A Brief Study of Discrete and Fast Fourier Transforms
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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. -
Convolution! (CDT-14) Luciano Da Fontoura Costa
Convolution! (CDT-14) Luciano da Fontoura Costa To cite this version: Luciano da Fontoura Costa. Convolution! (CDT-14). 2019. hal-02334910 HAL Id: hal-02334910 https://hal.archives-ouvertes.fr/hal-02334910 Preprint submitted on 27 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Convolution! (CDT-14) Luciano da Fontoura Costa [email protected] S~aoCarlos Institute of Physics { DFCM/USP October 22, 2019 Abstract The convolution between two functions, yielding a third function, is a particularly important concept in several areas including physics, engineering, statistics, and mathematics, to name but a few. Yet, it is not often so easy to be conceptually understood, as a consequence of its seemingly intricate definition. In this text, we develop a conceptual framework aimed at hopefully providing a more complete and integrated conceptual understanding of this important operation. In particular, we adopt an alternative graphical interpretation in the time domain, allowing the shift implied in the convolution to proceed over free variable instead of the additive inverse of this variable. In addition, we discuss two possible conceptual interpretations of the convolution of two functions as: (i) the `blending' of these functions, and (ii) as a quantification of `matching' between those functions. -
The Convolution Theorem
Module 2 : Signals in Frequency Domain Lecture 18 : The Convolution Theorem Objectives In this lecture you will learn the following We shall prove the most important theorem regarding the Fourier Transform- the Convolution Theorem We are going to learn about filters. Proof of 'the Convolution theorem for the Fourier Transform'. The Dual version of the Convolution Theorem Parseval's theorem The Convolution Theorem We shall in this lecture prove the most important theorem regarding the Fourier Transform- the Convolution Theorem. It is this theorem that links the Fourier Transform to LSI systems, and opens up a wide range of applications for the Fourier Transform. We shall inspire its importance with an application example. Modulation Modulation refers to the process of embedding an information-bearing signal into a second carrier signal. Extracting the information- bearing signal is called demodulation. Modulation allows us to transmit information signals efficiently. It also makes possible the simultaneous transmission of more than one signal with overlapping spectra over the same channel. That is why we can have so many channels being broadcast on radio at the same time which would have been impossible without modulation There are several ways in which modulation is done. One technique is amplitude modulation or AM in which the information signal is used to modulate the amplitude of the carrier signal. Another important technique is frequency modulation or FM, in which the information signal is used to vary the frequency of the carrier signal. Let us consider a very simple example of AM. Consider the signal x(t) which has the spectrum X(f) as shown : Why such a spectrum ? Because it's the simplest possible multi-valued function. -
B1. Fourier Analysis of Discrete Time Signals
B1. Fourier Analysis of Discrete Time Signals Objectives • Introduce discrete time periodic signals • Define the Discrete Fourier Series (DFS) expansion of periodic signals • Define the Discrete Fourier Transform (DFT) of signals with finite length • Determine the Discrete Fourier Transform of a complex exponential 1. Introduction In the previous chapter we defined the concept of a signal both in continuous time (analog) and discrete time (digital). Although the time domain is the most natural, since everything (including our own lives) evolves in time, it is not the only possible representation. In this chapter we introduce the concept of expanding a generic signal in terms of elementary signals, such as complex exponentials and sinusoids. This leads to the frequency domain representation of a signal in terms of its Fourier Transform and the concept of frequency spectrum so that we characterize a signal in terms of its frequency components. First we begin with the introduction of periodic signals, which keep repeating in time. For these signals it is fairly easy to determine an expansion in terms of sinusoids and complex exponentials, since these are just particular cases of periodic signals. This is extended to signals of a finite duration which becomes the Discrete Fourier Transform (DFT), one of the most widely used algorithms in Signal Processing. The concepts introduced in this chapter are at the basis of spectral estimation of signals addressed in the next two chapters. 2. Periodic Signals VIDEO: Periodic Signals (19:45) http://faculty.nps.edu/rcristi/eo3404/b-discrete-fourier-transform/videos/chapter1-seg1_media/chapter1-seg1- 0.wmv http://faculty.nps.edu/rcristi/eo3404/b-discrete-fourier-transform/videos/b1_02_periodicSignals.mp4 In this section we define a class of discrete time signals called Periodic Signals. -
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. -
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................................ -
The Discrete-Time Fourier Transform and Convolution Theorems: a Brief Tutorial
The Discrete-Time Fourier Transform and Convolution Theorems: A Brief Tutorial Yi-Wen Liu 25 Feb 2013 (Revised 21 Sep 2015) 1 Definitions and interpretation 1.1 Units Throughout this semester, we will use the integer-valued variable n as the time variable for discrete-time signal processing; that is, n = −∞, ..., −1, 0, 1, ..., ∞. In this convention, the unit of n is dimensionless, but be aware that in reality the sampling period is 1/fs, where fs denotes the sampling rate. In some text books, 1/fs is denoted as T , and its unit is second. In this course we do not do this, thereby favoring conciseness of notations. 1.2 Definition of DTFT The discrete-time Fourier transform (DTFT) of a discrete-time signal x[n] is a function of frequency ω defined as follows: ∞ ∆ X(ω) = x[n]e−jωn. (1) n=−∞ X Conceptually, the DTFT allows us to check how much of a tonal component at fre- quency ω is in x[n]. The DTFT of a signal is often also called a spectrum. Note that X(ω) is complex-valued. So, the absolute value |X(ω)| represents the tonal component’s magnitude, and the angle ∠X(ω) tells us the phase of the tonal component. Also note that Eq. (1) defines the DTFT as the inner product between the signal and the complex exponential tone e−jωn. Because complex exponentials {e−jωn,ω ∈ [−π, π]} form a complete family of orthogonal bases for (practically) all signals of interest, what Eq. (1) essentially does is to project x[n] onto the space spanned by all complex exponen- tials. -
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 -
Fourier Analysis
Chapter 1 Fourier analysis In this chapter we review some basic results from signal analysis and processing. We shall not go into detail and assume the reader has some basic background in signal analysis and processing. As basis for signal analysis, we use the Fourier transform. We start with the continuous Fourier transformation. But in applications on the computer we deal with a discrete Fourier transformation, which introduces the special effect known as aliasing. We use the Fourier transformation for processes such as convolution, correlation and filtering. Some special attention is given to deconvolution, the inverse process of convolution, since it is needed in later chapters of these lecture notes. 1.1 Continuous Fourier Transform. The Fourier transformation is a special case of an integral transformation: the transforma- tion decomposes the signal in weigthed basis functions. In our case these basis functions are the cosine and sine (remember exp(iφ) = cos(φ) + i sin(φ)). The result will be the weight functions of each basis function. When we have a function which is a function of the independent variable t, then we can transform this independent variable to the independent variable frequency f via: +1 A(f) = a(t) exp( 2πift)dt (1.1) −∞ − Z In order to go back to the independent variable t, we define the inverse transform as: +1 a(t) = A(f) exp(2πift)df (1.2) Z−∞ Notice that for the function in the time domain, we use lower-case letters, while for the frequency-domain expression the corresponding uppercase letters are used. A(f) is called the spectrum of a(t).