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Questions tagged [fourier-transform]

continuous Fourier transform, discrete Fourier transform (DFT), discrete cosine transform (DCT), discrete sine transform (DST)

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Tough integral from particle physics

I have been struggling to check if it's even possible to calculate the following integral $$ \iiint_{\Bbb R^3} d^3 \textbf{q} {1\over \sqrt{E_{\textbf{p}+\textbf{q}}(E_{\textbf{p}+\textbf{q}}+m)}}e^{-...
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Questions about fourier transformation

I understand to purpose of the fourier transformation is to transfer a PDE into an ODE. By solving the ODE and then take the inverse transform, I can get the answer. I have the following PDE ($\phi $ ...
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14 views

Compare R Programming with Sagemath for Fourier analysis? [on hold]

For Fourier mathematical simulation harmonics which one we will use R or Sagemath.
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44 views

Show that $\lim\limits_{R \uparrow \infty}\frac{1}{2\pi}\int_{-R}^{R}e^{-i\mu x}\hat{f}_{ab}(\mu)d\mu = f_{ab}(x)$

Show that$$(1)\lim\limits_{R \uparrow \infty}\frac{1}{2\pi}\int_{-R}^{R}e^{-i\mu x}\hat{f}_{ab}(\mu)d\mu = f_{ab}(x)$$ where $f_{ab}(x)$ and $\hat{f}_{ab}(\mu)$ are: More clarifications: Let the ...
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How do i get fourier series of signals given below [on hold]

how can i calculate and plot $Ae^{-at}$ amplitude and phase spectrum? how can i determine the trigonometric fourier series of the signal given below ? Your help is appreciated !! Thanks
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Reference for a (non-standard) table of characteristic functions

I am searching for a table of characteristic functions of probability distributions with more than the standard distributions. At a pinch a table of fourier transforms will do it too. But I computed ...
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23 views

The Fourier Transform of a periodic function

I am confused about this thing The definition of the Fourier transform of a function $f \in \mathbb{L}^1( \mathbb{R})$ (integrable function ): $$F(f)(x)=\int_{-\infty}^{\infty}f(x)e^{-i2\pi xt}dt$$ ...
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Evaluation of integral $\int_{-\infty}^{\infty} e^{itx} \frac{1- e^{-\frac 1 2 t^2}}{\frac 1 2 t^2} \text d t$ / specific characteristic Function

I want to calculate the value of $$I(x) :=\int_{-\infty}^{\infty} e^{itx} \frac{1- e^{-\frac 1 2 t^2}}{\frac 1 2 t^2} \text d t$$ where $x\in \Bbb R$. Of course we can write $$I(x) = \int_{-\infty}^{\...
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How to determine solution to the Laplace Equation using Fourier Transform

$\nabla \phi_1 = 0 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ on~~~ 0<y<H$ $d\phi_1 / dy = dn_1 / dx ~~~~~~~~~~~~~~~~~~~~~~~~~~ on~~~ y = H$ $d\phi_1 /dx + n_1 / F^2 + P(x) = 0 ~~~~~~~ on~~~y =...
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Show that $\int_{-\infty}^{\infty}f(x)\overline {g(x)}dx = \frac{1}{2\pi}\int_{-\infty}^{\infty} \hat{f}(\mu)\overline{\hat{g}(\mu)}d\mu.$

Given: Show that if f(x) is defined as: The Fourier transform $\hat(\mu)$ of a function $f(x)$ specified on $\mathbb R$ is often defined by the formula: $$\hat{f}(\mu) = \int_{-\infty}^{\infty}e^{i\...
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Extracting a fundamental period.

There is given sequence $$a_{1},a_{2},a_{3},...$$ Where $a_{n}\in\{1,2,...,N\}$ for every natural $n$. It is also known that this sequence is periodic. Find the fundamental period the sequence. ...
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+150

Show that $\int_{-\infty}^{\infty}|f(x)|^2dx = \frac{1}{2\pi}\int_{-\infty}^{\infty}|\hat{f}(\mu)|^2d\mu$

Given: Let $a_1 \lt b_1 \le a_2 \lt b_2 \le ... \le a_{n-1} \lt b_{n-1} \le a_n \lt b_n$ and let $$f(x) = \sum_{j=1}^nc_jf_{a_jb_j}(x).$$ Show that, $$(*)\int_{-\infty}^{\infty}|f(x)|^2dx = \frac{1}...
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If $ 0 \le t \lt 1$ then, $\lim\limits_{R \uparrow \infty}\int_{0}^{\pi}e^{-R\sin\theta}R^t\text d\theta = 0$

If $ 0 \le t \lt 1$ then, $$\lim\limits_{R \uparrow \infty}\int_{0}^{\pi}e^{-R\sin\theta}R^t\,\text d\theta = 0$$ I'm not sure how to solve this, I thought about starting to show that $ 0 \le \sin\...
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meaning of double headed arrow

I am studying properties of discrete time Fourier transform and I encountered a notation shown highlighted in attached photo What is meant by this notation? Is it meaning equality? screenshot of ...
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28 views

What is the Fourier transform of $\frac{x}{(x^2+y^2)^{n/2}}$?

We have the following Fourier transforms: $$ {\cal F}\left[\frac{1}{(x^2+y^2)^{1/2}}\right] = 1/\sqrt{k_x^2+k_y^2} $$ $$ {\cal F}\left[\frac{1}{(x^2+y^2)^{3/2}}\right] = -\sqrt{k_x^2+k_y^2} $$ $$ {\...
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Is there a connection between homomorphism and diagonalization?

The definition of homomorphism is https://en.wikipedia.org/wiki/Homomorphism and definition of diagonalization needs to be understood in the context described in Fourier transform as diagonalization ...
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31 views

Comparing energy and power signal

I am trying to define and graph the signal to noise ratio of my system in the frequency domain. The system is LTI, excited by a pulse and subjected to filtered ZMWN. The only relevant portion of my ...
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Fractional derivative question

This might be a duplicate but I could not find the answer here. I want to prove the following relation for the fractional derivative $a \in \mathbb{R}$: $$ (i\partial_x)^a = (ix)^{-a} \frac{\Gamma(a-x\...
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Analytical expression for PSD

Is it possible to obtain an analytical expression for the PSD? The PSD is defined as follows, $S(\omega) = lim_{T \rightarrow \infty} \frac{1}{T} |Y(\omega)|^2 $ Assuming there is $Y(\omega)$ ...
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1answer
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fourier series meaning complex radii meaning

I'm talking about the incredible answer from Mark Eichenlaub. Fourier transform for dummies He said : "we must allow the circles to have complex radii. It's the same thing as saying the ...
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Fourier transform unitary operator

I'm trying to understand why there is a continous extenstion of $\mathcal{F}_0f$ to a unitary operator in $L^2$. So let $\mathcal{F}_0f: S(\mathbb{R}^m) \to S(\mathbb{R}^m).$ This is a bijective ...
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Showing $\int_{-\infty}^{\infty}\frac{\sin^2(\omega)}{\omega^2}d\omega=\pi$

Given the function $f$ with $f(t)=1$ for $|t|<1$ and $f(t)=0$ otherwise, I have to calculate its Fourier-transform, the convolution of $f$ with itself and from that I have to show that $$\int_{-\...
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1answer
119 views

Inverse Fourier Transform of a Constant

The Fourier transform and its inverse can be defined as $$\mathcal{F}(f(x))=F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} f(x)e^{-ikx} \ dx \ \ \text{and} \ \ \mathcal{F}^{-1}(F(k))=\frac{1}{\sqrt{...
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Inverse Fourier Transform of a Spiral?

I am trying to minimize a free energy by plugging in this variation spin structure (a vector field): $$\mathbf{S}(\mathbf{r}) = {1 \over \sqrt{2}} \left( \mathbf{S_k} e^{i\mathbf{k \cdot r}} + \mathbf{...
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DFT is not a sampling of FT?

From wikipedia: The discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time ...
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1answer
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Showing that an orthonormal set becomes a basis for the Hilbert space

This is an exercise from Folland Real Analysis Chapter 8 that I am stuck at. I am actually stuck at (b). I succeeded in showing that $H_a$ is a Hilbert space and the given set is an orthonormal set of ...
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even/odd signals and their FFT

I want to use the following property of the Fourier transform: Even functions have even transforms; odd functions have odd transforms. in mathematical terms: if $f(t)$ is a function that has an ...
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$L^2$ functions with compactly supported Fourier transforms form a Hilbert space

Given a fixed compact subset of $\mathbb{R}$, I want to show that square integrable functions on the real line whose fourier transforms are supported in the given compact set form a Hilbert space in ...
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What is the Fourier cosine transform in complex notation and what is the conjugate of the Fourier cosine transform?

Suppose the Fourier cosine transform is given by: \begin{align} F_c(k)=\mathcal{F}(f(x))&=\sqrt{\frac{2}{\pi}}\int_0^{\infty} f(x) \cos(kx) \end{align} or any other form I'm not particular ...
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1answer
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What is the decomposition of $H^{T}H$, when $H$ is a circulant matrix?

Since $H$ is a circulant matrix, the decomposition using Fourier transform matrix $F$ $$H = F^{-1} \Lambda F$$ where $\Lambda$ is the diagonal matrix with the eigenvalues of $H$. If I plug in the ...
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Not understanding about the sign of Fourier transform of power spectrum

I don't understand with the below equations the affirmation that $FT(\Delta(\vec{k})$ is the Fourier transform of $\Delta(\vec{r})$ : $$\left\{\begin{array}{l}{\Delta(\vec{r})=\frac{V}{(2 \pi)^{3}} \...
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Density of a class of function in $L^2(\mathbb{R}, e^x\,dx)$

Consider the class of function defined by $$\mathcal{G}=\operatorname{Span}\left\{e^{-\frac{(x+a)^2}{2}}-e^{-x}e^{-\frac{(x+a)^2}{2}}\mid a\in\mathbb{R}\right\}.$$ Is $\mathcal{G}$ dense in $L^2(\...
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Fourier Transform of Rectangular Impulse Function possible in specific Form?

Rectangular Impulse Function The above Rectangular Impulse Function is given. It's height is $A$ and it's width is $T$. The question is the following: If the Fourier Transform of the above function ...
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1answer
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Fourier-Transformation of $\exp(-|t|)$ [closed]

Using the Fourier-inversion-theorem I have to show that $$\frac{\pi}{2}\exp(-|t|)=\int_0^\infty\frac{\cos(\omega t)}{1+\omega^2}\mathrm d\omega$$ Can anyone give me a hint on how to show it? ...
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Fourier transform and characteristic function of an interval

If we define the Fourier transform of a function $f$ as the function $$\mathcal{F}(f)(t) = \frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}e^{-ixt}f(x)\,dx,$$ how can I prove that $$\mathcal{F}(h)(t)=\chi_{[-b,...
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A question on discrete Fourier transform of some function

Let $\sigma(n) = \sum_{d|n} d$ and $\tau(n) = $ number of divisors of $n$. For each $k, 0 \le k \le n-1$ we can look at the discrete Fourier transform of the numbers $\sigma(\gcd(n,k))$ given by: $$\...
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Fourier transform with branch cut 2

How to evaluate the following integral by using residue theorem and how we choose the contour. $$W=\int_{-\infty}^{\infty}\frac{d\lambda}{\sqrt{i\lambda}} \frac{e^{-ib\lambda}}{\lambda^2-a}$$, where ...
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Fourier transformation for $f(x) = \frac{b^2}{(x^2+b^2)^2}$

Given the function \begin{align} f(x) = \frac{b^2}{(x^2+b^2)^2} \end{align} where $b>0$. I want to compute the Fourier Transformation using \begin{align} \hat{f}(k) = \frac{1}{\sqrt{2\pi}}\int_{-...
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Fourier Transform of $1$ using Tempered Distributions

I'd like to show that the Fourier Transform of $1$ is $\delta$ using the concept of tempered distributions : $$ \hat{1}(\phi)=1(\hat{\phi})=\int_{-\infty}^{+\infty}\hat{\phi}=\phi(0)=\delta(\phi)$$...
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Sketch Fourier Transform Analysis Equation

First of all,thank you for your attention. The exercise is very straightforward and it's easy to understand what is asking. I basically have two functions. I calculated (correctly) the fourier ...
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I solved the fourier transform of 1/t, but now I need to calculate for 1/t^alpha+1 and I should not be using contour integral formula.

So, I have f(t)=1/t^(alpha+1). with alpha >= 1. I am not allowed to use a contour integral formula and I can only use Γ(n+1)=n! identity if required. I tried solving with integration by parts but ...
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Graph Fourier Transform - Intuition of Eigenvalues of the Laplacian

first of all I don't have a solid math background, so a less general, but easier answer is preferable. In "The Emerging Field of Signal Processing on Graphs" by Shuman et. al. they describe how the ...
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Find the fourier transform of $e^{-ax^2}\cos(bx)$

I am trying to find the Fourier transform of the function $e^{-ax^2}\cos(bx)$. I am changing $\cos(bx)$ to its exponential form but after that I am getting stuck: can you guide me showing how to find ...
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Proving $\mathcal{F}\left(\frac{d^n}{dx^n} f(x)\right)=(ik)^n\mathcal{F}(f(x))$.

I am trying to prove the $n^{th}$ transform of the Fourier transform: $$\mathcal{F}\left(\frac{d^n}{dx^n} f(x)\right)=(ik)^n\mathcal{F}(f(x)) \tag{1}.$$ I have solved the problem for the case $n=1$ ...
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On a simple discrete Fourier transform

Given an integer N, and two non-negative integer valued variables x,y which take values in ${0,1,...,N-1}$. Is it possible to obtain a close form for the following summation? $$f(x,y)=\frac{1}{N}\...
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2D convolution with Gaussian using Fourier transform

I was solving 2D diffusion equation with initial condition \chi(x,0)=1 in the circle centered at origin with radius r. Equation I want to solve To solve this equation efficiently, I need to use ...
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1answer
21 views

Finding the Fourier transform of shifted rect function

This is my procedure but I am unsure if I did the right thing after talking with some of my peers. $$f(x) =\mathrm{rect}\bigg|\frac{x-x_0}{a} \bigg|= \begin{cases} 0 & \bigg|\frac{x-x_0}{...
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Why is the denominator and numerator of Fourier coefficient (in Fourier series(exponential) expansion) have a conjugate of the exponential set?

In the exponential Fourier series, the coefficient Fn takes both the conjugate and original of the exponential set in its denominator and numerator. This doesn't make any sense when seen from the ...
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1answer
34 views

Inverse Fourier transform of $F(k) = 1/(k^2+a^2), a>0$

I need help finding the Inverse Fourier transform of: $$F(k) = \frac1{ k^2 + a^2 },~ a>0$$ Here is what I have so far: Singular points at $k^2 = a^2$, namely, at $k = \pm ia$. The inverse ...
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1answer
35 views

Discretising the Fourier Integral gives a high condition number

I have the following integral equation $$ \int_0^1 e^{-2\pi i s t} f(t)\, \text{d} t = g(s), \hspace{3em} -1/2 \leq s \leq 1/2$$ where $f$ is to be found, and $g$ is known. I believe this problem is ...