## Fourier transform of derivative of correlations

I am trying to obtain spectra of derivative of a two-point correlation. What I know is that spectra of two-point correlations are computed as follows

$$phi_{a{b}} = FFT(a) times conj(FFT(b))$$

where $$phi_{a{b}}$$ is spectra of correlation of $$a$$ and $$b$$, and $$a$$ and $$b$$ are two signals collected at two different locations.

I also know that FFT of derivative of a function is

$$FFT(partial f/partial x) = i times kappa times FFT( f)$$

where $$kappa$$ is wavenumber and $$f$$ is a function.

My question is that is the following statement correct or not.

$$phi_{partial a{b}/partial x} = i times kappa times FFT(a) times conj(FFT(b))$$

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## differential equations – Approximate solution of a nonlinear ODE in the form of a Fourier series containing the coefficients of the initial ODE

In this topic we considering nonlinear ODE:

$$frac{dx}{dt}= (x^4) cdot a_1 cdot sin(omega_1 cdot t)-a_1 cdot sin(omega_1 cdot t + frac{pi}{2})$$ – Chini ODE

And system of nonlinears ODE:

$$frac{dx}{dt}= (x^4+y^4) cdot a_1 cdot sin(omega_1 cdot t)-a_1 cdot sin(omega_1 cdot t + frac{pi}{2})$$

$$frac{dy}{dt}= (x^4+y^4) cdot a_2 cdot sin(omega_2 cdot t)-a_2 cdot sin(omega_2 cdot t + frac{pi}{2})$$

Chini ODE’s NDSolve in Mathematica:

pars = {a1 = 0.25, (Omega)1 = 1}
sol1 = NDSolve({x'(t) == (x(t)^4) a1 Sin((Omega)1 t) - a1 Cos((Omega)1 t), x(0) == 1}, {x}, {t, 0, 200})
Plot(Evaluate(x(t) /. sol1), {t, 0, 200}, PlotRange -> Full)


System of Chini ODE’s NDSolve in Mathematica:

pars = {a1 = 0.25, (Omega)1 = 3, a2 = 0.2, (Omega)2 = 4}
sol2 = NDSolve({x'(t) == (x(t)^4 + y(t)^4) a1 Sin((Omega)1 t) - a1 Cos((Omega)1 t), y'(t) == (x(t)^4 + y(t)^4) a2 Sin((Omega)2 t) - a2 Cos((Omega)2 t), x(0) == 1, y(0) == -1}, {x, y}, {t, 0, 250})
Plot(Evaluate({x(t), y(t)} /. sol2), {t, 0, 250}, PlotRange -> Full)


There is no exact solution to these equations, therefore, the task is to obtain an approximate solution.

Using AsymptoticDSolveValue was ineffective, because the solution is not expanded anywhere except point 0.

The numerical solution contains a strong periodic component; moreover, it is necessary to evaluate the oscillation parameters. Earlier, we solved this problem with some users as numerically:
Estimation of parameters of limit cycles for systems of high-order differential equations (n> = 3)

How to approximate the solution of the equation by the Fourier series so that it contains the parameters of the original differential equation in symbolic form, namely $$a_1$$, $$omega_1$$, $$a_2$$ and $$omega_2$$.

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## Structure factor of fourier transform of an image

In this article authors show crystal and liquid phase from two dimensional crystals by calculating structure factor (Fourier transform of 2D points). I have generated set of points in 2D that represent lattice points of a perfect triangular lattice and a non perfect lattice. (images below)

perfect lattice not perfect lattice I want plot similar results from figure 2 of the article (image below) for both cases perfect and not perfect lattice. Can I do this in mathematica? The plot here shows for totally random points (left), not perfect lattice (middle), and perfect lattice (right)

Thank you.

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## Why is a quasicrystal characterized by a distribution for which it and its Fourier Transform both have only discrete supports?

Why does a distribution where both it and its Fourier Transform have discrete supports characterize a quasicrystal (an ordered but aperiodic structure) ?

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## reference request – Fourier transform of $|sin(x)|/|x|$ in one dimension

We know that the function
$$g(x) := frac{|sin(x)|}{|x|}, qquad xin mathbb{R}$$
is an $$L^2(mathbb{R})$$ function. Let $$f$$ be its Fourier transform, namely,
$$f(xi) := mathcal{F} g(xi) = int_{mathbb{R}} e^{-i xxi} g(x) dx = 2int_0^infty cos(x xi) frac{|sin(x)|}{|x|} d x.$$

The questions are

1. Is there a formula for $$f$$? I guess there should not be a formula for this transform, otherwise, Mathematica usually can do the job.

2. If there is not an explicit formula for $$f$$, can one tell whether $$f$$ changes signs? We know that since $$g$$ is nonnegative, $$f$$ has to be nonnegative definite. But I am interested to see $$text{whether f is nonnegative?}$$
I suspect this is the case but cannot prove it.

This function seems quite standard in harmonic analysis and results related to this function have to be buried somewhere in the literature. Any hints are highly appreciated!

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## fast fourier transform – Butterfly diagram from cooley-tukey algorithm

I am trying to understand the logic of this algorithm so i can implement my own but i am not understanding this diagram i see appearing many times in a fair few articles on the topic, i am teaching myself so i don’t have a computer science degree to help. But i understand the general idea behind the algorithm.

But this butterfly diagram is confusing me. How do you interpret this diagram from left to right mathematically speaking? It’s so confusing.. what are the arrows telling me to do in terms of math operations on the functions?

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## Substitution of frequation in fourier transform

In wikipedia of ESD and PSD, a transformation as

$$hat x(f) = int_{-infty}^infty e^{-2pi ift}x(t){rm d}t$$

we know $$omega= 2pi f$$. how can we obtain then

$$hat x(omega) = frac{1}{sqrt{T}}int_0^T e^{-iomega t}x(t){rm d}t$$

where does the $$sqrt{T}$$ come from?

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## Difference between the Magnitude and the Phase Spectrum of the Fourier Transform?

What is the difference between the Magnitude and the Phase Spectrum of the Fourier
Transform?

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## partial differential equations – Solve $u_t$ + -$u__xx$ + $xu$ using Fourier Transform

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