# Tagged Questions

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

98 views

### Fourier transform of Exp[x]/x

Could you please explain why Mathematica gives the following expression when taking Fourier transform of $\exp(\lambda z)/\lambda$? $$\frac{-\log(-z)+\log(z)}{\sqrt{2\pi}}$$ Why the answer does not ...
100 views

### Image Filtering in the frequency domain

Here is a good starting to fiter image in the frequency domain lena = ColorSeparate[ImageResize[ExampleData[{"TestImage", "Lena"}],{121,121}]][[1]]; (Create a ...
41 views

### FourierSeries command , for arbitrary period T?

ExptoTrig[FourierSeries[Piecewise[{{-Pi,-2Pi<x<=0},{Pi,0<x<=2Pi}}],x,3]] You can see, the function goes from $(-2\pi,2\pi]$ But Wolfram Alpha gives a ...
446 views

### Using 2D Fourier transform of an image to detect typical wavelengths

I am new to Mathematica, and I have read this post to understand how to perform Fourier transform on an image. My mission is to extract information on the typical distance between the black patches in ...
8 views

### What does top right corner value in Fast Fourier Transformation (FFT) show? [migrated]

In case to apply fft2on matrix we get a matrix with sum of all matrix elements on left top corner but a number also show on top right corner.What does it show ? ...
238 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 ...
85 views

### FourierTransform a differental-equations [duplicate]

I want to use the command FourierTransform to transform a differential equations from time - domain to complex frequency domain but it don't works. My code is ...
76 views

### Computation of a Fresnel Diffraction pattern with Discrete Hankel Transform

In the next link: Computation of Hankel Transform using FFT (Fourier) Rainer implemented a great solution given in the next reference: Manuel Guizar-Sicairos and Julio C. Gutiérrez-Vega, "Computation ...
507 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 ...
15 views

### Théorie de Fourier in Sontags book [migrated]

I was reading Sontags In America and she mentions: "La théorie de Fourier sur les douze passions radicales.." What is this theorem about?
3k views

### Computation of Hankel Transform using FFT (Fourier)

To address circularly symmetric cases of 2-D Fourier Transformations, the so-called Hankel Transform can be applied (for a detailed derivation of the relation between the 2-D Fourier transform and the ...
75 views

### Find the function $f(x)$ by using its fourier expansion

It is easy to find the fourier coefficient and fourier expansion of $f(x)$ function. But I want solve the inverse problem by using Mathematica How to find the function $f(x)$, if I know its ...
61 views

### How to use a Fourier-transform on variable which is multiplied by a constant

I have tried to do this calculation: ℏ := 1.05457168*10^-34 FourierTransform [ψ[x], x, ℏ*p] but the result is: How can I do it right?
49 views

### What is the correct way to represent this sum?

The summation of Fourier series of Sin[x-x[i]] at three location x[1] to x[3] is: ...
48 views

### Inverse of a Fourier matrix [closed]

I am trying to find the inverse of a Fourier matrix but I am getting confusing results. If R is a Fourier matrix, then ...
101 views

### Why do I encounter 1/0 expression

I am trying to write my periodical function $f(t)$ in Fourier Series $$f(t)=\frac{a_0}{2}+\sum_{j=1}^Na_j\cos(j 2\pi t/\tau)+\sum_{j=1}^Nb_j\sin(j 2\pi t/\tau)$$ and I know that there is a really nice ...
74 views

### FourierCoefficient performance

I encountered weird performance issue while using FourierCoefficient[]. I narrowed it down to calculating $n$-th coefficient for $\cos(nx)$. ...
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 ...
63 views

### How are Fourier transformations calculated?

Lets say my function is $$f(t)=\left\{\begin{matrix} 1 &, -T_p/2\leq t \leq T_p/2\\ 0& , |t|>T_p/2 \end{matrix}\right.$$ than a Fourier transformation is calculated ...
94 views

### Plotting a periodic function [duplicate]

My question is the following: how can I plot a function repeatedly? By following some topics here I found some basic information about using Fourier series. This is what I made: ...
90 views

### Why is Cos[Abs[x]] Fourier transformed to 0 in mathematica?

I found that FourierTransform[Cos[Abs[x]], x, ξ, FourierParameters -> {1, -1}] gives 0, while ...
87 views

### Plotting amplitude spectra of modulated message signal

I'm trying to plot the amplitude spectrum of the modulated message signal, I have managed to plot it by working through the maths, expanding out each trig function, and then drawing onto a graph ...
129 views

### How do I make an explicit DFT?

So to gain a better understanding of how Discrete Fourier Transforms are done, I'm trying to program one in Mathematica so I can easily check my answers with the built-in Mathematica code. Based on ...
87 views

### How to remove the frequency noises in a Fourier Transformation on an experimental data？

This question is a follow-up to a previous question. I need to perform a Fourier Transformation on an experimental data in time-domain, as shown in Fig 1. With the kind help from Daniel Lichtblau ...
63 views

### Need help for Discrete Fourier Transform on an experimental data

I have an experimental data, which is basically a time-domain triangle function. Part I: In theory The time-domain figure is shown in Fig. 1(a), and the corresponding frequency-domain figure is ...
78 views

### Deriving the Fourier Coefficients Formulas with a Manual Procedure

I already know that there are built-in functions to compute the Fourier coefficients but I want to derive their formulas manually. In fact, I want to obtain the final well-known formulas for the ...
125 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 ...
72 views

### NFourierTransform do not show all points, even change PlotPoints and Exclusions

Needs["FourierSeries"] I try solve the numerical Fourier transform of this function: ...
44 views

### How can I plot an exponential Fourier series? [closed]

Iam triying to plot summatory of the next function: (e^(i*n*t*wo)*Sin[n*Pi/2])/(n*pi) I have no problem plotting a trigonometric fourier serie, so the only ...
97 views

### Getting correct amplitudes with Periodogram

I'm having some trouble with the function Periodogram, mainly with getting it to produce correct amplitudes for the spectrum it returns. My question is how to fix ...
80 views

### Is oscillation seen in data signal or noise? [closed]

I have some experimental data which is well-fitted by 40 terms of a Fourier series. I want to know if the oscillation exhibited is signal or noise. I evaluated the code shown below to plot my data in ...
61 views

### Engineering spectral densities using filtered white noise

My goal is to create time-spectrum noise signals with spectral densities $S_{xx}(f)$ for pretty much arbitrary spectral densities (think white noise, lorentzian, $1/f$). In finding advice on how to do ...
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 ...
64 views

### Constructing the spectral density of white noise

My goal is to find the spectral density $S_{XX}(f)$ from a time domain noise signal $X(t)$. The approach should work for general time series, but to illustrate what I've been doing I'll keep it ...
62 views

### InverseFourierTransform very slow

I determined a distribution dist = SmoothKernelDistribution[ListOfValues];. Then I determined the FT of the PDF, which is the ...
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 ...
264 views

### Using BandpassFilter

https://drive.google.com/file/d/0B7G3oUuZG1TwRHNNV1RFZjFWUXM/view?usp=sharing Above I have linked a data set containing a 2 seconds EEG sample I have recorded of someone "blinking". This data set was ...
148 views

### Why the Spectrogram can't plot correctly?

The same data lead two different graph. Plot[E^(-Pi*(t - 5)^2), {t, -10, 10}, PlotRange -> All] The method to plot the spectrogram use the build-in ...
176 views

### Inverse Fourier transform not evaluating [closed]

I'm trying to do this simple task: InverseFourierTransform[0.031622*Exp[0.4995 ω^2], ω, t] Mathematica 9.0 can't handle it for some reason (it returns with the ...
60 views

### Fourier-style solutions to differential equations, not piecewise polynomials

NDSolve returns piecewise polynomials. Is there any way I can get a single (non piecewise) function consisting of sines and cosines instead? Sort of a "Fourier approximation" to the solution of my ...
98 views

### Discrete Fourier Transform: help on how to convert x axis in to the frequency when I have a set of data samples [duplicate]

I have a huge data set (400000 sample) and when I plot them ListLinePlot[Au4, PlotRange -> {All, All}] I used the following code to Fourier transform it ...
74 views

### How to do the Fourier transform of a picture? [duplicate]

I want to generate Fourier transforms of a picture. the transform can represent the diffraction pattern of the picture. Basically, if I have a picture. Then, the result should look like this: For ...
128 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 : ...
63 views

### Lack of precision in solar declination leads to odd Fourier result

Here's the sun's declination for a 10 year period (roughly 2000-2010): ...
124 views

### Discrete Fourier Transform - Inadequate precision and wrong peak positions

I have a small problem with a piece of code. I searched the site but was not able to find an answer to my specific question. I am trying a quick test of the Fourier capabilities (on lists) of ...
134 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 ...
71 views

### Reconstruct signal using InverseFourier [closed]

I start from a signal, I do discrete Fourier transform of the signal, and I want to get back this same signal doing a discrete inverse Fourier transform. However, when I do so, I do not obtain the ...
62 views

### How can I analyse and manipulate k-space data of a DICOM imageset? [closed]

I'm very new to Mathematica. I've some MRI images as 3D DICOM data. I want to get the corresponding k-space data to identify all the signals with their corresponding amplitude, frequency and phase. ...
Consider the following expression: HeavisideTheta[x]HeavisideTheta[-x] Mathematica will not simplify it further. But $\Theta(x)\Theta(-x)$ is zero if considered ...