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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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4
votes
1answer
97 views

How to set up a Fourier transform?

I am having trouble figuring out how to set up the Fourier transform to: ...
4
votes
1answer
563 views

Fourier in Mathematica different from octave [duplicate]

First I create a table of sin(x) within 0, 10 ...
4
votes
2answers
723 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. ...
4
votes
1answer
191 views

Comparing ordinates between FourierTransform and Fourier

I have a test function ff[t_] := Exp[-t^2 - t]. I have calculated the Fourier transform with FourierTransform: ...
4
votes
3answers
119 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 ...
4
votes
1answer
323 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 function: ...
4
votes
1answer
839 views

Animated Wave Propagation using Fourier & InverseFourier

This is a continuation off of previous help on the first part of my project: fourier issue arising from input miscommunication Now I want to go one step further in the current code. Here's the code ...
4
votes
1answer
287 views

Fourier transformation of Hann window is wrong

From the documentation f = FourierTransform[HannWindow[x], x, w]; Plot[Log10@Abs[f], {w, -80, 80}, PlotRange -> All] which produces Now I was trying to do ...
4
votes
1answer
1k views

Why Fourier transform is symmetric no matter what interval I choose? [duplicate]

I'm solving a vibration problem. After solving two systems of ODEs (one valid in 0-0.1 and the other in 0.1-1) I get the plot of 3 functions, one of them is this, the plot of the variable $y_w$ (Image ...
4
votes
1answer
146 views

Decomposition of a time series into sinusoids

I've noticed this interesting study on metatrader 4; see the picture below: Where the financial timeseries is "decomposed" (deconvolved?) into a series of periodic sinusoidal functions. In the ...
4
votes
1answer
2k views

Fourier transform of interpolating function [closed]

I want to get a Fourier transform of an interpolating function and plot that Fourier transform: ...
4
votes
1answer
784 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 ...
4
votes
1answer
831 views

Averaging over a circle from FFT to plot the amplitude vs wavelength

I am trying to calculate the amplitude of FFT vs wavelength for some experiments. I had put a question here previously on how to compute the FFT of an image. Thanks to everyone who provided the ...
4
votes
1answer
147 views

Apparently incorrect Fourier Series

While trying to compute Fourier Series, i find that It doesn't evaluate both positive and negative parts and hence apparently giving wrong answer. Below is the code that i tried: ...
4
votes
1answer
221 views

Asymptotic behavior of Fourier transform of $\exp(-x^4)$ at large $k$

I am computing the Fourier transform $$\int_{-\infty}^{+\infty}\mathrm{d}x\,e^{-x^4+ikx}$$ for large $k$. Since $e^{-x^4}$ is $C^{\infty}$ on the whole real line, I expect its Fourier transform to ...
4
votes
1answer
659 views

NFourierTransform of an interpolating function

I have an interpolating function called rs22 - a solution from NDSolve - which is defined from 0 to about 1 ms. rs22 is a function that reaches a non-zero equilibrium value. (only shown here up to ...
4
votes
1answer
522 views

Sampling a musical instrument (or re-tuning MIDI)

I'd like to try to convert a piece of MIDI music made with SoundNote to another tuning (not microtonal or even split-black-keys, just a slightly different tuning for each note). I searched for pitch ...
4
votes
2answers
102 views

Is there a way to scale a graph from the command Periodogram?

Does anyone know if there is a way to scale a graph from the command Periodogram? More specifically in terms of the y axis. I am wondering if there is a way that I can make the highest value on the y ...
4
votes
1answer
936 views

Finding the dominant frequency in a large dataset with Fourier + periodogramm

I know there are dozens of questions about how to extract the correct frequency out of data, however I probably don't interpret my data correctly, or do not apply those answers correctly. Problem I ...
4
votes
1answer
3k views

Discrete Fourier Transform with x axis data?

Usually, Fourier Transform in mathematica is supplied by y data only, like the famous Sin[x] transform example. What if my data is a list of ...
4
votes
1answer
821 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 ...
4
votes
0answers
361 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 simple:...
4
votes
0answers
613 views

Help with Fourier analysis and filtering of an image

I am starting with the following image: For the curious, it is an STM scanned image of graphite. I am hoping to find some kind of periodic structure in the image related to the honeycomb structure of ...
4
votes
0answers
7k views

Fourier Transform of Absolute Value

If you ask Mathematica to provide the Fourier Transform of a singular functions it is likely to provide an answer that while nearly correct, is technically incorrect and it will do so without a word ...
4
votes
0answers
265 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
2answers
202 views

Fourier Transform to help guess with NonLinearModelFit

I imported some data in Mathematica which I cleaned a bit. This process went well. I also managed to plot it, and tried to take the Fourier transform of the data and plot it again. I'm not sure if I ...
3
votes
3answers
529 views

NIntegrate not working after interpolation

I used linear interpolation between points: ...
3
votes
2answers
3k 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 ...
3
votes
1answer
128 views

Indeterminate result from Sinc

I have an impulse train given by (1 + Csc[(π x)/(1 + R)] Sin[(π (1 + 2 R) x)/(1 + R)])/(2 + 2 R) Because of the Csc, ...
3
votes
2answers
174 views

How to make this higher resolution transform as fast as Fourier?

Given a table of data, the Fourier command gives the discrete frequency content of the data, where the resolution in frequency space is equal to 1/N, where N is the number of data points. Higher ...
3
votes
1answer
751 views

Why doesn't Mathematica evaluate FourierTransform[Exp[-t], t, w]?

I am having trouble getting Mathematica to evaluate a trivial Fourier transform. I would like to be able to evaluate the Fourier transform of the right-sided exponential decay function as shown ...
3
votes
2answers
134 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 ...
3
votes
2answers
301 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 ...
3
votes
2answers
610 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
1answer
2k views

How to do continuous Fourier transform?

I want to do a Fourier transform to the below function by Mathematica. How can I do it? Here $c$, $d$, $a$, $L$ are constants. $$ w(r)= \left\{ \begin{array}{ll} -\frac{c}{\epsilon r} & ...
3
votes
1answer
2k views

Plot output of FourierTransform in mathematica [closed]

I'm a too newbie in mathematica.I've gotton my output correctly as Shown below: ...
3
votes
2answers
210 views

How do I numerically compute the $L^1$ norm of the Fourier transform of a bump function?

I'm trying to compute the following. Let $$f(t) := \exp\left(\frac{1}{t^2 - 1}\right)1_{-1 < t < 1}.$$ This function looks like (it's the standard example of the bump function on Wikipedia). ...
3
votes
1answer
101 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
3answers
986 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
728 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 ...
3
votes
1answer
77 views

Inverse Fourier transformation discrepancy between Wolfram|Alpha's solution and mine [closed]

I searched for the inverse fourier transformation of $$ \mathcal{F(\omega)} = \frac{2}{(1+i\cdot \omega)^2 +4} \rightarrow i^2=-1 $$ My solution (compliant with the solution from my textbook): $$ \...
3
votes
1answer
108 views

Fourier cosine series of displacement

I'm trying to calculate a Fourier Cosine Series of displacement for general 4-bar mechanism. My code not working. I don't understand why. I need the result as this form ...
3
votes
1answer
2k views

Fourier Transform on a Sound

I don't really know much above Fourier Transform , or how it's used in Mathematica but I want to know how I can find out, out of which Sin-waves a (static) sound is made of. So for example I take "G" ...
3
votes
1answer
226 views

Fourier transformation of image in only 1 dimension

I have the following image: img = Import["https://i.imgur.com/Peu6mwd.png"] Each column shows the y coordinate of some objects (each row is 0.012 mm apart) and the ...
3
votes
1answer
712 views

Changing Function Domain of Fourier Series

If I write FourierSinSeries[x, x, 10] Mathematica will output the first 10 terms of the Fourier Sine Series f(x) = x defined for 0 < x < Pi. How do I output the the Fourier Sine Series for ...
3
votes
1answer
457 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 addition, ...
3
votes
1answer
243 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
1answer
3k 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 $[-\pi,\pi]$, ...
3
votes
1answer
44 views

How does the Fourier transform take its input?

I'm using the Fourier command and I'm wondering how does it take its imput. I know that it takes a list of numbers representing a function of time and it gives back a list of numbers representing a ...
3
votes
2answers
161 views

Fourier transform of a bivariate normal distribution

I can define a Multinormal distribution by p = PDF[MultinormalDistribution[ {mu1, mu2}, {{sig11^2, ρ sig11 sig22}, {ρ sig11 sig22, sig22^2}}], {x, y}] I would ...