Integral transforms | Fourier analysis | Unitary operators
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called analysis. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that frequency, and the argument of the complex value represents that complex sinusoid's phase offset. If a frequency is not present, the transform has a value of 0 for that frequency. The Fourier transform is not limited to functions of time, but the domain of the original function is commonly referred to as the time domain. The Fourier inversion theorem provides a synthesis process that recreates the original function from its frequency domain representation. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the . The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation. The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint. The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on R or Rn (viewed as groups under addition), notably includes the discrete-time Fourier transform (DTFT, group = Z), the discrete Fourier transform (DFT, group = Z mod N) and the Fourier series or circular Fourier transform (group = S1, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT. (Wikipedia).
The Fourier Transform and Derivatives
This video describes how the Fourier Transform can be used to accurately and efficiently compute derivatives, with implications for the numerical solution of differential equations. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow
From playlist Fourier
Electrical Engineering: Ch 19: Fourier Transform (2 of 45) What is a Fourier Transform? Math Def
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the mathematical definition and equation of a Fourier transform. Next video in this series can be seen at: https://youtu.be/yl6RtWp7y4k
From playlist ELECTRICAL ENGINEERING 18: THE FOURIER TRANSFORM
The Two-Dimensional Discrete Fourier Transform
The two-dimensional discrete Fourier transform (DFT) is the natural extension of the one-dimensional DFT and describes two-dimensional signals like images as a weighted sum of two dimensional sinusoids. Two-dimensional sinusoids have a horizontal frequency component and a vertical frequen
From playlist Fourier
What is the Fourier Transform?
In this video, we'll look at the fourier transform from a slightly different perspective than normal, and see how it can be used to estimate functions. Learn about the Fourier series here: http://youtu.be/kP02nBNtjrU
From playlist Fourier
Electrical Engineering: Ch 19: Fourier Transform (1 of 45) What is a Fourier Transform?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is a Fourier transform and how is it different from the Fourier series. Next video in this series can be seen at: https://youtu.be/fMHk6_1ZYEA
From playlist ELECTRICAL ENGINEERING 18: THE FOURIER TRANSFORM
What is a Fourier Transform and how does it relate to the Fourier Series? In this video, we discuss the idea of the Fourier Cosine Transform.
From playlist Mathematical Physics II Uploads
Electrical Engineering: Ch 19: Fourier Transform (3 of 45) What is a Fourier Transform? Simple Ex.
Visit http://ilectureonline.com for more math and science lectures! In this video I will solve F(w)=? of a simple example of a Fourier transform. Next video in this series can be seen at:
From playlist ELECTRICAL ENGINEERING 18: THE FOURIER TRANSFORM
To Understand the Fourier Transform, Start From Quantum Mechanics
Develop a deep understanding of the Fourier transform by appreciating the critical role it plays in quantum mechanics! Get the notes for free here: https://courses.physicswithelliot.com/notes-sign-up Sign up for my newsletter for additional physics lessons: https://www.physicswithelliot.c
From playlist Physics Mini Lessons
Introduction to the z-Transform
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Introduces the definition of the z-transform, the complex plane, and the relationship between the z-transform and the discrete-time Fourier transfor
From playlist The z-Transform
Lecture 13 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). In this lecture, Professor Osgood demonstrates Fourier transforms of a general distribution. The Fourier transform is a tool for solving physical problems. In t
From playlist Lecture Collection | The Fourier Transforms and Its Applications
The discrete-time Fourier transform
The Fourier transform is arguably the most important algorithm in signal processing and communications technology (not to mention neural time series data analysis!). This video provides an in-depth, step-by-step explanation of how the Fourier transform works. The video uses files you can
From playlist OLD ANTS #2) The discrete-time Fourier transform
Lecture 7 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood reintroduces the Fourier Transform and its inverse, then he goes into specific properties and transforms. The Fourier transform is a tool for s
From playlist Lecture Collection | The Fourier Transforms and Its Applications
Lec 8 | MIT RES.6-008 Digital Signal Processing, 1975
Lecture 8: The discrete Fourier series Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES6-008S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6-008 Digital Signal Processing, 1975
12: Spectral Analysis Part 2 - Intro to Neural Computation
MIT 9.40 Introduction to Neural Computation, Spring 2018 Instructor: Michale Fee View the complete course: https://ocw.mit.edu/9-40S18 YouTube Playlist: https://www.youtube.com/playlist?list=PLUl4u3cNGP61I4aI5T6OaFfRK2gihjiMm Covers Fourier transform pairs and power spectra, spectral esti
From playlist MIT 9.40 Introduction to Neural Computation, Spring 2018
ME565 Lecture 15: Properties of Fourier Transforms and Examples
ME565 Lecture 15 Engineering Mathematics at the University of Washington Properties of Fourier Transforms and Examples Notes: http://faculty.washington.edu/sbrunton/me565/pdf/L15.pdf Course Website: http://faculty.washington.edu/sbrunton/me565/ http://faculty.washington.edu/sbrunton/
From playlist Engineering Mathematics (UW ME564 and ME565)
Lecture 8 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood continues lecturing on the general properties of the Fourier Transforms by two paths. First, to develop specific transforms and second, to unders
From playlist Lecture Collection | The Fourier Transforms and Its Applications
Lecture 8, Continuous-Time Fourier Transform | MIT RES.6.007 Signals and Systems, Spring 2011
Lecture 8, Continuous-Time Fourier Transform Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES-6.007S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT RES.6.007 Signals and Systems, 1987
Lecture 21 | The Fourier Transforms and its Applications
Lecture by Professor Brad Osgood for the Electrical Engineering course, The Fourier Transforms and its Applications (EE 261). Professor Osgood continues his lecture on the properties of discrete Fourier transforms. The Fourier transform is a tool for solving physical problems. In this
From playlist Lecture Collection | The Fourier Transforms and Its Applications
This video will discuss the Fourier Transform, which is one of the most important coordinate transformations in all of science and engineering. Book Website: http://databookuw.com Book PDF: http://databookuw.com/databook.pdf These lectures follow Chapter 2 from: "Data-Driven Science an
From playlist Fourier
ME565 Lecture 21: The Laplace Transform
ME565 Lecture 21 Engineering Mathematics at the University of Washington Laplace Transform Notes: http://faculty.washington.edu/sbrunton/me565/pdf/L21.pdf Course Website: http://faculty.washington.edu/sbrunton/me565/ http://faculty.washington.edu/sbrunton/
From playlist Engineering Mathematics (UW ME564 and ME565)