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 don't know if this question should be posted here, but I'll give it a shot anyways.
I am trying to find f(x,y), which can be obtain by doing the backward Fourier integral to F(\omega_x, \omega_y). I have 2 questions.
1. Is there any Fortran code that could evaluate the (numerical)...
I have two signals one continuous oscillating at a high frequency and another one instantaneous at a lower frequency.
How can I use a Fourier transform to single out the low frequency one?
See at attached picture for what I am trying to do.
Edit:
Yeah by the way, data is collected in a...
Prove: FT^2(f(x))=f(-x) where FT is the Fourier transform.
I tried to change x into -x' but with no success.
Do I need to separate cases for even f and odd f?
Homework Statement
I'd like to prove a F/T pair and to confim if they are correct.
s(t) = A Sin[w0 t] * rect[t/T - T/2] ... (1)
it's Fourier transform is
S(f) = exp(-j w T)*T/2*A* {Sinc[(w+w0)T/2/Pi] + Sinc[(w-w0)T/2/Pi]} ...(2)
where rect is rectangular function
Homework...
I am solving laplaces equation in the half plane and I have the following boundary condition of which I need to find the Fourier transform in the x-direction
S_\epsilon(x) = sgn(x)exp(\epsilon|x|), \epsilon >0
sgn(x)=\left\{\begin{array}{cc}1,&\mbox{ if }
x<0\\-1, & \mbox{ if } x<0\\...
How can i solve this Fourier Transform question?
Homework Statement
f(t)= a/((a^2)+(t^2)) if a>0 find the Fourier transform
Homework Equations
Just give me a hint to solve or first step for solving. Then i will solve.
The Attempt at a Solution
Thanks for help.
If someone wouldn't mind checking if this answer is correct, that would be awesome.
x(t) = a triangle with points (-2,0), (0,1), and (2,0).
I am supposed to compute the Fourier transform ( \hat x (\omega) ) of x(t) .
"Solution"
m = \frac{y_2-y_1}{x_2-x_1}=\frac{1}{2}
Thus...
Homework Statement
f(t)=N(e^(-a(t^2)))
N and a are constants
Find the Fourier transform of this problem?
Homework Equations
http://mathworld.wolfram.com/FourierTransformGaussian.html
The Attempt at a Solution
http://mathworld.wolfram.com/FourierTransformGaussian.html...
I am stumped on this...
Given a discrete function, and transform pair: x(n) \leftrightarrow \hat x (e^{j\omega})
What is the transform of:
x_3(n) = (n-1)^2 x(n)
I really don't know how to do this. I have a table proprety for nx(n) [/tex], but nothing with n^2 x(n) . The only...
I have a practice exam I'm going through, and I am stumped on one of the basic problems.
How is this a transform pair?
10 X(jt) <-----> 20 \pi x (-\omega)
I don't see how one can make this relation. What is the 10 X (jt) .
thanks in advance
Homework Statement
I am trying to show given f(x)=(sinax)/x, a>0
that the transform is 0, |k|>a
(pi/2)^1/2, |k|<a
Homework Equations
The Attempt at a Solution
so far i have f transform =1/(2pi)^1/2.[integral from -inf to +inf]exp[-ikx](sinax)/x.dk...
Homework Statement
I need to have the Fourier transform of a Gaussian
Homework Equations
The Attempt at a Solution
∫(exp[-ax^2])(exp[-ikπx]) dx
I tried by braking the last exponential into sine and cosine terms.The sine term is odd and it cancels.Then,I cannot evaluate the...
Hello,
does anyone know how a converging lens forms the Fourier transform of an aperture when the obs. screen is at distance=f?
If each point emits a spherical wave, the lens should make it then parallel and the FT should be the interference resulting from that.
However, if we decompose...
Homework Statement
Hi, Can someone explain the following?
This is in the book but I wasn't clear. So, I am given the following formulas for constructing the "y" vector from the equation Fc= y, and since I'm doing a fast Fourier transform, I am supposed to construct the "y" from these...
Homework Statement
Fourier transform of a constant
Homework Equations
The Attempt at a Solution
I am trying to prove that Fourier Transform of a constant is a Dirac delta function.I have fed f(x)=1 in the formula of Forward Fourier transform and got F(k)=int{exp[-ik*pi*x]}dx
I...
hi every one!
i want to know the Fourier transform of x(t)
x(t)=exp(-t/a)*sin(a*t), where a ,b is constant
and can it be work out by matlab?
another question is :
how to proof the Fourier transform of x(t) who follows normal distribution n(u, sigm^2 ) is also normal...
Why is it that we don't use contour integration when we take the integral of a complex function to find the Fourier transform:
X(j\omega) = \int_{-\infty}^\infty x(t) e^{- j\omega t} dt
We have
\int \frac{d^3 \texbf{q}}{(2 \pi)^3} \frac {e^{i \texbf{q} \dot \texbf{r}}} {q^2 + K^2} = \frac {e^{-Kr}} {4 \pi r}
How do we get from the left hand side to the right hand side?
I've tried regular Fourier transform of the function under the complex exponential I think that gives...
When we study physics at the faculty we are told that any non-sinusoidal wave can be regarded as a combination of sinusoidal waves of different frecuencies, with the ‘weight’ of the different frecuencies given by the Fourier transform. On the other hand, if we have an electromagnetic wave, we...
Hi, I got a problem in which I have to find the Fourier Transform of a function f(t) defined:
f(t) = { 1 - |t|, |t| < 1
0, |t| > 1 }
Well , I found the Fourier transform by working out the integral f(t)e^(-iwt) with the limits being -inf to +inf (and I...
Hi all,
I'm working in an exercise of advanced optics related to diffraction, in Fraunhoffer's aproximation.
I need to calculate the FT of a gaussian multiplied by a rectangle function, i.e, FT(exp(-x^2)*rect(x/a)), and I can't obtain a result expressed using analytical common functions. I...
I am reading something about Electromagnetic wave and Antenna, and come across some equations that the author says are "Azimuth Fourier Transform" and "Inverse Azimuth Fourier Transform". While I am somewhat familiar with Fourier Transform in the time/frequency domains, "Azimuth Fourier...
My notes (from a physics course) justifies the following equality by invoking the convolution thm:
\int_{-\infty}^{+\infty} \chi(\omega)\vec{E}_0(\omega)e^{-i\omega t}d\omega=\int_{-\infty}^{+\infty} \chi(\tau)\vec{E}_0(\tau-t)d\tau
From a mathematical standpoint (i.e. without reference to...
ok, i have a wave packet which is defined between (-pi/(2b)) and (pi/(2b)) as cos(bx), and it's zero everywhere else. here's what I've done so far:
i normalized and solved for b, getting pi/2. so now I'm thinking i should calculate the a0, an, and bn series, and add, right? and here is...
how would i calculate Fourier transform of functions such as 1/(1+t^2)?
because if you try to integrate the product of the above function and e^(-jxt), you would realize it's nonintegrable or something
at least my ti-89 does not calculate it for me.
any other way?
Hi all,
I'm new to Matlab, and I'm trying to evaluate a function via fast Fourier transform using Matlab, then compare the values at each gridpoint with the exact value.
The function is
y1 = cos(x)-20*sin(5*x)+6*sin(12*x)
on the interval [-pi, pi], using n = 9 gridpoints.
I first tried...
Trigonometric Polynomials...
It's too difficult to understand...
Please tell me how a complex trigonometric polynomial works. I think real trigonometric polynomial is good enough.
T_{N}=\sum^N_{n=0}a_n cos(nx) +i\sum^N_{n=0}a_n*sin(nx)
T_{N} is postion at time x of an object moving along a...
Hi,
I am very new to Matlab, and I'm supposed to use the built-in FFT function to do discrete Fourier Transform for f(x) = sin x + 4 cos(5x) + (sin(6x))^2 on the interval [-pi, pi] with a uniform partition for the interval with n = 9. Then I have to
(a) Plot the magnitudes of the Fourier...
Hi, I'm using the following definition for the Fourier transform.
F\left( q \right) = \int\limits_{ - \infty }^\infty {e^{iqx} f\left( x \right)dx}
(I used a capital F instead of f with a squiggle on top because the tex code doesn't seem to be working the way I intended it to.)
I...
Hello everyone ^^
Why I can say "The Fourier transform tells us " how much sinusoid" there is in the waveform at a given frequency "w""
Form Linear circuit analysis by Artice M. David
thanks a lot
Im trying to get the Fourier transform of x(t)=u(t)-u(t-1)
from what i know the FT of u(t) is pi*delta(omega)+1/jw
so for the u(t-1) would we have to use the time shifting property of Fourier transforms so that it becomes pi*delta(omega)+1/jw*(exp(-jw_o)??
:rolleyes: :cool: I have a question..yesterday at Wikipedia i heard about the "Hermite Polynomials2 as Eigenfunctions of Fourier (complex?) transform with Eigenvalues i^{n} and i^{-n}...could someone explain what it refers with that?...when it says "Eigenfunctions-values" it refers to the...
I am given this signal:
x(t) = sin(4(t-1))
and I need to find X(jw), i.e. it's FT, so I am confused whether I shift by 1 or by 4, in other words whether I multiply F{sin(4t)} by e^(4jw) or by e^(1jw)
which one is it? I am thinking it's 4jw... is it right?
Hi,
I'm solving an exercise in optics (Fraunhofer diffraction) and reached a mathematical difficulty - I need to find the Fourier transform of a phase function, of the form exp[-i f(x)]. I can't seem to be able to do this. I have an idea that the result should be a series of delta functions...
I'm trying to uniquely determine a complex function given pairs of real valued functions derived from it. For example, if you have its real and imaginary parts, or phase and the magnitude, the function is uniquely determined from them.
But what if you have the magnitude of the function and...
Unfortunately I haven't really been able to find anything to help me on how to interpret Fourier transforms and I need some help.
This is the question I am trying to do.
The superposition of two signals of the same amplitude but different frequencies and phase is responsible for the acoustic...
Find the solution (in integral form) of the equation:
u(x+1,t) - 2u(x,t) + u(x-1,t) = u_t
u(x,0) = f(x)
Hint: Use the shift formula
F[f(ax-b)] = \frac{\exp{i\omega b/a}}{|a|} \overline{f}(\omega/a)
So I took the Fourier transform of each term using the shift formula:
\exp{(-i\omega)}...
If we define the function:
F[w]=\int_{-\infty}^{\infty}dxe^{-iwx}g(x)
my question is..what would be the criterion to decide if F[w] has all the roots real (w=w*) and how is derived?..thanks.
Hi, how do I find the Fourier transform of this function sin x / x, i.e.,
f* = Integral( sin x / x * exp( i*w*x) dx from -infinity to +infinity ).
I've been using Jordan's Lemma up to this point, but it doesn't seem to
apply here as a way to evaluate the integral.
Thanks for any help.
What are the differences?
I mean when we will make a decision "hmm now i must use laplace transform or now i must use Fourier transform".
What are the absences in laplace transform so Fourier design a new transfom?
I want to know these transforms' main idea, differences.
I am looking...
Hi, I want to know how to get rid of the time part of the homogeneous wave equation:
\newcommand{\pd}[3]{ \frac{ \partial^{#3}{#1} }{ \partial {#2}^{#3} } }
\nabla^2\psi-c^{-2}\pd{\psi}{t}{2} = 0
I've read that this can be done using a Fourier transform, with the following given as the...
suppose we have this discreet function:
x(k)=rect(k+N/2)= 1 ; when -N/2=<K<=N/2-1
x(k)=0; otherwise
This is discreet function(not continuous) of k shifted forward by N/2, we need to find Fourier transform for it ..
anyway let N=6 for simplicity, then:
x(k)=rect(k+3)= 1 ; when...
Here's my problem:
I've got the Fourier transform of f(x) as
F(K) = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^\infty f(x) e^{-ikx} dx
Likewise for g(x) I have
G(K) = \frac{1}{\sqrt{2\pi}} \int\limits_{-\infty}^\infty g(x) e^{-ikx} dx
for F(K/lambda) I have
F(K/\lambda) =...
Fourier transform --> power spectrum
Hey all!
I've been learning about the discrete Fourier transform (and FFT too) recently. What I don't understand is why applying it to a signal gives its power spectrum. I am not really good in physics, so to me it just seems like a magical formulae, one...
ok so here's the question, show explicitly that
the integral from -inf to inf of |f(k)|^2=1
where f(x) = \frac{N}{\sqrt{\sigma}}*e^{\frac{-x^2}{2\sigma^2}}
When doing the integral for the forier transform, I was going to use the gaussian integral to simplify it, but I don't htink I can do...
I am trying to solve this Fourier problem where I have to integrate
∫f(x) * exp(-i§x) dx from -∞ to ∞ , where f(x) = exp(-sgn(x))
I tried breaking the function into two pieces where x is from -∞ to 0 and from 0 to ∞ where f(x) would then be exp(x) and exp(-x) and integrating two functions...
Can someone help me out in understanding this Fourier Transform and how can it be applied to different situations? Any Useful links or information would be highly appreciated.
Regards,
FuRy
I want to take an audio recording of a sound, perform a Fourier Transform on this sound, and then use the amplitude/frequency/phase information provided by this transform to set the amplitude/frequency/phase of an set of sine wave oscillators, in order to resynthesize the sound.
I need to know...