Transform Definition and 1000 Threads

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...

of course y and z terms are invariant but for the x and t terms I am getting an additional factor of 1/1-v^2/c^2

I am trying to understand the last block of equations in the picture (after 3.31). In the first line of that block, he transforms the spinor ##\psi## which I have no problem with. What I have a problem with is the ##\gamma ^{\mu} \partial _{\mu}##. They form a Lorentz scalar, so they should not...

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??

I'm trying to get transform the larger circuit into the smaller one and then from there calculate power. My plan was to do the transform and then use kirchhoffs laws to find the current tofind the power. My work so far: Is the sequence of steps I used valid? I'm not focusing on the calculations...

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...

transform the given equation into a system of first order equation$$u''+0.5u'+2u=0$$ok from examples it looks all we do is get rid of some of the primes and this is done by substitutionso if $u_1=u$ and $u_2=u'_1$ then $u_2=u'$ and $u'_2=u''$ then we have $u'_2+0.5u_2 +2u_1 = 0$then isolate...

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...

Summary: A 1963 paper by Michael Wertheim uses a Laplace transform in spherical coordinates. How is the resulting equation obtained? In 1963, Michael Wertheim published a paper (relevant page attached here), where he presented the following equation (Eq. 1): $$ y(\bar{r}) = 1 + n...

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! Fluid lies in a 2D rectangular channel and oscillates from a disturbance. I have several .csv files, each corresponding to a moment in time, where within each are two lists of numbers: the ##x## and ##y## position of a fluid interface. I'd like to decompose the interface into it's...

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...

Hi, Is it possible to transform this equation $$ln⁡((p_1 C_1)/(p_1 C_2 ))+ln⁡((p_2 C_3)/(p_2 C_4 ))+ln⁡((p_3 C_5)/(p_3 C_6 ))$$ to $$ln⁡((p_1 C_1+p_2 C_3+p_3 C_5)/(p_1 C_2+p_2 C_4+p_3 C_6 ))$$ Thanks

Hi, I was trying to see if the following differential equation could be solved using Laplace transform; its solution is y=x^4/16. You can see below that I'm not able to proceed because I don't know the Laplace pair of xy^(1/2). Is it possible to solve the above equation using Laplace...

Summary: I'm stuck on this simple excersize, to show that in this coord transform, despite x = x', d/dx != d/dx' From "Intro to Smooth Manifolds" (this is a calculus excersize), The Problem I have is with showing d/dx != d/dx' When I write out the Jacobian matrix, I get exactly d/dx = d/dx'...

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...

My solution is in the file shown here

So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd...

Summary: Transform the periodic table of chemical elements (periodic table) into a universal way of storing and transmitting information using spectral analysis. I propose a concept in which the basis for working with information (conservation, transmission in networks) is to use spectral...

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...

If I had to guess what the complex matrix would look like, it would be: ##T(x+iy)=(xa-by)+i(ya+bx)=\begin{pmatrix} a+bi & 0 \\ 0 & -b+ai\end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix}## I'm not too sure on where everything goes; it's my first time fiddling with complex numbers, and moreover...

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...

It is obvious that there is a one-to-one relationship between real numbers (defined to include infinity) and their multiplicative inverses (assuming we map the inverse of zero to infinity and vice versa). Thus, one should be able to replace the distance between two points in space with it's...

In Mathematical Methods in the Physical Sciences by Mary Boas, the author defines the Laplace transform as... $${L(f)=}\int_0^\infty{f(t)}e^{-pt}{dt=F(p)}$$ The author then states that "...since we integrate from 0 to ##\infty##, ##{L(f)}## is the same no matter how ##{f(t)}## is defined for...

I have used Laplace transform during my EE studies to solve differential equations and in control system analysis, but we were taught that as a tool kit to make the math easier. The physical meaning was never explained. I know basic time and frequency domain concepts (thanks to Fourier series)...

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...

Vo(S) = [ N(s)Vi(s) + (- s2 + s - 2) ] / s3 + s2 + 1 ; can ignore (-s^2 + s - 2). From relevant equations: Vo(S) = [N(s)*Vi(s)]/(s^3 + s^2 + 1); -> (d3Vo(t)/dt3) + (d2Vo(t)/dt2) + Vo(t) = N(t)(dvi)/dt L[vi(t)] = t to s domain: [s3Vo(s) - s2Vo(0-) - SV'o(0-) - Vo''(0-)]Vo(s) + s2 - SVo -...

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 wrote the following code in MATLAB: t = [0:0.001:0.1]; noise = randn(1,size(t,2)); a = 15*10^9; b = 15*10^(-3); c = 7*10^8; y = a*exp(-t/b)+c+noise*100000000; fun = @(p,t)p(1)*exp(-t/p(2))+p(3); p0 = [15.5*10^9, 14*10^(-3), 6*10^8]; p = lsqcurvefit(fun, p0, t, y); t_fit = [0:0.0001:0.1]; y_fit...

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?

Can anyone help me with the Proof of Parseval Identity for Fourier Sine/Cosine transform : 2/π [integration 0 to ∞] Fs(s)•Gs(s) ds = [integration 0 to ∞] f(x)•g(x) dx I've successfully proved the Parseval Identity for Complex Fourier Transform, but I'm unable to figure out from where does the...

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...

Homework Statement L{sin(ωt)/[1+cos^2(ωt)]} = Homework Equations d {arctan[cos(ωt)]} /dt = - ω•sin(ωt)/[1+cos^2(ωt)] The Attempt at a Solution ∫e^(-st)•[sin(ωt)/(1+cos²(ωt)] dt = -(1/ω)•∫e^(-st)•{arctan[cos(ωt)]}' dt = = (integrating by parts and taking Re(s) > 0) = = π/(4ω) -(s/ω)•∫...

I apologize for the simplicity of the question. I have been reading a paper on the Legendre transform (https://arxiv.org/pdf/0806.1147.pdf), and I am not understanding a particular step in the discussion. In the paper, Equation 16, where ##\mathcal{H} = \sqrt{\vec{p}^2 + m^2} ##...

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 shown in the picture is the question and answer to it, but i don't understand how they're getting it. this is me just not understanding the maths and i know its not difficult but I've been stuck on it for a while, so can someone explain in detail how you get the last two...