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
Can someone tell me how to Fourier transform this quantity:
\Sigma (x_(j+1) - x_j)^2
where the sum is from j=1 to N
Homework Equations
Define the Fourier transform as
x_j = \Sigma A_k *exp(-iqkj)
**Where i is sqrt(-1)
**The Sum is from k=0 to (N-1)
**q =...
Homework Statement
I'm supposed to take the "spatial Fourier transform" of the partial differential equation
p_t = \frac{a^2}{2\tau}p_{xx} + 2g(p + xp_x)
for p = p(x,t).
Homework Equations
Well, I guess I eventually need something like
\phi(k,t) = \mathbb F(p(x,t)) =...
Homework Statement
From the definition of the Fourier transform, find the Fourier transform of rect(t-5).Homework Equations
G(w) = \int^{\infty}_{-\infty}g(t)e^{jwt}dtThe Attempt at a Solution
So, I sketched the function which has area 1 and centre at 5, with its lower bound @ 4.5 and upper at...
This isn't really a homework problem, but I am having trouble understanding why this is true:
A function with the following symmetry does not have any even harmonics in its spectrum.
I understand the concept based on odd/even symmetry properties, but can anyone provide a mathematical proof?
I have about 40 tabs open on this right now and something important is slipping my grasp. I know this has been covered a million and a half times, but for some reason I cannot seem to find a straight answer (or more probably realize and understand it when I see it).
When I take the Continuous...
a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous.
b) show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0.
i am a little stuck on part b). i am trying to use the dominated convergence theorem but i am a bit confused on what...
Hi,
if we consider a constant function f(x)=1, it is well-known that its Fourier transform is the delta function, in other words:
\int_{-\infty}^{+\infty}e^{-i\omega x}dx = \delta(\omega)
The constant function does not tend to zero at infinity, so I was wondering: are there other...
Homework Statement
I need to determine the first three terms in the Fourier series pictured in the attachment.
Did I define the peace-wise functions correctly?
I'm re-posting this with the tex code instead of the attached document.
Homework Equations
a_o=\frac{1}{2L}\int_{-L}^Lf(t)dt...
Hi, just got set a 3d Fourier transform to solve but I've never seen one before and can't find any examples online. once the integral is set up I should be fine but I'm not sure how to set it up;
What is Fourier transfrom (f(k)) of following 3d function for k=(kx, ky, kz)=(1,2,3)
for...
Hello,
my question arises from reading the section on Smoothness/Compactness from Bracewell's "The Fourier Transform and Its Applications" page 162.
I don't quite understand the following reasoning:
F(\omega) = \ldots = \frac{1}{i\omega}\int_{-\infty}^{+\infty}f'(x)e^{-i\omega x}dx
and...
Homework Statement
Let h(t) be impulse response of unity-gain ideal lowpass filter with bandwidth of 50[Hz] and a time delay of 5[ms]. Sketch magnitude and phase of Fourier transform of h(t).
The Attempt at a Solution
I know that the magnitude2 of H(f) is total power gain, so perhaps by...
I have the readings from a signal in a file (floating point values) that I wish to apply the Fourier Transform to.
The samples (mV) were taken every 4 milliseconds and I wish to transform them into the frequency domain.
How would I apply the FT to a set of values without knowing any...
I have been playing with the FFT and graphs. The easiest example I could think of for a transform was the top hat function (ie 0,0,0,0,0...1,1,1...0,0,0,0,0). When I transform this from the time domain to the frequency domain, it returns a sinc function when I take the absolute value squared of...
my question is the following
let be the Fourier transform \int_{-\infty}^{\infty}d^{4}p \frac{exp( ip*k)}{p^{2}+a^{2}}
here p^{2}= p_{0}^{2}+p_{1}^{2}+p_{2}^{2}+p_{3}^{2}
is the modulus of vector 'p' , here * means scalar product
for the scalar product i can use the definition...
Homework Statement
"Solve for t > 0 the one-dimensional wave equation
\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}
with x > 0, with the use of Fourier transformation.
The boundary condition in x = 0 is u(0,t) = 0.
Assume that the initial values u(x,0) and...
Hello, I am trying to do some self-studying in Byron & Fuller mathematical methods for classical and quantum physics. I have slightly ran aground on this one task of finding 3d Fourier transforms and I can't find the info in the book itself to free me. Google has neither been very fruitfull...
Does it make sense to take the Fourier transform of a function that blows up at some point? For example the Fourier transform of f(x)=1/x, which blows up at zero?
Doesn't the integral:
\int^{\infty}_{-\infty} \frac{dx}{x} e^{-ikx}
not converge because of x=0?
Yet for some reason analytical...
Hello.
I understand that in the form of \int_{\mathbb R} f(x) \exp{2 \pi i tx} \mathrm d x the function f: \mathbb R \to \mathbb C: x \to \frac{1}{x} doesn't have a Fourier transform (because the function is not integrable).
But in my analysis course, there is a theorem that states that in...
Hey guys, I have a quick question about Fourier transforms.
I have been told that the Fourier transform of a function tells us the minimum components required to support that function and that a real pulse may have extra frequencies, but not too few frequencies.
I don't understand why...
Homework Statement
Compute the discrete Fourier transform of the ecg signal, graph the amplitude and phase response.
The problem gives data in the form of a ecg.mat file. Contained are two double variables: voltage data for the ecg and a time vector for the ecg. both the voltage and time...
I am trying to compute the inverse Fourier transform numerically (using a DFT) for some complicated characteristic functions in order to compute their corresponding probability distribution functions. As a test case I thought I would invert the characteristic function for the simple exponential...
Hello,
Thanks at first. If anyone can understand, then I would like to know how do I get to equation 4.15. Its a laplacian equation in which I want to apply the Fourier transform.
Thanks again.
Homework Statement
So we have a string of N particles connected by springs like so:
*...*...*...*...*
A corresponding Hamiltonian that looks like:
H= 1/2* \Sigma P_j^2 + (x_j - x_(j+1) )^2
Where x is transverse position of the particle as measured from the equilibrium position, and...
I need to deduce that \hat{\hat{f}}(x)=2\pif(-x) using the inversion formula for the Fourier transform, I was wondering if someone could explain why there's f(-x) because i just can't get started on this problem!
Does anyone know a good free library to do Fourier Transforms (FFT or DFT). I know FFTW but I'm having some problems with it. I want an alternative that do FFT in two dimensions with complex numbers. The libraries I have found doesn't fulfill this requirements.
Thank you
Homework Statement
Okay so i am applying a FT to an image of particles that are forming a lattice, and i need to find the average distance between the particles
because its not a perfect lattice, I am getting an airy pattern and i believe that the distance to the first ring is the average...
The Fourier transform relates spacetime domain to momentum-energy (wave number - frequency) domain. For example, a generic function f(x, t) is transformed as given by photo
I can't understant What does this theorem guarantee about the quantum systems?Hot to find the representation of...
Hi,
Can anyone tell me if there is a convolution theorem for the Fourier transform of:
\int^{t}_{0}f(t-\tau)g(\tau)d\tau
I know the convolution theorem for the Fourier Transform of:
\int^{\infty}_{-\infty}f(t-\tau)g(\tau)d\tau
But I can't seem to find (or proove!) anything...
Find the Fourier transform of the following aperodic digital signal
x[n] = 3
for -2<n<2
3. Not to surer where to start on this one any help would be great thanks
Homework Statement
x(t) = t*exp(a)*exp(-a*t)*u(t-1) - exp(a)*exp(-a*t)*u(t-1)
I need to find X(jw)...
Homework Equations
how to apply properties of Fourier transform to get an answer? Because i know that the only effective method for this..
The Attempt at a Solution
For...
Hello!
Can someone help me with this.
Evaluate:
the integral from zero to infinite of ((xcos(x)-sin(x))/x^3)cos(x/2)dx
I think it has to do with Fouriers Transform but I am just stuck.
Any help would be appreciated!
Thank You
Hi,
I have several sets of stochastic signals that oscillate about the x-axis over time. I would like to transform these signals into the frequency domain (make a periodogram) so that I can which signal has the most stable frequency. I was thinking about using taking the Fourier transform...
Homework Statement
I am new to FT and dirac delta function. Given the following signal:
x\left(t\right)=cos\left(2\pi5t\right)+cos\left(2\pi10t\right)+cos\left(2\pi20t\right)+cos\left(2\pi50t\right)
I use the online calculator to find me the FT of the signal, which is...
Hi there,
A quick question concerning the FFT. Let's say I explicitly know a 2D function \tilde{f}\left(\xi_1,\xi_2 \right) in the frequency domain.
If I want to know the values of f\left(x_1,x_2 \right) in the time domain at some specific times, I can calculate \tilde{f} at N_jdiscrete...
Homework Statement
Express the Fourier Transform of the following function
ae^{2\pi iabx}f(ax-c)
terms of the Fourier Transform of f . (Here a, b, c are positive constants.)Homework Equations
Define the following operators acting on function f(x):
T_{a}(f)(x)=f(x+a)
M_{b}(f)(x)=e^{-2\pi...
Hi there!
I need to calculate the Fourier transform of a continuous function in C++. To do this I need to use the Dft, but what is the relation between the Dft and the continuous Fourier transform? I mean, how can I get the continuous Fourier transform from the Dft?
I was wondering if this is correct:
\phi(k-a)=\phi(k)-\phi(a)
Where k=p/h (h bar that is) and a is some constant and \phi is the Fourier transform of a wave function (momentum function).
I know that if I had some real formula for \phi I could just test this but the problem isn't like...
Homework Statement
I have to find the Fourier transform of
f(x)=\frac{\beta^2}{\beta^2+x^2}
Homework Equations
Fourier Transform is given by
F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}f(x) dx
The Attempt at a Solution
I'm having trouble with the integration...
Homework Statement
Let \phi (k) be the Fourier transform of the function \psi (x). Determine the Fourier transform of e^{iax} \psi (x) and discuss the physical interpretation of this result.Homework Equations
(1) \tilde{f} (k) = \frac{1}{\sqrt{2 \pi}} \int{f (x) e^{-ikx} dx}
(2) \psi...
[PLAIN]http://img716.imageshack.us/img716/3663/semttulont.png
f(x) = 0 (|x| > 1)
= x² (|x| < 1)
I know that thing on integral is [F(x)]^2, but I have no clue what to do now.
Homework Statement
Show that Fourier transform does not violate causality. In other words, let \hat{E}(\omega) be Fourier transform of function {E}(t). Show that E(t_1), as evaluated from inverse Fourier transform formula using \hat{E}(\omega), does not depend on E(t_2) for
t_2>t_1...
Hi!
I'm trying to understand how do i get the phase spectrum from a Fourier Transform. From this site
http://sepwww.stanford.edu/public/docs/sep72/lin4/paper_html/node4.html#lin4_swhfactm
this statement
"The phase spectrum is usually calculated by taking the arctangent of the ratio...
I am learning about adv quantum and field theory and i have run across something unfamiliar mathematically. In several instances the author simpy expands the field or a wave function as a Fourier transform. that is they assume the field or wave function is simply the transform of two other...
Suppose a function f(k) has a power series expansion:
f(k)=\Sigma a_i k^i
Is it possible to inverse Fourier transform any such function?
For example:
f(k)=\Sigma a_i k^{i+2}\frac{1}{k^2}
Since g(k)=1/k^2 should have a well-defined inverse Fourier transform, and the inverse Fourier...
Hello Everybody.
I gave a quick look onto the internet but i couldn't get anything interesting.
Heres my problem.
Im solving the differential equation given by:
(-\Delta+k^2)^2u=\delta
Where \delta is the dirac delta distribuiton (and u is thought as a distribution as well)
The...
So its been awhile since I've taken PDE, and forgot a lot about Fourier transforms. Anyways I'm trying to understand what the inverse of the Fourier transform actually represents. I understand perfectly how the infinite sum of periodic functions can be used to create any periodic function when...