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

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17
votes
4answers
7k views

Plotting Fourier spectrum versus frequency of a signal

I have looked around here, and i am sure this has been answered, but i don't understand it. The thing is, I have taken a introductory signal processing course, and we had to use Mathematica, and i had ...
3
votes
0answers
95 views

Why does setting $Assumptions make my Fourier transforms slow?

Consider these Fourier transforms ...
8
votes
2answers
827 views

Correct fourier scaling and high-resolution frequency identification

I have a dataset of amplitude versus time $(t,A(t))$ and I need to extract the dominant frequency and amplitude, and also get the amplitude at one other specific frequency. My data looks like this: <...
4
votes
1answer
251 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 ...
1
vote
2answers
434 views

2D Fourier transform of a table - ArrayPlot issue

Recently, I have encountered an issue with ArrayPlot after performing a Fourier transform of a table. Picture presented above is an ArrayPlot of a 2D table. Using Fourier on this 2D table I obtain ...
3
votes
0answers
156 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 ...
5
votes
1answer
706 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, ...
16
votes
6answers
5k views

How do you find the frequency and amplitude from Fourier?

Edit This is a popular question with several answers. This edit organises the answers and gives links. Fourier just gives y values (ordinates) if you wish to read off frequencies and amplitudes you ...
3
votes
1answer
2k views

Plot Fourier transform of $\sin (2 t)$

I'm trying to plot the Fourier transform of $\sin (2 t)$. I tried using the FourierTransform Function (I'm expecting a peak at w = 2) but it gives me undefined at <...
0
votes
0answers
114 views

Recognizing a generalized function as zero

Consider the following expression: HeavisideTheta[x]HeavisideTheta[-x] Mathematica will not simplify it further. But $\Theta(x)\Theta(-x)$ is zero if considered ...
5
votes
1answer
389 views

Extracting frequencies with Fouriertransformation

I know this question came up so many times, but after browsing through many of the threads, I'm still not sure, if I'm doing it correctly with my dataset, so I'll give it a try. If have some sample ...
4
votes
1answer
877 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 ...
0
votes
2answers
148 views

How to obtain the form of sin(nx) and cos(nx) from the result of FourierSeries [closed]

How to obtain the form of sin(nx) and cos(nx) from the result of FourierSeries[]. For Example, $$\text{FourierSeries}[x,x,5]=i e^{-i x}-i e^{i x}-\frac{1}{2} i e^{-2 i x}+\frac{1}{2} i e^{2 i x}+\...
0
votes
1answer
309 views

ListLinePlot “Cannot Take Positions” error with Fourier of WAV file

When I try to do a Fourier analysis of a WAV file with a sample rate of 22050 I get an error "Cannot take positions". What am doing wrong? ...
12
votes
0answers
210 views

Possible bug with FourierTransform linearity

Each component is easily transformed but the sum is not: FourierTransform[f''[x], x, k] FourierTransform[f'[x], x, k] ...
2
votes
1answer
1k 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 $[-\pi,\pi]$, ...
5
votes
1answer
298 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: $ ...
12
votes
2answers
4k views

Finding fourier transform of data and hence frequencies

I looked at all the other questions related to mine before posting this, and they didn't solve my problem. I have a large data set which can be downloaded from here. I'm using the following code to ...
2
votes
1answer
687 views

Integrate message: can't prove Integration limits are real

I'm working on solving differential equations through Fourier series, I made a function to help me calculate the coefficients that looks like this: ...
1
vote
1answer
112 views

Two dimensional NIntegration failure - non-numerical values

I've got this function: E0[x0_, y0_, z_]:= A/w[z]*Exp[-(x0^2 + y0^2)/(w[z]*w[z])]*Exp[(I*2*Pi*(x0^2 + y0^2))/(λ*2*R[z])]*Exp[I*ϕ[z]]; Where the ...
0
votes
1answer
176 views

Fourier-Analysis How to display every period correctly? [duplicate]

I want to express a function as a sum of fourier series. I tried the code below. It works. But only get one period (from 0 to 1). How to display every period correctly? ...
11
votes
1answer
455 views

Inconsistent results of second derivative of inverse fourier transform

Bug introduced in 9.0 or earlier and persisting through 10.2.0 I am trying to get the Green's function of a toy diffusion equation $$\frac{\partial^2 u(x,t)}{\partial x^2} = \frac{1}{\alpha^2}\...
2
votes
1answer
330 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
3answers
105 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
1answer
597 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 ...
0
votes
2answers
123 views

fourier issue arising from input miscommunication

xdomain = Table[i , {i, -10, 10, .1}]; ListPlot[InverseFourier[ Fourier[E^-#^2 & /@ xdomain]*E^#^2 & /@ xdomain]] So I want to numerically fourier ...
8
votes
0answers
158 views

The same analytical expression gets inconsistent FourierTransform results

I have an expression which is just a linear combination of plane waves, and I'd like to calculate FT of it. I know what I will get should be a bunch of delta functions, but it turns out Mathematica ...
1
vote
1answer
565 views

Plotting partial sums of Fourier sine series

How do I plot this on Mathematica version 5.2? $\frac{4}{\pi} \sin{x} + \frac{4}{3 \pi} \sin{3 x} + \cdots + \frac{4}{(2 N+1) \pi} \sin{(2 N+1) x}$ over $x \in [-\pi,\pi]$ for $N= 3, 6, 12, 24$. I ...
18
votes
2answers
4k views

What's the correct way to shift zero frequency to the center of a Fourier Transform?

I'm trying to apply a Fourier transform of a one dimensional list of a time history of some quantity using the Fourier function. I'm interested in the frequency ...
6
votes
2answers
1k views

Band-pass filter an image

Ok so i have some HR-TEM images. In this technique fringes appear whereever a crystalline structure is present. This fringes are separated by a certain distance depending on the material, this is our ...
5
votes
2answers
1k views

Frequencies represented in a DFT list (Fourier[])

I have a sound sample (sample rate: 44100). When I have a list l with n successive elements from the sample, ...
11
votes
1answer
573 views

Why does Mathematica return a Fourier transform for a function for which it is not defined?

The following function $$g(x) = (1 + x^{1/a} )^a $$ should NOT have a Fourier transform, as far as I am aware, for any real values of $a$ since $g(x)$ is not nice in the sense of decays quickly ...
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 ...
10
votes
1answer
236 views

Fourier transform over a custom dimension

I have a multidimensional array A. Fourier[A] finds the discrete Fourier transform over all dimensions. How to find discrete ...
8
votes
2answers
723 views

Why Fourier doesn't show me the peaks?

I'm trying to identify the frequencies in my time history samples, and I can see a frequency in the time history, but can't see it in its Fourier transform. Here it is : the sample data: ...
1
vote
1answer
633 views

Problem with Fourier and InverseFourier function

I have the following code Image[img, ImageSize -> 300] data = ImageData[img]; fdct = Fourier[data] tdata = InverseFourier[fdct]; Image[tdata] but it seems to ...
2
votes
2answers
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 ...
9
votes
1answer
1k views

FourierTransform and Partial Derivatives?

I have a function of three variables $f(x,y,z)$, and it obeys a linear partial differential equation. I'm checking my by-hand calculations with Mathematica. I want to convert the PDE into the ...
13
votes
3answers
654 views

Next highly composite number?

R language has this function 'nextn' (link) which computes the next highly composite number greater than a given one, which is used to find the optimal padding size for the subsequent FFT operation. ...
17
votes
4answers
2k views

Digital filter of image in Fourier space

Consider the contour plot below, of a function f(x,y) - which is calculated numerically by solving some equations. If one looks carefully, one can observe two types of waves, to the left and right of ...
14
votes
1answer
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 ...
2
votes
1answer
126 views

Tough Inverse Fourier Integral: why does the sign matter?

I'm computing the Inverse Fourier Integral of ((a^2 + omega ^2) c^2) /((b^2 + omega^2) ((r^2 + omega^2 - omegaInt^2)^2 + (2 omegaInt r)^2)) ...
1
vote
1answer
526 views

How to manipulate a Fourier sine series?

I would like to plot the Fourier sine series for $f(x) = x$. I would like to manipulate the parameter n from $0$ to $10$ and plot the sine waves over $(0,1)$. My ...
11
votes
1answer
489 views

FourierSeries for rational function looks wrong

I have the following code: ser[x_] = FourierSeries[(\[Pi]^2 + a)/(3 x^2 + a), x, 10] // N // Chop It gives me some series, which I then try to plot. And ...
2
votes
1answer
1k views

Fourier transform of sampled data

I have got some impulse response data that I would like to transform via Fourier to get the amplitude-frequency characteristics of the performing loudspeaker. The ...
0
votes
1answer
177 views

Error messages when using NInverseFourierTransform

I have two functions that I need to inverse Fourier transform and I was trying to get Mathematica to help me. I tried simply using theInverseFourierTransform ...
1
vote
0answers
645 views

Doing local FFT on huge 3D vector data cell mesh and visualizing it spatially?

Simulation type: I'm running a simulation with the OOMMF micromagnetics package http://math.nist.gov/oommf/ where are magnet is represented by a mesh of 3 million cells, it gets excited by a ...
1
vote
2answers
2k views

Discrete FFT of non-periodic signal excited by short pulse

Say I have a signal F(t) which represents a excitation by a pulse (so it is not periodic and declining to zero amplitude). According to this answer, for instance, ...
0
votes
1answer
1k views

How can I find the Fourier series from discrete data?

I would like to define the function with Fourier series when I just have discrette data. (I don't have a specific function). I can draw the graph using the data, but I don't have a idea how can I get ...
11
votes
3answers
2k views

How to approximate a given WAV file with trigonometric series?

I'd like (together with a few people) to prepare a presentation about Fourier series for middle/high school students. I thought it might be quite cool to play a violin sound from, say, a WAV file, ...