Fourier transform - Fourier transform From Physics Notes...

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Fourier transform From Physics Notes The Fourier transform is one of the most important tools for analyzing functions. The basic idea is that a function can be expressed as a linear combination of more elementary "basis functions", which are sinusoidal waves. In this way, a function originally defined in a spatial domain, , becomes associated with a "Fourier transformed" counterpart , which is defined in a wave-number domain . As we'll see, this is useful because certain mathematical problems (specifically differential equations) are easier to solve in the wave-number domain. Contents 1 Fourier series 1.1 Complex Fourier series and inverse relations 1.2 Example: Fourier series of a square wave 2 Fourier transform 2.1 A simple example 2.2 Fourier transforms for time-domain functions 2.3 Basic properties 2.4 Fourier transforms of differential equations 3 Common Fourier transforms 3.1 Damped waves 3.2 Gaussian wave-packets 4 The delta function 5 Fourier transforms in multiple dimensions 6 Exercises Fourier series We begin by discussing the Fourier series , which is used to analyze functions which are periodic in their inputs. A periodic function is a function of a real variable which satisfies for all . The constant is called the period . The value of could be either real or complex, but in either case we assume that is real. In physics, Fourier series are commonly applied to functions that are either periodic in space or in time. For the moment, let's think of as a spatial coordinate. (For functions of time, there's an important difference in sign conventions, which is discussed below). A periodic function can also be regarded as a function that is defined over a finite domain of length , say . Then the periodicity condition is equivalent to imposing "periodic boundary conditions"—i.e., joining the edges of the finite domain to form a "ring". This is illustrated in Fig. 1. Now, consider what it means to fully specify an arbitrary periodic function . The most straightforward way is to state the value of for every . That's an uncountably infinite set of numbers, which is somewhat cumbersome to deal with. Another way to specify the function would be to write the function as a linear combination of simpler oscillations— specifically, sines and cosines:
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Fig. 1: A periodic function can be regarded as a function defined over a finite, ring-like domain. This is called a Fourier series . The coefficients are real numbers if is a real function, or they can be complex numbers if is complex. Note a slight difference in the sums: starts from 1, but starts from 0. (This is because the term with would just be zero for all .) What makes the Fourier series work is that the sine and cosine functions are designed to be periodic, with period : Hence, any linear combination of these sine and cosine functions will satisfy automatically.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern