Questions tagged [fourier-analysis]

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

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Why does this Fourier transform not match the formula?

From this page: So I'm trying to get that result in Mathematica. From the fourier formula above, we can see that {a, b} = {1,-1}. However, the code below returns a different result. ...
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Can I take the Fourier transform of a PDE in both sides? [duplicate]

I want to know if it is possible to use a PDE in the function FourierTransform instead of a function. Consider for example the simple case for the heat equation ...
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Nonlinear Schrödinger Equation With Periodically Varying Function

I am trying to solve Equation. 42 from https://labsites.rochester.edu/agrawal/wp-content/uploads/2019/08/paper_2019_03.pdf The equation I am trying to solve is a nonlinear Schrödinger equation More ...
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2 votes
2 answers
204 views

Collision of two waves with phase difference

I am trying to produce collisional figures from this paper https://arxiv.org/pdf/1803.07165.pdf. But failed to see the phase effect. In my case it just passes through one another. It basically solve a ...
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1 vote
1 answer
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Inverse Fourier Transform and the sign of the Shifted Delta

I am using the sign convention of FourierParameters->{0, -2 Pi} for calculating the inverse FT of $Aexp(-2\iota\pi f K)$, where $A$, $K$ are real numbers >0 ...
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I am trying to solve nonlinear Schrödinger equation with dipolar interaction

I am trying to solve numerically Equation number (29) with the help of Eq.(32) and (34) from this paper https://arxiv.org/pdf/1506.03283.pdf. for ...
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1 answer
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Convolution theorem if one function depends on the reciprocal space in a non-trivial way

I would like to write the following integral in terms of a convolution: \begin{equation} H(x_1,x_2) = \int dt e^{\mathrm{i} t (x_1+x_2)/2} f(t,x_1,x_2) g(t), \end{equation} where $f(t,x_1,x_2)$ ...
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Implementing the Shift Property of the Fourier Transform in Mathematica

I am trying to determine the Fourier tranform of a time shifted (t-b) Lorentzian function in Mathematica. With the zero centered Lorentzian (b=0), the Integrate function with the conditions that all ...
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Obtaining Inverse Fourier Transforms by "FourierTransform" vs. "Integrate" Option [duplicate]

I am trying to calculate the inverse Fourier transform of "1" using two approaches. One using the Integrate option and the other with InverseFourierTransform, so the function looks like the ...
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How to use FindSequenceFunction to obtain the general expression of Fourier series?

I want to get cosine series of the following functions. $f(x)=\left\{\begin{array}{cc}\cos x, & 0 \leqslant x<\frac{\pi}{2} \\ 0, & \frac{\pi}{2} \leqslant x \leqslant \pi\end{array}\right.$...
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Why does the following integral not evaluate? [closed]

I am a complete beginner to Mathematica. I am trying to evaluate a Fourier integral (see below). Here $\mu(\vec{r})$ is a mass density with units $[\frac{u}{mm^3}]$, and its Fourier transform is ...
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Why would the computation time be hugely different for the same computation expressed in different but totally equivalent forms?

I was trying to plot an interpolation function from fourier data. The plot was very fast and straightforward when I used the following code: ...
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Mathematica `fourier` get different result with python `np.fft` [closed]

In python,I run fft as below: ...
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Integrate does not converge while Fourier transform does

Let us say I want Mathematica to compute the integral $$\int dx \, \text{sign} (x) \, \frac{1}{1+x^2} = 0 \, .$$ Indeed ...
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TransformedField between "Cartesian" and "Polar"

When I do f[x_, y_] := Exp[-Sqrt[x^2 + y^2]] ft = FourierTransform[f[x, y], {x,y}, {u,v}, FourierParameters -> {0, 2*Pi} ]] The result is $$\frac{2 \pi }...
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fast fourier transformation question

I was testing fft on mathematica using simple functions and ran into problems. I first tried a simple cos function with the following code: ...
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What integrals can `FourierTransform` evaluate that `Integrate` cannot?

FourierTransform can make sense of integrals that diverge according to Integrate. ...
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How to fourier transform a numerical solution?

I numerically solved a wave equation and want to fourier transform the solution uwave1(t,x,z) at a particular instant of time t and position z. The code is as follows: ...
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1 answer
160 views

How to calculate Fourier series for an arbitray non-periodic function defined in arbitray interval by setting FourierParemters or by rescale? How?

I am struggling to understand the Fourier series in Mathematica. I don't find a way to calculate Fourier series for a general function which is not periodic and defined on a interval but not on ...
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3 votes
1 answer
194 views

generating 1/f noise

the spectrum of noise δε(t) is S(ω)=<|δε(ω)|²>, where δε(ω) is a Fourier coef. of δε(t). And the relationship between the variance σ² of a zero-mean δε(t) and the spectrum is ……(I), where ωl(ωh)...
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1 answer
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How to get the Inverse Fast Fourier Transform (ifft) of a dataset? [closed]

I'm looking at the inverse fast Fourier transform as calculated by Matlab. In the question "What's the correct way to shift zero frequency to the center of a Fourier Transform?" the way to ...
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Mathematica Fourier Transform not responding

I am trying to do Fourier Transformation on the following function Izzn, which is essentially just a lot of trigonometric function, whose Fourier transformation are all well defined. However, when I ...
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12 votes
2 answers
1k views

How can I use fast Fourier transform (FFT) to solve a PDE (heat equation)?

I'm trying to solve a one-dimensional heat equation (PDE) with the Fourier transform numerically, in the way it was done here. The equation: , is subject to the initial condition: , where U(x,t) is ...
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Why does the reported primary frequency with Mathematica's discrete Fourier transform scale with the length of the input signal?

I am testing Mathematica's DFT algorithm using one of the test cases provided in the documentation, of a noisy sine function with a frequency of 0.15 cycles per sample: ...
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3 votes
1 answer
115 views

How to conciliate results between Fourier and FourierTransform?

I can't match the analytical results of a continuous-time Fourier transformation with the results I get from a discrete Fourier transform using Fourier[] in ...
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1 answer
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How can I get the power spectrum density and separate low and high frequencies?

Good day everyone, I'm new in mathematica. I got these values over time (each value was obtained every 100 milliseconds): ...
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Spectrogram/SpectrogramArray output

Is there/ What is the trick to get the output of the STFT as {Fourier coeff., freq, time} format from Spectrogram/SpectrogramArray/anything else?
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4 votes
1 answer
133 views

How to solve Coupled a Parabolic and Elliptic PDE in NDSolve?

I want to solve a mixed PDE Parabolic-Elliptic system in 3-dimension (rectangular coordinate), as shown below: The respective code version with parameters value, boundary and initial conditions is, <...
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5 votes
1 answer
296 views

What is 0 dB in Periodogram?

Consider some sound: ViolinNote = Sound[SoundNote["G", 1, "Violin"]] We may plot the periodogram: ...
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4 votes
2 answers
187 views

How does Fourier work with arbitrary FourierParameters?

Mathematica's Fourier function allows you to insert an arbitrary real number in the exponent of the discrete Fourier transform, via ...
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2 votes
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Analyzing anomalies of Bluetooth audio generation with Mathematica

I' m working on an audio generator for physics lab, using an Android device and Bluetooth . However, the response of the Bluetooth speaker is wrong. When programmed to deliver, say 3150 Hz, it may ...
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9 votes
1 answer
263 views

How to visualize overtones in Mathematica?

I would like to visualize the difference in sounds extracted by different musical instruments, which is due to different overtones. In order to do this, I define the following samples: ...
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1 answer
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Need help plotting a wavefunction generated from a sum of eigenstates [closed]

I am a beginner in Mathematica. I am trying to plot the time evolution of a non-eigenstate wavefunction as a sum of eigenstates. So I start with an eigenstate of an infinite square well, then expand ...
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5 votes
2 answers
260 views

Fourier coefficient of $\frac{1}{5 + 4\cos(x)}$ wrong

I've been trying to use Mathematica to compute the Fourier coefficient of $$ f(x) = \frac{1}{5 + 4\cos(x)}, $$ which is a $2\pi$ periodic function. Inputting ...
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1 vote
1 answer
55 views

How to symbolically manipulate the closed form series output from the easyFourier package?

EasyFourier by @xzczd is a nice package to obtain a Fourier series in closed form, e.g. f = x^2 easyFourierTrigSeries[f, {x, -\[Pi], \[Pi]}, \[Infinity]] However, ...
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7 votes
3 answers
440 views

extracting phase from sinusoidal data

I am dealing with data which are essentially are sinusoidal with various degrees of noise. I have an interest in extracting the phase of the data and rather than simply fitting a sine wave, I would ...
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1 answer
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Power spectrum of a signal in negative frequency domain

I have a signal with fc = 308 kHz and B = 30 kHz and some white noise added. The signal is sampled at samplFreq = 4fc. ...
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0 answers
70 views

Fourier transform of $ \exp{\left(\text irx/\sqrt{1-\text ix}\right)}/(1-\text ix) $

I want to calculate the following Fourier transform: $$u(r,t) = \int_{-\infty}^{\infty}\text d\omega\ \frac{1}{1-\text i\omega}\, \exp\left\{\frac{\text ir \omega}{\sqrt{1-\text i \omega}} +\text i\...
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Is there a function in Mathematica to combine/decompose sinusoids? [closed]

Lets say I have some combination of an arbitrary number of sinusoids with potentially varying magnitudes, phase angles and frequencies that I would like to compose into a singular sinusoid, is there a ...
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1 answer
92 views

Microfluctuations analysis and Power spectrum

I measured a parameter over time and obtained the values below: 0,627896 0,205004 0,259237 1,059125 0,832184 0,587992 0,565537 0,527323 0,460228 0,471958 0,26696 0,75367 ...
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3 votes
0 answers
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2D Poisson equation with Piecewise RHS - symbolic solution?

I'm trying to solve the following 2D Poisson equation symbolically: $$ -\nabla^2 u(x,y) = \begin{cases} 1-\text{sech}\left(\frac{w}{2 d}\right) \cosh \left(\frac{x}{d}\right) & -w/2 \leq ...
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0 votes
1 answer
51 views

Debugging code with Plot function. Full details

This is some code: The first plot (with the f[x,3]) gives me a graph. The second plot (with the 'numerical' form of f[x,3] given) does give me a plot How can I fix this so that the first line gives me ...
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4 votes
3 answers
467 views

Useless result of Integrate

Considering a Fourier sine series, I calculate int = Integrate[Log[1 + Sin[x]]*Sin[i*x], {x, 0, Pi}, Assumptions -> i ∈ PositiveIntegers] ...
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2 votes
0 answers
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Raising awareness about a Fourier analysis issue [closed]

I have just posted a query on Wolfram's own forum: https://community.wolfram.com/groups/-/m/t/2299913?p_p_auth=HBTdYN1c This is because it allows me to share the notebook I am working on and therefore ...
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  • 103
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1 answer
96 views

Issues to compute and plot a function with some Fourier analysis

I'm currently in an internship and I have to construct a model that involve Fourier transform of 2 orbitals. My issue is that when I run the programm, the function evluateton non numerical values : &...
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1 vote
3 answers
145 views

Find FT and r.m.s noise of a signal

I have a signal : ...
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1 answer
108 views

How to obtain a normalized fourier transform?

In my code I solved a differential equation system, then I made a fourier transform of the solution in differents times, this is my code ...
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8 votes
4 answers
331 views

Challenge: Computing the Fourier Series of a List Plot!

I have a bunch of data points (copied from Mathematica below), which I have 'ListPlotted' I would now like to compute the Fourier series of this function (after doing a Fourier transform I assume? Or ...
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  • 103
1 vote
0 answers
120 views

Fourier transform not working! [closed]

The FourierTransform doesn't seem to work on the following function: ...
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2 votes
1 answer
128 views

Fourier transform of DawsonF not recovered by using Erfi

Bug introduced in 8 or earlier and fixed in 12.2 I wish to compute the Fourier transform related to the Dawson function: FourierTransform[1/u DawsonF[1/u], u, x] ...
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