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

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6
votes
1answer
869 views

What do the X and Y axis stand for in the Fourier transform domain?

In Wolfram Mathematica the function Fourier has the following declaration Fourier[list] And after a list is given to the ...
17
votes
2answers
3k views

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

I'm trying to apply a Fourier transform of a one dimensional list of a time history of some quantity using the Fourier function. I'm interested in the frequency ...
12
votes
2answers
11k 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 ...
1
vote
2answers
7k views

Calculate the 2D Fourier transform of an Image

I am new to Mathematica, and using version 8.0. I would like to calculate the 2D Fourier Transform of an Image with Mathematica and plot the magnitude and phase ...
13
votes
1answer
1k views

Problem with Fourier coefficients

I have just started using Mathematica with v9.0. I am trying to follow a computation from a book on Fourier series with the function $f(x)=x$ on the interval $-\pi < x < \pi$. Here is the code ...
5
votes
2answers
1k 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, ...
9
votes
1answer
178 views

Workarounds for a possible bug in the linearity of FourierTransform

This is somewhat similar to this question, except the problem I am encountering is to do with Fourier transforms of scalar multiples of functions and their derivatives. I wish to input ...
16
votes
4answers
6k 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 ...
18
votes
4answers
6k 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 ...
12
votes
2answers
4k 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
544 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 ...
10
votes
3answers
7k 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. ...
16
votes
3answers
459 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
3answers
2k 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, ...
16
votes
3answers
1k views

How to calculate the Fourier Transform of a Gaussian pulse with a nonlinear chirp: Cos[ t + Exp[t^2] ]?

I have an optical pulse in time domain: Exp[-t^2] Cos[50 t - Exp[-2 t^2] 8 π]. The figure of this formula is I hope to calculate the Fourier Transform of ...
8
votes
2answers
765 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: ...
20
votes
2answers
3k 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 ...
20
votes
3answers
3k 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 ...
15
votes
6answers
4k views

How do you find the frequency and amplitude from Fourier?

Edit This is a popular question with several answers. This edit organises the answers and gives links. Fourier just gives y values (ordinates) if you wish to read off frequencies and amplitudes you ...
9
votes
1answer
1k 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 ...
7
votes
1answer
576 views

Calculation of 2D FFT for an image

I am trying to obtain 2D FFT of an image. From the 2D FFT, I wish to calculate the amplitude of the FFT averaged over a circle vs the distance in pixels by varying the radius of the pixels which is ...
13
votes
2answers
328 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 ...
12
votes
0answers
201 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] ...
3
votes
1answer
341 views

Differences between FFT and analytical Fourier Transform

In MATLAB, if I perform an FFT on a simple Gaussian spot with a phase grating: ...
3
votes
2answers
2k views

Obtaining the Fourier transform of an operator

We know that if we have a function $f(x)$, and we call $g(\omega)$ its Fourier transform, then the Fourier transform of $x f(x)$ is $$\imath \frac{\mathrm{d} g(\omega))}{\mathrm{d}\omega} $$ and ...
2
votes
1answer
1k views

Plot Fourier transform of $\sin (2 t)$

I'm trying to plot the Fourier transform of $\sin (2 t)$. I tried using the FourierTransform Function (I'm expecting a peak at w = 2) but it gives me undefined at ...
2
votes
1answer
306 views

Why does FourierTransform[] converge while same integral manually written does not?

I'm studying the quantum mechanics of an infinite square well from a computational standpoint. My eigenfunctions are defined as $u_n(x)=\sqrt{2/L}\sin\left(n\pi x/L\right),\quad 0 \le x \le L, \quad ...
4
votes
4answers
839 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
1answer
290 views

Approximate $h$ in $F(\theta)=\sin \theta \int_{-L}^{+L}h(z)e^{-ikz\cos \theta} \,dz$

Consider $$F(\theta)=\sin \theta \int_{-L}^{+L}h(z)e^{-ikz\cos \theta} \,dz$$ $$|z|\le L$$ $$0 \le \theta \le \pi$$ By having knowledge of $F(\theta)$, how can one approximate $h(z)$? In ...
2
votes
1answer
1k views

Fourier Series of “Split” Defined Function

Not sure how to phrase this in a concise way, anyway it seems like the function FourierSeries assumes the interval for which to compute the Fourier coefficients of a given function is $[-\pi,\pi]$, ...
1
vote
1answer
257 views

Inverse Fourier transform

I am dealing with an inverse Fourier transform of a sampled complex function, $H(\omega)$. I want to obtain a real transformed function $h(t)$, which means that $H(-\omega)=H^{\ast}(\omega)$ has to be ...
0
votes
1answer
122 views
14
votes
1answer
449 views

How to use 2D Fourier analysis to clean the noise in an image

I have some other method to do this.But I desire to know how to use Fourier analysis to realize it. I found a base post about Fourier-analysis to use in Image-Processing--Calculate the 2D Fourier ...
7
votes
2answers
6k 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.
4
votes
1answer
120 views

Using 2 D FFT for filtering unwanted texture on an image

I have an image that spurious waves texture appears: I want to use 2D FFT to remove this texture, so I applied the following code from Calculation of 2D FFT for an image : ...
2
votes
2answers
352 views

Using Periodogram to investigate the seasonality of the data

After asking how to get rid of the seasonality of this set of data, I'd like to ask how to investigate the seasonality in data using Mathematica. Which tool would you use to study seasonality? I.e. ...
12
votes
0answers
406 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 ...
11
votes
1answer
485 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 ...
6
votes
1answer
540 views

Angle between two areas of an image of a 2D FFT

I'm trying to obtain the angle between the red line and the area surrounded by the red circle, related to the middle of the image. Usually the area surrounded by the red circle is lighter than the ...
5
votes
1answer
468 views

Find a Fourier series from discrete data

I am trying to fit a Fourier cosine series to discrete data. I read through this, but I was not able to understand it. The data should be periodic with a period of $0.2 \text{ m}$, and it is the ...
4
votes
1answer
128 views

Phase in discrete Fourier transformation [duplicate]

I asked a question some days ago and it was very well answer. However, I have a new one. I don't understand why the phase of the discrete fourier transformation is not correct. It has the correct ...
1
vote
2answers
120 views

How to realize a Fourier Transform on a non-uniform sampling data

I have a non-uniform sampling data (in time domain) from a Michelson interference experiment, as shown in Fig 1. But it was not evenly sampled (the step length was not uniform), because of the ...
1
vote
1answer
597 views

Problem with Fourier and InverseFourier function

I have the following code Image[img, ImageSize -> 300] data = ImageData[img]; fdct = Fourier[data] tdata = InverseFourier[fdct]; Image[tdata] but it seems to ...
1
vote
1answer
706 views

Fourier transformation of solution of differential equation

I'm trying to plot a Fourier transform of solution of differential equation. I have tried with: ...
1
vote
1answer
1k views

discrete Fourier transform

I generate a periodic discrete data representing a time dependent function and I want to apply a discrete Fourier transform to this data using just one period of it. I think something is wrong with ...
11
votes
1answer
260 views

How can I use Mathematica to numerically compute a Wigner spectrogram of an optical pulse?

This question was inspired by this question where it is necessary to numerically compute the Fourier transform of a Gaussian optical pulse with a Gaussian chirp function. $$E(t)=e^{-t^2} \cos(50 t - ...
11
votes
1answer
449 views

Inconsistent results of second derivative of inverse fourier transform

Bug introduced in 9.0 or earlier and persisting through 10.2.0 I am trying to get the Green's function of a toy diffusion equation $$\frac{\partial^2 u(x,t)}{\partial x^2} = ...
8
votes
0answers
158 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 ...
3
votes
1answer
160 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
2answers
286 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 ...