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

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5
votes
1answer
495 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 ...
5
votes
1answer
678 views

Fit data with linear combinations of Cos and Sin

Recall that if $f$ is a piecewise continuous function on the interval $[-\pi, \pi]$, then the Fourier series of $f$ is $$f(x) = \sum_{n=0}^{\infty} (a_n \cos(nx) + b_n \sin(nx)).$$ See for example, ...
5
votes
1answer
296 views

Prove an identity in quantum harmonic oscillator

Problem: In the context of quantum harmonic oscillator the eigenfunctions are given by: $ u_n(x) = (N_n/\sqrt{b}) H_n(x/b) \exp\left[-x^2/(2b^2)\right] $, where $N_n$ is the normalization factor: $ ...
5
votes
2answers
1k views

How to draw a graph about the FourierTransform of a 2D Function?

Here is a 2D function, and it cannot be calculated by the FourierTransfrom ...
5
votes
1answer
369 views

2D Fourier transform

I want to calculate the 2D Fourier transformation of a circular shaped aperture, but Mathematica won't finish the calculation: ...
5
votes
0answers
211 views

Periodogram for irregularly spaced data

I am trying to replicate some existing work on the length of the cycles in certain economic data. Specificially, I am using the Schularick and Taylor (2012) data set to replicate the periodogram shown ...
4
votes
4answers
846 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: ...
4
votes
1answer
244 views

Programming a recursive FFT with Mathematica

I'm trying to program a FFT using a recursive function in Mathematica, though my program is not getting me anywhere at the moment. Could you say what's wrong with it and what I can do to make it work? ...
4
votes
1answer
244 views

Reconciling results from Fourier with those form FourierTransform

I need to use the discrete Fourier transform for function that represented as list of values. I started with an easy task to check my understanding. I tried to get the amplitude values for ...
4
votes
1answer
124 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 : ...
4
votes
1answer
142 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
255 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
132 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
405 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
0answers
275 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 ...
4
votes
0answers
99 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
155 views

Compare two shifted signals

I have one set of original data (signal A) and a second set of derived data (signal B) which was computed from A by position rotation (circular shift). Here are the data for the signals (A and B): ...
3
votes
2answers
326 views

Integration and Fourier transform

I would like to calculate the distribution function from the characteristic function. There is a formula given as $$F_X(x)=\frac{1}{2}+\frac{1}{2\pi}\int_{0}^\infty \frac{e^{i w x}\phi(-w)-e^{-i w ...
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 ...
3
votes
1answer
73 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 ...
3
votes
2answers
99 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
123 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
1answer
376 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
1answer
801 views

How do you use fourier transforms to perform a deconvolution

I've been trying to teach myself about deconvolution through fourier transforms, but I seem to be missing something simple, as my results are garbage. I start by defining a test function and a window ...
3
votes
3answers
104 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 ...
3
votes
2answers
292 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
3answers
298 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
575 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 ...
3
votes
1answer
66 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
292 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 ...
3
votes
1answer
162 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
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 ...
3
votes
1answer
95 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 ...
3
votes
0answers
74 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 ...
3
votes
0answers
95 views

Using NIntegrate and DiscretePlot to visualize pseudodifferential operators

In harmonic analysis, pseudodifferential operators are a way to generalize the notions of derivatives, through the use of Fourier transforms. The basic idea being, Let ...
3
votes
0answers
80 views

Analytical Fourier Transforms with convolution rule [closed]

The task is the following, how to extend Mathematica's FourierTransform command in order to be able to analytically deal with non-linear differential equations. To do so it should be able to deal with ...
3
votes
0answers
124 views

Different FourierParameters in each dimension for discrete Fourier transform

I have a query about the Fourier utility, that does discrete Fourier transforms. I am unable to find any information on whether different FourierParameters can be used in each dimension. This is a ...
3
votes
0answers
181 views

Calling CUDA cuFFT

Following ybeltukov's excellent answers (see here and here) about how to access the CUDA library functions and interact with Mathematica through CUDAMemory, I tried to compile the code given in ...
3
votes
0answers
228 views

How to do Fast Fourier transform (FFT) for singular functions?

I want to do a 3-dimensional FFT on this function $\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left((1.0001+\sin (y)+\cos (z))^2+(0.0001+\cos (x)+\sin (z))^2+(0.0001+\sin (x)+\cos ...
3
votes
0answers
92 views

Why does setting $Assumptions make my Fourier transforms slow?

Consider these Fourier transforms ...
3
votes
0answers
154 views

Fourier transformation of HeavisideTheta functions

I want to find 2D-Fourier transformation of the function given below f = HeavisideTheta[y1]*HeavisideTheta[y2 - y1] For the purpose, I use built-in function in ...
2
votes
3answers
263 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 ...
2
votes
2answers
2k views

Computing the minimum number of terms required in a Fourier series to achieve a particular upper bound on the error

In a Fourier series, the maximum error bound is the difference of the function and the partial sum of its Fourier series. Within an interval, as we increase the number of terms of partial sums, the ...
2
votes
2answers
357 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. ...
2
votes
1answer
331 views
2
votes
3answers
442 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 ...
2
votes
1answer
310 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 ...
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 ...
2
votes
1answer
314 views

Is there any function in Wolfram Mathematica simmilar to fft.fftfreq (Python)

Here is an explanation of fft.fftfreq I have data with random noise: ...
2
votes
1answer
301 views

2-Dimensional NFourierTransform

Mathematica FourierSeries package contains the NFourierTransform function for calculating 1-D Fourier integral numerically. ...