Harvey Mudd College, MATH 197
Excerpt: ... What is DFT? Fast Fourier Transform Fast Fourier Transforms for the Symmetric Group Masanori Koyama Harvey Mudd College March 31, 2007 Masanori Koyama Fast Fourier Transforms for the Symmetric Group What is DFT? Fast Fourier Transform Discrete Fourier Transform Masanori Koyama Fast Fourier Transforms for the Symmetric Group What is DFT? Fast Fourier Transform Group Algebra CG If G is a group, CG is a vector space with elements in G as a standard basis. Masanori Koyama Fast Fourier Transforms for the Symmetric Group What is DFT? Fast Fourier Transform Example : An element of CS3 1(1) + 2(12) + 8(132) + (7 + i)(123) In the basis {(1), (12), (23), (123), (132), (13)} this can be written as 1 2 8 0 7 + i 0 0 Masanori Koyama Fast Fourier Transforms for the Symmetric Group What is DFT? Fast Fourier Transform Convolution We can multiply two elements in CG cg g g G and g G dg g . Example (1(12) + 3(23) ( ...
San Diego State, MATH 15
Excerpt: ... n+k (x) = sin(kx), 1 2 k (x) = cos(kx), 0 (x) = k = 1, . . . , n k = 1, . . . , n - 1 mahaffy@math.sdsu.edu http:/www-rohan.sdsu.edu/jmahaffy n < m [-, ] 2m xj = - + (j/m), j = 0, 1, . . . , (2m - 1). n = m $Id: lecture.tex,v 1.15 2007/12/05 23:05:11 blomgren Exp $ Approximation Theory: The Fast Fourier Transform, with Applications p. 1/40 Approximation Theory: The Fast Fourier Transform, with Applications p. 2/40 ...
Johns Hopkins, MATH 443
Excerpt: ... LECTURE 35. THE FAST FOURIER TRANSFORM FOURIER ANALYSIS (110.443) PROF. QIAO ZHANG Let f be a function defined over Z/N Z. Assume that we have obtained the function values f (0), f (1), . . . , f (N - 1) and want to compute its Fourier transform, which is equivalent to computing all the Fourier coefficients. 1. The Naive Approach Direct computations from the definition 1 ^ f (m) = N N -1 f (n) -mn . n=0 requires N + 2N 2 operations to obtain all the Fourier coefficients. 2. The Fast Fourier Transform Theorem 2.1 (FFT). Let N = 2n . There exists an algorithm that calculates all the Fourier coefficients of a function on Z/N Z with 4n 2n operations. More precisely, let f be a function defined over Z/N Z = Z/2n Z, then we have inductively 1 ^ -m ^ ^ (2.1) f (m) = (f0 (m) + 2n f1 (m), 2 where (2.2) f0 (n) = f (2n), f1 (n) = f (2n + 1). 3. Summary In this lecture, we have learned to apply the fast Fourier transform; analyze the complexity of the fast Fourier transform. Today's homework is the following e ...
Georgia Tech, VISIBLE 3
Excerpt: ... Lab 19: The Fast Fourier Transform The goal of this lab is to introduce the Fast Fourier Transform (FFT) for efficient computer calculation of the Fourier Transform and then to investigate some of the Fourier Tranform's properties. ...
Montana, CS 531
Excerpt: ... CS 531, Algorithms Assignment 7 April 6, 2007 Alden Wright, University of Montana c 2007 Due date: Problems 1, 2: Friday, April 6. Problems 3, 4, 5: Friday, April 13. Reading: Read chapter 5 of the text with emphasis on the topics covered in lecture. 1. Chapter 5, exercise 5 (continued from assignment 6). Hint: Sort the lines by slope. Divide the lines into two groups, those with slope less than or equal to the median slope, and those with slope greater than the median slop. Then recursively find the visible lines in each group, and figure out how to combine these solutions 2. Carry out the Fast Fourier Transform to find the product of the polynomials A(x) = 2 + 3x and B(x) = 1 - 2x (or the example that I gave in class which I can't seem to find right now). 3. Implement the Recursive Fast Fourier transform in a language of your choice. MATLAB might be a good choice. 4. Prove the lemma at the end of the Fast Fourier Transform handout. 5. Recall that for the weighted interval selection problem, p[j] = largest ...
Colorado, ASEN 5519
Excerpt: ... . THE FAST FOURIER TRANSFORM 3 31 Chapter 3: THE FAST FOURIER TRANSFORM 32 3.1 FFT IN A NUTSHELL Even though this course is not about any numerical recipe, in order for us to utilize the numerical tools in intelligent ways, it pays to understand the essnetial features of the tools we would frequently use. the Fast Fourier Transform (FFT) is one such numerical tool. According to G. Strang, FFT is perhaps the most widely used numerical tool today in scientic and engineering computations. To start our discussion of FFT, let us recall MATLAB c version the Fourier transform: N 1 n=0 X (k) = x(s)e j2kn/N , k = 0, 1, . . . , N 1 (3.1) where the denominator 1/N is omitted for brevity. The one thing we have taken for granted is that for every kthe Fourier coefcient term, X (k), one must perform N-complex term multiplications and N additions, thus requiring N 2 multiplications in order to obtain the complete coefcients. Hence, for a measurement data set of 100 with 1,024 sample ...
Cornell, ECE 2200
Excerpt: ... The sampling frequencies proposed are: 16 kHz, 14 kHz, 12 kHz, 8 kHz, 5 kHz, 4 kHz. Problem 4 Consider the sequence x[n] = cos( 2 n) for n = 0, 1, ., 31. 16 (a) For what values of k is the DFT X[k] of the sequence x[n] non zero? Hint: Note that the other way of posing this problem is to consider the sequence x[n] as a vector x = (x[n])31 and the DFT transform as a matrix F32 . Then the DFT of x is n=0 X = F32 x. The question asks which components X[k] of the vector X are non zero. (b) Assume that in (a) we set all odd numbered frequency samples X[k] (odd components of X) to zero and call the new frequency sequence as Xodd [k] (or vector Xodd ). (The length of the vector Xodd remains 32). What is the corresponding to Xodd [k] the time sequence xodd [n]? (c) Repeat (b) when the even samples of X[k] are set to zero. 2 Problem 5 There exist digital circuits which implement Fast Fourier Transforms of a fixed size. Those are designed for ultimate speed that cannot be achieved in software. Assume that you ...
Maple Springs, CSE 4080
Excerpt: ... CSE 4080.03 Computer Science Project : Advanced Algorithm Design and Analysis Project description: The goal of this project is to survey the key theory topics that are appropriate for advanced computer science students. The student will study broad range of algorithms while attending graduate course CSE 6111. In about two weeks for each selected topic, we will gain insights into the basics and study several examples in depth. These might include: - Divide and Conquer: fast Fourier transformations - Recursion: parsing - Greedy Algorithms: matroids - Amortized Analysis: union find - Dynamic Programming: parsing CFG - Network Flow: steepest assent, Edmonds-Karp, matching - Randomized Algorithms: Chernoff bounds, primes - NP-completeness: reductions - Approximation Algorithms: knapsack - Linear Programming: what to put in a hotdog - Distributed Systems: mud on forehead & common knowledge - Computability: reducibilities and oracle computations - Concurrency Theory: - Cryptography: RSA - Stru ...
Stanford, ENERGY 211
Excerpt: ... in half, and each half is sorted individually The two sorted halves are then merged in O(n) comparisons, for O(n log2n) n n comparisons overall Limiting case of recursion: an array with one element is already sorted! 5 Fast Fourier Transform The Fast Fourier Transform (FFT) computes the discrete Fourier transform of a vector of functions values It recursively transforms odd-numbered and even-numbered points, and then merges the results O(n2) work reduced to O(n log2n) n n For efficiency, should use pointers that take larger and larger steps to access elements! 6 3 Debuggers A debugger is a tool that performs stepby-step execution of a program, allowing the programmer to study its behavior and find bugs IDEs, like MDS and Microsoft Visual Studio, typically include debuggers A debugger runs your program, stopping execution at user-specified breakpoints to allow inspection 7 The MDS Debugger Before debugging, use F9 to set breakpoints in your program F5 begins exe ...
Berkeley, HISTORY 190
Excerpt: ... Physics H190 Spring 2003 Homework 6 Due Wednesday, March 12, 2003 Reading Assignment: Read pp. 5460 of the book, and study lecture notes for Mar. 5. There are not many pages of the book for this week, but the book goes over the material fairly super ...
Sanford-Brown Institute, CSCI 2560
Excerpt: ... CS256 Applied Theory of Computation Memory Hierarchy Tradeoffs III John E Savage Overview Application of the Hong-Kung Bound to the Fast Fourier Transform Lecture 18 Memory Hierachies III CS256 @John E Savage 2 Fast Fourier Transform Definition The S-span of DAG G, (S,G), is the maximum number of vertices of G that can be pebbled with S red pebbles in red pebble game maximized over all initial placements of S red pebbles. (Initialization rule is disallowed.) Theorem For every pebbling P of G = (V,E) in the red-blue pebble game with S red pebbles, the I/O time used, T2(2)(S,G,P) satisfies Lecture 18 Memory Hierachies III CS256 @John E Savage 3 Fast Fourier Transform (FFT) We derive matching upper and lower bounds on I/O complexity and number of computation steps to pebble the FFT graph F(d) on n = 2d inputs with S red pebbles. Lecture 18 Memory Hierachies III CS256 @John E Savage 4 Fast Fourier Transform Lemma The S-span of the FFT graph F(d) on n = 2d inputs satisfies (S,F(d) 2S log2 S ...
CSU Long Beach, CECS 570
Excerpt: ... ssages (various protocols, including pointto-point, synchronous, asynchronous, blocking, non-blocking, and group message passing) MPI support for both aspects Evaluating parallel programs Analytical evaluation (find computational time for sequential solution and parallel solution, and communication time for parallel solution, then assess speedup factor based on these timing figures) Empirical evaluation (use visualization tools, measure execution times) Chapter 4 and lecture notes: Partitioning and divide and conquer Example problems that can be solved using divide and conquer The Summation problem (with or without result communicated to all processes, performance comparison when different communication structures are used) The Prefix Sum problem Fast Fourier Transform, and its application to polynomial multiplication Bucket Sort Numerical Integration The Gravitational N-Body problem Chapter 5: Pipelined computations Three types of computations where pipelining can be beneficial Type 1: more tha ...
W. Alabama, CS 770
Excerpt: ... Discrete Fourier Transform CS 770G - Parallel Algorithms in Scientific Computing Fast Fourier Transform Let i=sqrt(-1) and index matrices and vectors from 0. The DFT of a vector v of dimension n is: F*v where F is the n*n matrix defined as: F[j,k] = (j*k) is: = e (-2i/n) = cos(2/n) - i*sin(2/n) This is a complex number with whose nth power is 1 and is therefore called the nth root of unity. 2 July 3 , 2001 Lecture 12 Fast Fourier Transform Generally attributed to Cooley and Turkey (1965), but FFT algorithm in Gauss notes (1805) Several different algorithms available: Decimation in time Decimation in frequency Prime factor algorithm Bluestein approach . Application of FFT Numerical integration Spectral methods for the solution of PDE Fast Poisson solvers Image processing Digital filtering It reduces the computational complexity from O(n^2) to O(n log n) References: Van Loan "Computational Frameworks for the FFT", SIAM Briggs and Henson " The DFT", SIAM 3 4 1 Image Comp ...
University of Illinois, Urbana Champaign, ECE 410
Excerpt: ... Connexions module: m10783 1 The Fast Fourier Transform Version 2.5: 2004/06/02 10:07:40 GMT-5 (FFT) Justin Romberg This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract This module describes the fast Fourier Transform (FFT). The Fast Fourier Transform (FFT) 1 Introduction The Fast Fourier Transform (FFT) is an ecient O(NlogN) algorithm for calculating DFTs originally discovered by Gauss in the early 1800's rediscovered by Cooley and Tukey at IBM in the 1960's C.S. Burrus, Rice University's very own Dean of Engineering, literally "wrote the book" on fast DFT algorithms. The FFT 1 exploits symmetries in the W matrix to take a "divide and conquer" approach. We won't talk about the actual FFT algorithm here, see these notes2 if you are interested in reading a little more on the idea behind FFT. 2 Speed Comparison How much better is O(NlogN) than O( N 2 )? N N2 N logN 10 100 1 100 104 200 1000 106 3000 106 1012 6 106 ...