What is inverse fourier: Definition and 53 Discussions
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.
The theorem says that if we have a function
f
:
R
→
C
{\displaystyle f:\mathbb {R} \to \mathbb {C} }
satisfying certain conditions, and we use the convention for the Fourier transform that
{\displaystyle f(x)=\int _{\mathbb {R} }e^{2\pi ix\cdot \xi }\,({\mathcal {F}}f)(\xi )\,d\xi .}
In other words, the theorem says that
f
(
x
)
=
∬
R
2
e
2
π
i
(
x
−
y
)
⋅
ξ
f
(
y
)
d
y
d
ξ
.
{\displaystyle f(x)=\iint _{\mathbb {R} ^{2}}e^{2\pi i(x-y)\cdot \xi }\,f(y)\,dy\,d\xi .}
This last equation is called the Fourier integral theorem.
Another way to state the theorem is that if
R
{\displaystyle R}
is the flip operator i.e.
(
R
f
)
(
x
)
:=
f
(
−
x
)
{\displaystyle (Rf)(x):=f(-x)}
, then
F
−
1
=
F
R
=
R
F
.
{\displaystyle {\mathcal {F}}^{-1}={\mathcal {F}}R=R{\mathcal {F}}.}
The theorem holds if both
f
{\displaystyle f}
and its Fourier transform are absolutely integrable (in the Lebesgue sense) and
f
{\displaystyle f}
is continuous at the point
x
{\displaystyle x}
. However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.
In my system I am trying to represent two lenses. L1 with focal length f1=910mm and the other lens, L2 with focal length f2=40mm. These lenses are space such that there is a distance of f1+f2 between the lenses. I have a unit amplitude plane wave incident on L1. My goal is to find the...
Hi,
This thread is an extension of this discussion where @DrClaude helped me. I thought that it'd be better to separate this question.
I couldn't find any other way to post my work other than as images so if any of the embedded images are not clear, just click on them. It'd make them clearer...
Hi, I've been looking all over the net for good examples but I've only found some intro but no examples being solved.
If you know of good resources (both theories and problems) please let me know!
a) Calculate Fourier and inverse Fourier transform of f(t).
b) Calculate the limit.
My...
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...
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
F(t) = sqrt(π/2)e-t for t>0
F(t) = sqrt(π/2)et for t<0
In other words the question asks to solve this integral: 1/sqrt(2π) ∫F(t)eitxdt and show that it equals 1/(1+x2)
Homework Equations
F(t) = sqrt(π/2)e-t for t>0
F(t) = sqrt(π/2)et for t<0
1/sqrt(2π) ∫F(t)eitxdt
The Attempt...
Homework Statement
Solve ut+3ux=0, where -infinity < x < infinity, t>0, and u(x,0)=f(x).Homework Equations
Fourier Transform where (U=fourier transform of u)
Convolution Theorem
The Attempt at a Solution
I've used Fourier transform to get that Ut-3iwU=0 and that U=F(w)e3iwt. However, I'm...
Hi,
I have a a Fourier transformed variable \hat{\eta}(k) defined as the following:
\hat{\eta}(k)=\frac{e^{-k^{2}}\tanh k}{kU^{2}+(-B+\Omega U+E_{b}|k|-k^{2})\tanh k}
The parameters U,B,\Omega,E_{b} have all been defined previously. I have naively tried the following:
\eta...
Homework Statement
From the derivation of v(x,t) and i(x,t) I am stuck on how the inverse Fourier transform of e^(-jwx/u) was calculated. I am trying to understand how the PDE was fully solved here: http://fourier.eng.hmc.edu/e84/lectures/transmission_line/node1.htmlHomework Equations
Not...
Fourier transform is defined as
$$F(jw)=\int_{-\infty}^{\infty}f(t)e^{-jwt}dt.$$
Inverse Fourier transform is defined as
$$f(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}F(jw)e^{jwt}dw.$$
Let ##f(t)=e^{-at}h(t),a>0##, where ##h(t)## is heaviside function and ##a## is real constant.
Fourier...
Homework Statement
[/B]
I was using the Fourier transform to solve the following IVP:
\frac{\partial^2 u}{\partial t \partial x} = \frac{\partial^3u}{\partial x^3} \\
u(x,0)=e^{-|x|}
Homework Equations
[/B]
f(x) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\hat{f}(\omega)e^{i\omega...
Homework Statement
This comes up in the context of Poisson's equation
Solve for ##\mathbf{x} \in \mathbb{R}^n ## $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$
Homework Equations
$$\int_0^\pi \sin\theta e^{ikr \cos\theta}\mathop{dk} = \int_{-1}^1 e^{ikr \cos\theta}\mathop{d\cos \theta
}$$...
Homework Statement
Find the inverse Fourier transform of
X(ejw = 1/(1-ae-jw)2
using the convolution theorem.
Homework EquationsThe Attempt at a Solution
I tried finding the partial fraction coefficients but without success.
Hello, everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field:
\tilde{p}=-\frac{i}{{{k}^{2}}}\mathbf{F}\centerdot \mathbf{k}
where i is the imaginary unit...
Here we will use the following transforms: $\displaystyle \begin{align*} \mathcal{F}^{-1} \left\{ \frac{n!}{ \left( a + \mathrm{i}\,\omega \right) ^{n+1} } \right\} = t^n\,\mathrm{e}^{-a\,t}\,\mathrm{H}(t) \end{align*}$ and $\displaystyle \begin{align*} \mathcal{F}^{-1} \left\{...
Homework Statement
\int_{-\infty}^{\infty} |k|^{2\lambda} e^{ikx} dkHomework Equations
The Attempt at a Solution
As you can guess, this is the inverse Fourier transform of |k|^{2\lambda}. I've tried splitting it from -infinity to 0 and 0 to infinity. I've tried noting that |k| is even, cos is...
Homework Statement
Take the inverse Fourier Transform of
5[\delta(f+100)+\delta(f-100)]\bigg(\frac{180+j2\pi f*0.0135}{1680+j2\pi f*0.0135}\bigg)Homework Equations
g(t)=\int_{-\infty}^{\infty} G(f)e^{j2\pi ft}dt
The Attempt at a Solution
g(t)=\int_{-\infty}^{\infty}...
Homework Statement
Find the inverse Fourier transform of f(w)=1 Hint: Denote by f(x) the inverse Fourier transform of 1 and consider convolution of f with an arbitrary function.
Homework Equations
From my textbook the inverse Fourier transform of f(w)=\int F(w)e^-iwt dw
The...
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...
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...
Homework Statement
For a physics problem I must take the inverse Fourier transform of 2 functions.
Namely I must compute the integral ##\frac{1}{\sqrt{2\pi}}\int_{-\infty} ^\infty [A\cos (ckt)+B\sin (ckt)]e^{ikx}dk##.Homework Equations
Already given.
i is the complex number. t is greater or...
I am having trouble with this homework problem, I know how to get started but I just don't know how to carry through the completion of the problem:
Question: Given the Fourier transform of an aperiodic signal
X(ω) = 2*sin(3(ω-2π))/ω-2π
(a)find its inverse Fourier transform x(t) using...
Hi, I am taking a random process class and I came across a problem that has stumped me. I believe I know the end result but I would like to know how it is solved. I have been out of college for a while and I am a little rusty with integration.
Homework Statement
What I need is to find out...
Homework Statement
The argument of the kernel of the Fourier transform has a different sign for the forward and inverse transform. For a general function, show how the original function isn’t recovered upon inverse transformation if the sign of the argument is the same for both the forward and...
Hi all,
I've been trying to solve the following
I = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{x}{(x^2+y^2+d^2)^{\frac{5}{2}}} e^{-i(kx+\ell y)} \ dx \ dy
where d,k,\ell are constants. I haven't been able to put this into a tractable analytic form and I figured I'd consult all...
I have written some routines that compute the 2D inverse Fourier transform, if anyone thinks that this may be useful at all then please let me know and I will gladly post the code.
Hi everyone,
I'm trying to solve an exercise in which I need to find x(t) considering that X(ω) = cos(4ω). So, I need to find the Inverse Fourier Transform of cos(4ω), but I don't have the inverse Fourier transform table.
So, I thought about applying the duality property. If x(t) <-->...
Homework Statement
I can't figure out what the limits of integration should be;
if a transfer function is given as follows:
h(ω)=1 if 1<|ω|<2, 0 otherwise
1) find the impulse response
2) if the input is white noise of intensity σ² find the variance of the output signal
3)state...
Homework Statement
Inverse Fourier of:
[ jω+2 ] / [ (jω)2 +5jω+9 ] where j = sqrt(-1)
I tried using partial fractions but the denominator can't be factored...I tried completing the square on the denominator but I get a sum of squares.
What can I try? I am sure I don't have to use the formal...
I am having problem with the inverse transformation of a Fourier transformed function which should give the function itself.
Let
f=f(x) and let f be Fourier transformable (whatever that implies)
Let
\tilde{f}(k)=∫^{\infty}_{-\infty}dx e^{-ikx}f(x) (1)
then we should have...
Homework Statement
For α > 0, determine u(x) by the inverse Fourier transform
u(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\ \frac{e^{ikx}}{ik+\alpha}\ dk
Homework Equations
The Attempt at a Solution
This seemed like a relatively simple residue problem. You just note that...
Homework Statement
(part of a problem)
Find the inverse Fourier of F(w) = (3jw+9)/((jw)^2+6jw+8)
where w is the angular frequency, w=2pi * f = 2*pi/T
Homework Equations
The fourier transfrom and its properties i guess.
Also the exponential FT common pair exp(-at)u(t) <-> 1/(jw+a)
where...
Homework Statement
Hi!
I tried to get the inverse Fourier transform of the function:
X(j\omega)=1/(jw+a)
for a>0, using the integral:
x(t)=(1/2\pi)\int_{-\infty}^{+\infty} X(j\omega)e^{j\omega t}d\omega
I know that the inverse Fourier transform of X(j\omega) is...
well, i have to prove that the inv. Fourier transform of a gaussian (e^(-(k^2/2)) is a gaussian, i know some elementary complex analysis(never actually taken a class in it), not well enough, it seems, to find the solution to this. I tried to integrate over a circular contour, and let the radius...
Hi everyone, this is not a homework question but from my reading of a signals processing paper.
This paper says if f(t) is the inverse Fourier transform of a function
f(\lambda) = e^{-2i\pi\lambda d}
then we can "easily see" that f(t) will have a peak d.
Part of the issue here is...
I've been using the Hilbert transform a bit as part of my research work (to analyse time series) and found http://personal.atl.bellsouth.net/p/h/physics/hilberttransforms.pdf document that explains some of the theory in a way that I can understand. I'm just having a problem showing that the...
I have been trying to solve the inverse Fourier transform:
\int_{-\infty}^{\infty}\left[e^{-j2\pi ft_0}e^{j\theta}\right]e^{j2\pi ft}df
I know that the Fourier transform pair says
e^{-j2\pi ft_0}e^{j\theta} \leftrightarrow \delta(t-t_0)
but the extra phase term e^{j\theta} makes...
Hi there. I have some trouble with this. I have to find the inverse Fourier transform for: \frac{e^{i 6\omega}}{\omega}
So I'm using a table, then:
F^{-1}\left ( \frac{e^{i 6\omega}}{\omega}\right )=F^{-1}\left ( e^{i 6\omega}\right ) * F^{-1}\left ( \frac{1}{\omega}\right )=2\pi\left[ \delta...
[Solved] Inverse Fourier Transform
Homework Statement
If
F(\omega)=e^{-|\omega|\alpha}\,(\alpha>0),
determine the inverse Fourier transform of F(\omega). The answer is \frac{2\alpha}{x^{2}+\alpha^{2}}Homework Equations
Inverse Fourier Transform is defined as...
1. find the inverse FT of 1/(iw+3)3
2. well partial fractions gave the same thing back... I'm not sure how to transform this as there's no property that deals with cubics.
3. i tried using the differentiation property but it doesn't work as it increases the power of 3 to 4 and so...
1. find the inverse Fourier transform of f(w)=e-i5wsinc(2w)
2. I set up the integral to be from defn of sinc: 1/2pi*integral from -infinity to infinity (sin(2w)/2w)*e^-5w
3. i have no idea how to solve this integral, is there a better way to do this?
i know that rect(t) has a...
Hi i am stuck with a program in MATLAB to find the time domain impulse response of a system from its continuous transfer function in the frequency domain.
Here is the program-
delt=1.5625e-9; %definition of delta t(sampling time).To be taken sufficiently small depending upon the time...
Homework Statement
The problem is to obtain the inverse Fourier transform of the following 2D functions
F(\mathbf{k})=\frac{k_{x}k_{y}}{k^{2}}
Homework Equations
The relevant equations are the 2d Fourier transform formulas described...
Homework Statement
We are asked to prove that if F(\omega ) is the Fourier transform of f(x) then prove that the inverse Fourier transform of e^{i\omega \beta}F(\omega) is f(x-\beta )
Homework Equations
F(\omega)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(x)e^{i\omega x}dx...
Hi all,
I'm having a bit trouble computing the Inverse Fourier Transform of the following:
\frac{\alpha}{2\pi}\exp\left(\frac{1}{2} \alpha^2 C^2(K) \tau \omega^2\right)
Here, C^2(K), \alpha and \tau can be assumed to be constant. Hence, we have an integral with respect to \omega.
Who...
Hello,
I am working through Signals and Systems Demystified, but I am at an impasse.
I would like to take the inverse Fourier transform of
H(f)=\begin{cases}
-j&\text{if } f > 0\\
j&\text{if } f<0\end{cases}
So
h(t) = \int_{-\infty}^{0} je^{j2\pi f t}df +...
Homework Statement
calculate the inverse Fourier transform of \left( a^2 + \left( bk \right)^2 \right)^{-1}
The Attempt at a Solution
I know that FT[e^{-|x|)}](k) = ( \pi (k^2 + 1 ) )^{-1}. I've tried to to concatenate the shift FT or the strech FT, but the "+1" in the known FT is in the...
I would like to do an inverse Fourier transform using MATLAB's IFFT. I am confused by MATLAB'S single input of X for its IFFT function. Has anyone had experience using MATLAB for these tranforms?
I would like to do an inversion of Fourier transform for my function y(iw) at some value real...