0
votes
0answers
6 views

What's the best way to recognize a shape o a function with N-points

I've many shapes with points in theirs countours, how is the best way to recognize a shape? I think the DTF is available but i don't know whether this is the optimal way. P.S. I think if i will ...
0
votes
1answer
45 views

Where is the symmetry of Fourier transform in its implementation in Maxima and Wolfram Alpha?

From Wikipedia I saw that there is a symmetry of the Fourier transformation $F(F(f))(x) = f(-x)$ This matches the graphical explanation of the (German) Youtube video (9:15 to 9:45). I tried to see ...
0
votes
0answers
14 views

Is there a way to characterize the range of a chebyshev series through its coefficients?

Let $f$ be a Chebyshev series of order $n$ $$ f(x) = \sum_{i=0}^n a_i \cos\left( i \arccos\left(x\right)\right), x \in \left[-1, 1\right]. $$ Is it possible to characterize all the $\lbrace a_i ...
1
vote
1answer
34 views

Relative error when computing derivatives via FFT

I want to compute a discrete derivative via the FFT. This amounts to multiplication by the wave number in Fourier space, as detailed in the stack exchange answer here. When I increase the ...
0
votes
0answers
14 views

Calculating h-ellipticity

How do we calculate h-ellipticity $E_{h}$ of standard five point discrete Laplacian of two dimensional partial differential equation?
3
votes
1answer
89 views

Fourier Transformation: an Animated GIF

Here I found the animated GIF below. I don't get it! Would someone explain it please?
0
votes
0answers
42 views

Autocorrelation Function and Power spectrum from ACF

In my assignment I am required to write or use a C code to find the autocorrelation function of a given function and then find the power spectrum from it. The function is as follows: $$f(t) = \cos(10 ...
1
vote
1answer
122 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
0
votes
0answers
32 views

Split Step Fourier Algorithm

Consider the NLSE (Nonlinear Schroedinger equation) that can be written as the following partial differential equation: $$ \frac{\partial{A}}{\partial{z}}=({\cal{L+N}})A\quad\quad(1) $$ where $A: ...
1
vote
2answers
49 views

fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
2
votes
0answers
67 views

(newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
0
votes
0answers
11 views

von Neumann analysis

When performing a von Neumann analysis on a PDE, we end up getting an expression in Fourier-space on the form (k is a "wave number", as in $\exp(ikr)$) $$ A_{n+1}(k) = G(k)A_{n}(k) $$ where $G$ is our ...
0
votes
0answers
52 views

A numerical inverse Fourier transform

I am doing an research in phase noise, And I'm troubled in a paper. In that paper,it says when perform a inverse Fourier transform on ...
2
votes
0answers
37 views

Approximating the Fourier transform with DFT/FFT

Suppose I have a continuous function $f(x)$, $x\in[-L/2,L/2]$. Its $L-$periodic Fourier coefficients are given by $$ \hat{f}[k]=\frac{1}{L}\int_{-L/2}^{L/2}f(x)\exp(-2\pi ikx/L)dx $$ If I apply ...
1
vote
2answers
32 views

Which function satisfies the conditions?

I'm solving a problem and, in order to run test case, I need a function $ b(x,y) $ that satisfies: $$ \int_0^L \int_0^H b(x,y) \, dx \, dy = 0 $$ and $$ \int_0^L \int_0^H b(x,y) \cos \left(\frac{n ...
0
votes
1answer
23 views

von Neumann stability analysis for irregular meshes

All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be ...
0
votes
0answers
37 views

What is the error when approximating $L^2([0,1])$ by a finite dimensional space?

Let $X \subset L^2([0,1])$ such that $f([0,1]) \subset [-M,M]$, for some constant $M$, and any $f \in X$. By choosing a finite dimensional basis $V=\left(v_i\right)_{i=0}^n$, where each $v_i \in X$, ...
0
votes
0answers
144 views

FFT for Fourier Integrals of analytic Functions

So, I'm trying to implement a fourier-transform of an analytic function through DFT or FFT as I'm only interested in a certain frequency range. So far I've tested Numerical Recipes' FFT algorithm ...
2
votes
0answers
113 views

Solution of an implicit Fourier transform equation

How does one solve the following equation ($\hat{a}(k)$ denotes the Fourier transform of $a(x)$ and $q$ is real positive): $$\hat{a}(k)=f(k)\widehat{a^q}(k).$$ This equation appeared in some paper. ...
0
votes
0answers
142 views

Bluestein Algorithm for Fast Fourier Transform

Can anyone demonstrate the full algorithm of Fast Fourier Transform? Because from Wikipeida and other internet sources, I saw that there are different ways of padding. So can anyone tell me when the ...
3
votes
1answer
117 views

Numerical computation of continuous Fourier transform

Are there any algorithms that numerically compute the continuos Fourier transform of a given function f? I find plenty of implementations of the discrete Fourier transform, using FFT, but, if I´m not ...
2
votes
2answers
710 views

Numerical Approximation of the Continuous Fourier Transform

Given a function $F(k)$ in frequency space (sufficiently nice enough, eg. a Gaussian), I would like to compute its Fourier inverse \begin{equation}f(x) = ...
1
vote
1answer
235 views

Von Neumann Stability Analysis

I came across the following task recently: Use the von-Neumann stability analysis to investigate the stability of the discrete form of $\frac{\partial c}{\partial x} = \frac{\partial^2 c}{\partial ...
5
votes
0answers
287 views

How can I solve the Poisson PDE efficiently and fast in cylindrical coordinates?

I am trying to numerically solve the Possion PDE in cylindrical coordinate system. $$\Delta f = {1 \over \rho} {\partial \over \partial \rho} \left(\rho {\partial f \over \partial \rho} \right) + {1 ...
1
vote
1answer
361 views

Discretization of differential equation via FFT routine

I just have a question related to the following problem: Find a discrete approximation to the differential equation $u^{\prime \prime} + 2u^{\prime} + 2u = 3\cos(6t)$ using Equation 3.12 for these ...
2
votes
1answer
296 views

FFT Algorithm for an interpolating polynomial

I'm trying to use the Fast Fourier transform algorithm to determine the trigonometric interpolating polynomial of degree $16$ for $f(x) = x^2\cos(x)$ on $[-\pi,\pi]$ I see a computer result in my ...
2
votes
2answers
229 views

Integral of $f(x) \exp(ikx)$ with finite bounds calculated using Fourier transform, and its derivative

I have an integral which I need to calculate numerically along the lines of $$ I(k)=\int_0^{L} \exp(i k x)f(x) dx $$ where $x$ and $L$ are real. $f(x)$ is not necessarily periodic and differentiable ...
1
vote
1answer
100 views

Need numerical approximation for Fourier{max(0,f(x,y))} given Fourier{f(x,y)}

Given $\mathscr{F}\{f(x,y)\}$ is there a way to numerically approximate $\mathscr{F}\{max(0,f(x,y))\}$ ? I am not necessarily looking for a closed formula. Even some iterative method would be ...
2
votes
0answers
409 views

Understanding Fourier Transform and FFT

First off, I'm sorry if this is a repost. I am currently writing my thesis, and I've been thrown into some Fourier analysis, which I know nothing of. So, even if this question has been posted before, ...
0
votes
0answers
118 views

Taking inverse Fourier transform of complicated multipart equation

Define $\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\frac{1}{{\pi Q(\tau ')}}}} - 1} ...
2
votes
0answers
75 views

A question on algorithm complexity

It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$ for a signal with bandwidth $N$. How to see or show that the fast Fourier transform ...
2
votes
0answers
242 views

Fourier transform for Neumann boundary condition

I need to solve system of two coupled partial differential equations numerically. $\frac{\partial x_1}{\partial t} = c_1\nabla ^2 x_1 + f_1(x_1,x_2) \\$ $\frac{\partial x_2}{\partial t} = ...
1
vote
2answers
548 views

Fourier transform of heat equation

I need to solve following partial differential equation with Fourier transform numerically. $ \frac{\partial T}{\partial t} = \nabla(c\nabla T) $ where T is temperature, c heat conductivity and t is ...
2
votes
1answer
260 views

Fourier integral/ Fourier transformation of an oscillatory function with FFT

$f(x) = \cos(x^2)$ and $g(k) = \sqrt\pi \cos((\pi k)^2 - \pi/4)$ are a Fourier pair. I want to reproduce $g(k)$ by Fourier integrating $f(x)$ using FFT, i.e. approximating ...
1
vote
1answer
578 views

How to perform deterministic deconvolution?

Here is my problem: I have a random variable $A$ that is the sum of independent random variables $B$ and $C$, i.e. $A=B+C$. All three random variables are on the real domain. $B$ is a Gaussian with ...
4
votes
1answer
278 views

Fourier (Hankel?) transform of a discrete set of radial points (question from a chemist!)

I'm sorry because I'm not a mathematician so that my question may look a little bit messy. I have tabulated values [1] of a 3 dimensional radial function $f(r)$: ...
3
votes
2answers
1k views

Python Numpy FFT problem, FFT of a Gaussian should be a Gaussian but its not what I'm getting

I am trying to utilize Numpy's fft function, however when I give the function a simple Gaussian function the FFT of that Gaussian function is not a Gaussian, its ...