# Polynomials and the Fast Fourier Transform (FFT) Battle Plan

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Polynomials and the Fast Fourier Transform (FFT) Algorithm Design and Analysis (Wee 7) 1 Polynomials Battle Plan Algorithms to add, multiply and evaluate polynomials Coefficient and point-value representation Fourier Transform Discrete Fourier Transform (DFT) and inverse DFT to translate between polynomial representations A Short Digression on Complex Roots of Unity Fast Fourier Transform (FFT) is a divide-and-conquer algorithm based on properties of complex roots of unity 2

2 Polynomials A polynomial in the variable x is a representation of a function A x = a n 1 x n a 2 x 2 + a 1 x + a 0 as a formal sum A x = n 1 j=0 a j x j. We call the values a 0, a 1,, a n 1 the coefficients of the polynomial A x is said to have degree if its highest nonzero coefficient is a. Any integer strictly greater than the degree of a polynomial is a degree-bound of that polynomial 3 A x = x 3 2x 1 Examples A(x) has degree 3 A(x) has degree-bounds 4, 5, 6, or all values > degree A(x) has coefficients ( 1, 2, 0, 1) B x = x 3 + x B(x) has degree 3 B(x) has degree bounds 4, 5, 6, or all values > degree B(x) has coefficients (1, 0, 1, 1) 4

3 Coefficient Representation A coefficient representation of a polynomial A x = n 1 j=0 a j x j of degree-bound n is a vector of coefficients a = a 0, a 1,, a n 1. More examples A x = 6x 3 + 7x 2 10x + 9 (9, 10, 7, 6) B x = 2x 3 + 4x 5 ( 5, 4, 0, 2) The operation of evaluating the polynomial A(x) at point x 0 consists of computing the value of A x 0. Evaluation taes time Θ(n) using Horner s rule A(x 0 ) = a 0 + x 0 (a 1 + x 0 (a x 0 a n 2 + x 0 a n 1 )) 5 Adding Polynomials Adding two polynomials represented by the coefficient vectors a = (a 0, a 1,, a n 1 ) and b = (b 0, b 1,, b n 1 ) taes time Θ(n). Sum is the coefficient vector c = (c 0, c 1,, c n 1 ), where c j = a j + b j for j = 0,1,, n 1. Example A x = 6x 3 + 7x 2 10x + 9 (9, 10, 7, 6) B x = 2x 3 + 4x 5 ( 5, 4, 0, 2) C(x) = 4x 3 + 7x 2 6x + 4 (4, 6,7,4) 6

4 Multiplying Polynomials For polynomial multiplication, if A(x) and B(x) are polynomials of degree-bound n, we say their product C(x) is a polynomial of degree-bound 2n 1. Example 6x 3 + 7x 2 10x + 9 2x 3 + 4x 5 30x 3 35x x 45 24x x 3 40x x 12x 6 14x x 4 18x 3 12x 6 14x x 4 20x 3 75x x 45 7 Multiplying Polynomials Multiplication of two degree-bound n polynomials A(x) and B(x) taes time Θ n 2, since each coefficient in vector a must be multiplied by each coefficient in vector b. Another way to express the product C(x) is 2n 1 j=0 c j x j j, where c j = =0 a b j. The resulting coefficient vector c = (c 0, c 1, c 2n 1 ) is also called the convolution of the input vectors a and b, denoted as c = a b. 8

5 Point-Value Representation A point-value representation of a polynomial A(x) of degree-bound n is a set of n point-value pairs x 0, y 0, x 1, y 1,, x n 1, y n 1 such that all of the x are distinct and y = A(x ) for = 0, 1,, n 1. Example A x = x 3 2x + 1 x 0, 1, 2, 3 A(x ) 1, 0, 5, 22 Using Horner s method, n-point evaluation taes time Θ(n 2 ). * 0, 1, 1, 0, 2, 5, 3, Point-Value Representation The inverse of evaluation is called interpolation determines coefficient form of polynomial from pointvalue representation For any set * x 0, y 0, x 1, y 1,, x n 1, y n 1 + of n pointvalue pairs such that all the x values are distinct, there is a unique polynomial A(x) of degree-bound n such that y = A(x ) for = 0, 1,, n 1. Lagrange s formula n 1 j (x x j ) A x = y j (x x j ) =0 Using Lagrange s formula, interpolation taes time Θ(n 2 ). 10

6 Example Using Lagrange s formula, we interpolate the pointvalue representation 0, 1, 1, 0, 2, 5, 3, x 1 (x 2)(x 3) = x3 6x 2 +11x 6 (0 1)(0 2)(0 3) 6 0 (x 0)(x 2)(x 3) (1 0)(1 2)(1 3) = 0 = x3 +6x 2 11x (x 0)(x 1)(x 3) = 5 x3 4x 2 +3x = 15x3 +60x 2 45x (2 0)(2 1)(2 3) (x 0)(x 1)(x 2) = 22 x3 3x 2 +2x = 22x3 66x 2 +44x (3 0)(3 1)(3 2) x3 + 0x 2 12x + 6 x 3 2x Adding Polynomials In point-value form, addition C x = A x + B(x) is given by C x = A x + B(x ) for any point x. A: x 0, y 0, x 1, y 1,, x n 1, y n 1 B: x 0, y 0, x 1, y 1,, x n 1, y n 1 C: * x 0, y 0 + y 0, x 1, y 1 + y 1,, x n 1, y n 1 + y n 1 + A and B are evaluated for the same n points. The time to add two polynomials of degree-bound n in point-value form is Θ(n). 12

7 We add C x Example = A x + B(x) in point-value form A x = x 3 2x + 1 B x = x 3 + x x = (0, 1, 2, 3) A: 0, 1, 1, 0, 2, 5, 3, 22 B: * 0, 1, 1, 3, 2, 13, 3, 37 + C: * 0, 2, 1, 3, 2, 18, 3, Multiplying Polynomials In point-value form, multiplication C x = A x B(x) is given by C x = A x B(x ) for any point x. Problem: if A and B are of degree-bound n, then C is of degree-bound 2n. Need to start with extended point-value forms for A and B consisting of 2n point-value pairs each. A: x 0, y 0, x 1, y 1,, x 2n 1, y 2n 1 B: x 0, y 0, x 1, y 1,, x 2n 1, y 2n 1 C: * x 0, y 0 y 0, x 1, y 1 y 1,, x 2n 1, y 2n 1 y 2n 1 + The time to multiply two polynomials of degreebound n in point-value form is Θ(n). 14

8 We multiply C x Example = A x B(x) in point-value form A x = x 3 2x + 1 B x = x 3 + x x = ( 3, 2, 1, 0, 1, 2, 3) We need 7 coefficients! A: 3, 17, 2, 3, 1,1, 0, 1, 1, 0, 2, 5, 3, 22 B: * 3, 20, 2, 3, 1, 2, 0, 1, 1, 3, 2, 13, 3, 37 + C: * 3,340, 2,9, 1,2, 0, 1, 1, 0, 2, 65, 3, The Road So Far a 0, a 1,, a n 1 b 0, b 1,, a n 1 Ordinary multiplication Time Θ(n 2 ) c 0, c 1,, c n 1 Coefficient representations Evaluation Time Θ(n 2 ) Interpolation Time Θ(n 2 ) A x 0, B x 0 A x 1, B x 1 A x 2n 1, B(x 2n 1 ) Pointwise multiplication Time Θ(n) C x 0 C x 1 C(x 2n 1 ) Point-value representations Can we do better? Using Fast Fourier Transform (FFT) and its inverse, we can do evaluation and interpolation in time Θ(n log n). The product of two polynomials of degree-bound n can be computed in time Θ(n log n), with both the input and output in coefficient form. 16

9 Amplitude Weight Fourier Transform Fourier Transforms originate from signal processing Transform signal from time domain to frequency domain Time Frequency Input signal is a function mapping time to amplitude Output is a weighted sum of phase-shifted sinusoids of varying frequencies 17 Fast Multiplication of Polynomials Using complex roots of unity Evaluation by taing the Discrete Fourier Transform (DFT) of a coefficient vector Interpolation by taing the inverse DFT of point-value pairs, yielding a coefficient vector Fast Fourier Transform (FFT) can perform DFT and inverse DFT in time Θ(n log n) Algorithm 1. Add n higher-order zero coefficients to A(x) and B(x) 2. Evaluate A(x) and B(x) using FFT for 2n points 3. Pointwise multiplication of point-value forms 4. Interpolate C(x) using FFT to compute inverse DFT 18

10 Complex Roots of Unity A complex n th root of unity (1) is a complex number ω such that ω n = 1. There are exactly n complex n th root of unity e 2πi n for = 0, 1,, n 1 where e iu = cos u + i sin (u). Here u represents an angle in radians. Using e 2πi n = cos 2π n + i sin( 2π n), we can chec that it is a root e 2πi n n = e 2πi = cos(2π) +i sin (2π) = Examples The complex 4 th roots of unity are 1, 1, i, i where 1 = i. The complex 8 th roots of unity are all of the above, plus four more 1 + i, 1 i, 1 + i, and 1 i For example i 2 2 = i 2 + i2 2 = i 20

11 The value Principal n th Root of Unity ω n = e 2πi n is called the principal n th root of unity. All of the other complex n th roots of unity are powers of ω n. The n complex n th roots of unity, ω n 0, ω n 1,, ω n n 1, form a group under multiplication that has the same structure as (Z n, +) modulo n. ω n n = ω n 0 = 1 implies ω n j ω n = ω n j+ = ω n j+ mod n ω n 1 = ω n n 1 21 Visualizing 8 Complex 8 th Roots of Unity imaginary ω 8 2 = i i ω 8 3 = i 2 ω 8 1 = i 2 ω 8 4 = 1 ω 8 0 = ω 8 8 = real ω 8 5 = 1 2 i 2 ω 8 7 = 1 2 i 2 i ω 8 6 = i 22

12 Cancellation Lemma For any integers n 0, 0, and b > 0, Proof ω d dn = e 2πi dn ω d dn = ω n. d = e 2πi n = ωn n For any even integer n > 0, ω 2 n = ω 2 = 1. Example ω 6 24 = ω 4 ω 6 24 = e 2πi 24 6 = e 2πi 6 24 = e 2πi 4 = ω 4 23 Halving Lemma If n > 0 is even, then the squares of the n complex n th roots of unity are the n 2 complex n 2 th roots of unity. Proof By the cancellation lemma, we have ω 2 n = ωn 2 for any nonnegative integer. If we square all of the complex n th roots of unity, then each n 2 th root of unity is obtained exactly twice ω n +n 2 2 = ωn 2+n = ω n 2 ω n n = ω n 2 = ω n 2 Thus, ω n and ω n +n 2 have the same square 24

13 Summation Lemma For any integer n 1 and nonzero integer not divisible by n, n 1 j=0 ω n j = 0. Proof Geometric series n 1 j=0 x j ω j n = ω n n n 1 1 j=0 ω n 1 = xn 1 x 1 = ω n n 1 = 1 1 = 0 ω n 1 ω n 1 Requiring that not be divisible by n ensures that the denominator is not 0, since ω n = 1 only when is divisible by n 25 Discrete Fourier Transform (DFT) Evaluate a polynomial A(x) of degree-bound n at the n complex n th roots of unity, ω n 0, ω n 1, ω n 2,, ω n n 1. assume that n is a power of 2 assume A is given in coefficient form a = (a 0, a 1,, a n 1 ) We define the results y, for = 0, 1,, n 1, by y = A ω n 1 j n = j=0 a j ω n. The vector y = (y 0, y 1,, y n 1 ) is the Discrete Fourier Transform (DFT) of the coefficient vector a = a 0, a 1,, a n 1, denoted as y = DFT n (a). 26

14 Fast Fourier Transform (FFT) Fast Fourier Transform (FFT) taes advantage of the special properties of the complex roots of unity to compute DFT n (a) in time Θ(n log n). Divide-and-conquer strategy define two new polynomials of degree-bound n 2, using even-index and odd-index coefficients of A(x) separately A 0 x = a 0 + a 2 x + a 4 x a n 2 x n 2 1 A 1 x = a 1 + a 3 x + a 5 x a n 1 x n 2 1 A x = A 0 x 2 + xa 1 (x 2 ) 27 Fast Fourier Transform (FFT) The problem of evaluating A(x) at ω n 0, ω n 1,, ω n n 1 reduces to 1. evaluating the degree-bound n 2 polynomials A 0 (x) and A 1 (x) at the points ω n 0 2, ω n 1 2,, ω n n combining the results by A x = A 0 x 2 + xa 1 (x 2 ) Why bother? The list ω n 0 2, ω n 1 2,, ω n n 1 2 does not contain n distinct values, but n 2 complex n 2 th roots of unity Polynomials A 0 and A 1 are recursively evaluated at the n 2 complex n 2 th roots of unity Subproblems have exactly the same form as the original problem, but are half the size 28

15 Example A x = a 0 + a 1 x + a 2 x 2 + a 3 x 3 of degree-bound 4 A ω 0 4 = A 1 = a 0 + a 1 + a 2 + a 3 A ω 1 4 = A i = a o + a 1 i a 2 a 3 i A ω 2 4 = A 1 = a 0 a 1 + a 2 a 3 A ω 3 4 = A i = a 0 a 1 i + a 2 + a 3 i Using A x = A 0 x 2 + xa 1 x 2 A x = a 0 + a 2 x 2 + x a 1 + a 3 x 2 A ω 4 0 = A 1 = a 0 + a 2 + 1(a 1 + a 3 ) A ω 4 1 = A i = a 0 a 2 + i(a 1 a 3 ) A ω 2 4 = A 1 = a 0 + a 2 1 a 1 + a 3 A ω 3 4 = A i = a 0 a 2 i(a 1 a 3 ) 29 Recursive FFT Algorithm RECURSIVE-FFT a 1 n length,a- n is a power of 2 2 if n = 1 3 then return a basis of recursion 4 ω n e 2πi n ω n is principal n th root of unity 5 ω 1 6 a 0 (a 0, a 2,, a n 2 ) 7 a 1 (a 1, a 3,, a n 1 ) 8 y 0 RECURSIVE-FFT a 0 y 0 = A 0 ωn 2 = A 0 (ω 2 n ) 9 y 1 RECURSIVE-FFT a 1 y 1 = A 1 ωn 2 = A 1 (ω 2 n ) 10 for 0 to n do y y ωy 12 y +( n 2 ) y 0 1 ωy since ω + n = ω n 2 n 13 ω ωω n compute ω n iteratively 14 return y 30

16 Why Does It Wor? For y 0, y 1, yn 2 1 (line 11) y = y 0 + ω 1 n y = A 0 2 ω n + ω n A 1 2 ω n = A ω n For yn 2, yn 2 +1,, y n 1 (line 12) y + n 2 = y 0 ω 1 n y = y 0 + ω n + n 2 y 1 = A 0 ω n 2 + ω n + n 2 A 1 ω n 2 = A 0 ω n 2+n + ω n + n 2 A 1 ω n 2+n = A ω n + n 2 since ω n = ω n + n 2 since ω n 2+n = ω n 2 31 Input Vector Tree of RECURSIVEFFT(a) (a 0, a 1, a 2, a 3, a 4, a 5, a 6, a 7 ) (a 0, a 2, a 4, a 6 ) (a 1, a 3, a 5, a 7 ) (a 0, a 4 ) (a 2, a 6 ) (a 1, a 5 ) (a 3, a 7 ) (a 0 ) (a 4 ) (a 2 ) (a 6 ) (a 1 ) (a 5 ) (a 3 ) (a 7 ) 32

17 Interpolation Interpolation by computing the inverse DFT, denoted by a = DFT n 1 (y). By modifying the FFT algorithm, we can compute DFT n 1 in time Θ(n log n). switch the roles of a and y replace ω n by ω n 1 divide each element of the result by n 33

### Lecture 4: Polynomial Algorithms

Lecture 4: Polynomial Algorithms Piotr Sankowski Stefano Leonardi sankowski@dis.uniroma1.it leon@dis.uniroma1.it Theoretical Computer Science 29.10.2009 - p. 1/39 Lecture Overview Introduction Point Polynomial

### Discrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes

Discrete Fourier Series & Discrete Fourier Transform Chapter Intended Learning Outcomes (i) Understanding the relationships between the transform, discrete-time Fourier transform (DTFT), discrete Fourier

### MATH : HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS

MATH 16300-33: HONORS CALCULUS-3 HOMEWORK 6: SOLUTIONS 25-1 Find the absolute value and argument(s) of each of the following. (ii) (3 + 4i) 1 (iv) 7 3 + 4i (ii) Put z = 3 + 4i. From z 1 z = 1, we have

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### Chapter 3 Discrete-Time Fourier Series. by the French mathematician Jean Baptiste Joseph Fourier in the early 1800 s. The

Chapter 3 Discrete-Time Fourier Series 3.1 Introduction The Fourier series and Fourier transforms are mathematical correlations between the time and frequency domains. They are the result of the heat-transfer

MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

### Curve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.

Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections Least-Squares Regression Linear Regression General Linear Least-Squares Nonlinear

### This is the 23 rd lecture on DSP and our topic for today and a few more lectures to come will be analog filter design. (Refer Slide Time: 01:13-01:14)

Digital Signal Processing Prof. S.C. Dutta Roy Department of Electrical Electronics Indian Institute of Technology, Delhi Lecture - 23 Analog Filter Design This is the 23 rd lecture on DSP and our topic

### Lecture 11 Fast Fourier Transform (FFT)

Lecture 11 Fast Fourier Transform (FFT) Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University, tieli@pku.edu.cn

### 1.4 Fast Fourier Transform (FFT) Algorithm

74 CHAPTER AALYSIS OF DISCRETE-TIME LIEAR TIME-IVARIAT SYSTEMS 4 Fast Fourier Transform (FFT Algorithm Fast Fourier Transform, or FFT, is any algorithm for computing the -point DFT with a computational

### Steven Skiena. skiena

Lecture 3: Recurrence Relations (1997) Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Argue the solution to T(n) =

### FFT Algorithms. Chapter 6. Contents 6.1

Chapter 6 FFT Algorithms Contents Efficient computation of the DFT............................................ 6.2 Applications of FFT................................................... 6.6 Computing DFT

### Part I Preliminaries

Part I Preliminaries Chapter An Elementary Introduction to the Discrete Fourier ransform his chapter is intended to provide a brief introduction to the discrete Fourier transform (DF) It is not intended

### Lecture 2: Analysis of Algorithms (CS )

Lecture 2: Analysis of Algorithms (CS583-002) Amarda Shehu September 03 & 10, 2014 1 Outline of Today s Class 2 Big-Oh, Big-Omega, Theta Techniques for Finding Asymptotic Relationships Some more Fun with

### Outline of Complex Fourier Analysis. 1 Complex Fourier Series

Outline of Complex Fourier Analysis This handout is a summary of three types of Fourier analysis that use complex numbers: Complex Fourier Series, the Discrete Fourier Transform, and the (continuous) Fourier

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### Math 56 Homework 4 Michael Downs

Fourier series theory. As in lecture, let be the L 2 -norm on (, 2π). (a) Find a unit-norm function orthogonal to f(x) = x on x < 2π. [Hint: do what you d do in vector-land, eg pick some simple new function

### Recurrence Relations

Recurrence Relations Introduction Determining the running time of a recursive algorithm often requires one to determine the big-o growth of a function T (n) that is defined in terms of a recurrence relation.

### Final Year Project Progress Report. Frequency-Domain Adaptive Filtering. Myles Friel. Supervisor: Dr.Edward Jones

Final Year Project Progress Report Frequency-Domain Adaptive Filtering Myles Friel 01510401 Supervisor: Dr.Edward Jones Abstract The Final Year Project is an important part of the final year of the Electronic

### PROPERTIES AND EXAMPLES OF Z-TRANSFORMS

PROPERTIES AD EXAMPLES OF Z-TRASFORMS I. Introduction In the last lecture we reviewed the basic properties of the z-transform and the corresponding region of convergence. In this lecture we will cover

### STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an

### Polynomials can be added or subtracted simply by adding or subtracting the corresponding terms, e.g., if

1. Polynomials 1.1. Definitions A polynomial in x is an expression obtained by taking powers of x, multiplying them by constants, and adding them. It can be written in the form c 0 x n + c 1 x n 1 + c

### Complex Numbers. Misha Lavrov. ARML Practice 10/7/2012

Complex Numbers Misha Lavrov ARML Practice 10/7/2012 A short theorem Theorem (Complex numbers are weird) 1 = 1. Proof. The obvious identity 1 = 1 can be rewritten as 1 1 = 1 1. Distributing the square

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### Recurrences (CLRS )

Recurrences (CLRS 4.1-4.2) Last time we discussed divide-and-conquer algorithms Divide and Conquer To Solve P: 1. Divide P into smaller problems P 1,P 2,P 3...P k. 2. Conquer by solving the (smaller) subproblems

### Understand the principles of operation and characterization of digital filters

Digital Filters 1.0 Aim Understand the principles of operation and characterization of digital filters 2.0 Learning Outcomes You will be able to: Implement a digital filter in MATLAB. Investigate how commercial

### HOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!

Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following

### Digital image processing

Digital image processing The two-dimensional discrete Fourier transform and applications: image filtering in the frequency domain Introduction Frequency domain filtering modifies the brightness values

### The Discrete Fourier Transform

The Discrete Fourier Transform Introduction The discrete Fourier transform (DFT) is a fundamental transform in digital signal processing, with applications in frequency analysis, fast convolution, image

### Divide-and-conquer algorithms

Chapter 2 Divide-and-conquer algorithms The divide-and-conquer strategy solves a problem by: 1 Breaking it into subproblems that are themselves smaller instances of the same type of problem 2 Recursively

### Markov Chains, part I

Markov Chains, part I December 8, 2010 1 Introduction A Markov Chain is a sequence of random variables X 0, X 1,, where each X i S, such that P(X i+1 = s i+1 X i = s i, X i 1 = s i 1,, X 0 = s 0 ) = P(X

### Montana Common Core Standard

Algebra 2 Grade Level: 10(with Recommendation), 11, 12 Length: 1 Year Period(s) Per Day: 1 Credit: 1 Credit Requirement Fulfilled: Mathematics Course Description This course covers the main theories in

### Frequency Response of FIR Filters

Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input

### 1.2. Mathematical Models: A Catalog of Essential Functions

1.2. Mathematical Models: A Catalog of Essential Functions Mathematical model A mathematical model is a mathematical description (often by means of a function or an equation) of a real-world phenomenon

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### THE FAST FOURIER TRANSFORM

THE FAST FOURIER TRASFORM LOG CHE ABSTRACT. Fast Fourier transform (FFT) is a fast algorithm to compute the discrete Fourier transform in O( log ) operations for an array of size 2 J. It is based on the

MODULAR ARITHMETIC KEITH CONRAD. Introduction We will define the notion of congruent integers (with respect to a modulus) and develop some basic ideas of modular arithmetic. Applications of modular arithmetic

### COMPLEX NUMBERS. -2+2i

COMPLEX NUMBERS Cartesian Form of Complex Numbers The fundamental complex number is i, a number whose square is 1; that is, i is defined as a number satisfying i 1. The complex number system is all numbers

### Sampling and Reconstruction of Analog Signals

Sampling and Reconstruction of Analog Signals Chapter Intended Learning Outcomes: (i) Ability to convert an analog signal to a discrete-time sequence via sampling (ii) Ability to construct an analog signal

### Exercises with solutions (1)

Exercises with solutions (). Investigate the relationship between independence and correlation. (a) Two random variables X and Y are said to be correlated if and only if their covariance C XY is not equal

### MTH 06 LECTURE NOTES (Ojakian) Topic 2: Functions

MTH 06 LECTURE NOTES (Ojakian) Topic 2: Functions OUTLINE (References: Iyer Textbook - pages 4,6,17,18,40,79,80,81,82,103,104,109,110) 1. Definition of Function and Function Notation 2. Domain and Range

### Class XI Chapter 5 Complex Numbers and Quadratic Equations Maths. Exercise 5.1. Page 1 of 34

Question 1: Exercise 5.1 Express the given complex number in the form a + ib: Question 2: Express the given complex number in the form a + ib: i 9 + i 19 Question 3: Express the given complex number in

### Fast Fourier Transform

Swiss Federal Institute of Technology Zurich Fast Fourier Transform umerical Analysis Seminar Stefan Wörner Contents 1 Historical Introduction 1 2 Continuous and Discrete Fourier Transform 2 2.1 Continuous

### A Tutorial on Fourier Analysis

A Tutorial on Fourier Analysis Douglas Eck University of Montreal NYU March 26 1.5 A fundamental and three odd harmonics (3,5,7) fund (freq 1) 3rd harm 5th harm 7th harmm.5 1 2 4 6 8 1 12 14 16 18 2 1.5

### 7.1 Introduction. CSci 335 Software Design and Analysis III Chapter 7 Sorting. Prof. Stewart Weiss

Chapter 7 Sorting 7.1 Introduction Insertion sort is the sorting algorithm that splits an array into a sorted and an unsorted region, and repeatedly picks the lowest index element of the unsorted region

### Quantum Computing Lecture 7. Quantum Factoring. Anuj Dawar

Quantum Computing Lecture 7 Quantum Factoring Anuj Dawar Quantum Factoring A polynomial time quantum algorithm for factoring numbers was published by Peter Shor in 1994. polynomial time here means that

### Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

### Recurrence Relations

Recurrence Relations Time complexity for Recursive Algorithms Can be more difficult to solve than for standard algorithms because we need to know complexity for the sub-recursions of decreasing size Recurrence

### Roots of Real Numbers

Roots of Real Numbers Math 97 Supplement LEARNING OBJECTIVES. Calculate the exact and approximate value of the square root of a real number.. Calculate the exact and approximate value of the cube root

### Homework Set 2 (Due in class on Thursday, Sept. 25) (late papers accepted until 1:00 Friday)

Math 202 Homework Set 2 (Due in class on Thursday, Sept. 25) (late papers accepted until 1:00 Friday) Jerry L. Kazdan The problem numbers refer to the D Angelo - West text. 1. [# 1.8] In the morning section

### A matrix over a field F is a rectangular array of elements from F. The symbol

Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted

### SUBGROUPS OF CYCLIC GROUPS. 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by

SUBGROUPS OF CYCLIC GROUPS KEITH CONRAD 1. Introduction In a group G, we denote the (cyclic) group of powers of some g G by g = {g k : k Z}. If G = g, then G itself is cyclic, with g as a generator. Examples

### 1 The Dirichlet Problem. 2 The Poisson kernel. Math 857 Fall 2015

Math 857 Fall 2015 1 The Dirichlet Problem Before continuing to Fourier integrals, we consider first an application of Fourier series. Let Ω R 2 be open and connected (region). Recall from complex analysis

### Geometry and Arithmetic

Geometry and Arithmetic Alex Tao 10 June 2008 1 Rational Points on Conics We begin by stating a few definitions: A rational number is a quotient of two integers and the whole set of rational numbers is

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Digital Filter Design (FIR) Using Frequency Sampling Method

Digital Filter Design (FIR) Using Frequency Sampling Method Amer Ali Ammar Dr. Mohamed. K. Julboub Dr.Ahmed. A. Elmghairbi Electric and Electronic Engineering Department, Faculty of Engineering Zawia University

### Speech Signal Processing Handout 4 Spectral Analysis & the DFT

Speech Signal Processing Handout 4 Spectral Analysis & the DFT Signal: N samples Spectrum: DFT Result: N complex values Frequency resolution = 1 (Hz) NT Magnitude 1 Symmetric about Hz 2T frequency (Hz)

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +

### Module 4. Contents. Digital Filters - Implementation and Design. Signal Flow Graphs. Digital Filter Structures. FIR and IIR Filter Design Techniques

Module 4 Digital Filters - Implementation and Design Digital Signal Processing. Slide 4.1 Contents Signal Flow Graphs Basic filtering operations Digital Filter Structures Direct form FIR and IIR filters

### 1 Quiz on Linear Equations with Answers. This is Theorem II.3.1. There are two statements of this theorem in the text.

1 Quiz on Linear Equations with Answers (a) State the defining equations for linear transformations. (i) L(u + v) = L(u) + L(v), vectors u and v. (ii) L(Av) = AL(v), vectors v and numbers A. or combine

### The continuous and discrete Fourier transforms

FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

### Section 1.2. Angles and the Dot Product. The Calculus of Functions of Several Variables

The Calculus of Functions of Several Variables Section 1.2 Angles and the Dot Product Suppose x = (x 1, x 2 ) and y = (y 1, y 2 ) are two vectors in R 2, neither of which is the zero vector 0. Let α and

### n th roots of complex numbers

n th roots of complex numbers Nathan Pflueger 1 October 014 This note describes how to solve equations of the form z n = c, where c is a complex number. These problems serve to illustrate the use of polar

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### Complex Numbers and the Complex Exponential

Complex Numbers and the Complex Exponential Frank R. Kschischang The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto September 5, 2005 Numbers and Equations

### CHAPTER III - MARKOV CHAINS

CHAPTER III - MARKOV CHAINS JOSEPH G. CONLON 1. General Theory of Markov Chains We have already discussed the standard random walk on the integers Z. A Markov Chain can be viewed as a generalization of

### FAST FOURIER TRANSFORM ALGORITHMS WITH APPLICATIONS. A Dissertation Presented to the Graduate School of Clemson University

FAST FOURIER TRANSFORM ALGORITHMS WITH APPLICATIONS A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

### MATH 289 PROBLEM SET 4: NUMBER THEORY

MATH 289 PROBLEM SET 4: NUMBER THEORY 1. The greatest common divisor If d and n are integers, then we say that d divides n if and only if there exists an integer q such that n = qd. Notice that if d divides

### The divide and conquer strategy has three basic parts. For a given problem of size n,

1 Divide & Conquer One strategy for designing efficient algorithms is the divide and conquer approach, which is also called, more simply, a recursive approach. The analysis of recursive algorithms often

### 4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn

I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

### Level Sets of Arbitrary Dimension Polynomials with Positive Coefficients and Real Exponents

Level Sets of Arbitrary Dimension Polynomials with Positive Coefficients and Real Exponents Spencer Greenberg April 20, 2006 Abstract In this paper we consider the set of positive points at which a polynomial

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

Albuquerque School of Excellence Math Curriculum Overview Grade 11- Algebra II Module Polynomial, Rational, and Radical Relationships( Module Trigonometric Functions Module Functions Module Inferences

### A simple and fast algorithm for computing exponentials of power series

A simple and fast algorithm for computing exponentials of power series Alin Bostan Algorithms Project, INRIA Paris-Rocquencourt 7815 Le Chesnay Cedex France and Éric Schost ORCCA and Computer Science Department,

### The Dirichlet Unit Theorem

Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if

### PURE MATHEMATICS AM 27

AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

### Interpolating Polynomials Handout March 7, 2012

Interpolating Polynomials Handout March 7, 212 Again we work over our favorite field F (such as R, Q, C or F p ) We wish to find a polynomial y = f(x) passing through n specified data points (x 1,y 1 ),

### PURE MATHEMATICS AM 27

AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

### Lecture 1: Jan 14, 2015

E0 309: Topics in Complexity Theory Spring 2015 Lecture 1: Jan 14, 2015 Lecturer: Neeraj Kayal Scribe: Sumant Hegde and Abhijat Sharma 11 Introduction The theme of the course in this semester is Algebraic

### ASS.PROF.DR Thamer Information Theory 4th Class in Communication. Finite Field Arithmetic. (Galois field)

Finite Field Arithmetic (Galois field) Introduction: A finite field is also often known as a Galois field, after the French mathematician Pierre Galois. A Galois field in which the elements can take q

### MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS

MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a

### Design and Analysis of Algorithms

Design and Analysis of Algorithms CSE 5311 Lecture 3 Divide-and-Conquer Junzhou Huang, Ph.D. Department of Computer Science and Engineering CSE5311 Design and Analysis of Algorithms 1 Reviewing: Θ-notation

### Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

### Tangent and normal lines to conics

4.B. Tangent and normal lines to conics Apollonius work on conics includes a study of tangent and normal lines to these curves. The purpose of this document is to relate his approaches to the modern viewpoints

### 10 k + pm pm. 10 n p q = 2n 5 n p 2 a 5 b q = p

Week 7 Summary Lecture 13 Suppose that p and q are integers with gcd(p, q) = 1 (so that the fraction p/q is in its lowest terms) and 0 < p < q (so that 0 < p/q < 1), and suppose that q is not divisible

### THE 2-ADIC, BINARY AND DECIMAL PERIODS OF 1/3 k APPROACH FULL COMPLEXITY FOR INCREASING k

#A28 INTEGERS 12 (2012) THE 2-ADIC BINARY AND DECIMAL PERIODS OF 1/ k APPROACH FULL COMPLEXITY FOR INCREASING k Josefina López Villa Aecia Sud Cinti Chuquisaca Bolivia josefinapedro@hotmailcom Peter Stoll

### Department of Electronics and Communication Engineering 1

DHANALAKSHMI COLLEGE OF ENGINEERING, CHENNAI DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING III Year ECE / V Semester EC 6502 PRINCIPLES OF DIGITAL SIGNAL PROCESSING QUESTION BANK Department of

### Math 4 Review Problems

Topics for Review #1 Functions function concept [section 1. of textbook] function representations: graph, table, f(x) formula domain and range Vertical Line Test (for whether a graph is a function) evaluating

### Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions

Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective

### SMT 2014 Algebra Test Solutions February 15, 2014

1. Alice and Bob are painting a house. If Alice and Bob do not take any breaks, they will finish painting the house in 20 hours. If, however, Bob stops painting once the house is half-finished, then the

### Solving recurrences. CS 4407, Algorithms University College Cork, Gregory M. Provan

Solving recurrences The analysis of divide and conquer algorithms require us to solve a recurrence. Recurrences are a major tool for analysis of algorithms MergeSort A L G O R I T H M S A L G O R I T H

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Chapter 1 Quadratic Equations in One Unknown (I)

Tin Ka Ping Secondary School 015-016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real

### Problem Set #1 Solutions

CS161: Design and Analysis of Algorithms Summer 2004 Problem Set #1 Solutions General Notes Regrade Policy: If you believe an error has been made in the grading of your problem set, you may resubmit it

### Week 9-10: Recurrence Relations and Generating Functions

Week 9-10: Recurrence Relations and Generating Functions March 31, 2015 1 Some number sequences An infinite sequence (or just a sequence for short is an ordered array a 0, a 1, a 2,..., a n,... of countably

### Laboratory Assignment 4. Fourier Sound Synthesis

Laboratory Assignment 4 Fourier Sound Synthesis PURPOSE This lab investigates how to use a computer to evaluate the Fourier series for periodic signals and to synthesize audio signals from Fourier series

### On Polynomial Multiplication Complexity in Chebyshev Basis

On Polynomial Multiplication Complexity in Chebyshev Basis Pascal Giorgi INRIA Rocquencourt, Algorithms Project s Seminar November 29, 2010. Outline 1 Polynomial multiplication in monomial basis 2 Polynomial