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.
I have an exercise with a function of the form:
h(t) = f(t)g(t)
and f(t) and g(t) both have discrete Fourier series, which implies that h does too. I want to find the Fourier series of h, so my teacher said I should apply the convolution theorem which would turn the product above into a...
Homework Statement
Part (a): State inverse Fourier transform. Show Fourier transform is:
Part (b): Show Fourier transform is:
Part (c): By transforming LHS and RHS, show the solution is:
Part(d): Using inverse Fourier transform, find an expression for T(x,t)
Homework Equations
The Attempt...
Homework Statement
The exercise is a) in the attached trial. I have attached my attempt at a solution, but there are some issues. First of all: Isn't the example result wrong? As I demonstrate you get a delta function which yields the sum I have written (as far as I can see), not the sum...
http://calclab.math.tamu.edu/~fulling/m412/f07/airywkb.pdf
Can someone walk me through this derivation of the Airy integral by Fourier transform?
I have tried it but failed
I'm trying to prove that the discrete form of the Fourier transform is a unitary transformation
So I used the equation for the discrete Fourier transform:
##y_k=\frac{1}{\sqrt{N}}\sum^{N-1}_{j=0}{x_je^{i2\pi\frac{jk}{N}}}##
and I put the Fourier transform into a N-1 by N-1 matrix form...
From -infinity to infinity at the extreme ends do Fourier transforms always converge to 0? I know in the case of signals, you can never have an infinite signal so it does go to 0, but speaking in general if you are taking the Fourier transform of f(x)
If you do integration by parts, you get a...
Homework Statement
What is the Fourier transform of a single short pulse and of a sequence of pulses?
The Attempt at a Solution
In class we haven't dealt with the mathematics of a Fourier transform, however my professor has simple stated that a Fourier transform is simply a equation...
I have a tutorial question for maths involving the heat equation and Fourier transform.
{\frac{∂u}{∂t}} = {\frac{∂^2u}{∂x^2}}
you are given the initial condition:
u(x,0) = 70e^{-{\frac{1}{2}}{x^2}}
the answer is:
u(x,t) = {\frac{70}{\sqrt{1+2t}}}{e^{-{\frac{x^2}{2+4t}}}}
In this course...
I learned how to integrate it using the complex plane and semi circle contours but I was wondering if there is a way using Fourier transforms. I know that the Fourier transform of the rectangle wave form is the sinc function so I was thinking maybe i could do an inverse Fourier on sinc x and get...
Hello,
Recently I've learned about Fourier Transform, and the uncertainty principle that is arose from it.
According to Fourier Transform, if there is only one pulse in a signal, then it is composed from a lot more frequencies, compared to the number of frequencies that are building a...
Fourier transform of RF signal with a "prism"?
We can use a prism to decompose visible light into components of different frequencies. This is a Fourier transform by nature. For an ideal prism, the energy is conserved in the process.
How about RF signals? There is no fundamental difference...
\mathcal{F}\{f(r)\}=\int e^{i\vec{k}\cdot \vec{r}}f(r)d\vec{r}
in spherical polar coordinates
\mathcal{F}\{f(r)\}=\int^{\infty}_0r^2dr\int^{\pi}_0\sin\theta d\theta\int^{\pi}_0d\varphi e^{ikr\cos \theta}f(r)
Why could I take ##e^{ikr\cos \theta}## and to take that ##\theta## is angle which goes...
Homework Statement
Finding the Fourier Transform using Transform Pair and Properties
x(t) = 2[u(t+1)- t^{3}e^{6t}u(t)]
Homework Equations
The Attempt at a Solution
For the first problem, I got
u(t) \leftrightarrow ∏δ(ω)+\frac{1}{jω}
F(at-t_{0}) \leftrightarrow...
The windowed Fourier transform on R
Defi nition-Proposition-Theorems (Plancherel formula-Parseval formula-inversion formula-Calderon's formula)
http://www.4shared.com/office/b2Ho5n7H/The_windowed_Fourier_transform.html
I'm working on some research with a professor, and we're looking at data collected by an x-band radar array looking at ocean waves as they approach the coast (the radar is on land, and we can see about 3 miles out).
What we're trying to do is perform an fft on the signal using Matlab, and...
I'm given a Gaussian function to apply a Fourier transform to.
$$f(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{ik_ox}e^{-\frac{x^2}{2a^2}}$$
Not the most appetizing integral...
$$g(k)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{a\sqrt{\pi}}}\int_{-\infty}^{\infty}e^{ik_ox}e^{-\frac{x^2}{2a^2}}e^{-ikx}dx$$...
I understand the Fourier transform conceptually, but I am unable to reproduce it mathematically; I am very familiar with calculus and integration, but I am taking a QM course and I need to know how to apply it. No websites or videos are able to give me a good explanation as to how I can use it...
When the rod is infinite or semi-infinite, I was taught to use Fourier transform.
But I don't know when should the full Fourier transform or sine/cosine transform be used.
how's the B.C. related to the choice of the transform ?
Hi all,
Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.)
Now I have a doubt on CTFT (continuous time Fourier transform).
This is a concept I got from my...
I've been assigned the following homework:
I have to compute the spectral density of a QFT and in order to do so I have to compute Fourier tranform of the following quantity (in Minkowsky signature, mostly minus)
\rho\left(p\right) = \int \frac{1}{\left(-x^2 + i \epsilon...
Somehow I have really hard time wrapping my head around the concept.I mean,I get it,but I can't seem to solve any questions regarding it.
Here are some examples ,and I just get stuck.Its a part of test,so I think it shouldn't be that hard to solve,and if it looks hard,I know there are some...
Hi,
I'm following the proof of the "Scaling Property of the Fourier Transform" from here:
http://www.thefouriertransform.com/transform/properties.php
...but don't understand how they went from the integral to the right hand term here:
The definition of the Fourier Trasform they...
Consider \(u_t = -u_{nxxx} - 3(u^2)_{nx}\).
The Fourier Transform is linear so taking the Inverse Fourier transform of the Fourier Transform on the RHS we have
\begin{align}
-\mathcal{F}^{-1}\left[\mathcal{F}\left[u_{nxxx} - 3(u^2)_{nx}\right]\right] &= -\mathcal{F}^{-1}...
Does every FFT have \(i\) in it?
Given \(u_t = -(u_{xxx} + 6uu_x)\).
\(f'''(x) = \mathcal{F}^{-1}\left[(ik)^3\mathcal{F}(f(x))\right]\)
\(f'(x) = \mathcal{F}^{-1}\left[(ik)\mathcal{F}(f(x))\right]\)
The only equation I have used the pseudo-spectral method on was the NLS which is
\(u_t =...
Today I found a program, which does Fourier transforms on pictures and tried it on some basic patterns. One of those was a lattice of dots and I have attached this and its Fourier transform to the thread.
I would very much like if someone in basic details could explain what is going on. Why...
Finite Fourier Transform on a 2d wave
How does the finite Fourier transform work exactly?
The transform of f(x) is
\widetilde{f}(\lambda_{n}) =\int^{L}_{0} f(x) X_{n} dx
If I had a 3d wave equation pde and I applied Finite Fourier transform on the pde for
z(x,y,t)=X(x)Y(y)T(t)...
Hi all, as a physics student, I seldom use Fourier transform but from my understanding, given a periodic function you can decompose the function into sine function with different frequencies. Also, to get a ultra short pulse in time domain, this would require mixing many frequencies. I would...
Hi, I have the following question:
A signal x(t) which is band-limited to 10kHz is sampled with a sampling frequency of 20kHz. The DFT (Discrete Fourier Transform) of N= 1000 samples of x(n) is then computed. To what analogue frequency does the index k=120 respond to?
I'm trying to...
Hi all,
I want to calculate \int_0^{\infty}e^{-a t^2}\cos(2xt)dt=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{\frac{-x^2}{a}}. The answer is known from the literature, but I don't know how to do it step by step. Any one has a clue? Thanks.
Jo
Hi there,
I am trying to get some practice with Fourier Transforms, there is a long way to go.
For example, let me consider the function $$ \gamma (t) = \int_{-\infty}^{t} C(t-\tau) \sigma(\tau) \mathrm{d}{\tau}$$
Defining the Fourier Transform as
$$ \gamma(\omega) = \frac{1}{2 \pi}...
I was looking through some examples which applied the duality principle while studying for an up and coming exam when it hit me that the transform applied 4 times gives you back the same function.
So is there some theory that uses this? perhaps some sort of operator?
I thought it...
Homework Statement
Compute the Fourier transform of a function of norm f(\norm{x}).
Homework Equations
\mathbb{F}{\frac{1}{1+\norm{x}}
The Attempt at a Solution
Attempt at using Cauchy theorem and the contour integral with the contour [(-R,R),(R,R+ip),(R+ip,-R+ip),(-R+ip,-R)] does...
Checked around a buch and could not find any help. But I needed help with:
Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant image/data i.e. for every tick on the axis, how do I know the spatial length?
More detailed...
Hello everyone:
I have some question using the FFT in MATLAB for data interpolating. I don't know what the relation between the normal Fourier series and the real, image number.
For example, given a set of measurement data, I can use the curve fitting toolbox to fit a curve.
The general...
Homework Statement
Hi, this is not a homework question per se, but something I'm wondering. Let C be a circulant n x n matrix, let x, b, be vectors such that
C x = b.
We would like to find a solution x. One way is to use the DFT: According to section 5, In Linear Equations, in the wikipedia...
I have been studying Fourier Optics and I have a basic conceptual question. I understand the mathematics of how to perform Fourier Transforms however the part of this topic I seem to have missed is why the action of a lens on light is the same as performing a Fourier Transform on the functional...
Hi all,
I'm a complete novice when it comes to describing images in frequency space and i understand that it is a way of representing images as being composed of a series of sinusoids. So a horizontal striped pattern with a single spatial frequency would have a magnitude image in frequency...
Homework Statement
I have am doing a two dimensional discrete Fourier transform on an image (using MATLAB). What are the units associated with each pixel of the image in the frequency domain?
Homework Equations
The Attempt at a Solution
I thought that the frequency should be...
Homework Statement
Find the inverse Fourier transform of F(jω) = cos(4ω + pi/3)Homework Equations
δ(t) <--> 1
δ(t - to) <--> exp(-j*ωo*t)
cos(x) = 1/2 (exp(jx) + exp(-jx))The Attempt at a Solution
So first I turned the given equation into its complex form using Euler's Formula.
F(jω) = 1/2...
can you use Fourier transform to find a moving average on a data set?
so, you do a Fourier transform on your one dimensional data set.
next remove high order harmonics from FT result.
do reverse Fourier transform on new FT result.
And, vola! smoothed out data set.
Hi,
I was wondering what would the Fourier transform of a signal like below give:
s(t) = sin(2πt*10) ; t in [0s,5s]
= sin(2πt*20) ; t in [5s,10s]
I certainly did not expect it to give me 2 sharp peaks at frequencies 10Hz and 20Hz - because I understand that the addition of...
Hallo,
I really don't understand Fourier transform.
Do somebody know a good book for beginners?
Something like Fourier transform for dummies or so?
I need it just for physics.
So it don't have to be to mathematical. ^^
THX
Fourier Transform on the "connected part" of QFT transition prob.
Homework Statement
Calculate ⟨0|T[ϕ(x₁)ϕ(x₂)ϕ(x₃)ϕ(x₄)]|0⟩ up to order λ from the generating functional Z[J] of λϕ⁴-theory.
Using the connected part, derive the T-matrixelement for the reaction a(p₁) + a(p₂) → a(p₃) +...
Homework Statement
In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that:
F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv)
where F_{l,v}(g(l)) is the Fourier transform of g(l) and...
Hello,
Consider I have a linear time-invariant (LTI) system, with ##x(t)##, ##y(t)##, and ##h(t)##, as input, output, and impulse response functions, respectively.
I have two choices to write the convolution integral to get ##y(t)##:
$$ 1)\ \ \ y(t) = \int_{0}^{t} h(t-t')x(t')dt' $$
and...
Homework Statement
Find the Fourier transform of (1/p)e^{[(-pi*x^2)/p^2]} for any p > 0
Homework Equations
The Fourier transform of e^{-pi*x^2} is e^{-pi*u^2}.
The scaling property is given to be f(px) ----> (1/p)f(u/p)
The Attempt at a Solution
Using the information above, I got...
Find the DTFT of:
h[n]=(-1)^{n}\frac{sin(\frac{\pi}{2}n}{sin(\pi n}
useful properties:
x[n]y[n] --> X[Ω]*Y[Ω]
\frac{sin(\frac{\pi}{2}n}{sin(\pi n} --> rect[\frac{2Ω}{\pi}
I have no clue how to deal with the (-1)[itex]^{n}[\itex] the DTFT of that doesn't converge. . .
any help...