Fourier transform Definition and 950 Threads

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...

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...

If I understand correctly (a big caveat), one shows that if one can get from one function to the other via a Fourier transform and multiplication by a constant, then the width of the corresponding Gaussian wave of one gets larger as that of the other gets smaller, and vice-versa, and by a bit...

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...

When considering the forward FFT of a mathematical function sampled at times ##t = 0, \Delta, \ldots, (N-1) \Delta##, following the usual convention, we have something like $$ H(f) = \int_{-\infty}^{+\infty} h(t) e^{-2 \pi i f t} dt \quad \Rightarrow \quad H_k = \sum_{n=0}^{N-1} h_n e^{-2 \pi i...

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

In my QFT homework I was asked to prove that $$\int d^3x \int \frac{d^3k}{(2\pi)^3} e^{i\mathbf{k} \cdot (\mathbf{x} - \mathbf{y})} k_j f(\mathbf{x}) = i \frac{df}{dx_j}(\mathbf{y}) $$ Using ##\frac{\partial e^{i\mathbf{k} \cdot (\mathbf{x} - \mathbf{y})}}{\partial x^j} = i k_j e^{i\mathbf{k}...

Hi, I've been looking all over the net for good examples but I've only found some intro but no examples being solved. If you know of good resources (both theories and problems) please let me know! a) Calculate Fourier and inverse Fourier transform of f(t). b) Calculate the limit. My...

Hello, everyone. :) I'm trying to do a certain problem regarding Fourier transforms (but one that's supposedly easy, because of just using tables, rather than fully computing stuff), and I know how to do it, but I don't know why it works. Here's the problem statement.: "Compute the Fourier...

Summary:: My TI-89 is not evaluating the Fourier transform? Change angle to radians and retry. Hello, I discovered this forum trying to answer the question: Why is my TI-89 not properly evaluating the Fourier transform? I found no answer, by chance I experimented and found that the calculator...

I'm confused on how units work with regards to the Fourier Transform (CTFT). I was reading the Wikipedia article on spectral density. In an example, they use Parseval's equation, along with the units calculated on the time side, to determine the units on the frequency domain side. The units of...

Given a function F(t) $$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$ I am looking for a proof of the following: $$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d...

In order to obtain equation (3), I think I have to do the Fourier transform in the x direction: \begin{equation} \tilde{G}(k,y,x_0,y_0) = \int_{- \infty}^{\infty} G(x,y,x_0,y_0) e^{-i k x} dx \end{equation} So I have: \begin{equation} -k ^2 \tilde{G}(k,y,x_0,y_0) + \frac{\partial^2...

Dear all. I'm learning about the discrete Fourier transform. ##I(\nu) \equiv \int_{-\infty}^{\infty} i(t) e^{2 \pi \nu i t} d t=\frac{N}{T} \sum_{\ell=-\infty}^{\infty} \delta\left(\nu-\ell \frac{N}{T}\right)## this ##i(t)## is comb function ##i(t)=\sum_{k=-\infty}^{\infty}...

Here it goes. I have been taught that a finite pulse of light does not have a single frequency. By finite pulse I was given an example of a source of light that has been emitted during a finite amount of time and, consequently, covers a finite region of space. Then I was taught that you can...

Dear all. I can't understand how to derive Eq.(2.3a). Fourier coefficients, ##A_j## and ##B_j## are described by summation in this paper as (2.2).　I think this is weird. Because this paper said "In this section 2.1 ,the Fourier transform is introduced in very general terms". and I understand...

Homework Statement: I don't know how can I derivation Eq.(2.2) Homework Equations: Fourier coefficients Homework Statement: I don't know how can I derivation Eq.(2.2) Homework Equations: Fourier coefficients Dear all. I don't know how can I derivation Eq.(2.2). Where Σk is come from??

Given that the wave function represented in momentum space is a Fourier transform of the wave function in configuration space, is the conjugate of the wave function in p-space is the conjugate of the whole transformation integral?

Hi All. I hope this question makes sense. In the case of Fourier Transforms one has the complex exponentials exp(2..π i. ξ.x) In 3-D, if we single out where the complex exponentials are equal to 1 (zero phase), which is when ξ.x is an integer, a given ( ξ1,ξ2,ξ3).defines a family ξ.x= integer...

Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w)) Hello to my Math Fellows, Problem: I am looking for a way to calculate w-derivative of Fourier transform,d/dw (F{x(t)}), in terms of regular Fourier transform,X(w)=F{x(t)}. Definition Based Solution (not good enough): from...

Hi PF! I'm following a tutorial in MATLAB, shown here t = 0:.001:.25; x = sin(2*pi*50*t) + sin(2*pi*120*t); y = x + 2*randn(size(t)); Y = fft(y,251); Pyy = Y.*conj(Y)/251; f = 1000/251*(0:127); plot(f,Pyy(1:128)) title('Power spectral density') xlabel('Frequency (Hz)') I read the...

Hi PF! Suppose we take a drop of fluid and let it sit on a substrate, and then vibrate the substrate. Doing this excites different modes. If someone where to analyze the vibrations, would they take an FFT of the interface, basically reconstructing it from basis functions (harmonics), where the...

Hi, I tried to apply different forms of Fourier transform, exponential and trigonometric forms, to the same function, f(t)=a⋅e^-(bt)⋅u(t). The result reached using exponential form is correct. Please notice that while appling the trigonometric form of Fourier transform, the factor of 1/π was...

First, the inverse quantum Fourier transform is ##\left | k \right > =\frac 1 {\sqrt {2^n}} \sum_{j=0}^{2^n-1} e^{-2 \pi ijk / 2^n} \left | j \right >##, and it is equal to ##\left | k_1 , k_2 , \dots , k_n \right > \rightarrow \frac { \left ( \left | 0 \right > + e^{-2 \pi i 0.k_n} \left...

Well what I did was first use the inverse Fourier transform: $$u(x,t)=\frac{1}{2\pi }\int_{-\infty }^{\infty }\tilde{u}(\xi ,t)e^{-i\xi x}d\xi$$ I substitute the equation that was given to me by obtaining:$$u(x,t)=\frac{1}{2\pi }\left \{ \int_{-\infty }^{\infty}\tilde{f}(\xi)cos(c\xi...

Attached is a personal problem that I spent last night working on for about 2 hours and something is going wrong, I just can not figure it out what. The answer by the big X is what I wound up with but it's obviously not correct. Could someone please guide me through solving this? The starting...

Hi all, I need to calculate Fourier transform of the following function: sin(a*t)*exp(-t/b), where 'a' and 'b' are constants. I used WolphramAlpha site to find the solution, it gave the result that you can see following the link...

By applying the Fourier transform equation, and expanding the dot product, I get a sum of terms of the form: $$V(k)=\sigma_1^x\nabla_1^x\sigma_2^y\nabla_2^y\frac{1}{|\vec{r_2}-\vec{r_1}|}e^{-m|\vec{r_2}-\vec{r_1}|}e^{-ik(r_2-r_1)} =...

I used a matrix to calculate the Fourier transform of a lorentzian and it did generate a decaying exponential but that was followed by the mirror image of the exponential going up. I am referring to the real part of the exponential. If I use an fft instead I also see this. Shouldn't the...

I am struggling to figure out how to approach this problem. I've only solved a homogenous heat equation $$u_t = u_{xx}$$ using a Fourier transform, where I can take the Fourier transform of both sides then solve the general solution in Fourier terms then inverse transform. However, since this...

Hi all :oldbiggrin: Yesterday I was thinking about the central limit theorem, and in doing so, I reached a conclusion that I found surprising. It could just be that my arguments are wrong, but this was my process: 1. First, define a continuous probability distribution X. 2. Define a new...

I am having trouble with doing the inverse Fourier transform. Although I can find some solutions online, I don't really understand what was going on, especially the part that inverse Fourier transform of cosine function somehow becomes some dirac delta. I've been stuck on it for 2 hrs...

I've been exposed to this notion in multiple classes (namely math and physics) but can't find any details about how one would actually calculate something using this principle: Diffraction in optics is closely related to Fourier transforms and finding the Fraunhofer diffraction of an aperture...

Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?

I understand that the Fourier transform is changing the domain (time/space) to frequency domain and provides the sin waves. I have seen the visualizations of Fourier transform and they are all showing the transform results as the list of frequencies and their amplitude. My question is, what if...

I run sympy.fourier_transform. from sympy import fourier_transform, exp,symbols from sympy.abc import x, k a=fourier_transform(exp(-x**2), x, k) s=symbols('s') Ori=(s)*exp(-(x**2)/(s**2)) FT=fourier_transform(Ori,x,k) a.subs({k:1}).evalf()...

I understand that the Uncertainty Principle relates the variances of Fourier conjugates. I am having trouble finding: 1) the mathematical relationship between the expectation values of Fourier conjugates generally; 2) and then specifically for a normalized Gaussian. Any suggestions or insights?

I have come across a paper where it is stated that if the infinity assumption in the FT is removed, the uncertainty doesn't hold. Is this a sensible argument? Thank you.

I am having trouble following a step in a book. So we are given that $$\varphi (x) = \int \frac {d^3k}{(2\pi)^3 2\omega} [a(\textbf{k})e^{ikx} + a^*(\textbf{k})e^{-ikx}] $$ where the k in the measure is the spatial (vector) part of the four-momentum k=(##\omega##,##\textbf{k}##) and the k in the...

Homework Statement By using Fourier transform, I want to calculate power of signal. I confuse that f(x) in attached equation represents voltage or power. Is that possible when f(x) means power to use Fourier transform. Homework Equations The Attempt at a Solution

I've a system of partial diff. eqs. in thermo-elasticity, I can solve it using normal mode analysis method but I need to solve it using laplace or Fourier

Homework Statement Calculate ##F(\frac 1 {1+x^4})##. Homework Equations ##\hat f (ξ) = \int_ℝ \frac 1 {1+x^4} e^{-2\pi i ξ x} dx## and Residue Theorem The Attempt at a Solution I know the function has to be real and even because ##\frac 1 {1+x^4}## is real and even, but I can't work out the...

Hi, I have to show that if ##f \in L^1(ℝ^n)## then: $$ ||\hat f||_{C^0(ℝ^n)} \le ||f||_{L^1(ℝ^n)}$$ Since ##|f(y)e^{-2 \pi i ξ ⋅y}| \le |f(y)|##, using the dominated convergence theorem, it is possible to show that ##\hat f \in C^0(ℝ^n)## but now I don't know how to go on. Thanks is advance.

It is often reported that the Fourier transform of a constant is δ(f) : that δ denotes the dirac delta function. ƒ{c} = δ(f) : c ∈ R & f => Fourier transform however i cannot prove this Here is my attempt:(assume integrals are limits to [-∞,∞]) ƒ{c} = ∫ce-2πftdt = c∫e-2πftdt = c∫ƒ{δ(f)}e-2πftdt...

Fourier Transform problem with f(t)=cos(at) for |t|<1 and same f(t)=0 for |t|>1. I have an answer with me as F(w)=[sin(w-a)/(w-a)]+[sin(w+a)/(w+a)]. But I can't show it.