Appendix A Wirtinger Calculus
|
|
- Julianna Wilkins
- 5 years ago
- Views:
Transcription
1 Precoding and Signal Shaping for Digital Transmission. Robert F. H. Fischer Copyright John Wiley & Sons, Inc. ISBN: Appendix A Wirtinger Calculus T he optimization of system parameters is a very common problem in communications and engineering. For example, the optimal tap weights of an equalizer should be adapted for minimum deviation (e.g., measured by the mean-squared error or the peak distortion) of the output signal from the desired (reference) signal. For an analytical solution, a cost function is set up and the partial derivatives with respect to the adjustable parameters are set to zero. Solving this set of equations results in the desired optimal solution. Often, however, the problem is formulated using complex-valued parameters. In digital communications, signals and systems are preferably treated in the equivalent complex baseband [Fra69, Tre7 1, Pro0 11. For solving such optimization problems, derivation with respect to a complex variable is required. Starting from well-known principles, this Appendix derives a smart and easily remembered calculus, sometimes known as the Wirtinger Calculus [FL88, Rem
2 406 WIRTINGER CALCULUS A.l REAL AND COMPLEX DERIVATIVES First, we consider a real-valued function of a real variable: f : IR 3 x t+ y = f(x) E IR. (A.l.l) The point xopt, for which f(x) is maximum is obtained by taking the derivative of f with respect to x and setting it to zero. For xopt the following equation has to be valid: (A. 1.2) Here we assume f(x) to be continuous in some region R, and the derivative to exist. Whether the solution of the above equation actually gives a minimum or maximum point has to be checked via additional considerations or by inspecting higher-order derivatives. Analogous to real functions, a derivative can be defined for complex functions of a complex variable as well: f: C 3 zt+ w =f(z) E c (A.1.3) (A.1.4) The above limit has to exist for the infinitely many series {zn} which approach ZO, i.e., lim z, = 20. If f (z) exists in a region R C C, the function f(z) is called n+ 03 analytic, holomorphic, or regular in R. In the following, the relations between real and complex derivatives are discussed. A complex function can be decomposed into two real functions, each depending on two real variables x and y, the real and imaginary parts of z: f(z) = f(x +j y) 2 u(x, y) + ju(x, y), z = x +j y. (A.1.5) It can be shown that in order for f(z) to be holomorphic, the component functions u(x, y) and u(x, y) have to meet the Cauchy-Riemann differential equations, which read (e.g., [FL88, Rem891): (A.1.6a) (A. 1.6b) The same considerations are valid for minimization.
3 WlRTlNGER CALCULUS 407 The complex derivative of a holomorphic function f(z) can then be expressed by the partial derivatives of the real functions u(x, y) and ~ (x, y): (A.1.7) The complex derivative of a complex function plays an important role in complex analysis-in communications it has almost no significance. In fact, a more common problem is the optimization of real functions, depending on complex parameters. Complex cost functions are of no interest, because in the field of complex numbers no ordering (relations < and >) is defined and thus minimization or maximization makes no sense. A.2 WlRTlNGER CALCULUS As already stated, we have to treat real functions of one or more complex variables. Thus, let us now consider functions f : Q: 3 z =x+j y c-) w = f(z) = u(zly) E IR. (A.2.1) Since ~ (x, y) = 0 holds (cf. (A.lS)), f(z) generally is not holomorphic. A real function would only be regular if, according to (A. 1.6), - 0 and &@& 8Y = - 0 are valid. But this only holds for a real constant, and hence can be disregarded. The straightforward solution to the optimization of the above function is as follows: Instead of regarding f(z) as a real function of one complex variable, we view f(z) = u(z, y) as a function of two real variables. Thus optimization can be done as for multidimensional real functions. We want to find which requires f(z) -+ opt. E u(x,y) -+ opt., WX,Y) I dx - 0 and ~ du(z,y) I(). 8Y (A.2.2) In order to obtain a more compact representation, both of the above real-valued equations for the optimal components xopt and yopt can be linearly combined into one complex-valued equation: (A.2.3) where, for the moment, a1 and a2 are arbitrary real and nonzero constants. Equations (A.2.2) and (A.2.3) are equivalent (and hence, of course, result in the same solution) because real and imaginary part are orthogonal. As already stated, this procedure is mainly intended to get a compact representation.
4 408 WlRTlNGER CALCULUS Writing real part and imaginary part of z = x + j y as the tuple (z, y), we can define the following differential operator: (A.2.4) This operator can, of course, also be applied to complex functions (A. 1.5). This is reasonable, because real cost functions are often composed of complex components, e.g., f(z) = 1zI2 = z. z* fl(z). f ~(z), with an obvious definition of fl,z(z) E C. Note, z* 2 x - j y denotes the complex conjugate of z = x + j y. The remaining task is to chose suitable constants a1 and u2. The main aim is to obtain a calculus that is easily remembered and easy to apply. As will be shown later, the choice a1 = $ and a2 = -$ meets all requirements. To honor the work of the Austrian mathematician Wilhelrn Wirtinger ( ) who established this differential calculus, we call it Wirtinger Calculus. Definition A. 1 : Wirfinger Calculus The partial derivatives of a (complex) function f (z) of a complex variable z = 2 + j y E C, 5, y E R, with respect to z and z*, respectively, are defined as: and - af a 1. a.f dz --,-.a.f - 2 (ax By) (A.2.5) (A.2.6) A.2.1 Examples We now study some important examples. First, let f(z) = cz, where c 6 C is a constant. Derivation of f(z) yields and + j "(>;J ") = 1 (c + j (j c)) = 0. (A.2.8) az* 2 2 Similarly, for f (z) = cz*, we arrive at: az 2 dx -j -Jy)) = (c- j (-j c)) = 0, ay (A.2.9)
5 WlRTlNGER CALCULUS 409 and +jdc(xd~jy)) = -(c+j(-jc)) 1 =c. (A.2.10) dz* 2 dx 2 Next, we consider the function f = zz* = 1zI2 = x2 + y2. Here the derivatives read: a a 1 d(x2+ y2) - + y2)) = (22- j2y) = zz* z*, (A.2.11) = - ( -j and az 2 dx d 1 d(x2+y2) - zz* = - ( dz* 2 dx ay +j dy To summarize, the correspondences in Table A.l are valid. y2)) = f (2x + j 2y) = z. (A.2.12) Table A. I Wirtinger derivatives of some important functions CZ CZ* ZZ* C 0 0 c z* z Note that using the Wirtinger Calculus differentiation is formally done as with real functions. Moreover, and somewhat unexpected, z* is formally considered as a constant when derivating with respect to z and vice versa. It is also easy to show that the sum, product, and quotient rules still hold. For example, given f(z) = fl(z). f2(z), we obtain a -f1(z).f2(z) = az - j afl(.)f2(z) - j j l(z)~) dy (A.2.13)
6 410 WIRTINGER CALCULUS Finally, for f(z) = h(g(z)) 5 h(w), g : C ++ C, the following chain rules hold [FL88, Rem891: A.2.2 Discussion The Wirtinger derivative can be considered to lie inbetween the real derivative of a real function and the complex derivative of a complex function. Rewriting (A.2.5) and (A.2.6), we arrive at: af- = o dz* (A.2.15a) (A.2.15 b) On the one hand, equation (A.2.15a) states that for holomorphic functions the Wirtinger derivative with respect to z agrees with the ordinary derivativeof a complex function (cf. (A.l.7)). On the other hand, (A.2.15b) can be interpreted in the way that holomorphic functions do not formally depend on z*. Contrary to the usual complex derivative, the Wirtinger derivative exists for all functions, in particular nonholomorphic ones, such as real functions. Since both operators and & are merely a compact notation incorporating two real differential quotients, they can be applied to arbitrary functions of complex variables. For nonholomorphic functions, $ # 0 usually holds, and thus either the derivative with respect to z or z* can be used for optimization. The actual cost functions determines
7 GRADIENTS which one is more advantageous; if quadratic forms are considered, the operator is preferable. To summarize, it should again be emphasized, that, because of its compact notation, Wirtinger Calculus is very well suited for optimization in engineering. It circumvents a separate inspection of real part and imaginary part of the cost function. Because of the simple arithmetic rules-mostly it can be calculated as known from real functions-the Wirtinger Calculus is very clear. A.3 GRADIENTS For the majority of applications the cost function does not only depend on one, but on many variables, e.g., we have f : C" 3 z = [zl, z2,..., znit c-) w = f(z) E IR. For optimization, all n partial derivatives with respect to the complex variables z1 = jy1 through z, = 2, + jy, have to be calculated. Usually, these derivatives are again combined into a vector, the so called gradient: A which, in the optimum, has to equal the zero vector 0 = [0, 0,..., 0IT. Wirtinger Calculus is especially well suited for such multidimensional functions, because here only with a great effort can the real part and the imaginary part be separated and inspected independently. Using the above definitions of the partial derivatives ((A.2.5) and (A.2.6)), we arrive at simple arithmetic rules, now expressed using vectors and matrices. A.3.1 Examples We now again study some important examples. First, let f(z) = ctz = c:=l c,z, or n f(r) = ctz* = c,=l czz:, respectively, with c = [c~, c2,..., c,it and c, constant. It is easy to prove the following properties for gradients: -' T c z=c, ' T -c z*=o, az 8% L T az* c z=o, Z c T z*=c. az* (A.3.2) *Here we use column vectors, but the same considerations also apply to row vectors.
8 4 f 2 WlRTfNGER CALCULUS Finally, considering the quadratic form f (z) = zhmz, where M is a constant n x n matrix, derivation results in: d T - zhmz = (zhm) 8% az* and for f(z) = zhz we arrive at: -' z H z=z*, 8% a - zhmz = MZ d H z z=z. dz* To summarize, the correspondences in Table A.2 are valid. (A.3.3) (A.3.4) Table A.2 Wirtinger derivatives (gradients) of some important functions. A.3.2 Discussion The gradient with respect to the Wirtinger derivatives, is related to the gradient (A.3.5) which is frequently used, e.g., in [Hay961 by or since f(z) is real-valued, af (z) (Vf(Z))* = 2-. 8% (A.3.7) The first disadvantage of definition (A.3.6) compared to Wirtinger Calculus is that an undesired factor of 2 occurs. In particular, if the chain rule is applied 1 times, the result is artificially multiplied by 2l. Second, the calculus is not very elegant, because the gradient of, e.g., f (z) = ctz is V(cTz) = 0, but that of f(z) = ctz* calculates to V(cTz*) = 2%. Hence, because of its much clearer arithmetic rules, we exclusively apply the Wirtinger Calculus.
9 GRADIENTS 413 REFERENCES [FL881 [Fra69] [Hay961 [Pro011 [Rem89] [Tre711 W. Fischer and 1. Lieb. Funktionentheorie. Vieweg-Verlag, Braunschweig, Germany, (In German.) L. E. Franks. Signal Theory. Prentice-Hall, Inc., Englewood Cliffs, NJ, S. Haykin. Adaptive Filter Theory. Prentice-Hall, Inc., Englewood Cliffs, NJ, 3rd edition, J. G. Proakis. Digital Communications. McGraw-Hill, New York, 4th edition, R. Remmert. Funktionentheorie 1. Springer Verlag, Berlin, Heidelberg, (In German.) H. L. van Trees. Detection, Estimation, and Modulation Theory-Part 111: Radar-Sonar Signal Processing and Gaussian Signals in Noise. John Wiley & Sons, Inc., New York, 1971.
(z 0 ) = lim. = lim. = f. Similarly along a vertical line, we fix x = x 0 and vary y. Setting z = x 0 + iy, we get. = lim. = i f
. Holomorphic Harmonic Functions Basic notation. Considering C as R, with coordinates x y, z = x + iy denotes the stard complex coordinate, in the usual way. Definition.1. Let f : U C be a complex valued
More information2. Complex Analytic Functions
2. Complex Analytic Functions John Douglas Moore July 6, 2011 Recall that if A and B are sets, a function f : A B is a rule which assigns to each element a A a unique element f(a) B. In this course, we
More informationMath Homework 2
Math 73 Homework Due: September 8, 6 Suppose that f is holomorphic in a region Ω, ie an open connected set Prove that in any of the following cases (a) R(f) is constant; (b) I(f) is constant; (c) f is
More information1 Introduction. 1.1 Introduction to the Book
1 Introduction 1.1 Introduction to the Book To solve increasingly complicated open research problems, it is crucial to develop useful mathematical tools. Often, the task of a researcher or an engineer
More informationSPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS
SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly
More information26.2. Cauchy-Riemann Equations and Conformal Mapping. Introduction. Prerequisites. Learning Outcomes
Cauchy-Riemann Equations and Conformal Mapping 26.2 Introduction In this Section we consider two important features of complex functions. The Cauchy-Riemann equations provide a necessary and sufficient
More informationR- and C-Differentiability
Lecture 2 R- and C-Differentiability Let z = x + iy = (x,y ) be a point in C and f a function defined on a neighbourhood of z (e.g., on an open disk (z,r) for some r > ) with values in C. Write f (z) =
More informationMath 185 Homework Exercises II
Math 185 Homework Exercises II Instructor: Andrés E. Caicedo Due: July 10, 2002 1. Verify that if f H(Ω) C 2 (Ω) is never zero, then ln f is harmonic in Ω. 2. Let f = u+iv H(Ω) C 2 (Ω). Let p 2 be an integer.
More informationQ You mentioned that in complex analysis we study analytic functions, or, in another name, holomorphic functions. Pray tell me, what are they?
COMPLEX ANALYSIS PART 2: ANALYTIC FUNCTIONS Q You mentioned that in complex analysis we study analytic functions, or, in another name, holomorphic functions. Pray tell me, what are they? A There are many
More informationMATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic.
MATH 45 SAMPLE 3 SOLUTIONS May 3, 06. (0 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. Because f is holomorphic, u and v satisfy the Cauchy-Riemann equations:
More informationMath 61CM - Solutions to homework 2
Math 61CM - Solutions to homework 2 Cédric De Groote October 5 th, 2018 Problem 1: Let V be the vector space of polynomials of degree at most 5, with coefficients in a field F Let U be the subspace of
More informationSynopsis of Complex Analysis. Ryan D. Reece
Synopsis of Complex Analysis Ryan D. Reece December 7, 2006 Chapter Complex Numbers. The Parts of a Complex Number A complex number, z, is an ordered pair of real numbers similar to the points in the real
More informationAPPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2.
APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the n-dimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product
More informationLecture Notes on Metric Spaces
Lecture Notes on Metric Spaces Math 117: Summer 2007 John Douglas Moore Our goal of these notes is to explain a few facts regarding metric spaces not included in the first few chapters of the text [1],
More informationf (n) (z 0 ) Theorem [Morera s Theorem] Suppose f is continuous on a domain U, and satisfies that for any closed curve γ in U, γ
Remarks. 1. So far we have seen that holomorphic is equivalent to analytic. Thus, if f is complex differentiable in an open set, then it is infinitely many times complex differentiable in that set. This
More information22.3. Repeated Eigenvalues and Symmetric Matrices. Introduction. Prerequisites. Learning Outcomes
Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one
More informationSOLUTION GUIDE TO MATH GRE FORM GR9367
SOLUTION GUIDE TO MATH GRE FORM GR9367 IAN COLEY The questions for this solution guide can be found here. Solution. (D) We have f(g(x)) = g(x) + 3 = 5 for all x, so g(x) = for all x R. Solution. (C) We
More informationHere are brief notes about topics covered in class on complex numbers, focusing on what is not covered in the textbook.
Phys374, Spring 2008, Prof. Ted Jacobson Department of Physics, University of Maryland Complex numbers version 5/21/08 Here are brief notes about topics covered in class on complex numbers, focusing on
More information2 Complex Functions and the Cauchy-Riemann Equations
2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)
More informationf (x) dx = F (b) F (a), where F is any function whose derivative is
Chapter 7 Riemann Integration 7.1 Introduction The notion of integral calculus is closely related to the notion of area. The earliest evidence of integral calculus can be found in the works of Greek geometers
More informationSOLUTION OF EQUATIONS BY MATRIX METHODS
APPENDIX B SOLUTION OF EQUATIONS BY MATRIX METHODS B.1 INTRODUCTION As stated in Appendix A, an advantage offered by matrix algebra is its adaptability to computer use. Using matrix algebra, large systems
More informationTheorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r
2. A harmonic conjugate always exists locally: if u is a harmonic function in an open set U, then for any disk D(z 0, r) U, there is f, which is analytic in D(z 0, r) and satisfies that Re f u. Since such
More informationMathematical Constraint on Functions with Continuous Second Partial Derivatives
1 Mathematical Constraint on Functions with Continuous Second Partial Derivatives J.D. Franson Physics Department, University of Maryland, Baltimore County, Baltimore, MD 15 Abstract A new integral identity
More informationwhich are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar.
It follows that S is linearly dependent since the equation is satisfied by which are not all zero. The proof in the case where some vector other than combination of the other vectors in S is similar. is
More informationTHEORY OF MIMO BIORTHOGONAL PARTNERS AND THEIR APPLICATION IN CHANNEL EQUALIZATION. Bojan Vrcelj and P. P. Vaidyanathan
THEORY OF MIMO BIORTHOGONAL PARTNERS AND THEIR APPLICATION IN CHANNEL EQUALIZATION Bojan Vrcelj and P P Vaidyanathan Dept of Electrical Engr 136-93, Caltech, Pasadena, CA 91125, USA E-mail: bojan@systemscaltechedu,
More informationThe Complex Gradient Operator and the CR-Calculus
arxiv:0906.4835v1 [math.oc] 26 Jun 2009 1 Introduction The Complex Gradient Operator and the CR-Calculus June 25, 2009 Ken Kreutz Delgado Electrical and Computer Engineering Jacobs School of Engineering
More informationABEL S THEOREM BEN DRIBUS
ABEL S THEOREM BEN DRIBUS Abstract. Abel s Theorem is a classical result in the theory of Riemann surfaces. Important in its own right, Abel s Theorem and related ideas generalize to shed light on subjects
More informationMA22S3 Summary Sheet: Ordinary Differential Equations
MA22S3 Summary Sheet: Ordinary Differential Equations December 14, 2017 Kreyszig s textbook is a suitable guide for this part of the module. Contents 1 Terminology 1 2 First order separable 2 2.1 Separable
More informationA = 3 B = A 1 1 matrix is the same as a number or scalar, 3 = [3].
Appendix : A Very Brief Linear ALgebra Review Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics Very often in this course we study the shapes
More informationSome Notes on Linear Algebra
Some Notes on Linear Algebra prepared for a first course in differential equations Thomas L Scofield Department of Mathematics and Statistics Calvin College 1998 1 The purpose of these notes is to present
More informationChapter 13: Complex Numbers
Sections 13.3 & 13.4 1. A (single-valued) function f of a complex variable z is such that for every z in the domain of definition D of f, there is a unique complex number w such that w = f (z). The real
More informationChapter 1. Complex Numbers. Dr. Pulak Sahoo
Chapter 1 Complex Numbers BY Dr. Pulak Sahoo Assistant Professor Department of Mathematics University Of Kalyani West Bengal, India E-mail : sahoopulak1@gmail.com 1 Module-1: Basic Ideas 1 Introduction
More informationAccumulation constants of iterated function systems with Bloch target domains
Accumulation constants of iterated function systems with Bloch target domains September 29, 2005 1 Introduction Linda Keen and Nikola Lakic 1 Suppose that we are given a random sequence of holomorphic
More informationx 1 x 2. x 1, x 2,..., x n R. x n
WEEK In general terms, our aim in this first part of the course is to use vector space theory to study the geometry of Euclidean space A good knowledge of the subject matter of the Matrix Applications
More informationSyllabus For II nd Semester Courses in MATHEMATICS
St. Xavier s College Autonomous Mumbai Syllabus For II nd Semester Courses in MATHEMATICS Contents: (November 2016 onwards) Theory Syllabus for Courses: S.MAT.2.01 : Calculus II. S.MAT.2.02 : Linear Algebra.
More informationChapter 2. Error Correcting Codes. 2.1 Basic Notions
Chapter 2 Error Correcting Codes The identification number schemes we discussed in the previous chapter give us the ability to determine if an error has been made in recording or transmitting information.
More informationF (z) =f(z). f(z) = a n (z z 0 ) n. F (z) = a n (z z 0 ) n
6 Chapter 2. CAUCHY S THEOREM AND ITS APPLICATIONS Theorem 5.6 (Schwarz reflection principle) Suppose that f is a holomorphic function in Ω + that extends continuously to I and such that f is real-valued
More informationOR MSc Maths Revision Course
OR MSc Maths Revision Course Tom Byrne School of Mathematics University of Edinburgh t.m.byrne@sms.ed.ac.uk 15 September 2017 General Information Today JCMB Lecture Theatre A, 09:30-12:30 Mathematics revision
More informationCHAPTER 2. CONFORMAL MAPPINGS 58
CHAPTER 2. CONFORMAL MAPPINGS 58 We prove that a strong form of converse of the above statement also holds. Please note we could apply the Theorem 1.11.3 to prove the theorem. But we prefer to apply the
More informationQuasi-conformal maps and Beltrami equation
Chapter 7 Quasi-conformal maps and Beltrami equation 7. Linear distortion Assume that f(x + iy) =u(x + iy)+iv(x + iy) be a (real) linear map from C C that is orientation preserving. Let z = x + iy and
More informationChapter 2 Wiener Filtering
Chapter 2 Wiener Filtering Abstract Before moving to the actual adaptive filtering problem, we need to solve the optimum linear filtering problem (particularly, in the mean-square-error sense). We start
More informationEstimates for probabilities of independent events and infinite series
Estimates for probabilities of independent events and infinite series Jürgen Grahl and Shahar evo September 9, 06 arxiv:609.0894v [math.pr] 8 Sep 06 Abstract This paper deals with finite or infinite sequences
More informationOn Cauchy s theorem and Green s theorem
MA 525 On Cauchy s theorem and Green s theorem 1. Introduction No doubt the most important result in this course is Cauchy s theorem. There are many ways to formulate it, but the most simple, direct and
More informationLecture 1: Systems of linear equations and their solutions
Lecture 1: Systems of linear equations and their solutions Course overview Topics to be covered this semester: Systems of linear equations and Gaussian elimination: Solving linear equations and applications
More informationJUST THE MATHS UNIT NUMBER 6.1. COMPLEX NUMBERS 1 (Definitions and algebra) A.J.Hobson
JUST THE MATHS UNIT NUMBER 6.1 COMPLEX NUMBERS 1 (Definitions and algebra) by A.J.Hobson 6.1.1 The definition of a complex number 6.1.2 The algebra of complex numbers 6.1.3 Exercises 6.1.4 Answers to exercises
More informationRepeated Eigenvalues and Symmetric Matrices
Repeated Eigenvalues and Symmetric Matrices. Introduction In this Section we further develop the theory of eigenvalues and eigenvectors in two distinct directions. Firstly we look at matrices where one
More informationMath Homework 1. The homework consists mostly of a selection of problems from the suggested books. 1 ± i ) 2 = 1, 2.
Math 70300 Homework 1 September 1, 006 The homework consists mostly of a selection of problems from the suggested books. 1. (a) Find the value of (1 + i) n + (1 i) n for every n N. We will use the polar
More informationAdvanced Digital Signal Processing -Introduction
Advanced Digital Signal Processing -Introduction LECTURE-2 1 AP9211- ADVANCED DIGITAL SIGNAL PROCESSING UNIT I DISCRETE RANDOM SIGNAL PROCESSING Discrete Random Processes- Ensemble Averages, Stationary
More informationCompression on the digital unit sphere
16th Conference on Applied Mathematics, Univ. of Central Oklahoma, Electronic Journal of Differential Equations, Conf. 07, 001, pp. 1 4. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
More informationSIMON FRASER UNIVERSITY School of Engineering Science
SIMON FRASER UNIVERSITY School of Engineering Science Course Outline ENSC 810-3 Digital Signal Processing Calendar Description This course covers advanced digital signal processing techniques. The main
More informationNORTH MAHARASHTRA UNIVERSITY JALGAON.
NORTH MAHARASHTRA UNIVERSITY JALGAON. Syllabus for S.Y.B.Sc. (Mathematics) With effect from June 013. (Semester system). The pattern of examination of theory papers is semester system. Each theory course
More informationMODULE 8 Topics: Null space, range, column space, row space and rank of a matrix
MODULE 8 Topics: Null space, range, column space, row space and rank of a matrix Definition: Let L : V 1 V 2 be a linear operator. The null space N (L) of L is the subspace of V 1 defined by N (L) = {x
More informationCongruent Numbers, Elliptic Curves, and Elliptic Functions
Congruent Numbers, Elliptic Curves, and Elliptic Functions Seongjin Cho (Josh) June 6, 203 Contents Introduction 2 2 Congruent Numbers 2 2. A certain cubic equation..................... 4 3 Congruent Numbers
More informationSolutions to Complex Analysis Prelims Ben Strasser
Solutions to Complex Analysis Prelims Ben Strasser In preparation for the complex analysis prelim, I typed up solutions to some old exams. This document includes complete solutions to both exams in 23,
More informationA RAPID INTRODUCTION TO COMPLEX ANALYSIS
A RAPID INTRODUCTION TO COMPLEX ANALYSIS AKHIL MATHEW ABSTRACT. These notes give a rapid introduction to some of the basic results in complex analysis, assuming familiarity from the reader with Stokes
More informationVector calculus background
Vector calculus background Jiří Lebl January 18, 2017 This class is really the vector calculus that you haven t really gotten to in Calc III. Let us start with a very quick review of the concepts from
More informationSYLLABUS UNDER AUTONOMY MATHEMATICS
SYLLABUS UNDER AUTONOMY SEMESTER III Calculus and Analysis MATHEMATICS COURSE: A.MAT.3.01 [45 LECTURES] LEARNING OBJECTIVES : To learn about i) lub axiom of R and its consequences ii) Convergence of sequences
More informationA f = A f (x)dx, 55 M F ds = M F,T ds, 204 M F N dv n 1, 199 !, 197. M M F,N ds = M F ds, 199 (Δ,')! = '(Δ)!, 187
References 1. T.M. Apostol; Mathematical Analysis, 2nd edition, Addison-Wesley Publishing Co., Reading, Mass. London Don Mills, Ont., 1974. 2. T.M. Apostol; Calculus Vol. 2: Multi-variable Calculus and
More informationMATH MIDTERM 1 SOLUTION. 1. (5 points) Determine whether the following statements are true of false, no justification is required.
MATH 185-4 MIDTERM 1 SOLUTION 1. (5 points Determine whether the following statements are true of false, no justification is required. (1 (1pointTheprincipalbranchoflogarithmfunctionf(z = Logz iscontinuous
More informationTotal Variation Image Edge Detection
Total Variation Image Edge Detection PETER NDAJAH Graduate School of Science and Technology, Niigata University, 8050, Ikarashi 2-no-cho, Nishi-ku, Niigata, 950-28, JAPAN ndajah@telecom0.eng.niigata-u.ac.jp
More informationAppendix C. Modal Analysis of a Uniform Cantilever with a Tip Mass. C.1 Transverse Vibrations. Boundary-Value Problem
Appendix C Modal Analysis of a Uniform Cantilever with a Tip Mass C.1 Transverse Vibrations The following analytical modal analysis is given for the linear transverse vibrations of an undamped Euler Bernoulli
More informationWeek 4: Differentiation for Functions of Several Variables
Week 4: Differentiation for Functions of Several Variables Introduction A functions of several variables f : U R n R is a rule that assigns a real number to each point in U, a subset of R n, For the next
More informationAnnouncements Wednesday, November 7
Announcements Wednesday, November 7 The third midterm is on Friday, November 16 That is one week from this Friday The exam covers 45, 51, 52 53, 61, 62, 64, 65 (through today s material) WeBWorK 61, 62
More informationArticle (peer-reviewed)
Title Author(s) Influence of noise intensity on the spectrum of an oscillator Swain, Rabi Sankar; Gleeson, James P.; Kennedy, Michael Peter Publication date 2005-11 Original citation Type of publication
More information01 Harmonic Oscillations
Utah State University DigitalCommons@USU Foundations of Wave Phenomena Library Digital Monographs 8-2014 01 Harmonic Oscillations Charles G. Torre Department of Physics, Utah State University, Charles.Torre@usu.edu
More informationSome of the different forms of a signal, obtained by transformations, are shown in the figure. jwt e z. jwt z e
Transform methods Some of the different forms of a signal, obtained by transformations, are shown in the figure. X(s) X(t) L - L F - F jw s s jw X(jw) X*(t) F - F X*(jw) jwt e z jwt z e X(nT) Z - Z X(z)
More information21.4. Engineering Applications of z-transforms. Introduction. Prerequisites. Learning Outcomes
Engineering Applications of z-transforms 21.4 Introduction In this Section we shall apply the basic theory of z-transforms to help us to obtain the response or output sequence for a discrete system. This
More informationA VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 2010
A VERY BRIEF LINEAR ALGEBRA REVIEW for MAP 5485 Introduction to Mathematical Biophysics Fall 00 Introduction Linear Algebra, also known as matrix theory, is an important element of all branches of mathematics
More informationNOTES ON MATRICES OF FULL COLUMN (ROW) RANK. Shayle R. Searle ABSTRACT
NOTES ON MATRICES OF FULL COLUMN (ROW) RANK Shayle R. Searle Biometrics Unit, Cornell University, Ithaca, N.Y. 14853 BU-1361-M August 1996 ABSTRACT A useful left (right) inverse of a full column (row)
More informationAppendix A: Matrices
Appendix A: Matrices A matrix is a rectangular array of numbers Such arrays have rows and columns The numbers of rows and columns are referred to as the dimensions of a matrix A matrix with, say, 5 rows
More information1 Fourier Transformation.
Fourier Transformation. Before stating the inversion theorem for the Fourier transformation on L 2 (R ) recall that this is the space of Lebesgue measurable functions whose absolute value is square integrable.
More informationHow to Use Calculus Like a Physicist
How to Use Calculus Like a Physicist Physics A300 Fall 2004 The purpose of these notes is to make contact between the abstract descriptions you may have seen in your calculus classes and the applications
More informationMath 126: Course Summary
Math 126: Course Summary Rich Schwartz August 19, 2009 General Information: Math 126 is a course on complex analysis. You might say that complex analysis is the study of what happens when you combine calculus
More informationMath 52: Course Summary
Math 52: Course Summary Rich Schwartz September 2, 2009 General Information: Math 52 is a first course in linear algebra. It is a transition between the lower level calculus courses and the upper level
More informationB Elements of Complex Analysis
Fourier Transform Methods in Finance By Umberto Cherubini Giovanni Della Lunga Sabrina Mulinacci Pietro Rossi Copyright 21 John Wiley & Sons Ltd B Elements of Complex Analysis B.1 COMPLEX NUMBERS The purpose
More informationComplex Variables. Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit.
Instructions Solve any eight of the following ten problems. Explain your reasoning in complete sentences to maximize credit. 1. The TI-89 calculator says, reasonably enough, that x 1) 1/3 1 ) 3 = 8. lim
More informationUNCERTAINTY PRINCIPLES FOR THE FOCK SPACE
UNCERTAINTY PRINCIPLES FOR THE FOCK SPACE KEHE ZHU ABSTRACT. We prove several versions of the uncertainty principle for the Fock space F 2 in the complex plane. In particular, for any unit vector f in
More informationLinear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations.
POLI 7 - Mathematical and Statistical Foundations Prof S Saiegh Fall Lecture Notes - Class 4 October 4, Linear Algebra The analysis of many models in the social sciences reduces to the study of systems
More informationLS.2 Homogeneous Linear Systems with Constant Coefficients
LS2 Homogeneous Linear Systems with Constant Coefficients Using matrices to solve linear systems The naive way to solve a linear system of ODE s with constant coefficients is by eliminating variables,
More informationODEs and Redefining the Concept of Elementary Functions
ODEs and Redefining the Concept of Elementary Functions Alexander Gofen The Smith-Kettlewell Eye Research Institute, 2318 Fillmore St., San Francisco, CA 94102, USA galex@ski.org, www.ski.org/gofen Abstract.
More informationCIRCUIT ANALYSIS TECHNIQUES
APPENDI B CIRCUIT ANALSIS TECHNIQUES The following methods can be used to combine impedances to simplify the topology of an electric circuit. Also, formulae are given for voltage and current division across/through
More informationTutorials in Optimization. Richard Socher
Tutorials in Optimization Richard Socher July 20, 2008 CONTENTS 1 Contents 1 Linear Algebra: Bilinear Form - A Simple Optimization Problem 2 1.1 Definitions........................................ 2 1.2
More informationLectures. Variance-based sensitivity analysis in the presence of correlated input variables. Thomas Most. Source:
Lectures Variance-based sensitivity analysis in the presence of correlated input variables Thomas Most Source: www.dynardo.de/en/library Variance-based sensitivity analysis in the presence of correlated
More informationIII.2. Analytic Functions
III.2. Analytic Functions 1 III.2. Analytic Functions Recall. When you hear analytic function, think power series representation! Definition. If G is an open set in C and f : G C, then f is differentiable
More informationMatrix Operations and Equations
C H A P T ER Matrix Operations and Equations 200 Carnegie Learning, Inc. Shoe stores stock various sizes and widths of each style to accommodate buyers with different shaped feet. You will use matrix operations
More information5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns
5.7 Cramer's Rule 1. Using Determinants to Solve Systems Assumes the system of two equations in two unknowns (1) possesses the solution and provided that.. The numerators and denominators are recognized
More informationLinear Algebra: Matrix Eigenvalue Problems
CHAPTER8 Linear Algebra: Matrix Eigenvalue Problems Chapter 8 p1 A matrix eigenvalue problem considers the vector equation (1) Ax = λx. 8.0 Linear Algebra: Matrix Eigenvalue Problems Here A is a given
More informationCITY UNIVERSITY LONDON. BEng (Hons) in Electrical and Electronic Engineering PART 2 EXAMINATION. ENGINEERING MATHEMATICS 2 (resit) EX2003
No: CITY UNIVERSITY LONDON BEng (Hons) in Electrical and Electronic Engineering PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2003 Date: August 2004 Time: 3 hours Attempt Five out of EIGHT questions
More informationWilliam Stallings Copyright 2010
A PPENDIX E B ASIC C ONCEPTS FROM L INEAR A LGEBRA William Stallings Copyright 2010 E.1 OPERATIONS ON VECTORS AND MATRICES...2 Arithmetic...2 Determinants...4 Inverse of a Matrix...5 E.2 LINEAR ALGEBRA
More informationLecture 7 - Separable Equations
Lecture 7 - Separable Equations Separable equations is a very special type of differential equations where you can separate the terms involving only y on one side of the equation and terms involving only
More information송석호 ( 물리학과 )
http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Introduction to Electrodynamics, David J. Griffiths Review: 1. Vector analysis 2. Electrostatics 3. Special techniques 4. Electric fields in mater 5. Magnetostatics
More informationVector analysis and vector identities by means of cartesian tensors
Vector analysis and vector identities by means of cartesian tensors Kenneth H. Carpenter August 29, 2001 1 The cartesian tensor concept 1.1 Introduction The cartesian tensor approach to vector analysis
More informationMTH 215: Introduction to Linear Algebra
MTH 215: Introduction to Linear Algebra Lecture 5 Jonathan A. Chávez Casillas 1 1 University of Rhode Island Department of Mathematics September 20, 2017 1 LU Factorization 2 3 4 Triangular Matrices Definition
More informationn=0 ( 1)n /(n + 1) converges, but not
Math 07H Topics for the third exam (and beyond) (Technically, everything covered on the first two exams plus...) Absolute convergence and alternating series A series a n converges absolutely if a n converges.
More informationOn Information Maximization and Blind Signal Deconvolution
On Information Maximization and Blind Signal Deconvolution A Röbel Technical University of Berlin, Institute of Communication Sciences email: roebel@kgwtu-berlinde Abstract: In the following paper we investigate
More informationLINEAR SYSTEMS AND MATRICES
CHAPTER 3 LINEAR SYSTEMS AND MATRICES SECTION 3. INTRODUCTION TO LINEAR SYSTEMS This initial section takes account of the fact that some students remember only hazily the method of elimination for and
More information. D Matrix Calculus D 1
D Matrix Calculus D 1 Appendix D: MATRIX CALCULUS D 2 In this Appendix we collect some useful formulas of matrix calculus that often appear in finite element derivations D1 THE DERIVATIVES OF VECTOR FUNCTIONS
More informationLecture 7. Econ August 18
Lecture 7 Econ 2001 2015 August 18 Lecture 7 Outline First, the theorem of the maximum, an amazing result about continuity in optimization problems. Then, we start linear algebra, mostly looking at familiar
More informationis a new metric on X, for reference, see [1, 3, 6]. Since x 1+x
THE TEACHING OF MATHEMATICS 016, Vol. XIX, No., pp. 68 75 STRICT MONOTONICITY OF NONNEGATIVE STRICTLY CONCAVE FUNCTION VANISHING AT THE ORIGIN Yuanhong Zhi Abstract. In this paper we prove that every nonnegative
More informationPolynomial Solutions of the Laguerre Equation and Other Differential Equations Near a Singular
Polynomial Solutions of the Laguerre Equation and Other Differential Equations Near a Singular Point Abstract Lawrence E. Levine Ray Maleh Department of Mathematical Sciences Stevens Institute of Technology
More information