Fast Fourier Transform and MATLAB Implementation
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The Fast Fourier Transform
The Fast Fourier Transform Derek L. Smith SIAM Seminar on Algorithms - Fall 2014 University of California, Santa Barbara October 15, 2014 Table of Contents History of the FFT The Discrete Fourier Transform The Fast Fourier Transform MP3 Compression via the DFT The Fourier Transform in Mathematics Table of Contents History of the FFT The Discrete Fourier Transform The Fast Fourier Transform MP3 Compression via the DFT The Fourier Transform in Mathematics Navigating the Origins of the FFT The Royal Observatory, Greenwich, in London has a stainless steel strip on the ground marking the original location of the prime meridian. There's also a plaque stating that the GPS reference meridian is now 100m to the east. This photo is the culmination of hundreds of years of mathematical tricks which answer the question: How to construct a more accurate clock? Or map? Or star chart? Time, Location and the Stars The answer involves a naturally occurring reference system. Throughout history, humans have measured their location on earth, in order to more accurately describe the position of astronomical bodies, in order to build better time-keeping devices, to more successfully navigate the earth, to more accurately record the stars... and so on... and so on... Time, Location and the Stars Transoceanic exploration previously required a vessel stocked with maps, star charts and a highly accurate clock. Institutions such as the Royal Observatory primarily existed to improve a nations' navigation capabilities. The current state-of-the-art includes atomic clocks, GPS and computerized maps, as well as a whole constellation of government organizations. -
Fourier Series
Academic Press Encyclopedia of Physical Science and Technology Fourier Series James S. Walker Department of Mathematics University of Wisconsin–Eau Claire Eau Claire, WI 54702–4004 Phone: 715–836–3301 Fax: 715–836–2924 e-mail: [email protected] 1 2 Encyclopedia of Physical Science and Technology I. Introduction II. Historical background III. Definition of Fourier series IV. Convergence of Fourier series V. Convergence in norm VI. Summability of Fourier series VII. Generalized Fourier series VIII. Discrete Fourier series IX. Conclusion GLOSSARY ¢¤£¦¥¨§ Bounded variation: A function has bounded variation on a closed interval ¡ ¢ if there exists a positive constant © such that, for all finite sets of points "! "! $#&% (' #*) © ¥ , the inequality is satisfied. Jordan proved that a function has bounded variation if and only if it can be expressed as the difference of two non-decreasing functions. Countably infinite set: A set is countably infinite if it can be put into one-to-one £0/"£ correspondence with the set of natural numbers ( +,£¦-.£ ). Examples: The integers and the rational numbers are countably infinite sets. "! "!;: # # 123547698 Continuous function: If , then the function is continuous at the point : . Such a point is called a continuity point for . A function which is continuous at all points is simply referred to as continuous. Lebesgue measure zero: A set < of real numbers is said to have Lebesgue measure ! $#¨CED B ¢ £¦¥ zero if, for each =?>A@ , there exists a collection of open intervals such ! ! D D J# K% $#L) ¢ £¦¥ ¥ ¢ = that <GFIH and . Examples: All finite sets, and all countably infinite sets, have Lebesgue measure zero. "! "! % # % # Odd and even functions: A function is odd if for all in its "! "! % # # domain. -
FOURIER TRANSFORM Very Broadly Speaking, the Fourier Transform Is a Systematic Way to Decompose “Generic” Functions Into
FOURIER TRANSFORM TERENCE TAO Very broadly speaking, the Fourier transform is a systematic way to decompose “generic” functions into a superposition of “symmetric” functions. These symmetric functions are usually quite explicit (such as a trigonometric function sin(nx) or cos(nx)), and are often associated with physical concepts such as frequency or energy. What “symmetric” means here will be left vague, but it will usually be associated with some sort of group G, which is usually (though not always) abelian. Indeed, the Fourier transform is a fundamental tool in the study of groups (and more precisely in the representation theory of groups, which roughly speaking describes how a group can define a notion of symmetry). The Fourier transform is also related to topics in linear algebra, such as the representation of a vector as linear combinations of an orthonormal basis, or as linear combinations of eigenvectors of a matrix (or a linear operator). To give a very simple prototype of the Fourier transform, consider a real-valued function f : R → R. Recall that such a function f(x) is even if f(−x) = f(x) for all x ∈ R, and is odd if f(−x) = −f(x) for all x ∈ R. A typical function f, such as f(x) = x3 + 3x2 + 3x + 1, will be neither even nor odd. However, one can always write f as the superposition f = fe + fo of an even function fe and an odd function fo by the formulae f(x) + f(−x) f(x) − f(−x) f (x) := ; f (x) := . e 2 o 2 3 2 2 3 For instance, if f(x) = x + 3x + 3x + 1, then fe(x) = 3x + 1 and fo(x) = x + 3x. -
Finding Pi Project
Name: ________________________ Finding Pi - Activity Objective: You may already know that pi (π) is a number that is approximately equal to 3.14. But do you know where the number comes from? Let's measure some round objects and find out. Materials: • 6 circular objects Some examples include a bicycle wheel, kiddie pool, trash can lid, DVD, steering wheel, or clock face. Be sure each object you choose is shaped like a perfect circle. • metric tape measure Be sure your tape measure has centimeters on it. • calculator It will save you some time because dividing with decimals can be tricky. • “Finding Pi - Table” worksheet It may be attached to this page, or on the back. What to do: Step 1: Choose one of your circular objects. Write the name of the object on the “Finding Pi” table. Step 2: With the centimeter side of your tape measure, accurately measure the distance around the outside of the circle (the circumference). Record your measurement on the table. Step 3: Next, measure the distance across the middle of the object (the diameter). Record your measurement on the table. Step 4: Use your calculator to divide the circumference by the diameter. Write the answer on the table. If you measured carefully, the answer should be about 3.14, or π. Repeat steps 1 through 4 for each object. Super Teacher Worksheets - www.superteacherworksheets.com Name: ________________________ “Finding Pi” Table Measure circular objects and complete the table below. If your measurements are accurate, you should be able to calculate the number pi (3.14). Is your answer name of circumference diameter circumference ÷ approximately circular object measurement (cm) measurement (cm) diameter equal to π? 1. -
An Introduction to Fourier Analysis Fourier Series, Partial Differential Equations and Fourier Transforms
An Introduction to Fourier Analysis Fourier Series, Partial Differential Equations and Fourier Transforms Notes prepared for MA3139 Arthur L. Schoenstadt Department of Applied Mathematics Naval Postgraduate School Code MA/Zh Monterey, California 93943 August 18, 2005 c 1992 - Professor Arthur L. Schoenstadt 1 Contents 1 Infinite Sequences, Infinite Series and Improper Integrals 1 1.1Introduction.................................... 1 1.2FunctionsandSequences............................. 2 1.3Limits....................................... 5 1.4TheOrderNotation................................ 8 1.5 Infinite Series . ................................ 11 1.6ConvergenceTests................................ 13 1.7ErrorEstimates.................................. 15 1.8SequencesofFunctions.............................. 18 2 Fourier Series 25 2.1Introduction.................................... 25 2.2DerivationoftheFourierSeriesCoefficients.................. 26 2.3OddandEvenFunctions............................. 35 2.4ConvergencePropertiesofFourierSeries.................... 40 2.5InterpretationoftheFourierCoefficients.................... 48 2.6TheComplexFormoftheFourierSeries.................... 53 2.7FourierSeriesandOrdinaryDifferentialEquations............... 56 2.8FourierSeriesandDigitalDataTransmission.................. 60 3 The One-Dimensional Wave Equation 70 3.1Introduction.................................... 70 3.2TheOne-DimensionalWaveEquation...................... 70 3.3 Boundary Conditions ............................... 76 3.4InitialConditions................................ -
Lecture 11 : Discrete Cosine Transform Moving Into the Frequency Domain
Lecture 11 : Discrete Cosine Transform Moving into the Frequency Domain Frequency domains can be obtained through the transformation from one (time or spatial) domain to the other (frequency) via Fourier Transform (FT) (see Lecture 3) — MPEG Audio. Discrete Cosine Transform (DCT) (new ) — Heart of JPEG and MPEG Video, MPEG Audio. Note : We mention some image (and video) examples in this section with DCT (in particular) but also the FT is commonly applied to filter multimedia data. External Link: MIT OCW 8.03 Lecture 11 Fourier Analysis Video Recap: Fourier Transform The tool which converts a spatial (real space) description of audio/image data into one in terms of its frequency components is called the Fourier transform. The new version is usually referred to as the Fourier space description of the data. We then essentially process the data: E.g . for filtering basically this means attenuating or setting certain frequencies to zero We then need to convert data back to real audio/imagery to use in our applications. The corresponding inverse transformation which turns a Fourier space description back into a real space one is called the inverse Fourier transform. What do Frequencies Mean in an Image? Large values at high frequency components mean the data is changing rapidly on a short distance scale. E.g .: a page of small font text, brick wall, vegetation. Large low frequency components then the large scale features of the picture are more important. E.g . a single fairly simple object which occupies most of the image. The Road to Compression How do we achieve compression? Low pass filter — ignore high frequency noise components Only store lower frequency components High pass filter — spot gradual changes If changes are too low/slow — eye does not respond so ignore? Low Pass Image Compression Example MATLAB demo, dctdemo.m, (uses DCT) to Load an image Low pass filter in frequency (DCT) space Tune compression via a single slider value n to select coefficients Inverse DCT, subtract input and filtered image to see compression artefacts. -
Parallel Fast Fourier Transform Transforms
Parallel Fast Fourier Parallel Fast Fourier Transform Transforms Massimiliano Guarrasi – [email protected] MassimilianoSuper Guarrasi Computing Applications and Innovation Department [email protected] Fourier Transforms ∞ H ()f = h(t)e2πift dt ∫−∞ ∞ h(t) = H ( f )e−2πift df ∫−∞ Frequency Domain Time Domain Real Space Reciprocal Space 2 of 49 Discrete Fourier Transform (DFT) In many application contexts the Fourier transform is approximated with a Discrete Fourier Transform (DFT): N −1 N −1 ∞ π π π H ()f = h(t)e2 if nt dt ≈ h e2 if ntk ∆ = ∆ h e2 if ntk n ∫−∞ ∑ k ∑ k k =0 k=0 f = n / ∆ = ∆ n tk k / N = ∆ fn n / N −1 () = ∆ 2πikn / N H fn ∑ hk e k =0 The last expression is periodic, with period N. It define a ∆∆∆ between 2 sets of numbers , Hn & hk ( H(f n) = Hn ) 3 of 49 Discrete Fourier Transforms (DFT) N −1 N −1 = 2πikn / N = 1 −2πikn / N H n ∑ hk e hk ∑ H ne k =0 N n=0 frequencies from 0 to fc (maximum frequency) are mapped in the values with index from 0 to N/2-1, while negative ones are up to -fc mapped with index values of N / 2 to N Scale like N*N 4 of 49 Fast Fourier Transform (FFT) The DFT can be calculated very efficiently using the algorithm known as the FFT, which uses symmetry properties of the DFT s um. 5 of 49 Fast Fourier Transform (FFT) exp(2 πi/N) DFT of even terms DFT of odd terms 6 of 49 Fast Fourier Transform (FFT) Now Iterate: Fe = F ee + Wk/2 Feo Fo = F oe + Wk/2 Foo You obtain a series for each value of f n oeoeooeo..oe F = f n Scale like N*logN (binary tree) 7 of 49 How to compute a FFT on a distributed memory system 8 of 49 Introduction • On a 1D array: – Algorithm limits: • All the tasks must know the whole initial array • No advantages in using distributed memory systems – Solutions: • Using OpenMP it is possible to increase the performance on shared memory systems • On a Multi-Dimensional array: – It is possible to use distributed memory systems 9 of 49 Multi-dimensional FFT( an example ) 1) For each value of j and k z Apply FFT to h( 1.. -
Evaluating Fourier Transforms with MATLAB
ECE 460 – Introduction to Communication Systems MATLAB Tutorial #2 Evaluating Fourier Transforms with MATLAB In class we study the analytic approach for determining the Fourier transform of a continuous time signal. In this tutorial numerical methods are used for finding the Fourier transform of continuous time signals with MATLAB are presented. Using MATLAB to Plot the Fourier Transform of a Time Function The aperiodic pulse shown below: x(t) 1 t -2 2 has a Fourier transform: X ( jf ) = 4sinc(4π f ) This can be found using the Table of Fourier Transforms. We can use MATLAB to plot this transform. MATLAB has a built-in sinc function. However, the definition of the MATLAB sinc function is slightly different than the one used in class and on the Fourier transform table. In MATLAB: sin(π x) sinc(x) = π x Thus, in MATLAB we write the transform, X, using sinc(4f), since the π factor is built in to the function. The following MATLAB commands will plot this Fourier Transform: >> f=-5:.01:5; >> X=4*sinc(4*f); >> plot(f,X) In this case, the Fourier transform is a purely real function. Thus, we can plot it as shown above. In general, Fourier transforms are complex functions and we need to plot the amplitude and phase spectrum separately. This can be done using the following commands: >> plot(f,abs(X)) >> plot(f,angle(X)) Note that the angle is either zero or π. This reflects the positive and negative values of the transform function. Performing the Fourier Integral Numerically For the pulse presented above, the Fourier transform can be found easily using the table. -
Fourier Transforms & the Convolution Theorem
Convolution, Correlation, & Fourier Transforms James R. Graham 11/25/2009 Introduction • A large class of signal processing techniques fall under the category of Fourier transform methods – These methods fall into two broad categories • Efficient method for accomplishing common data manipulations • Problems related to the Fourier transform or the power spectrum Time & Frequency Domains • A physical process can be described in two ways – In the time domain, by h as a function of time t, that is h(t), -∞ < t < ∞ – In the frequency domain, by H that gives its amplitude and phase as a function of frequency f, that is H(f), with -∞ < f < ∞ • In general h and H are complex numbers • It is useful to think of h(t) and H(f) as two different representations of the same function – One goes back and forth between these two representations by Fourier transforms Fourier Transforms ∞ H( f )= ∫ h(t)e−2πift dt −∞ ∞ h(t)= ∫ H ( f )e2πift df −∞ • If t is measured in seconds, then f is in cycles per second or Hz • Other units – E.g, if h=h(x) and x is in meters, then H is a function of spatial frequency measured in cycles per meter Fourier Transforms • The Fourier transform is a linear operator – The transform of the sum of two functions is the sum of the transforms h12 = h1 + h2 ∞ H ( f ) h e−2πift dt 12 = ∫ 12 −∞ ∞ ∞ ∞ h h e−2πift dt h e−2πift dt h e−2πift dt = ∫ ( 1 + 2 ) = ∫ 1 + ∫ 2 −∞ −∞ −∞ = H1 + H 2 Fourier Transforms • h(t) may have some special properties – Real, imaginary – Even: h(t) = h(-t) – Odd: h(t) = -h(-t) • In the frequency domain these -
STATISTICAL FOURIER ANALYSIS: CLARIFICATIONS and INTERPRETATIONS by DSG Pollock
STATISTICAL FOURIER ANALYSIS: CLARIFICATIONS AND INTERPRETATIONS by D.S.G. Pollock (University of Leicester) Email: stephen [email protected] This paper expounds some of the results of Fourier theory that are es- sential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose the structure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite data sequences. An ideal filter with a gain of unity throughout the pass band and a gain of zero throughout the stop band is commonly regarded as incapable of being realised in finite samples. It is shown here that, to the contrary, such a filter can be realised both in the time domain and in the frequency domain. The algebra of circulant matrices is also helpful in revealing the nature of statistical processes that are band limited in the frequency domain. In order to apply the conventional techniques of autoregressive moving-average modelling, the data generated by such processes must be subjected to anti- aliasing filtering and sub sampling. These techniques are also described. It is argued that band-limited processes are more prevalent in statis- tical and econometric time series than is commonly recognised. 1 D.S.G. POLLOCK: Statistical Fourier Analysis 1. Introduction Statistical Fourier analysis is an important part of modern time-series analysis, yet it frequently poses an impediment that prevents a full understanding of temporal stochastic processes and of the manipulations to which their data are amenable. This paper provides a survey of the theory that is not overburdened by inessential complications, and it addresses some enduring misapprehensions. -
Fourier Transform, Convolution Theorem, and Linear Dynamical Systems April 28, 2016
Mathematical Tools for Neuroscience (NEU 314) Princeton University, Spring 2016 Jonathan Pillow Lecture 23: Fourier Transform, Convolution Theorem, and Linear Dynamical Systems April 28, 2016. Discrete Fourier Transform (DFT) We will focus on the discrete Fourier transform, which applies to discretely sampled signals (i.e., vectors). Linear algebra provides a simple way to think about the Fourier transform: it is simply a change of basis, specifically a mapping from the time domain to a representation in terms of a weighted combination of sinusoids of different frequencies. The discrete Fourier transform is therefore equiv- alent to multiplying by an orthogonal (or \unitary", which is the same concept when the entries are complex-valued) matrix1. For a vector of length N, the matrix that performs the DFT (i.e., that maps it to a basis of sinusoids) is an N × N matrix. The k'th row of this matrix is given by exp(−2πikt), for k 2 [0; :::; N − 1] (where we assume indexing starts at 0 instead of 1), and t is a row vector t=0:N-1;. Recall that exp(iθ) = cos(θ) + i sin(θ), so this gives us a compact way to represent the signal with a linear superposition of sines and cosines. The first row of the DFT matrix is all ones (since exp(0) = 1), and so the first element of the DFT corresponds to the sum of the elements of the signal. It is often known as the \DC component". The next row is a complex sinusoid that completes one cycle over the length of the signal, and each subsequent row has a frequency that is an integer multiple of this \fundamental" frequency. -
A Hilbert Space Theory of Generalized Graph Signal Processing
1 A Hilbert Space Theory of Generalized Graph Signal Processing Feng Ji and Wee Peng Tay, Senior Member, IEEE Abstract Graph signal processing (GSP) has become an important tool in many areas such as image pro- cessing, networking learning and analysis of social network data. In this paper, we propose a broader framework that not only encompasses traditional GSP as a special case, but also includes a hybrid framework of graph and classical signal processing over a continuous domain. Our framework relies extensively on concepts and tools from functional analysis to generalize traditional GSP to graph signals in a separable Hilbert space with infinite dimensions. We develop a concept analogous to Fourier transform for generalized GSP and the theory of filtering and sampling such signals. Index Terms Graph signal proceesing, Hilbert space, generalized graph signals, F-transform, filtering, sampling I. INTRODUCTION Since its emergence, the theory and applications of graph signal processing (GSP) have rapidly developed (see for example, [1]–[10]). Traditional GSP theory is essentially based on a change of orthonormal basis in a finite dimensional vector space. Suppose G = (V; E) is a weighted, arXiv:1904.11655v2 [eess.SP] 16 Sep 2019 undirected graph with V the vertex set of size n and E the set of edges. Recall that a graph signal f assigns a complex number to each vertex, and hence f can be regarded as an element of n C , where C is the set of complex numbers. The heart of the theory is a shift operator AG that is usually defined using a property of the graph.