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

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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 ...
1
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1answer
98 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 ...
1
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2answers
40 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 ...
0
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0answers
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 ? ...
0
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0answers
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 ...
3
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0answers
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 ...
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0answers
15 views

Théorie de Fourier in Sontag`s book [migrated]

I was reading Sontag`s In America and she mentions: "La théorie de Fourier sur les douze passions radicales.." What is this theorem about?
3
votes
1answer
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 ...
0
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1answer
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?
2
votes
1answer
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: ...
0
votes
1answer
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 ...
3
votes
2answers
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 ...
1
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1answer
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 ...
1
vote
1answer
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: ...
2
votes
0answers
74 views

FourierCoefficient performance

I encountered weird performance issue while using FourierCoefficient[]. I narrowed it down to calculating $n$-th coefficient for $\cos(nx)$. ...
1
vote
2answers
85 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 ...
3
votes
1answer
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 ...
7
votes
1answer
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 ...
2
votes
1answer
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 ...
2
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1answer
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 ...
14
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1answer
503 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 ...
1
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1answer
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 ...
1
vote
2answers
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 ...
1
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1answer
71 views

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

Needs["FourierSeries`"] I try solve the numerical Fourier transform of this function: ...
0
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1answer
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 ...
3
votes
1answer
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 ...
1
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0answers
79 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 ...
0
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0answers
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 ...
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0answers
63 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 ...
0
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1answer
62 views

InverseFourierTransform very slow

I determined a distribution dist = SmoothKernelDistribution[ListOfValues];. Then I determined the FT of the PDF, which is the ...
2
votes
3answers
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 ...
0
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0answers
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 ...
0
votes
1answer
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 ...
2
votes
0answers
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 ...
4
votes
1answer
127 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 : ...
0
votes
1answer
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): ...
3
votes
2answers
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 ...
4
votes
1answer
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 ...
4
votes
1answer
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 ...
2
votes
1answer
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 ...
2
votes
0answers
61 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. ...
0
votes
1answer
98 views

Fourier Series and epicycles - How to find the radii and angular velocities from a function's Fourier Series expansion

The following code is a bit clumsy, but it constructs a variable number (n) of epicycles using the values from the radii and angularVel tables that follows the path of a parametric given as a sum of ...
0
votes
1answer
75 views

Discrete Fourier Transformation

Assuming I have a function $f(x)$. I can use a numerical Fourier transformation: NFourierTransform[] to find some points of $F(\omega)$. How can I do the same ...
11
votes
1answer
275 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 - ...
17
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 ...
1
vote
2answers
107 views

Parseval identity in one side Fourier Transform

Is there any posibility to apply Parseval identity(energy in time domain is equal to freq. domain) on One/Single-side Fourier transform? In other word for "standard" FT the test Mathematica can be ...
1
vote
2answers
177 views

How do I extract peak frequencies? [closed]

Suppose I have large text file filled with numerical data. Then I do Fourier operation with this file and plot the result: ...
9
votes
1answer
166 views

How can I make FourierCoefficient et al. handle the zero-order term correctly?

Say that I would like to use Mathematica to step through one of the most famous solutions to the Basel Problem, using Parseval's Identity, $$ \sum_{n = -\infty}^\infty \left|c_n\right|^2 = ...
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votes
1answer
107 views

A piecewise function with way too many pieces

I'm currently calculating a periodic square wave current function to feed into a particle accelerator. It's a square wave that ramps up to a certain frequency and then drops back to the base frequency ...
2
votes
1answer
51 views

TrigExpand-ing too far

Something simple: I'd like TrigExpand[Sin[3n-3]] to return Sin[3n]Cos[3]-Cos[3n]Sin[3] instead of ...