Questions on the discrete and continuous Fourier analysis functions of Mathematica, as well as the FourierSeries` package.

learn more… | top users | synonyms

20
votes
2answers
2k views

How to make Fourier behave like FourierTransform?

I'm not very experienced with Fourier Transforms, so there may be something inherently wrong with attempting to do this, but how can I make the discrete Fourier behave like the continuous ...
18
votes
3answers
2k views

Can one find the beat of a tune with Fourier analysis?

I'm trying to find out if it's possible to find the beat of a tune by Fourier analysis with Mathematica. I'm taking a 44.1 kHz sample sound and hoping that I might get a nice peak for a frequency ...
16
votes
4answers
3k views

Numerical Fourier transform of a complicated function

Say I have a function $f(x)$ that is given explicitly in its functional form, and I want to find its Fourier transform[1]. If $f$ is too complicated to have an analytic expression for $\hat f(k)$, how ...
16
votes
4answers
907 views

Digital filter of image in Fourier space

Consider the contour plot below, of a function f(x,y) - which is calculated numerically by solving some equations. If one looks carefully, one can observe two types of waves, to the left and right of ...
13
votes
4answers
2k views

Plotting Fourier spectrum versus frequency of a signal

I have looked around here, and i am sure this has been answered, but i don't understand it. The thing is, I have taken a introductory signal processing course, and we had to use Mathematica, and i had ...
12
votes
1answer
482 views

How can I obtain a zero-filled Fourier transform, as produced by MATLAB's fft?

The MATLAB FFT function, fft(X,n), can be used to return the n-point DFT. This effectively extends the original signal, X, to ...
12
votes
1answer
3k views

Coulomb potential as a Fourier transform

It is well known from theory that the Coulomb potential can be obtained as a Fourier transform in the following way: $$ \int \frac{\mathrm{d}^3p}{\left( 2 \pi \right)^3} \frac{e^{\mathrm{i} ...
12
votes
2answers
281 views

Parallelization of distinct array write access from subkernels

I'm working on an implementation of a multivariate FFT, which is (or at least should be) highly parallelizable due to the row-column-algorithm. However, i can't figure out how to implement that. The ...
11
votes
3answers
300 views

Disentangling the data

An experiment yields a functional dependence on two variables $F(x,y)$, can be imagined as a 2D map. It is known that the function can be factored as follows $F(x,y)=fx(x) gy(y) +gx(x) fy(y)$. Given ...
11
votes
2answers
204 views

Rotational Averaging of Image Periodograms (FFTS)

I'm writing a code to do rotational averages of image FFT's. I have a working version but it is pretty slow. Wondering if anyone has any thoughts on a way to do this more efficiently. Data looks ...
10
votes
2answers
1k views

Finding fourier transform of data and hence frequencies

I looked at all the other questions related to mine before posting this, and they didn't solve my problem. I have a large data set which can be downloaded from here. I'm using the following code to ...
10
votes
1answer
915 views

What's the correct way to shift zero frequency to the center of a Fourier Transform?

I'm trying to Fourier transform a one dimensional list of a time history of some quantity using the Fourier function. I'm interested in the frequency spectrum, but ...
10
votes
1answer
378 views

Why does Mathematica return a Fourier transform for a function for which it is not defined?

The following function $$g(x) = (1 + x^{1/a} )^a $$ should NOT have a Fourier transform, as far as I am aware, for any real values of $a$ since $g(x)$ is not nice in the sense of decays quickly ...
9
votes
2answers
5k views

Plotting the frequency spectrum of a data series using Fourier

testData = Table[N@Sin[500 x], {x, 0, 100}]; ListLinePlot[Abs[Fourier[testData]], PlotRange -> Full] Gives me Which I do not expect because the Fourier ...
9
votes
1answer
551 views

FourierTransform and Partial Derivatives?

I have a function of three variables $f(x,y,z)$, and it obeys a linear partial differential equation. I'm checking my by-hand calculations with Mathematica. I want to convert the PDE into the ...
9
votes
1answer
409 views

FourierSeries for rational function looks wrong

I have the following code: ser[x_] = FourierSeries[(\[Pi]^2 + a)/(3 x^2 + a), x, 10] // N // Chop It gives me some series, which I then try to plot. And ...
9
votes
0answers
319 views

Proving (self) similarity with Mathematica - Reccurrence Plots, Similarity Plots etc

I posted this question in math.se but given the sheer tumultuous number of questions that keep appearing on math.se and also given that I am trying to accomplish this in Mathematica, I thought I'd ...
8
votes
3answers
4k views

Plotting Partial Sums of Fourier Series

I'd like to plot some partial sums for a Fourier Series problem, but I am not sure if the output I am getting is correct. I want to be able to plot the partial sums and the function on the same graph. ...
8
votes
2answers
425 views

Why Fourier doesn't show me the peaks?

I'm trying to identify the frequencies in my time history samples, and I can see a frequency in the time history, but can't see it in its Fourier transform. Here it is : the sample data: ...
8
votes
2answers
1k views

How to approximate a given WAV file with trigonometric series?

I'd like (together with a few people) to prepare a presentation about Fourier series for middle/high school students. I thought it might be quite cool to play a violin sound from, say, a WAV file, ...
8
votes
1answer
1k views

Computation of Hankel Transform using FFT (Fourier)

To address circular symmetric cases of 2D Fourier Transformations the so called Hankel Transform can be applied (for a detailed derivation of the relation between the 2D Fourier transform and the 1D ...
8
votes
2answers
385 views

Next highly composite number?

R language has this function 'nextn' (link) which computes the next highly composite number greater than a given one, which is used to find the optimal padding size for the subsequent FFT operation. ...
7
votes
2answers
4k views

Does Mathematica implement the fast Fourier transform?

Is there a fast Fourier transform in Mathematica? Although looking in the help I could not find one. I am looking to implement the equivalent of fft in MATLAB.
7
votes
2answers
2k views

Fourier Transform of a Step Function

I'm trying to obtain the form of a sinc function that I know I'm supposed to get in Mathematica. I'm doing this because I intend to do a lot with Fourier transforms (FT) and I'd like to know I'm not ...
7
votes
2answers
189 views

Speed up Fourier for Booleans

I'm calculating a bit of the power spectrum of a two color cellular automaton using: ...
7
votes
1answer
518 views

Speeding up numerical Fourier Transform

I wrote this function NFourierTransform, which takes a function $f(k)$ and numerically calculates the fourier transform integral for discrete values of $k \in ...
7
votes
0answers
132 views

Possible bug with FourierTransform linearity

Each component is easily transformed but the sum is not: FourierTransform[f''[x], x, k] FourierTransform[f'[x], x, k] ...
6
votes
3answers
738 views

Why this gives 0?

Why this gives 0? I am in shock, it should be Sinh[x]. FullSimplify[I InverseFourierTransform[FourierTransform[Cosh[t],t,w]/w,w,x]]
6
votes
2answers
587 views

Band-pass filter an image

Ok so i have some HR-TEM images. In this technique fringes appear whereever a crystalline structure is present. This fringes are separated by a certain distance depending on the material, this is our ...
6
votes
0answers
139 views

Fourier transform over a custom dimension

I have a multidimensional array A. Fourier[A] finds the discrete Fourier transform over all dimensions. How to find discrete ...
5
votes
1answer
284 views

Fit data with linear combinations of Cos and Sin

Recall that if $f$ is a piecewise continuous function on the interval $[-\pi, \pi]$, then the Fourier series of $f$ is $$f(x) = \sum_{n=0}^{\infty} (a_n \cos(nx) + b_n \sin(nx)).$$ See for example, ...
5
votes
1answer
209 views

Prove an identity in quantum harmonic oscillator

Problem: In the context of quantum harmonic oscillator the eigenfunctions are given by: $ u_n(x) = (N_n/\sqrt{b}) H_n(x/b) \exp\left[-x^2/(2b^2)\right] $, where $N_n$ is the normalization factor: $ ...
5
votes
2answers
572 views

How to draw a graph about the FourierTransform of a 2D Function?

Here is a 2D function, and it cannot be calculated by the FourierTransfrom ...
5
votes
1answer
122 views

How to plot stability region of Fourier analysis

Stability limits for Runge–Kutta methods in the complex ΩΔt-plane are needed to be plotted like this: For example, let P=1+z where ...
5
votes
1answer
165 views

calling CUDA library functions

Mathematica has some predefined CUDA functions, like CUDAMap, CUDAFourier, etc. My question is how can one call functions which are part of CUDA library but not part of predefined Mathematica ...
5
votes
0answers
114 views

The same analytical expression gets inconsistent FourierTransform results

I have an expression which is just a linear combination of plane waves, and I'd like to calculate FT of it. I know what I will get should be a bunch of delta functions, but it turns out Mathematica ...
4
votes
2answers
570 views

Frequencies represented in a DFT list (Fourier[])

I have a sound sample (sample rate: 44100). When I have a list l with n successive elements from the sample, ...
4
votes
1answer
182 views

Reconciling results from Fourier with those form FourierTransform

I need to use the discrete Fourier transform for function that represented as list of values. I started with an easy task to check my understanding. I tried to get the amplitude values for ...
4
votes
1answer
239 views

Extracting frequencies with Fouriertransformation

I know this question came up so many times, but after browsing through many of the threads, I'm still not sure, if I'm doing it correctly with my dataset, so I'll give it a try. If have some sample ...
3
votes
4answers
445 views

Why does Fourier give a shifted frequency?

I have a signal that I want to identify the frequencies in it, I used the Fourier function but I can't get the frequency correctly. Here is a simplified example: ...
3
votes
2answers
196 views

Integration and Fourier transform

I would like to calculate the distribution function from the characteristic function. There is a formula given as $$F_X(x)=\frac{1}{2}+\frac{1}{2\pi}\int_{0}^\infty \frac{e^{i w x}\phi(-w)-e^{-i w ...
3
votes
1answer
267 views

Correct fourier scaling and high-resolution frequency identification

I have a dataset of amplitude versus time $(t,A(t))$ and I need to extract the dominant frequency and amplitude, and also get the amplitude at one other specific frequency. My data looks like this: ...
3
votes
1answer
299 views

How do you use fourier transforms to perform a deconvolution

I've been trying to teach myself about deconvolution through fourier transforms, but I seem to be missing something simple, as my results are garbage. I start by defining a test function and a window ...
3
votes
3answers
95 views

Filling in explicit zeros in a (x,y) list for nonexisting y values for discrete Fourier transform in Mathematica

I have a list of (x,y) values in Mathematica for various discrete x values, as in ...
3
votes
2answers
138 views

Fourier Analysis: How to get rid of a discontinuity

When I compute the phase error of a spatial series data using Fourier analysis in Mathematica there's a discontinuity @ parameter ...
3
votes
3answers
151 views

Fitting the Plot by fourier Series of either sine or cosine

I have list of data and its plot and I don't know how to fit the plot with fourier series of either sine or cosine. Full Data: ...
3
votes
1answer
53 views

How to create a 1D vector by selecting a straight line in a given angle of 2D array

Suppose I have an image created by the code provided below. I want to select a line (1D vector) $ \left(y-y_{0}\right)=n\left(x-x_{0}\right) $ from the output image (2D matrix) by specifying $n$ and ...
3
votes
0answers
109 views

How to do Fast Fourier transform (FFT) for singular functions?

I want to do a 3-dimensional FFT on this function $\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left((1.0001+\sin (y)+\cos (z))^2+(0.0001+\cos (x)+\sin (z))^2+(0.0001+\sin (x)+\cos ...
3
votes
0answers
44 views

What are good/best practices to take the Fourier transform of an InterpolatingFunction?

I have a function which I have obtained from numerical integration of a differential equation, and I would like to take its Fourier transform. What are good practices for doing this? To make things ...
3
votes
0answers
115 views

Fourier transformation of HeavisideTheta functions

I want to find 2D-Fourier transformation of the function given below f = HeavisideTheta[y1]*HeavisideTheta[y2 - y1] For the purpose, I use built-in function in ...