Polynomial Multiplication and Fast Fourier Transform
|
|
- Claud Wilkinson
- 6 years ago
- Views:
Transcription
1 Polyomial Multiplicatio ad Fast Fourier Trasform Com S 477/577 Notes Ya-Bi Jia Sep 19, 2017 I this lecture we will describe the famous algorithm of fast Fourier trasform FFT, which has revolutioized digital sigal processig ad i may ways chaged our life It was listed by the Sciece magazie as oe of the te greatest algorithms i the 20th cetury Here we will lear FFT i the cotext of polyomial multiplicatio, ad later o ito the semester reveal its coectio to Fourier trasform Suppose we are give two polyomials: Their product is defied by where px = a 0 +a 1 x+ +a 1 x 1, qx = b 0 +b 1 x+ +b 1 x 1 px qx = c 0 +c 1 x+ c 2 2 x 2 2 c i = max{0,i 1} k mi{i, 1} a k b i k I computig the product polyomial, every a i is multiplied with every b j, for 0 i,j 1 So there are at most 2 multiplicatios, give that some of the coefficiets may be zero Obtaiig every c i ivolves oe fewer additios tha multiplicatios Sothere are at most 2 2+1additios ivolved I short, the umber of arithmetic operatios is O 2 This is hardly efficiet But ca we obtai the product more efficietly? The aswer is yes, by the use of a well-kow method called fast Fourier trasform, or simply, FFT 1 Discrete Fourier Trasform Let us start with itroducig the discrete Fourier trasform DFT problem Deote by ω a th complex root of 1, that is, ω = e i2π, where i 2 = 1 DFT is the mappig betwee two vectors: a 0 â 0 a 1 a = â = â 1 a 1 â 1 1
2 such that 1 â j = a k ω jk, k=0 It ca also be writte as a matrix equatio: ω ω 2 ω 1 1 ω 1 ω 2 1 ω 12 j = 0,, 1 a 0 a 1 a 1 The matrix above is a Vadermode matrix ad deoted by V Essetially, DFT evaluates the polyomial px = a 0 +a 1 x+ +a 1 x 1 = â 0 â 1 â 1 at poits ω 0,ω1,,ω 1 ; i other words, â k = pω k for 0 k 1 From ow o we assume that is a power of 2 If ot, we ca always add i higher order terms with zero coefficiets a = a +1 = = a 2 log 2 1 = 0 The powers of ω are illustrated i the complex plae i the followig figure yi ω /4+1 i ω /4 1 ω /2 1 ω 1 ω /2+1 1 ω 1 x ω 3/4 1 i ω 3/4+1 The fast Fourier trasform algorithm cleverly makes use of the followig properties about ω : ω = 1, ω +k = ω k, ω 2 = 1, ω 2 +k = ω k 2
3 It uses a divide-ad-coquer strategy More specifically, it divides px ito two polyomials p 0 x ad p 1 x, both of degree 2 1; amely, Hece p 0 x = a 0 +a 2 x+ +a 2 x 2 1, p 1 x = a 1 +a 3 x+ +a 1 x 2 1 I this way the problem of evaluatig px at ω 0,,ω 1 1 evaluatig p 0 x ad p 1 x at ω 0 2, ω 1 2,, ω 1 2, 2 combiig the resultig accordig to 1 px = p 0 x 2 +xp 1 x 2 1 breaks dow ito two steps: Note that the list ω 02, ω 12,, ω 1 2 cosists of oly 2 complex roots of uity, ie, ω,ω 0,,ω 2 2 So the subproblems of evaluatig p 0 x ad p 1 x have exactly the same form as the origial problem of evaluatig px, oly at half the size This decompositio forms the basis for the recursive FFT algorithm preseted below Recursive-DFTa, 1 if = 1 2 the retur a 3 w e i2π 4 w 1 5 a [0] a 0,a 2,,a 2 6 a [1] a 1,a 3,,a 1 7 â [0] Recursive-DFTa [0], 2 8 â [1] Recursive-DFTa [1], 2 9 for k = 0 to 2 1 do 10 â k â [0] k +ωâ[1] k 11 â k+ â[0] 2 k ωâ[1] k 12 ω ωω 13 retur â 0,â 1,,â 1 To verify the correctess, we here uderstad lie 11 i the procedure Recursive-DFT: â k+ 2 = â[0] k ωâ[1] k At the kth iteratio of the for loop of lies 9 12, ω = ω k We have â k+ 2 = â [0] k ωk â [1] k = â [0] = p 0 = p 0 = p k +ωk+ 2 â [1] k ω 2k ω 2k+ ω k+ 2 +ω k+ 2 p 1 ω 2k +ω k+ 2 p 1, from 1 ω 2k+ 3
4 Let T be the ruig time of Recursive-DFT Steps 1 6 take time Θ Steps 7 ad 8 each takes time T 2 Steps 9 13 take time Θ So we ed up with the recurrece T = 2T +Θ, 2 which has the solutio T = Θlog 2 2 Iverse DFT Suppose we eed to compute the iverse Fourier trasform give by a = V 1 â Namely, we would like to determie the coefficiets of the polyomial px = a 0 + +a 1 x 1 give its values at ω,,ω 0 1 Ca we do it with the same efficiecy, that is, i time Θlog? The aswer is yes To see why, ote that the Vadermode matrix V has iverse V 1 = 1 1 ω 1 ω 2 ω 1 1 ω 1 ω 2 1 ω 12 To verify the above, make use of the equatio 1 j=0 ωk j = 0 for o-egative iteger k ot divisible by Based o the above observatio, we ca still apply Recursive-DFT by replacig a with â, â with a, ω with ω 1 that is, ω 1, ad scalig the result by 1 3 Fast Multiplicatio of Two Polyomials Let us ow go back to the two polyomials at the begiig: Their product px = a 0 +a 1 x+ +a 1 x 1, qx = b 0 +b 1 x+ +b 1 x 1 p qx = px qx = c 0 +c 1 x+ c 2 2 x 2 2 ca be computed by combiig FFT with iterpolatio The computatio takes time Θ log ad cosists of the followig three steps: 1 Evaluatepxadqxat2poitsω 0 2,,ω2 1 2 usigdftthissteptakestimeθlog 4
5 2 Obtai the values of pxqx at these 2 poits through poitwise multiplicatio This step takes time Θ p qω 0 2 = pω 0 2 qω 0 2, p qω 1 2 = pω1 2 qω1 2, p qω = pω qω Iterpolate the polyomial p q at the product values usig iverse DFT to obtai coefficiets c 0,c 1,,c 2 2 This last step requires time Θlog We ca also use FFT to compute the covolutio of two vectors a = a 0,,a 1 ad b = b 0,,b 1, which is defied as a vector c = c 0,,c 1 where j c j = a k b j k, k=0 j = 0,, 1 The ruig time is agai Θlog 4 History of FFT Moder FFT is widely credited to the paper [2] by Cooley ad Tukey But the algorithm had bee discovered idepedetly by a few idividuals i the past Oly the appearace of digital computers ad the wide applicatio of sigal processig made people realize the importace of fast computatio of large Fourier series A icomplete list of pioeers icludes Gauss 1805 the earliest kow origi of the FFT algorithm Ruge ad Köig 1924 the doublig algorithm Daielso ad Laczos 1942 divide-ad-coquer o DFTs Rudick 1960s the first computer program implemetatio with O log time Refereces [1] T H Corme et al Itroductio to Algorithms McGraw-Hill, Ic, 2d editio, 2001 [2] J W Cooley ad J W Tukey A algorithm for the machie calculatio of complex Fourier series Mathematics of Computatio, 1990: ,
Lecture 3: Divide and Conquer: Fast Fourier Transform
Lecture 3: Divide ad Coquer: Fast Fourier Trasform Polyomial Operatios vs. Represetatios Divide ad Coquer Algorithm Collapsig Samples / Roots of Uity FFT, IFFT, ad Polyomial Multiplicatio Polyomial operatios
More informationLet A(x) and B(x) be two polynomials of degree n 1:
MI-EVY (2011/2012) J. Holub: 4. DFT, FFT ad Patter Matchig p. 2/42 Operatios o polyomials MI-EVY (2011/2012) J. Holub: 4. DFT, FFT ad Patter Matchig p. 4/42 Efficiet Patter Matchig (MI-EVY) 4. DFT, FFT
More informationThe Discrete Fourier Transform
The Discrete Fourier Trasform Complex Fourier Series Represetatio Recall that a Fourier series has the form a 0 + a k cos(kt) + k=1 b k si(kt) This represetatio seems a bit awkward, sice it ivolves two
More informationAlgorithms and Data Structures 2014 Exercises and Solutions Week 13
Algorithms ad Data Structures 204 Exercises ad Solutios Week 3 Toom-Cook (cotiued) Durig the last lecture, two polyomials A(x) a 0 + a x ad B(x) b 0 + b x both of degree were multiplied, first by evaluatig
More informationFFTs in Graphics and Vision. The Fast Fourier Transform
FFTs i Graphics ad Visio The Fast Fourier Trasform 1 Outlie The FFT Algorithm Applicatios i 1D Multi-Dimesioal FFTs More Applicatios Real FFTs 2 Computatioal Complexity To compute the movig dot-product
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationMAXIMALLY FLAT FIR FILTERS
MAXIMALLY FLAT FIR FILTERS This sectio describes a family of maximally flat symmetric FIR filters first itroduced by Herrma [2]. The desig of these filters is particularly simple due to the availability
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationTMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods
TMA4205 Numerical Liear Algebra The Poisso problem i R 2 : diagoalizatio methods September 3, 2007 c Eiar M Røquist Departmet of Mathematical Scieces NTNU, N-749 Trodheim, Norway All rights reserved A
More informationIntensive Algorithms Lecture 11. DFT and DP. Lecturer: Daniel A. Spielman February 20, f(n) O(g(n) log c g(n)).
Itesive Algorithms Lecture 11 DFT ad DP Lecturer: Daiel A. Spielma February 20, 2018 11.1 Itroductio The purpose of this lecture is to lear how use the Discrete Fourier Trasform to save space i Dyamic
More informationFall 2011, EE123 Digital Signal Processing
Lecture 5 Miki Lustig, UCB September 14, 211 Miki Lustig, UCB Motivatios for Discrete Fourier Trasform Sampled represetatio i time ad frequecy umerical Fourier aalysis requires a Fourier represetatio that
More informationDivide & Conquer. Divide-and-conquer algorithms. Conventional product of polynomials. Conventional product of polynomials.
Divide-ad-coquer algorithms Divide & Coquer Strategy: Divide the problem ito smaller subproblems of the same type of problem Solve the subproblems recursively Combie the aswers to solve the origial problem
More informationCSE Introduction to Parallel Processing. Chapter 3. Parallel Algorithm Complexity
Dr. Izadi CSE-40533 Itroductio to Parallel Processig Chapter 3 Parallel Algorithm Complexity Review algorithm complexity ad various complexity classes Itroduce the otios of time ad time-cost optimality
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationLecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016
Lecture 3 Digital Sigal Processig Chapter 3 z-trasforms Mikael Swartlig Nedelko Grbic Begt Madersso rev. 06 Departmet of Electrical ad Iformatio Techology Lud Uiversity z-trasforms We defie the z-trasform
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationFrequency Domain Filtering
Frequecy Domai Filterig Raga Rodrigo October 19, 2010 Outlie Cotets 1 Itroductio 1 2 Fourier Represetatio of Fiite-Duratio Sequeces: The Discrete Fourier Trasform 1 3 The 2-D Discrete Fourier Trasform
More information4.3 Growth Rates of Solutions to Recurrences
4.3. GROWTH RATES OF SOLUTIONS TO RECURRENCES 81 4.3 Growth Rates of Solutios to Recurreces 4.3.1 Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer.
More informationRecursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
More informationACCELERATING CONVERGENCE OF SERIES
ACCELERATIG COVERGECE OF SERIES KEITH CORAD. Itroductio A ifiite series is the limit of its partial sums. However, it may take a large umber of terms to get eve a few correct digits for the series from
More informationAnalysis of Algorithms. Introduction. Contents
Itroductio The focus of this module is mathematical aspects of algorithms. Our mai focus is aalysis of algorithms, which meas evaluatig efficiecy of algorithms by aalytical ad mathematical methods. We
More informationRecursive Algorithm for Generating Partitions of an Integer. 1 Preliminary
Recursive Algorithm for Geeratig Partitios of a Iteger Sug-Hyuk Cha Computer Sciece Departmet, Pace Uiversity 1 Pace Plaza, New York, NY 10038 USA scha@pace.edu Abstract. This article first reviews the
More informationFinite-length Discrete Transforms. Chapter 5, Sections
Fiite-legth Discrete Trasforms Chapter 5, Sectios 5.2-50 5.0 Dr. Iyad djafar Outlie The Discrete Fourier Trasform (DFT) Matrix Represetatio of DFT Fiite-legth Sequeces Circular Covolutio DFT Symmetry Properties
More informationNICK DUFRESNE. 1 1 p(x). To determine some formulas for the generating function of the Schröder numbers, r(x) = a(x) =
AN INTRODUCTION TO SCHRÖDER AND UNKNOWN NUMBERS NICK DUFRESNE Abstract. I this article we will itroduce two types of lattice paths, Schröder paths ad Ukow paths. We will examie differet properties of each,
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationCSI 2101 Discrete Structures Winter Homework Assignment #4 (100 points, weight 5%) Due: Thursday, April 5, at 1:00pm (in lecture)
CSI 101 Discrete Structures Witer 01 Prof. Lucia Moura Uiversity of Ottawa Homework Assigmet #4 (100 poits, weight %) Due: Thursday, April, at 1:00pm (i lecture) Program verificatio, Recurrece Relatios
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationLESSON 2: SIMPLIFYING RADICALS
High School: Workig with Epressios LESSON : SIMPLIFYING RADICALS N.RN.. C N.RN.. B 5 5 C t t t t t E a b a a b N.RN.. 4 6 N.RN. 4. N.RN. 5. N.RN. 6. 7 8 N.RN. 7. A 7 N.RN. 8. 6 80 448 4 5 6 48 00 6 6 6
More informationAlgorithms Design & Analysis. Divide & Conquer
Algorithms Desig & Aalysis Divide & Coquer Recap Direct-accessible table Hash tables Hash fuctios Uiversal hashig Perfect Hashig Ope addressig 2 Today s topics The divide-ad-coquer desig paradigm Revised
More informationEllipsoid Method for Linear Programming made simple
Ellipsoid Method for Liear Programmig made simple Sajeev Saxea Dept. of Computer Sciece ad Egieerig, Idia Istitute of Techology, Kapur, INDIA-08 06 December 3, 07 Abstract I this paper, ellipsoid method
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationA widely used display of protein shapes is based on the coordinates of the alpha carbons - - C α
Nice plottig of proteis: I A widely used display of protei shapes is based o the coordiates of the alpha carbos - - C α -s. The coordiates of the C α -s are coected by a cotiuous curve that roughly follows
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of o-time jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationSection 1.1. Calculus: Areas And Tangents. Difference Equations to Differential Equations
Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More informationWarped, Chirp Z-Transform: Radar Signal Processing
arped, Chirp Z-Trasform: Radar Sigal Processig by Garimella Ramamurthy Report o: IIIT/TR// Cetre for Commuicatios Iteratioal Istitute of Iformatio Techology Hyderabad - 5 3, IDIA Jauary ARPED, CHIRP Z
More informationGenerating Functions for Laguerre Type Polynomials. Group Theoretic method
It. Joural of Math. Aalysis, Vol. 4, 2010, o. 48, 257-266 Geeratig Fuctios for Laguerre Type Polyomials α of Two Variables L ( xy, ) by Usig Group Theoretic method Ajay K. Shula* ad Sriata K. Meher** *Departmet
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationOptimum LMSE Discrete Transform
Image Trasformatio Two-dimesioal image trasforms are extremely importat areas of study i image processig. The image output i the trasformed space may be aalyzed, iterpreted, ad further processed for implemetig
More informationNumerical Conformal Mapping via a Fredholm Integral Equation using Fourier Method ABSTRACT INTRODUCTION
alaysia Joural of athematical Scieces 3(1): 83-93 (9) umerical Coformal appig via a Fredholm Itegral Equatio usig Fourier ethod 1 Ali Hassa ohamed urid ad Teh Yua Yig 1, Departmet of athematics, Faculty
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationSignals & Systems Chapter3
Sigals & Systems Chapter3 1.2 Discrete-Time (D-T) Sigals Electroic systems do most of the processig of a sigal usig a computer. A computer ca t directly process a C-T sigal but istead eeds a stream of
More informationChapter 8. DFT : The Discrete Fourier Transform
Chapter 8 DFT : The Discrete Fourier Trasform Roots of Uity Defiitio: A th root of uity is a complex umber x such that x The th roots of uity are: ω, ω,, ω - where ω e π /. Proof: (ω ) (e π / ) (e π )
More informationAlgebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
More informationSession 5. (1) Principal component analysis and Karhunen-Loève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad Karhue-Loève trasformatio Topic 2 of this course explais the image
More informationRADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify
Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL
More informationSome remarks for codes and lattices over imaginary quadratic
Some remarks for codes ad lattices over imagiary quadratic fields Toy Shaska Oaklad Uiversity, Rochester, MI, USA. Caleb Shor Wester New Eglad Uiversity, Sprigfield, MA, USA. shaska@oaklad.edu Abstract
More informationAlgorithms and Data Structures Lecture IV
Algorithms ad Data Structures Lecture IV Simoas Šalteis Aalborg Uiversity simas@cs.auc.dk September 5, 00 1 This Lecture Aalyzig the ruig time of recursive algorithms (such as divide-ad-coquer) Writig
More informationMath 104: Homework 2 solutions
Math 04: Homework solutios. A (0, ): Sice this is a ope iterval, the miimum is udefied, ad sice the set is ot bouded above, the maximum is also udefied. if A 0 ad sup A. B { m + : m, N}: This set does
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationEE422G Homework #13 (12 points)
EE422G Homework #1 (12 poits) 1. (5 poits) I this problem, you are asked to explore a importat applicatio of FFT: efficiet computatio of covolutio. The impulse respose of a system is give by h(t) (.9),1,2,,1
More informationClassroom. We investigate and further explore the problem of dividing x = n + m (m, n are coprime) sheep in
Classroom I this sectio of Resoace, we ivite readers to pose questios likely to be raised i a classroom situatio. We may suggest strategies for dealig with them, or ivite resposes, or both. Classroom is
More information2D DSP Basics: 2D Systems
- Digital Image Processig ad Compressio D DSP Basics: D Systems D Systems T[ ] y = T [ ] Liearity Additivity: If T y = T [ ] The + T y = y + y Homogeeity: If The T y = T [ ] a T y = ay = at [ ] Liearity
More informationCS 5150/6150: Assignment 1 Due: Sep 23, 2010
CS 5150/6150: Assigmet 1 Due: Sep 23, 2010 Wei Liu September 24, 2010 Q1: (1) Usig master theorem: a = 7, b = 4, f() = O(). Because f() = log b a ε holds whe ε = log b a = log 4 7, we ca apply the first
More informationPractical Spectral Anaysis (continue) (from Boaz Porat s book) Frequency Measurement
Practical Spectral Aaysis (cotiue) (from Boaz Porat s book) Frequecy Measuremet Oe of the most importat applicatios of the DFT is the measuremet of frequecies of periodic sigals (eg., siusoidal sigals),
More informationCS:3330 (Prof. Pemmaraju ): Assignment #1 Solutions. (b) For n = 3, we will have 3 men and 3 women with preferences as follows: m 1 : w 3 > w 1 > w 2
Shiyao Wag CS:3330 (Prof. Pemmaraju ): Assigmet #1 Solutios Problem 1 (a) Cosider iput with me m 1, m,..., m ad wome w 1, w,..., w with the followig prefereces: All me have the same prefereces for wome:
More informationThe Choquet Integral with Respect to Fuzzy-Valued Set Functions
The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationBijective Proofs of Gould s and Rothe s Identities
ESI The Erwi Schrödiger Iteratioal Boltzmagasse 9 Istitute for Mathematical Physics A-1090 Wie, Austria Bijective Proofs of Gould s ad Rothe s Idetities Victor J. W. Guo Viea, Preprit ESI 2072 (2008 November
More informationC. Complex Numbers. x 6x + 2 = 0. This equation was known to have three real roots, given by simple combinations of the expressions
C. Complex Numbers. Complex arithmetic. Most people thik that complex umbers arose from attempts to solve quadratic equatios, but actually it was i coectio with cubic equatios they first appeared. Everyoe
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More information11. FINITE FIELDS. Example 1: The following tables define addition and multiplication for a field of order 4.
11. FINITE FIELDS 11.1. A Field With 4 Elemets Probably the oly fiite fields which you ll kow about at this stage are the fields of itegers modulo a prime p, deoted by Z p. But there are others. Now although
More informationQ-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received:
More informationMath 155 (Lecture 3)
Math 55 (Lecture 3) September 8, I this lecture, we ll cosider the aswer to oe of the most basic coutig problems i combiatorics Questio How may ways are there to choose a -elemet subset of the set {,,,
More informationLecture 2: Monte Carlo Simulation
STAT/Q SCI 43: Itroductio to Resamplig ethods Sprig 27 Istructor: Ye-Chi Che Lecture 2: ote Carlo Simulatio 2 ote Carlo Itegratio Assume we wat to evaluate the followig itegratio: e x3 dx What ca we do?
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationIntroduction to Digital Signal Processing
Fakultät Iformatik Istitut für Systemarchitektur Professur Recheretze Itroductio to Digital Sigal Processig Walteegus Dargie Walteegus Dargie TU Dresde Chair of Computer Networks I 45 Miutes Refereces
More information, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)
Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationRay-triangle intersection
Ray-triagle itersectio ria urless October 2006 I this hadout, we explore the steps eeded to compute the itersectio of a ray with a triagle, ad the to compute the barycetric coordiates of that itersectio.
More informationA GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS. Mircea Merca
Idia J Pure Appl Math 45): 75-89 February 204 c Idia Natioal Sciece Academy A GENERALIZATION OF THE SYMMETRY BETWEEN COMPLETE AND ELEMENTARY SYMMETRIC FUNCTIONS Mircea Merca Departmet of Mathematics Uiversity
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationDiscrete Orthogonal Moment Features Using Chebyshev Polynomials
Discrete Orthogoal Momet Features Usig Chebyshev Polyomials R. Mukuda, 1 S.H.Og ad P.A. Lee 3 1 Faculty of Iformatio Sciece ad Techology, Multimedia Uiversity 75450 Malacca, Malaysia. Istitute of Mathematical
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationRight circulant matrices with ratio of the elements of Fibonacci and geometric sequence
Notes o Number Theory ad Discrete Mathematics Prit ISSN 1310 5132, Olie ISSN 2367 8275 Vol. 22, 2016, No. 3, 79 83 Right circulat matrices with ratio of the elemets of Fiboacci ad geometric sequece Aldous
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More informationFilter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and
Filter bas Separately, the lowpass ad highpass filters are ot ivertible T removes the highest frequecy / ad removes the lowest frequecy Together these filters separate the sigal ito low-frequecy ad high-frequecy
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More informationPolynomial identity testing and global minimum cut
CHAPTER 6 Polyomial idetity testig ad global miimum cut I this lecture we will cosider two further problems that ca be solved usig probabilistic algorithms. I the first half, we will cosider the problem
More informationThe Discrete Fourier Transform
The iscrete Fourier Trasform The discrete-time Fourier trasform (TFT) of a sequece is a cotiuous fuctio of!, ad repeats with period. I practice we usually wat to obtai the Fourier compoets usig digital
More informationCOMM 602: Digital Signal Processing
COMM 60: Digital Sigal Processig Lecture 4 -Properties of LTIS Usig Z-Trasform -Iverse Z-Trasform Properties of LTIS Usig Z-Trasform Properties of LTIS Usig Z-Trasform -ve +ve Properties of LTIS Usig Z-Trasform
More informationLecture 2 Clustering Part II
COMS 4995: Usupervised Learig (Summer 8) May 24, 208 Lecture 2 Clusterig Part II Istructor: Nakul Verma Scribes: Jie Li, Yadi Rozov Today, we will be talkig about the hardess results for k-meas. More specifically,
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationSection 1 of Unit 03 (Pure Mathematics 3) Algebra
Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course
More informationClassification of problem & problem solving strategies. classification of time complexities (linear, logarithmic etc)
Classificatio of problem & problem solvig strategies classificatio of time complexities (liear, arithmic etc) Problem subdivisio Divide ad Coquer strategy. Asymptotic otatios, lower boud ad upper boud:
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationExponents. Learning Objectives. Pre-Activity
Sectio. Pre-Activity Preparatio Epoets A Chai Letter Chai letters are geerated every day. If you sed a chai letter to three frieds ad they each sed it o to three frieds, who each sed it o to three frieds,
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationOn the Lebesgue constant for the Xu interpolation formula
O the Lebesgue costat for the Xu iterpolatio formula Le Bos Dept. of Mathematics ad Statistics, Uiversity of Calgary Caada Stefao De Marchi Dept. of Computer Sciece, Uiversity of Veroa Italy Marco Viaello
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationSOLUTIONS TO PRISM PROBLEMS Junior Level 2014
SOLUTIONS TO PRISM PROBLEMS Juior Level 04. (B) Sice 50% of 50 is 50 5 ad 50% of 40 is the secod by 5 0 5. 40 0, the first exceeds. (A) Oe way of comparig the magitudes of the umbers,,, 5 ad 0.7 is 4 5
More information