEMy purpose in this book is to treat linear transformations on finite-dimensional vector spaces by the methods of more general theories. Theidea is to emphasize the simple geometric notions common to many partsof mathematics and its applications, and to do so in a language that givesaway the trade secrets and tells the student what is in the back of the mindsof people proving theorems about integral equations and Hilbert spaces.The reader does not, however, have to share my prejudiced motivationExcept for an occasional reference to undergraduate mathematics the bookis self-contained and may be read by anyone who is trying to get a feelingfor the linear problems usually discussed in courses on matrix theory orhigher"algebra. The algebraic, coordinate-free methods do not lose powerand elegance by specialization to a finite number of dimensions, and theyare, in my belief, as elementary as the classical coordinatized treatmentI originally intended this book to contain a theorem if and only if aninfinite-dimensional generalization of it already exists, The temptingeasiness of some essentially finite-dimensional notions and results washowever, irresistible, and in the final result my initial intentions are justbarely visible. They are most clearly seen in the emphasis, throughout, ongeneralizable methods instead of sharpest possible results. The reader maysometimes see some obvious way of shortening the proofs i give In suchcases the chances are that the infinite-dimensional analogue of the shorterproof is either much longer or else non-existent.A preliminary edition of the book (Annals of Mathematics Studies,Number 7, first published by the Princeton University Press in 1942)hasbeen circulating for several years. In addition to some minor changes instyle and in order, the difference between the preceding version and thisone is that the latter contains the following new material:(1) a brief dis-cussion of fields, and, in the treatment of vector spaces with inner productsspecial attention to the real case.(2)a definition of determinants ininvariant terms, via the theory of multilinear forms. 3 ExercisesThe exercises(well over three hundred of them) constitute the mostsignificant addition; I hope that they will be found useful by both studentPREFACEand teacher. There are two things about them the reader should knowFirst, if an exercise is neither imperative "prove that.., )nor interrogtive("is it true that...?" )but merely declarative, then it is intendedas a challenge. For such exercises the reader is asked to discover if theassertion is true or false, prove it if true and construct a counterexample iffalse, and, most important of all, discuss such alterations of hypothesis andconclusion as will make the true ones false and the false ones true. Secondthe exercises, whatever their grammatical form, are not always placed 8oas to make their very position a hint to their solution. Frequently exer-cises are stated as soon as the statement makes sense, quite a bit beforemachinery for a quick solution has been developed. A reader who tries(even unsuccessfully) to solve such a"misplaced"exercise is likely to ap-preciate and to understand the subsequent developments much better forhis attempt. Having in mind possible future editions of the book, I askthe reader to let me know about errors in the exercises, and to suggest im-provements and additions. (Needless to say, the same goes for the text.)None of the theorems and only very few of the exercises are my discovery;most of them are known to most working mathematicians, and have beenknown for a long time. Although i do not give a detailed list of my sources,I am nevertheless deeply aware of my indebtedness to the books and papersfrom which I learned and to the friends and strangers who, before andafter the publication of the first version, gave me much valuable encourage-ment and criticism. Iam particularly grateful to three men: J. L. Dooband arlen Brown, who read the entire manuscript of the first and thesecond version, respectively, and made many useful suggestions, andJohn von Neumann, who was one of the originators of the modern spiritand methods that I have tried to present and whose teaching was theinspiration for this bookP、R.HCONTENTS的 FAPTERPAGRI SPACESI. Fields, 1; 2. Vector spaces, 3; 3. Examples, 4;4. Comments, 55. Linear dependence, 7; 6. Linear combinations. 9: 7. Bases, 108. Dimension, 13; 9. Isomorphism, 14; 10. Subspaces, 16; 11. Calculus of subspaces, 17; 12. Dimension of a subspace, 18; 13. Dualspaces, 20; 14. Brackets, 21; 15. Dual bases, 23; 16. Reflexivity, 24;17. Annihilators, 26; 18. Direct sums, 28: 19. Dimension of a directsum, 30; 20. Dual of a direct sum, 31; 21. Qguotient spaces, 33;22. Dimension of a quotient space, 34; 23. Bilinear forms, 3524. Tensor products, 38; 25. Product bases, 40 26. Permutations41; 27. Cycles,44; 28. Parity, 46; 29. Multilinear forms, 4830. Alternating formB, 50; 31. Alternating forms of maximal degree,52II. TRANSFORMATIONS32. Linear transformations, 55; 33. Transformations as vectors, 5634. Products, 58; 35. Polynomials, 59 36. Inverses, 61; 37. Mat-rices, 64; 38. Matrices of transformations, 67; 39. Invariance,7l;40. Reducibility, 72 41. Projections, 73 42. Combinations of pro-jections, 74; 43. Projections and invariance, 76; 44. Adjoints, 78;45. Adjoints of projections, 80; 46. Change of basis, 82 47. Similarity, 84; 48. Quotient transformations, 87; 49. Range and null-space, 88; 50. Rank and nullity, 90; 51. Transformations of rankone, 92 52. Tensor products of transformations, 95; 53. Determinants, 98 54. Proper values, 102; 55. Multiplicity, 104; 56. Triangular form, 106; 57. Nilpotence, 109; 58. Jordan form. 112III ORTHOGONALITY11859. Inner products, 118; 60. Complex inner products, 120; 61. Innerproduct spaces, 121; 62 Orthogonality, 122; 63. Completeness, 124;64. Schwarz e inequality, 125; 65. Complete orthonormal sets, 127;CONTENTS66. Projection theorem, 129; 67. Linear functionals, 130; 68. P aren, gBCHAPTERtheses versus brackets, 13169. Natural isomorphisms, 138;70. Self-adjoint transformations, 135: 71. Polarization, 13872. Positive transformations, 139; 73. Isometries, 142; 74. Changeof orthonormal basis, 144; 75. Perpendicular projections, 14676. Combinations of perpendicular projections, 148; 77. Com-plexification, 150; 78. Characterization of spectra, 158; 79. Spec-ptral theorem, 155; 80. normal transformations, 159; 81. Orthogonaltransformations, 162; 82. Functions of transformations, 16583. Polar decomposition, 169; 84. Commutativity, 171; 85. Self-adjoint transformations of rank one, 172IV. ANALYSIS....17586. Convergence of vectors, 175; 87. Norm, 176; 88. Expressions forthe norm, 178; 89. bounds of a self-adjoint transformation, 17990. Minimax principle, 181; 91. Convergence of linear transformations, 182 92. Ergodic theorem, 184 98. Power series, 186APPENDIX. HILBERT SPACERECOMMENDED READING, 195INDEX OF TERMS, 197INDEX OF SYMBOLS, 200CHAPTER ISPACES§L. FieldsIn what follows we shall have occasion to use various classes of numbers(such as the class of all real numbers or the class of all complex numbers)Because we should not at this early stage commit ourselves to any specificclass, we shall adopt the dodge of referring to numbers as scalars. Thereader will not lose anything essential if he consistently interprets scalarsas real numbers or as complex numbers in the examples that we shallstudy both classes will occur. To be specific(and also in order to operateat the proper level of generality) we proceed to list all the general factsabout scalars that we shall need to assume(A)To every pair, a and B, of scalars there corresponds a scalar a+called the sum of a and B, in such a way that(1) addition is commutative,a+β=β+a,(2)addition is associative, a+(8+y)=(a+B)+y(3 there exists a unique scalar o(called zero)such that a+0= a forevery scalar a, and(4)to every scalar a there corresponds a unique scalar -a such that十(0(B)To every pair, a and B, of scalars there corresponds a scalar aBcalled the product of a and B, in such a way that(1)multiplication is commutative, aB pa(2)multiplication is associative, a(Br)=(aB)Y,( )there exists a unique non-zero scalar 1 (called one)such that al afor every scalar a, and(4)to every non-zero scalar a there corresponds a unique scalar a-1or-such that aaSPACES(C)Multiplication is distributive with respect to addition, a(a+n)If addition and multiplication are defined within some set of objectsscalars) so that the conditions(A),B), and (c)are satisfied, then thatset(together with the given operations) is called a field. Thus, for examplethe set Q of all rational numbers(with the ordinary definitions of sumand product)is a field, and the same is true of the set of all real numberaand the set e of all complex numbersHHXERCISIS1. Almost all the laws of elementary arithmetic are consequences of the axiomsdefining a field. Prove, in particular, that if 5 is field and if a, and y belongto 5. then the following relations hold80+a=ab )Ifa+B=a+r, then p=yca+(B-a)=B (Here B-a=B+(a)(d)a0=0 c=0.(For clarity or emphasis we sometimes use the dot to indi-cate multiplication.()(-a)(-p)(g).If aB=0, then either a=0 or B=0(or both).2.(a)Is the set of all positive integers a field? (In familiar systems, such as theintegers, we shall almost always use the ordinary operations of addition and multi-lication. On the rare occasions when we depart from this convention, we shallgive ample warningAs for "positive, "by that word we mean, here and elsewherein this book, "greater than or equal to zero If 0 is to be excluded, we shall say"strictly positive(b)What about the set of all integers?(c) Can the answers to these questiong be changed by re-defining addition ormultiplication (or both)?3. Let m be an integer, m2 2, and let Zm be the set of all positive integers lessthan m, zm=10, 1, .. m-1). If a and B are in Zmy let a +p be the leastpositive remainder obtained by dividing the(ordinary) sum of a and B by m, andproduct of a and B by m.(Example: if m= 12, then 3+11=2 and 3. 11=9)a) Prove that i is a field if and only if m is a prime.(b What is -1 in Z5?(c) What is囊izr?4. The example of Z, (where p is a prime)shows that not quite all the laws ofelementary arithmetic hold in fields; in Z2, for instance, 1 +1 =0. Prove thatif is a field, then either the result of repeatedly adding 1 to itself is always dif-ferent from 0, or else the first time that it is equal to0 occurs when the numberof summands is a prime. (The characteristic of the field s is defined to be 0 in thefirst case and the crucial prime in the second)SEC. 2VECTOR SPACES35. Let Q(v2)be the set of all real numbers of the form a+Bv2, wherea and B are rational.(a)Ie(√2) a field?(b )What if a and B are required to be integer?6.(a)Does the set of all polynomials with integer coefficients form a feld?(b)What if the coeficients are allowed to be real numbers?7: Let g be the set of all(ordered) pairs(a, b)of real numbers(a) If addition and multiplication are defined by(a月)+(,6)=(a+y,B+6)and(a,B)(Y,8)=(ary,B6),does s become a field?(b )If addition and multiplication are defined by(α,月)+⑦,b)=(a+%,B+6)daB)(,b)=(ay-6a6+的y),is g a field then?(c)What happens (in both the preceding cases)if we consider ordered pairs ofcomplex numbers instead?§2. Vector spaceWe come now to the basic concept of this book. For the definitionthat follows we assume that we are given a particular field s; the scalarsto be used are to be elements of gDEFINITION. A vector space is a set o of elements called vectors satisfyingthe following axiomsQ (A)To every pair, a and g, of vectors in u there corresponds vectora t y, called the aum of a and y, in such a way that(1)& ddition is commutative,x十y=y十a(2)addition is associative, t+(y+2)=(+y)+a(3)there exists in V a unique vector 0(called the origin) such thata t0=s for every vector and(4)to every vector r in U there corresponds a unique vector -rthat c+(-x)=o(B)To every pair, a and E, where a is a scalar and a is a vector in u,there corresponds a vector at in 0, called the product of a and a, in sucha way that(1)multiplication by scalars is associative, a(Bx)=aB)=, and(2 lz a s for every vector xSPACESSFC B(C)(1)Multiplication by scalars is distributive with respect to vectorddition, a(+y=a+ ag, and2)multiplication by vectors is distributive with respect to scalar ad-dition, (a B )r s ac+ Bc.These axioms are not claimed to be logically independent; they aremerely a convenient characterization of the objects we wish to study. Therelation between a vector space V and the underlying field s is usuallydescribed by saying that v is a vector space over 5. If S is the field Rof real number, u is called a real vector space; similarly if s is Q or if gise, we speak of rational vector spaces or complex vector space§3. ExamplesBefore discussing the implications of the axioms, we give some examplesWe shall refer to these examples over and over again, and we shall use thenotation established here throughout the rest of our work.(1) Let e(= e)be the set of all complex numbers; if we interpretr+y and az as ordinary complex numerical addition and multiplicatione becomes a complex vector space2)Let o be the set of all polynomials, with complex coeficients, in avariable t. To make into a complex vector space, we interpret vectoraddition and scalar multiplication as the ordinary addition of two poly-nomials and the multiplication of a polynomial by a complex numberthe origin in o is the polynomial identically zeroExample(1)is too simple and example (2)is too complicated to betypical of the main contents of this book. We give now another exampleof complex vector spaces which(as we shall see later)is general enough forall our purposes.3)Let en,n= 1, 2,. be the set of all n-tuples of complex numbers.Ix=(1,…,轨)andy=(m1,…,n) are elements of e, we write,,bdefinitionz+y=〔1+叽,…十物m)0=(0,…,0),-inIt is easy to verify that all parts of our axioms(a),(B), and (C),52, aresatisfied, so that en is a complex vector space; it will be called n-dimenaionalcomplex coordinate space
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水声信号处理基础.pdf
水声信号处理基础内!提本书从统计糖点御述信她理的骞狸论。第一幸概述水产僧号处理的棋型及发展櫶況。第二、三章介绍信号分析、緞性系统和随机过程,是全书的基础部分。第四章到第六章紋迷信号统计处理的一般理论,包括最传线性滤、信号的统计检测和佔计理论初步。最后三章偏重于声纳十的信早处理间题一混喻干下的信号处理、相关接收阵和自适应处理技术。本书可作为水声工程专业有关信号处理户面的教材,或从事这方面研究工作的人员的参考书,也可作为雷,逍信专业在信号统计处题方面射参考书水声营号处理锅我璟陆根游轴萨“管B萨出新华带店北京魔行所发行各地新华书店楚督防工业出版社印倒厂印装78X10924/19张11/4403千字1981年9月第一叔981年9月第一孜印刷印敦:0,00一20肝篮→号15034-2175定价:180元前言木书是按教材的受求端写,以基袖知识和一般理论为主,但也惹虑了水的一些殊的要求。近十年来信号处理技术发展很快,理论上新月异,应用领域上不断扩大,通信、雷达,地震、控制,测量……都离不开信号处理。声纳往往在强干抗背景下工作,信号处飓技术的地位显得相当重耍。因此,这门课是水声工程专业重要主课之五十年代初期以维纳滤波为代表的最佳线性滤被得到迅速发展,五十年代后期,将统计决策理论应用到通信、控制中来,出现了信号检测理论,共十年代估计理论迅速发展,检测理论有点相形失色。卡曼滤波继承发展了维纳滤波,在估计理论中找到了自己的坚实理论基础。在过去教学中,曾一度突出了检测理论,现在看来不一定合。现我们仍旧把最生线性滤波列为专门的一章,但是加进了卡曼滤波的新内容。检测论也是专门的章,没有作更深入的分析,好在内已出版了这方面的专门著作。近来估计理论发展很快,内容很多,这本数材不可能全部包括进来。我们用了一章篇掘介绍估计理论初步,介绍了参量估计,也介绍了非参量佐计。为了学好最伟线性滤波,检测理论,估计理论,打好基础是必要的。尽管前面的课程已经学过信号分析和概率论,我们还是用了两章篇幅研究有规信号分析和随机信号分析,第一从教学角度来看适当重复是必要的;第二重复也不是简单的重复,而是向更深入方向发展,向离数的,多维的向发展,这种发展对后面分析是完全必要的混响干扰背景是水声信号处理的特殊问题,在第七章中作了专门的介绍,但是该章讨论的模糊度函数,信道的滤波器模型……对于水声以外的其他领域也是有用的。用基阵按收和发射信号也是声纳的特点,指关接收阵提高了被动声纳的性能,第八掌中对于这厅面闯题作了较详细的分析。承声信道复杂多变,加之,数字技术的发展,促使自适应技术应用到水声领域中,提高了声纳的性能,第九章中对自适应作了初步的介绍。在分析中要用到许多数学丁具,如积分方程,差分方程,矩阵……,我们认为不应回遍这些工具,但是在本教材中北不宜系统地全面地介绍这些工具,我们力图用一些简单易懂的例子使读者初步掌揸这些工具。木书申稿时,南京工学院水声工程教研组的黄建人、姚治国和周刚临三同志提出了许多宝贵意见,特表示感谢。由于我们的水平有限,错误一定很多,请读者批评指正。编者且录第一聋引论會會會曾會1伽會會身t會■血司■會自·自·會會5.}随机参量信号的检测…………1441.1基本的声纳系统的模型■面司唱■要85.4序列检测5612水声信号处理的发展概况………!习题會■山『■PP酽d普■■1■■b;b画4·4I§1.3主要符号表示………4第六章估计理论初步…………162第二章信号的解析表示和线性系统……5§6.16计的木慨念6§2.1,信号和信号分析……§6.2则叶斯估计……………,………l§2.2谱函数的性展…,…6.3递归线性最小方為佔计§2.线性系统■■q■■■口司■■司口6.4功率谱估计……§2.4釆样定理…………习题…………中“9s2.5离散线性系统4,35第七章混响干扰下的信号处理……202习题……………………………477,1模粉度效看1自中即■■:第三章随机过程…_q号看■唱■§7.2混疃的純计特…………………"235§3.1概率和随机变量9§P3混响于扰下的信号处理3.3随机变量的函数5了习§3.3期螺……第八章相关接收阵33,4随机过程6588.1引言…………225§3.5随机过程的功率…分8.2平方累积阵分析…晶中h2E5§36高斯随机过程■bd备A8§83乘积阵分析……………230习題………………………“…………7s8.4有板性处理的基阵公析………2?第四章最佳线性滤渡…………………y2§8.5最大信噪比滤欲阵的分忻…………2b4.1最佳线性滤波的标准………$_2习题………………………………24§4.2匹配滤戏器的分枥…93第九草岩适应处理技术…"245E43匹配滤波器的实现…………0惡9,1水声信道的滤波器模型84.4继纳滤波……………………19.2自适应概念和MS适应逐算…24684.5卡曼滤波初步……!59,3自遞应波束形成器……习恶…曾曾會P■224§9.4自逅应滤被…第五章信号的统计检测习题…………269§5.1惭述……………)6录有关的儿个矩阵公式n··"‘2§52高斯噪声背景下确知信号§1儿个有关的矩阵求逆公式…………*的检测+-+jFs2矩阵做分7第一章引论§1,1基本的声纳系统的模型声纳系统按用途来分有许多型号,但它的工作方式不外乎主动式和被动式两种,如图1-1-所声波故射体主动式声纳由发射机发射已舶信号,发射积目标通过换能器转换成声波信号,经月标反射后成回波,经过渐水中各式各样的散射⊥式/怀回效+散射回液(响体散射形成混响接收机不但接收到回波接收机和混响,而且还会接收到环境噪声,通常射哗产发声目标包括海洋噪声和本舰噪声。回波是信号境噪声信号其余都是干扰。在简单的检测模型中,回被动式波信号是完仝确知的,它和发射波形图1-1-1甚本的声纳系鸵模型样,只不过是时闻上平移了一下。但实际情况要复杂得多,于目标的运动,会引起多普泐( Doppler)频移,由于目标不是一个点,多径传揹,回波在时间轴上被犷展,波形和发射波形可態会大不相同,此外由于介质的随机性,也会造成回波信号的起伏。在主动声纳中,混响往往是主要的干扰。混响也是由发射信号产生的,它的许多方而的特件和阿波极为相似,使得抗混响相当闲难。本噪芦也是一种亚重的干扰:在本舰高速诺动时,更是如此。但是它是一个宽带信号,回波是一个窄带信号,可以利用功率谱的亲异来风分它们被动声纳是根据日标辐射的声液来检测目标的。这辐射的声波可能是目标上的声纳发射的声波,也可能是目标运动时辐射出的舰船噪声,也可能是目标引起的其它声音。目标辐射出的觑船噪声的功率漕和本舰噪声的功率都是宽带的,从功率谱上区分它们有一定的困难。但是本舰噪声和日标舰噪声在空间相关性上是不同的。可以利用空间相关性上的差异来区分它们。声纳系统的模型反映了声纳、介质和目标三者的关系。介质和目标的特性是我们无法控制的,而声纳,它的发射方式与接收方式是我们可以合理地选择的。使得它最佳地和介质、目标的特件匹配。早期的声纳系统,它的发射方式、接收方式是不变的,而介质和目标的特性却是因时、因地而异的,这就不可能实现最住地匹配。必须改变这种不变的发射方式、接牧方式,要根据当时当的介质、目标的特鉎,自动地调整发射方式、接收方式,以运到最住地匹配。这便是所谓的直适应技术。§1.2水声信号处理的发展概况市纳从它的诞生到现在有了很大的发展和变化。特别是近二十多年,更是如此。声纳的变化和水声信号处理的发展是一致的,或者说水声信号处理每出现一个新技术,都导致声纳的一次大变化。五十年代以前的声纳,发射波形是比较简单的,是一个正弦填充的方波,接收机采用窄带滤波。脉冲压缩我术的出现,导致脉冲压缩声纳逃出现,发射波形釆用线性调掘脉沖,戌者伪随机编码。在接收机中相应地出现了信号处理器(如 DELTIC系统等〕草题的声纳是利用换能器的自然方向性,用机械旋转的方法实现波束的旋转。随着换能器的越来越大、越来越笨重,机槭旋转越来越困难。利用相控阵孜术之后,可以用电的力法形成波束的旋转;还可以用电照方法同时形成多个波束,出现了多液束阵。利用信号有干扰的空间相关性的差别,将信号处理技术应用于基阵,出现了相关接收阵。由于铵收到的信号和干扰变化范围很大,给信号处埋带来困难,便出现了动态范压缩和归一化技术目前,自适应技术及各式各样的数字处理技术迣入了声纳领域。图1-2-1是目前的典型声纳发射机方框图。信号发生器的输出可具有多种形式(模拟的或数字的,正弦填充脉裨或线性调频脉冲,或几种信号同时共用),这取决了所考虑的系统的具体要求。信号发生器的犏出送到波束形成器矩阵去,波束形成器矩阵的用途是对信号进行适当的加权和延迟,以使发射阵产生所要求的声束图,将声能聚集到所要求的空间中去。程序器的用途是使多路或顺序发射同步。程序器信号束发生器形成矩臂发射阵图⊥-2-4声纳发射机户框图图1-2-2是目前的典型声纳接收机方框图。它比发射机复杂。这是由于在发射时信噪比是无限大的(或接近无限大),而在接收时,在大多数有实际意义的情况下,信噪比是小于1的。接收基阵和波束形成矩阵对应于发射机中发射基辉和波束形成矩阵。这两个基阵通常是共用的。但是波卓形成兔阵是有差别的,在发射机中,为了使发射能量最大,波東形成航阵具有最小的幅度加权。在接收机里,则使用幅度加权,抑制旁瓣或增大方向性指数。在发射机中用相控发射,在接收棚中采用自适应波束形成处理。程序器判决设舒一接收F态范信号形成矩阵和归处理器显示器昕觉指示网1-2-2声接收机了框图动态范围压缩和归一化(DRCN)与信号处理器具有共同的任务:对接收的信息进如工,以便适当地将其显示在视觉显示器或听党显示器上,或送给判决设备(可能是一个缴字计算机)。程序器是用来达到同步和自适应的。本课程的任务是介绍信号处理的基本理论及其实现方案。1.3主要符号表示x(t),f(t),g(:)等表示实信号x(扌)∫2(f),ga(t)等表示复数解析信号F()或ⅹ(的)表示实信号X(#)的频谱函数F(①)或X2(①)表示复数解析信号xa(4)的频函数Rx(℃)或R2x()或R(T)表示信号x(t)的自相关函数R2(τ)或R12(T)表示信号x(:)和y(t)的互相关函数H()表示时不变线性系统的频半响应函数h(T)表承时不变线性系统的冲激响应函数H(,t)表示时变线性系统的频率响应函数h(,T)表示时变线性系统的冲激响定函数xn}表示实信号序列Y(z)表示{x》}的z变换hn)表示线性系统刈单仕釆样序列("-k)的晌应h(n)表示线性位移不夾系统对单位采祥序列8(”)的响应正(z)裁示a(B)的之变换尸(A)表示事件A的粥率P(A.B)表不事牛丹出现条供下,惠件A出现的条件率F(a)或F(x)表示随机变量x的概率分布函数p()或P()表小陡机变量x的概率密度函数F()或F(x|y)表示条件概率分布函数Fx(妖}或F(K)衣爪随机向量X的慨率分布咝爨p()或户(x)表示随机向量X的概率密度区数px(夕(XY)表示随机向量的条件概率密度函数E〔x会具表示随机变量x的微学期望或统计平均E(x°)裘示随机变量x的n阶愿点矩E〔(x-μ)“)表示随机变量x的阶中心矩Mx(s)=Ee〕表示随机变量x的特征函数E(X)仝μx裴示数学期望或统计平均中x全FXX表示均方矩阵Vk≌EX-)(X-)2〕表示方差矩阵z(β)会E(XY-)或xy表条件均值vx()会E[(x-x)(X-x)Y=月或4y表示条件方差矩阵Mk(s)表示特征函数x(t,》或x(t)表示一维随机过程样本函数X(t,k或X(!)表示n雏随机过程样本函数EX(t,)!表示X(t,)的均值中x(t)表示X〔f,)的均方矩阵V2(t?表示X(t,k)方差矩阵φx(1,t:)表示X〔e)的相关矩阵vx(t1,f2)表示K(t,四)的协方差矩阵S(o)表示x(f:①)的功率谱第二章信号的解析表示和线性系统§2,1信号和信号分析信号信息传输系统的任务在于向接收者传輸消息。声纳是一种信息传输系统,它所传输的消息是目标的有无,日标的运动要素(距离、方向、形度……),目标的性质等。消息本身往往不便于传输,要將消忘放在某…个便于传输的物理量屮,这种带有消息的物理量叫做信号。在主动声纳中,回波的行无表示目标的有无,回波出现的方向,苌示目标的方向,回波出现的时刻反映了目标的距离,回波中多卜勒频偏反映了目标的相对速度,回波的频谱结构反映了目杯的类型。因此主动声纳中回波是信号。类似跑,日标的辐射噪声是被动声纳的信号。凡是妨碍接收者接收信号的物理量都是干扰。本翹噪声、混响、海洋噪声都是声纳的干扰。应当说明,干桄也带有消总,只不过是接收者不需要的消息。因此同一个勒理量对于这个接收者是信号,对于那一个接收者可能就是干扰。钶如编队航行时,甲觎主动声纳引起的日标回波是乙舰主动声纳的干扰,它会严重妨碍乙舰主动声纳的工作。再如对一个搜索潜艇的芦纳兵而言,本舰噪声是干扰,对一个测量本舰噪声的工作人员而言,本舰噪声是信号。由此可见,信号和干扰的差别仪仪在于接枚者是否需要。因此,研究信号的方法和工具,也适用于干扰。第二章、第三章介绍的内容既适用于信号也适用于干扰。信号的时域分析我们将信号用时间纟的函数表示,记作x(4)。如果时间连续地取值,信号叫做连续时间信号。如果时间f离散地取值,信号叫离散时间信号。在声纳系统中常见的信号有:1.脉冲信号矩形脉冲「A0≤f≤Tx(t)0其余的r它有两个参数A和TA称做脉御幅度,T称倣脉冲班度钟形脉冲x(t)=A·e(2-1-2)它也有两个叁数A和α:A称做脉冲幅虔。它是κ〔扌)的最大值。a反映了脉冲度。钟
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