Fourier Definition and 1000 Threads

In many articles, authors compare S-duality in physics to Fourier transforms. For example: Joseph Polchinski, in his article "String Duality" (hep-th/9607050v2), writes "Weak/strong duality [...] is similar to a Fourier transform, where a function which becomes spread out in position space...

I just want to make sure I am on the right track here (hence have not given the other information in the question). In taking the Fourier transform of the PDE above, I get: F{uxx} = iω^2*F{u}, F{uxt} = d/dt F{ux} = iω d/dt F{u} F{utt} = d^2/dt^2 F{u} Together the transformed PDE gives a second...

I cite an original report of a colleague : 1) I can't manage to proove that the statistical error is formulated like : ##\dfrac{\sigma (P (k))}{P(k)} = \sqrt{\dfrac {2}{N_{k} -1}}_{\text{with}} N_{k} \approx 4\pi \left(\dfrac{k}{dk}\right)^{2}## and why it is considered like a relative error ...

Would someone be able to explain like I am five years old, what is the precise relationship between Fourier series and Fourier transform? Could someone maybe offer a concrete example that clearly illustrates the relationship between the two? I found an old thread that discusses this, but I...

Consider the function ##f:[0,1]\rightarrow \mathbb{R}## given by $$f(x)=x^2$$ (1) The Fourier coefficients of ##f## are given by $$\hat{f}(0)=\int^1_0x^2dx=\Big[\frac{x^3}{3}\Big]^1_0=\frac{1}{3}$$ $$\hat{f}(k)=\int^1_0x^2e^{-2\pi i k x}dx$$ Can this second integral be evaluated?

The definition of Fourier transform (F.T.) that I am using is given as: $$f(\vec{x},t)=\frac{1}{\sqrt{2\pi}}\int e^{-i\omega t}\tilde{f}(\vec{x},\omega)\,\mathrm{d}\omega$$ I want to show that: $$\frac{1}{c\sqrt{2\pi}}\int e^{-i\omega t}\omega^2...

Continue reading...

Say we have a transform of a line profile that extends out to the Nyquist frequency such that you cannot see the noise level, what could you change in your spectrograph arrangement that would allow you to see the noise level in the Fourier domain? My thought is that we can apply a filter, P(s)...

Given ##f(\vec{x})##, where the Fourier transform ##\mathcal{F}(f(\vec{x}))= \hat{f}(\vec{k})##. Given ##\vec{x}=[x_1,x_2,x_3]## and ##\vec{k}=[k_1,k_2,k_3]##, is the following true? \begin{equation} \begin{split} \mathcal{F}(f(x_1))&= \hat{f}(k_1) \\ \mathcal{F}(f(x_2))&= \hat{f}(k_2) \\...

Hi I would like to transform the S-parameter responce, collected from a Vector Network Analyzer (VNA), in time domain by using the Inverse Fast Fourier Transform (IFFT) . I use MATLAB IFFT function to do this and the response looks correct, the problem is that I do not manage to the time scaling...

I have not studied the Fourier transform (FT) in great detail, but came across a problem in electrodynamics in which I assume it is needed. The problem goes as follows: Evaluate ##\chi (t)## for the model function...

Good day I really don't understand how they got this result? for me the sum of the Fourier serie of of f is equal to f(2)=log(3) any help would be highly appreciated! thanks in advance!

Hi, I am interested in understanding the relationships between Fourier series and Fourier transform better. My goal is 1) Start with a set of ordered numbers representing Fourier coefficients. I chose to create 70 coefficients and set the first 30 to the value 1 and the remaining to zero. 2)...

Convolving two signals, g and h, of lengths X and Y respectively, results in a signal with length X+Y-1. But through convolution theorem, g*h = F^{-1}{ F{g} F{h} }, where F and F^{-1} is the Fourier transform and its inverse, respectively. The Fourier transform is unitary, so the output signal...

The answer in the textbook writes: $$ f(x) = \frac{1}{4} +\frac{1}{\pi}(\frac{\cos(x)}{1}-\frac{\cos(3x)}{3}+\frac{\cos(5x)}{5} \dots) + \frac{1}{\pi}(\frac{\sin(x)}{1}-\frac{2\sin(2x)}{2}+\frac{\sin(3x)}{3} + \frac{\sin(5x)}{5}\dots)$$ I am ok with the two trigonometric series in the answer...

I have two questions regarding Fourier transformation. First of all is it ok to call Fourier transformation operator, or it should be distinct more? For instance, if I wrote F[f(x)]=\lambda f(y) is that eigenproblem, regardless of the different argument of function ##f##? Could I call ##F##...

If given a set of data points for the magnitude of a cepheid variable at a certain time (JD), how can we use Fourier series to find the period of the cepheid variable? I'm trying to do a math investigation (IB math investigation) on finding the period of the cepheid variable M31_V1 from data...

In August, "Quantum Information Processing" published an article describing a full FFT in the quantum domain - a so-called QFFT, not to be confused with the simpler QFT. According to the publication:

Hi, I was just thinking about different ways to use the Fourier transform in other areas of mathematics. I am not sure whether this is the correct forum, but it is related to probability so I thought I ought to put it here. Question: Is the following method an appropriate way to calculate the...

I am attempting to be able to do this problem with the help of Mathematica and Fourier transform. My professor gave us instructions for Fourier Transformation and Inverse Fourier, but I don't believe that my output in Mathematica is correct.

A beam of length L with fixed ends, has a concentrated force P applied in the center exactly in L / 2. In the differential equation: \[ \frac{d^4y(x)}{dx^4}=\frac{1}{\text{EI}}q(x) \] In which \[ q(x)= P \delta(x-\frac{L}{2}) \] P represents an infinitely concentrated charge distribution...

I was content with the understanding of the Fourier transform (FT) as a change of basis, from the time to the frequency basis or vice versa, an approach that I have often seen reflected in texts. It makes sense, since it is the usual trick so often done in Physics: you have a problem that is...

Hi, Question: If we have an initial condition, valid for -L \leq x \leq L : f(x) = \frac{40x}{L} how can I utilise a know Fourier series to get to the solution without doing the integration (I know the integral isn't tricky, but still this method might help out in other situations)? We are...

Hi, I was hoping to gain more insight into these window questions when looking at frequency spectra questions. I don't really know what makes windows better than one another. My attempt: In the question, we have f(t) = cos(\omega_0 t) and therefore its F.T is F(\omega ) = \pi \left(...

Hello, I am unfamiliar with Maxwell's equations' Fourier transform. Are there any materials talking about it?

Suppose I decompose a discrete audio signal in a set of frequency components. Now, if I would add the harmonics I got, I would get the original discrete signal. My question is: if I would randomize the phases of the harmonics first, and then add them, I would get a different signal, but would it...

From the sketch, I know that this function is an even function. So, I simplify the Fourier transform in the limit of the integration (but still in exponential form). Then, I try to find the exponential FOurier transform. Here what I get: $$g(a)=\frac{2}{2\pi} \int_{0}^{\infty} e^{-x} e^{-iax}...

First, I try to define the function in the figure above: ##V(t)=100\left[sin(120\{pi}\right]##. Then, I use the fact that absolute value function is an even function, so only Fourier series only contain cosine terms. In other words, ##b_n = 0## Next, I want to determine Fourier coefficient...

Hello Everyone! I want to learn about Fourier series (not Fourier transform), that is approximating a continuous periodic function with something like this ##a_0 \sum_{n=1}^{\infty} (a_n \cos nt + b_n \sin nt)##. I tried some videos and lecture notes that I could find with a google search but...

Summary:: If ##f(x)=-f(x+L/2)##, where L is the period of the periodic function ##f(x)##, then the coefficient of the even term of its Fourier series is zero. Hint: we can use the shifting property of the Fourier transform. So here's my attempt to this problem so far...

Hi, In class I have learned how to find the Fourier transform of rectangular pulses. However, how do I solve a problem when I should sketch the Fourier transform of a pulse that isn't exactly rectangular. For instance "Sketch the Fourier transform of the following 2 pulses" Thanks in advance...

Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex? For example, given \begin{equation} \begin{split} \hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...

I found this video on youtube which is trying to explain Fourier transform using the center of mass concept At 15:20 the expression of the x coordinate is given in the video. I believe it is wrong, and it should be: ##\frac{{\int g(t)e^{(-2 \pi ift)}.g(t).2 \pi f.dt}} { \int g(t).2 \pi...

Summary:: Discrete Fourier transform exam question Hi there, I'm not really sure how to do this question at all. Any help would be appreciated.

I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as \int_{0}^{2\pi} \sin((m + 1)...

Hi, A rectangular pulse having unit height and lasts from -T/2 to T/2. "T" is pulse width. Let's assume T=2π. The following is Fourier transform of the above mentioned pulse. F(ω)=2sin{(ωT)/2}/ω ; since T=2π ; therefore F(ω)=2sin(ωπ)/ω Magnitude of F(ω)=|F(ω)|=√[{2sin(ωπ)/ω}^2]=|2sin(ωπ)/ω|...

Hi, I was trying to find Fourier transform of two rectangular pulses as shown below. The inverted rectangular pulse has unit height, -1, and lasts from -π to 0. The other rectangular pulse has unit height, 1, and lasts from 0 to π. I was making use of Laplace transform and its time shifting...

In order to use the Second Shift Theorem, the function needs to be entirely of the form $\displaystyle f\left( t - 1 \right) $. To do this let $\displaystyle v = t - 1 \implies t = v + 1 $, then $\displaystyle \begin{align*} \mathrm{e}^{-2\,t} &= \mathrm{e}^{-2 \, \left( v + 1 \right) } \\ &=...

In order to factorise this quadratic, we will need to recognise that $\displaystyle \mathrm{i}^2 = -1 $, so we can rewrite this as $\displaystyle \begin{align*} \frac{1}{6 + 5\,\mathrm{i}\,\omega - \omega ^2 } &= \frac{1}{\mathrm{i}^2\,\omega ^2 + 5\,\mathrm{i}\,\omega + 6} \\ &= \frac{1}{...

Hi, This thread is an extension of this discussion where @DrClaude helped me. I thought that it'd be better to separate this question. I couldn't find any other way to post my work other than as images so if any of the embedded images are not clear, just click on them. It'd make them clearer...

Hey! :o When it is given that a signal $x(t)$ has a real-valued Fourier transformation $X(f)$ then is the signal necessarily real-valued? I have thought the following: $X_R(ω)=\frac{1}{2}[X(ω)+X^{\star}(ω)]⟺\frac{1}{2}[x(t)+x^{\star}(−t)]=x_e(t) \\ X_I(ω)=\frac{1}{2i} [X(ω)−X^{\star}(ω)]⟺...

Hey! :o A real periodic signal with period $T_0=2$ has the Fourier coefficients $$X_k=\left [2/3, \ 1/3e^{j\pi/4}, \ 1/3e^{-i\pi/3}, \ 1/4e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$. I want to calculate $\int_0^{T_0}x^2(t)\, dt$. I have done the following: It holds that...

I am trying to reproduce the results of a thesis that is 22 years old and I'm a bit stuck at solving the differential equations. Let's say you have the following equation $$\frac{\partial{\phi}}{\partial{t}}=f(\phi(r))\frac{{\nabla_x}^2{\nabla_y}^2}{{\nabla}^2}g(\phi(r))$$ where ##\phi,g,f## are...

In a numerical Fourier transform, we find the frequency that maximizes the value of the Fourier transform. However, let us consider an analytical Fourier transform, of ##\sin\Omega t##. It's Fourier transform is given by $$-i\pi\delta(\Omega-\omega)+i\pi\delta(\omega+\Omega)$$ Normally, to find...

i tired using complex identity equation for sin(pi*k/3) but it doesn't work out

Momentum ##\vec{p}## and position ##\vec{x}## are Fourier conjugates, as are energy ##E## and time ##t##. What is the Fourier conjugate of spin, i.e., intrinsic angular momentum? Angular position?

ANY AND ALL HELP IS GREATLY APPRECIATED :smile: I have found old posts for this question however after reading through them several times I am having a hard time knowing where to start. I am happy with the sketch that the function is correctly drawn and is neither odd nor even. It's title is...

I would like to know the equation of Fourier transform when the data has lack. like this sine wave.

This is written on Greiner's Classical Mechanics when solving a Tautochrone problem. Firstly,I don’t understand why we didn’t use the term ##m=0## and Sencondly, how the integrand helps us to fulfill the Dirichlet conditions. That means,how do we know that the period is 1?Thanks