What is Fourier transform: Definition and 1000 Discussions
In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. 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 of time is a complex-valued function of frequency, whose magnitude (absolute value) represents the amount of that frequency present in the original function, and whose argument is the phase offset of the basic sinusoid in 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. There is also an inverse Fourier transform that mathematically synthesizes the original function from its frequency domain representation, as proven by the Fourier inversion theorem.
Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency, so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are "simpler" in one or the other, and has deep connections to many areas of modern mathematics.
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 uncertainty principle. 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.
Homework Statement
I am suppose to write a program that compares the FFT (Fast Fourier Transform Diagrams) of a sampled signal without the use of a window function and with it. The window function should be as long as the signal and the signal should have N points, N chosen as to not cause...
Homework Statement
find the Fourier transform of the following function in two ways , once using direct computation , and second using the convolution therom .
Homework Equations
Acos(w0t)/(d2+t2)
The Attempt at a Solution
I tried first to solve directly . used Euler's identity and got...
I apologize ahead of time for the simplicity of the question, but this has really been bothering me.Given the de Broglie relation, assuming natural units, where ##\hbar = 1##:
\begin{equation}
\begin{split}
\vec{k} &= M \vec{v}
\end{split}
\end{equation}My question regards velocity and...
Hi All! I've been looking at this Fourier Transform integral and I've realized that I'm not sure how to integrate the exponential term to infinity. I would expect the result to be infinity but that wouldn't give me a very useful function. So I've taken it to be zero but I have no idea if you can...
Homework Statement
The Fourier transfrom of the wave function is given by
$$\Phi(p) = \frac{N}{(1+\frac{a_0^2p^2}{\hbar^2})^2}$$
where ##p:=|\vec{p}|## in 3 dimensions.
Find N, choosing N to be a positive real number.
Homework Equations
$$\int d^3\vec{p}|\Phi(p)|^2=1$$
, over all p in the 3...
Please, can anyone explain how formula (5) is obtained in J.J. Barton article ''Approximate translation of screened spherical waves" . Phys.Rew. A ,Vol.32,N2, 1985. ?
https://doi.org/10.1103/PhysRevA.32.1019
The same formula are given in the book Pendry J.B. "Low enrgy electron diffraction. The...
When you do a discrete Fourier transform (DFT) of a one-dimensional signal, I understand that the second half of the result is the complex conjugate of the first half. If you threw out the second half of the result, you're not actually losing any data and you would be able to recreate the entire...
Hi, outside the mathematical proof that shows that sines of different frequency are orthogonal... is there geometric interpretation/picture of this phenomena?
Homework Statement
Given a continuous non-periodic function, its Fourier transform is defined as:
$$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(x) e^{-ikx} dx$$
The problem is proving this is true by evaluating the...
A common use of the Fourier transform in physics is to transform between momentum-space and position-space. But what physical variables am I allowed to transform between? For instance can I use the Fourier transform to go from momentum space to frequency space or whatever?
These are from Griffith's:
My lecture note says that
I am having quite a confusion over here...Does the ##\Psi## in the expression ##\langle f_p|\Psi \rangle## equals to ##\Psi(x,t)##? I understand it as ##\Psi(x,t)## being the component of the position basis to form ##\Psi##, so...
Can all differential equations be turned into algebraic equations by Fourier transform (FT)? If not, what kind of differential equations can be solved by the FT technique?
The orientation of frequency components in the 2-D Fourier spectrum of an image reflect the orientation of the features they represent in the original image.
In techniques such as nonlinear microscopy, they use this idea to determine the preferred (i.e. average) orientation of certain features...
Homework Statement
Consider some unknown function f:R --> C. Denote its Fourier transform by F. Suppose we know |f(x)|2 for all x and |F(k)|2 for all k. Can we recover f(x) (for all x) from this information?
Homework Equations
None.
The Attempt at a Solution
None. It's a yes or no question...
Homework Statement
I am supposed to compute the Fourier transform of f(x) = integral (e-a|x|)
Homework Equations
Fourier transformation:
F(p) = 1/(2π) n/2 integral(f(x) e-ipx dx) from -infinity to +infinity
The Attempt at a Solution
My problem is, that I do not know how to handle that there...
Homework Statement
Given the Fourier transformation pair ##f(t) \implies F(jw)## where
##f(t) = e^{-|t|}## and ##F(jw)=\frac{2}{w^2+1}## find and make a graph of the Fourier transform of the following functions:
a) ##g(t)=\frac{2}{t^2+1}##
b) ##h(t) = \frac{2}{t^2+1}\cos (w_ot)##
Homework...
https://en.wikipedia.org/wiki/Discrete_Fourier_transform
Why is the signal obtained from a DFT periodic?
The time signal x[n] is finite and the number of sinusoids being correlated with it is finite, yet its said the frequency spectrum obtained after the DFT is periodic. I've also read the...
I would like to express that when I am viewing the repetitive Fourier transform on Internet I encounter that for instance twice Fourier transform may lead the same value at the end of first Fourier transform. When does repetitive( twice or third... consecutively)fourier transform be same with...
Given two probability amplitude wavefunctions, one in position space ##\psi(r,k)## and one in wavenumber space ##\phi(r,k)##, where ##r## and ##k## are Fourier conjugates, how is it possible for the modulus squared, i.e., probability density, of BOTH wavefunctions to be normalized? It seems...
Homework Statement
y(w)= 3/(iw-1)^2(-4+iw)
Homework Equations
N/A
The Attempt at a Solution
3/(iw-1)^2(-4+iw)
= A/iw-1 + B/(iw-1)^2 + C/-4+iw
for B iw = 1
B=3/-4+1 = -1
for C iw = 4
C= 3/(4-1)^2 = 1/3
I know the answer for A should be -1/3 however I am unsure how to obtain this as if the...
Homework Statement
I am reading the book of Gerry and Knight "Introductory Quantum Optics" (2004). In page 60, Chapter 3.7, there is two equation referring Fourier Transformation in the complex plane as follows:
$$g(u)=\int f(\alpha)e^{\alpha^{*}u-\alpha u^{*}}d^{2}\alpha, (3.94a)$$...
Hi i’m having problems with the following equations:
X(w)=2/(-1+iw)(-2+iw)(-3+iw)
This then becomes the following equation according the the tutorial, although there is no explanation as to how:
X(w)=1/-1+iw, -2/-2+iw, +1/-3+iw
The commas indicated the end of each fraction to make it easier...
Hi,
I'm struggling with a conceptual problem involving the Fourier transform of distributions. This could possibly have gone in Physics but I suspect what I'm not understanding is mathematical.
The inverse Fourier transform of a Cauchy distribution, or Lorentian function, is an exponentially...
Hi everybody. There has been a thread about this on physics forums, where the Fourier transform X(w) of x(t) volts (with time units in seconds) could be considered as volt second, or volt per Hz. So when we see tables of Fourier transform pairs, we might see Fourier transform plots associated...
Homework Statement
I am to solve an ODE using the Fourier Transform, however I am quite inexperienced in using this method so I'd like some advice:
Homework Equations
a) The Fourier Transform
b) The Inverse Fourier Transform
The Attempt at a Solution
I started by applying the Fourier...
1. The problem statement, all variables, and given/known data
Task begins by giving sample function and a corresponding Fourier transform $$f(t) = e^{-t^2 / 2} \quad \Longleftrightarrow \quad F(\omega) = \sqrt{2 \pi} e^{-\omega^2 / 2}$$ and then asks to find the Fourier transform of $$f_a(t) =...
Homework Statement
I am giving the following signal:
and asked to get the DSB-SC signal of this
Homework Equations
3. The Attempt at a Solution [/B]
So, I know a few things.
My prof writes that
I would have to multiply my original signal out and then take the Fourier transforms...
Homework Statement
I've written a program that calculates the discrete Fourier transform of a set of data in FORTRAN 90. To test it, I need to "generate a perfect sine wave of given period, calculate the DFT and write both data and DFT out to file. Plot the result- does it look like what you...
Hi,
I'm doing my third year project on diffraction gratings and patterns and analysaing them via computer software and Fourier transforms. I'm going to design my diffraction gratings this week and then create them via taking a picture with a disposable camera and using the negative film. What...
Homework Statement
Homework Equations
Scaling property and property of dual. I got the answer.
The Attempt at a Solution
I got the answer using scaling property and using property of dual.
x1(t)---> X2(W)----(another Fourier transform)--->2(3.14) x1(-w)
But I think the final answer should be...
Hi, let's take this ode:
y''(t) = f(t),y(0)=0, y'(0)=0.
using the FT it becomes:
-w^2 Y(w) = F(w)
Y(w)=( -1/w^2 )F(w)
so i can say that -1/w^2 is the Fourier transorm of the green's function(let's call it G(w)).
then
y(t) = g(t) * f(t)
where
g(t) = F^-1 (G(w)) (inverse Fourier transorm)
how can...
I'm doing a research project over the summer, and need some help understanding how to construct an inverse Fourier transform (I have v. little prior experience with them).
1. Homework Statement
I know the explicit form of ##q(x)##, where
$$ q(x) = \frac{M}{2 \pi} \int _{- \infty}^{\infty} dz...
Homework Statement
Homework Equations
if x(t) --> X(W)
then
x(-t) --> X(-W)
and
x(t+a)-->ejwX(W)
The Attempt at a Solution
I'm getting right answer for 1st part. For second part book says right answer is C.
Where am I wrong?[/B]
Homework Statement
Show, by completing the square in the exponent, that the Fourier transform of a Gaussian wavepacket ##a(t)## of width ##\tau## and centre (angular) frequency ##\omega_0##:
##a(t)=a_0e^{-i\omega_0t}e^{-(t/\tau)^2}##
is a Gaussian of width ##2/\tau##, centred on ##\omega_0##...
I am studying online course notes from University of Waterloo on 'Analytical mathematics in geology' in which the author describes a 'modified Fourier transform' which can be used to incorporate 3rd kind of boundary conditions. The formula is
## \Gamma \small[ f(x) \small] = \bar{f}(a) =...
Homework Statement
I have this function ##f(\theta)=cos(n \ sin(\frac{\theta}{2})\pi)## and I need to take the discrete Fourier transform (DFT) numerically. I did so and I attached the result for ##\theta \in [0,2\pi)## and n =2,4,8,16,32, together with the function for a given n. I need to...
Homework Statement
I need to calculate the derivative of a function using discrete Fourier transform (DFT). Below is a simplified version of my code (just for sin function) in python
Homework Equations
from __future__ import division
import numpy as np
from pylab import *
pi = np.pi
def...
Homework Statement
An LTI system has an impulse response h(t) = e-|t|
and input of x(t) = ejΩt
Homework Equations
Find y(t) the system output using convolution
Find the dominant frequency and maximum value of y(t)
Ω = 2rad/s
The Attempt at a Solution
I have tried using the Fourier transform...
Given that position and momentum are Fourier conjugates, what is the derivation for the equation ##\hbar \vec{k} = m \vec{v}##, where momentum-space momentum is defined as ##\hbar \vec{k}## and position-space momentum is defined as ##m \vec{v}##?
When you do a Fourier transform of spacetime.. what do you get? (or how does spacetime look in frequency domain? And what applications do this and what results are they looking or solving for?
Hi, this thread is an extension of this one: https://www.physicsforums.com/posts/5829265/
As I've realized that the problem is that I don't know how to properly use FFTW, from http://www.fftw.org.
I am trying to calculate a derivative using FFTW. I have ##u(x)=e^{\sin(x)}##, so...
Hi. I was checking the library for the discrete Fourier transform, fftw. So, I was using a functition ##f(x)=sin(kx)##, which when transformed must give a delta function in k. When I transform, and then transform back, I effectively recover the function, so I think I am doing something right...
Hi there - just a quick question about Fourier transforms:
When learning about quantum mechanics, I found that the Fourier transform and inverse Fourier transform were both defined with constants of ##{ \left( 2\pi \right) }^{ -d/2 }## in front of the integral. This is useful, as...
Hello buddies!
Please, check out these equations...
Tell me, please, are they mathematically correct or not?
I need a simple YES/NO answer.
I have not sufficient knowledge to understand them. I just need to know whether they are correct...
Thank you!
P.S. Am is amplitude; I guess it is a...
Homework Statement
Q/ in this inverse Fourier problem, how did he come with the results of integration of (Sinc) function and how did he come up with those results of integration with the inverse part (as in the attached picture)
here is the problem:
https://i.imgur.com/Ir3TQIN.png
Homework...
Homework Statement
Hello everyone,
am trying to solve this Fourier Trans. problem,
here is the original solution >> https://i.imgur.com/eJJ5FLF.pngQ/ How did he come up with this result and where is my mistake?
Homework Equations
All equation are in the above attached picture
The Attempt at a...
Could you explain a bit about the relationship between locality and uncertainty in Fourier pairs?
Many pages talk about uncertainty principle stating that the precision at which we can measure time duration of signal cannot unlimitedly grow without affecting precision on bandwidth.
Many other...