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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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23 views

Using Fourier transform to solve for pde

Consider the initial value problem for the wave equation $\frac{\partial^2}{\partial t^2}u(x,t) = c^2 \frac{\partial^2}{\partial x^2}u(x,t)$ , $-\infty < x < \infty$, t > 0 with initial ...
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21 views

Is it possible to use Fourier Transformation for that problem?

I am trying to solve a convection-diffusion problem for multi-stacked layer system. I obtained a system of second-order PDEs with variable coefficients. The independent variables are radius (r, {0 - ...
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29 views

Convolution of 3 functions

Assume that I have an equation like this: $f(t)=\int{g(\tau)h(2t-\tau)k(t-\tau)d\tau}$, in which g(t), h(t) and k(t) are three arbitrary functions. It can be seen that it is similar to convolution ...
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27 views

Limit of Series of Distributions

Let $\displaystyle \langle \psi_k,\phi\rangle = \int_{-\infty}^\infty \sin(kx)\phi(x)\,\mathrm{d}x,\ \phi \in \mathcal{C}_c^\infty(\Omega),\Omega\subset \mathbb{R}$ open. That means $\phi$ is a ...
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17 views

Fourier Transform of Indicator Function $\chi_{[-0.5,0.5]}$

I want to compute $\displaystyle(\mathcal{F}f)(k)=\int_\mathbb{R} \chi_{[-0.5,0.5]} e^{-ikx}\,\mathrm{d}x$ I proceeded as follows: $\displaystyle(\mathcal{F}f)(k) =\int_{-0.5}^{0.5}\cos(kx)+i\sin(...
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1answer
28 views

Fourier transform of derivative piecewise function

Consider the function $$f(x) = \begin{cases}(e^x-1)^2 & x < 0 \\ x^2 & x \geq 0\end{cases}$$ I would like to compute the Fourier transform of the fourth derivative, however, I find ...
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How to calculte the Fourier Transform of a sovable chaos waveform?

Recently I am stucking in frequency estimation of a solvable chaos waveform. Its analytic expression in time domain is $$ z(t)=s_m(u_m-s_m)e^{\beta(t-mT)}\cos(\omega_0 t+\varphi) $$ where $u_m \sim U(-...
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25 views

How to generate Fourier invariant closest norm sequence?

I am not sure if it is a useful question but would like to know if possible. Consider I have a random numbers $A=a_1, a_2,,a_n$. Now suppose, I do Fourier Transform of A (for example use fft command ...
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11 views

Sum of Convolutions, Laplace Transform?

Suppose that we have a function $g(t)$ that can be computed from a function $h(t)$ from $g(t) = h(t) + h(t)*h(t) + h(t)*h(t)*h(t) + ...$ where $*$ is convolution. We have the following conditions: ...
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1answer
37 views

Fourier Transform for Cosine-Squared

I'm having trouble finding the Fourier transform of $g(t) = \cos^2{a x}$. I know the answer has to be a summation of $3$ dirac delta functions, but I'm having trouble showing this. I'll show you ...
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Solutions of $\partial_t f =-\Delta^2 f$

I'm looking for references (or direct solutions :) ) of how to solve the following partial differential equation : $\partial_t f =-\Delta^2 f$ , with $\Delta f$ the Laplacian of $f$. I'm interested ...
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Completing the square to find the Fourier transform of a Gaussian [duplicate]

I'm trying to find the Fourier Transform of a Gaussian, and I end up having to complete the square in the argument of an exponential so I can use the standard Gaussian integral. I basically have: $$\...
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1answer
31 views

Difference real and complex fourier series

I'm working on fourier series and I'm trying to compute the fourier transformation for the $2\pi$-periodic function of $f(x)=x^2$ with $x \in [-\pi,\pi]$. Now with the real way, that is $$f(x) \sim \...
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Solve $f'(t)=0$ and $f'(t)=1$ using Fourier transform

I'm trying to solve $f'(t)=0$ and $f'(t)=1$ using Fourier transform, but no luck: a) $f'(t)=0$ $$ f'(t)=0 \Rightarrow jwF(w)=0 \Rightarrow \begin{cases}F(w)=0 ~ \text{if} ~ w \ne 0\\ F(0) = \text{...
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34 views

Derivative of Fourier Transform to phase for Gradient descent

My Problem is the following: I have the complex signal over time $$f(t) = r(t)\cdot exp(i\phi(t)) \in \mathbb{C}$$ with the corresponding discrete Fourier Transform in frequency space $$s(\omega) = ...
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Generalisation of the Fourier Transform

If we define a Phase-shift operator as follows $$ \Phi_{(a,b)}[f(x)] = e^{i b (x - a /2)} f(x-a) $$ and recall the following property of the Fourier transform $$ F[e^{i a (x+ b /2)} f(x+b)] = e^{ib (\...
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Fourier components of an incomplete wave

Say we have a periodic wave defined by the region $[0,4\pi]$. However, for some reason we only know what the wave is between $[0,\pi]$ and $[3\pi,4\pi]$. We also know that the amplitude of the wave at ...
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15 views

Soft Question: Highlights of Osculatory Integral Theory

I'm curious to look into Oscillatory integral operator theory and was wondering what are some of the highlights, main results, and historical development. Are there distributional characterizations ...
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1answer
75 views

Find the Fourier transform of $e^{- x^2}$.

I am trying to get the cosine Fourier transform of $e^{- x^2}$ where I am stuck in simplifying the integration further. The integration is between the limits $0$ to $\infty$!
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Vector scalar product and Fourier transform in the hyperbolic plane?

I study a set of points $r_j$ in the hyperbolic plane, let's say in the upper half plane $\{ z\in\mathbb{C} | \Im(z)>0\}$, and I try to define a structure factor in the same spirit as we would ...
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45 views

Fourier Transform on Musical Notes

I am trying to apply the Fourier transform analysis on music. So far, I am aware that the Fourier transform is essentially the breaking down of superposed sine waves, into its individual frequencies. ...
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1answer
33 views

How to prove this inequation in Fourier Analysis? [closed]

$|e^{2\pi i (x+h)\xi} - e^{2\pi i (x)\xi}| \leq 2sin(\pi |h\xi|)$ Original inequation is $\int \hat{f(\xi)}(e^{2\pi i (x+h)\xi} - e^{2\pi i (x)\xi})d\xi \leq \int |\hat{f(\xi)}|2sin(\pi |h\xi|)d\xi$ ...
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1answer
31 views

Fourier transform of a product of exponential and rectangular window

Is there any formula to calculate the Fourier Transform of a product of a exponential function and a rectangular windows? i.e. Formally, calculating $$F(x(t))=F[(e^{-at})\Pi((t-T/2)/T]$$ Where $\Pi$...
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21 views

Fourier Transform of a sequence

The FT of a discrete sequence $x$ is well known. However, is there a way to compute the FT of $x^{2}$ given $x$ and the FT of $x$? How does one go about deriving this? F(x) = $\sum x \;e^{(-j2\pi\...
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1answer
68 views

Taking a Fourier transform after taking Laplace transform

Consider the following PDE, $$\frac{\partial u}{\partial t} − κ \frac{\partial^2 u}{\partial x^2} = S_0δ(x)δ(t),$$ subject to the initial condition, u(x, 0) = δ(x), with κ > 0, and $S_0$ > 0. ...
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1answer
31 views

Analog of convolution theorem for rotation (rather than translation) of coordinates?

According to convolution theorem convolution between two functions $f(r), g(r)$ in real space (e.g. 2D,3D) can be calculated as product of Fourier images. Convolution $(f*g)(\vec R) = \int_{\vec r} f(...
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23 views

Fourier transform and fourth root

Given a well-behaved convex function $f(t):\mathbb{R}\to \mathbb{R}$, its Fourier transform (FT) $\hat{f}(\omega)=\mathcal{F}[f(t)](\omega)$ is positive (and decreasing) proof here. It follows that ...
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1answer
25 views

invalid proof positivity of fourier transforms

I just skimmed through this paper on positivity of fourier transforms where it is shown that the sine transform $$\int_0^\infty f(x) \sin(x t) \mathrm{d}x$$ is positive if $f$ is decreasing. The proof ...
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Convolution of two square pulses and the fourier transform of a triangular pulse

I would like some help clearing up some confusion. The fourier transform of a triangular pulse with base $2T_b$ is given as, $T_bsinc^2(\pi f T_b)$ You can also get this result by realizing that 2 ...
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16 views

Hybrid implicit FFT resampling, does it make sense?

In signal processing there exist so many different methods of interpolation that one could probably write a book about it. Or ten. Or a hundred. When learning about the Fast Fourier Transform, one of ...
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1answer
107 views

Fourier transform for Helmholtz equation

The Helmholtz equation takes the form, $$u_{xx} + u_{yy} + k^ 2u = f(x, y),$$ for $−∞ < x < ∞$, $−∞ < y < ∞$. i) Assuming that the functions $u(x, y)$ and $f(x, y)$ have Fourier ...
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Fourier inverse transform of $\frac1{(i\omega+a)(\omega^2-b^2)}$

How can I calculate the Inverse Fourier transform of $$f(\omega)=\frac{1}{(i\omega+a)(\omega^2-b^2)},\;a,b\in\mathbb{R}, a>0.$$ I guess that I cannot use the residual theorem, since the function ...
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Difference between NFFTs, NUFFTs, USFFTs (for CT purposes)

I am trying to understand for what cases and in what way NUFFTs would be useful for CT reconstruction. Therefore, I am trying to create an overview for myself that starts with the "easy" problems ...
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Find $f(\widehat{3x+5})$ when $\widehat{f(x)} = \hat{f}(w)$

Find $$f(\widehat{3x+5})$$ when $$\widehat{f(x)} = \hat{f}(w)$$ Im unsure exactly what they're asking and how to approach the problem. The hint I got is to use the change of variabes $y = 3x+5$
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What is the Discrete Fourier Transform of a periodic sequence?

Shor's algorithm utilises the properties of quantum computers to find $r$ in, $$ a^r = 1 \mod n $$ wherein $r$ is even and $a^{r/2} + 1\neq 0$. However I haven't been able to find any resources in how ...
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65 views

Folland - Real Analysis, Plancherel theorem proof

I have a problem in understanding the proof of Plancherel Theorem in Folland - Real Analysis. Theorem. If $f \in L^1 \cap L^2$, then $\hat{f} \in L^2$; and $\mathcal{F}|(L^1 \cap L^2)$ extend ...
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Boundary functions of elements in $H^\infty(U)$

Let $U$ the upper-half plane and let $H^\infty(U)$ the set of bounded holomorphic functions defined on $U$. From the theory of harmonic bounded functions in the upper-half plane, we know that: $$\...
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60 views

1D Fourier transform of 2D function with shift

Say you have a 2D function $f\left(x,t\right)$ whose 1D Fourier transform in time is \[F\left(x,\omega\right)=\int_{-\infty}^{\infty}f\left(x,t\right)e^{-i \omega t}dt.\] And you now introduce a time-...
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1answer
70 views

Fourier transform of a product in $L^1$

There is a little thing that I do not understand, about the Fourier transform of a product of functions in $L^1$ (and only in this space), with the relation ${\mathcal F}(f g)(\lambda) = \mathcal{F}(f)...
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What's known about these “almost” convolution operators?

Convolution operators like $$T_1 f(x) := \int_\mathbb R K_1(x-y)f(y)\, dy$$ or $$T_2 f(x) := \int_{\mathbb R/\mathbb Z} K_2(x-y)f(y)\, dy$$ are classical objects of study. Is there much known about ...
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Fourier transform of $x / \tanh(x)$

I have problems to calculate analytically the (inverse) Fourier transform of $x / \tanh(x)$: $$\frac{1}{\sqrt{2\pi}}\int_\mathbb{R} \frac{x}{\tanh(x)} \mathrm{e}^{- \mathrm{i} x k} \mathrm{d} x$$ ...
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1answer
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Questions derived from trying to prove Plancherel's Theorem

I'm going to try to derive it here and illustrate any questions I have about it in the process. I tried to understand it from here, and it's where my confusions are based off of. If you wonder why I ...
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1answer
67 views

Finding the Fourier Transform of $De^{-\lambda \lvert x\rvert}$

I'm having trouble computing the Fourier transform of the following function: $$y(x)=De^{-\lambda \lvert x\rvert}$$ It mainly has to do with the integration, I think, but I'll try to attempt it and ...
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1answer
51 views

Function on HP Prime calculator to get Symbolic expression of Fourier transform from Laplace transformation

I would like to have a simple algorithm (or function) on HP Prime calculator to get symbolic expression of the Fourier transform of a function. On my HP Prime, it is possible to get the Laplace ...
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1answer
46 views

Proving $\int^{\infty}_{-\infty}e^{-2\pi i k (x-a)}dk=\delta(x-a)$ with this method

I want to prove $$\int^{\infty}_{-\infty}e^{-2\pi i k (x-a)}dk=\delta(x-a)$$ By following the following logic: $$\int^{\infty}_{-\infty}e^{-2\pi i k (x-a)}dk$$ equals $0$ whenever $x\ne a$ and $\infty$...
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36 views

Given $f(x+iy):=\int_{-a}^{a}g(t)e^{2\pi i(x+iy)t}dt$ is it true that $\forall M>0, \sup_{y\in[-M,M]}|f(x+iy)|\rightarrow0, |x|\rightarrow\infty$?

If $a>0$, and $g\in L^1(-a,a)$ define: $$f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto \int_{-a}^{a}g(t)e^{2\pi izt}\operatorname{d}t.$$ I know from Riemann-Lebesgue lemma that: $$\forall y\in\mathbb{...
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1answer
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Evaluating $\int_{-\infty}^\infty e^{i(ax^2+bx+c)}\frac{\operatorname{sin}^2x}{x^2}dx$

Is there a closed form for the following integral? $$\int_{-\infty}^\infty e^{i(ax^2+bx+c)}\frac{\operatorname{sin}^2x}{x^2}dx$$ Nothing of this form seems to appear in Gradshteyn and Ryzhik.
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16 views

How is variable change established in this integral?

Regarding a Fourier transform formula, Im having difficulty to understand how is the below integral as a function of $\omega$ converted as a function of $f$. I only know that $\omega=2\pi f$. $$H(\...
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1answer
33 views

Inverse Fourier transform of 1/(1-1/4*e^(-jw)) [closed]

How can I calculate the inverse Fourier transform of the following $$\frac{1}{1-\frac{1}{4}e^{-jw}}$$ I have tried looking at the table of Fourier transforms but I can't find anything useful. Thanks ...
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0answers
26 views

Double 3-d Fourier transform

I have a nested Fourier transform as following $$\int\frac{d^3\mathbf{q_1}}{(2\pi)^3}\frac{d^3\mathbf{q_2}}{(2\pi)^3}\frac{1}{\mathbf{q_1}^2\mathbf{q_2}^2(\mathbf{q_1}^2+\mathbf{q_2}^2)}\frac{1}{|\...