Fourier transform Definition and 1000 Threads

Hi, I hope somebody can help me with this one. Homework Statement Compute the Fourier Transform of the distribution x-a Homework Equations The Fourier Transform of a distribution is just the distribution evaluated with the Fourier Transform of a test function.The Attempt at a Solution See...

I have an exercise with a function of the form: h(t) = f(t)g(t) and f(t) and g(t) both have discrete Fourier series, which implies that h does too. I want to find the Fourier series of h, so my teacher said I should apply the convolution theorem which would turn the product above into a...

Homework Statement Part (a): State inverse Fourier transform. Show Fourier transform is: Part (b): Show Fourier transform is: Part (c): By transforming LHS and RHS, show the solution is: Part(d): Using inverse Fourier transform, find an expression for T(x,t) Homework Equations The Attempt...

Homework Statement The exercise is a) in the attached trial. I have attached my attempt at a solution, but there are some issues. First of all: Isn't the example result wrong? As I demonstrate you get a delta function which yields the sum I have written (as far as I can see), not the sum...

http://calclab.math.tamu.edu/~fulling/m412/f07/airywkb.pdf Can someone walk me through this derivation of the Airy integral by Fourier transform? I have tried it but failed

I'm trying to prove that the discrete form of the Fourier transform is a unitary transformation So I used the equation for the discrete Fourier transform: ##y_k=\frac{1}{\sqrt{N}}\sum^{N-1}_{j=0}{x_je^{i2\pi\frac{jk}{N}}}## and I put the Fourier transform into a N-1 by N-1 matrix form...

From -infinity to infinity at the extreme ends do Fourier transforms always converge to 0? I know in the case of signals, you can never have an infinite signal so it does go to 0, but speaking in general if you are taking the Fourier transform of f(x) If you do integration by parts, you get a...

Homework Statement What is the Fourier transform of a single short pulse and of a sequence of pulses? The Attempt at a Solution In class we haven't dealt with the mathematics of a Fourier transform, however my professor has simple stated that a Fourier transform is simply a equation...

I have a tutorial question for maths involving the heat equation and Fourier transform. {\frac{∂u}{∂t}} = {\frac{∂^2u}{∂x^2}} you are given the initial condition: u(x,0) = 70e^{-{\frac{1}{2}}{x^2}} the answer is: u(x,t) = {\frac{70}{\sqrt{1+2t}}}{e^{-{\frac{x^2}{2+4t}}}} In this course...

I learned how to integrate it using the complex plane and semi circle contours but I was wondering if there is a way using Fourier transforms. I know that the Fourier transform of the rectangle wave form is the sinc function so I was thinking maybe i could do an inverse Fourier on sinc x and get...

Hello, Recently I've learned about Fourier Transform, and the uncertainty principle that is arose from it. According to Fourier Transform, if there is only one pulse in a signal, then it is composed from a lot more frequencies, compared to the number of frequencies that are building a...

Fourier transform of RF signal with a "prism"? We can use a prism to decompose visible light into components of different frequencies. This is a Fourier transform by nature. For an ideal prism, the energy is conserved in the process. How about RF signals? There is no fundamental difference...

Hi guys, I'm not sure how they got from first step to the second. Did they use integration by parts? I tried but I didn't arrive at the same result..

\mathcal{F}\{f(r)\}=\int e^{i\vec{k}\cdot \vec{r}}f(r)d\vec{r} in spherical polar coordinates \mathcal{F}\{f(r)\}=\int^{\infty}_0r^2dr\int^{\pi}_0\sin\theta d\theta\int^{\pi}_0d\varphi e^{ikr\cos \theta}f(r) Why could I take ##e^{ikr\cos \theta}## and to take that ##\theta## is angle which goes...

Homework Statement Finding the Fourier Transform using Transform Pair and Properties x(t) = 2[u(t+1)- t^{3}e^{6t}u(t)] Homework Equations The Attempt at a Solution For the first problem, I got u(t) \leftrightarrow ∏δ(ω)+\frac{1}{jω} F(at-t_{0}) \leftrightarrow...

The windowed Fourier transform on R Definition-Proposition-Theorems (Plancherel formula-Parseval formula-inversion formula-Calderon's formula) http://www.4shared.com/office/b2Ho5n7H/The_windowed_Fourier_transform.html

I'm working on some research with a professor, and we're looking at data collected by an x-band radar array looking at ocean waves as they approach the coast (the radar is on land, and we can see about 3 miles out). What we're trying to do is perform an fft on the signal using Matlab, and...

I'm given a Gaussian function to apply a Fourier transform to. $$f(x)=\frac{1}{\sqrt{a\sqrt{\pi}}}e^{ik_ox}e^{-\frac{x^2}{2a^2}}$$ Not the most appetizing integral... $$g(k)=\frac{1}{\sqrt{2\pi}}\frac{1}{\sqrt{a\sqrt{\pi}}}\int_{-\infty}^{\infty}e^{ik_ox}e^{-\frac{x^2}{2a^2}}e^{-ikx}dx$$...

I understand the Fourier transform conceptually, but I am unable to reproduce it mathematically; I am very familiar with calculus and integration, but I am taking a QM course and I need to know how to apply it. No websites or videos are able to give me a good explanation as to how I can use it...

When the rod is infinite or semi-infinite, I was taught to use Fourier transform. But I don't know when should the full Fourier transform or sine/cosine transform be used. how's the B.C. related to the choice of the transform ?

Hi all, Only few days back I got the idea of probability density function. (Till that day , I believed that pdf plot shows the probability. Now I know why it is density function.) Now I have a doubt on CTFT (continuous time Fourier transform). This is a concept I got from my...

I've been assigned the following homework: I have to compute the spectral density of a QFT and in order to do so I have to compute Fourier tranform of the following quantity (in Minkowsky signature, mostly minus) \rho\left(p\right) = \int \frac{1}{\left(-x^2 + i \epsilon...

Somehow I have really hard time wrapping my head around the concept.I mean,I get it,but I can't seem to solve any questions regarding it. Here are some examples ,and I just get stuck.Its a part of test,so I think it shouldn't be that hard to solve,and if it looks hard,I know there are some...

Hi, I'm following the proof of the "Scaling Property of the Fourier Transform" from here: http://www.thefouriertransform.com/transform/properties.php ...but don't understand how they went from the integral to the right hand term here: The definition of the Fourier Trasform they...

Consider \(u_t = -u_{nxxx} - 3(u^2)_{nx}\). The Fourier Transform is linear so taking the Inverse Fourier transform of the Fourier Transform on the RHS we have \begin{align} -\mathcal{F}^{-1}\left[\mathcal{F}\left[u_{nxxx} - 3(u^2)_{nx}\right]\right] &= -\mathcal{F}^{-1}...

Does every FFT have \(i\) in it? Given \(u_t = -(u_{xxx} + 6uu_x)\). \(f'''(x) = \mathcal{F}^{-1}\left[(ik)^3\mathcal{F}(f(x))\right]\) \(f'(x) = \mathcal{F}^{-1}\left[(ik)\mathcal{F}(f(x))\right]\) The only equation I have used the pseudo-spectral method on was the NLS which is \(u_t =...

Today I found a program, which does Fourier transforms on pictures and tried it on some basic patterns. One of those was a lattice of dots and I have attached this and its Fourier transform to the thread. I would very much like if someone in basic details could explain what is going on. Why...

Finite Fourier Transform on a 2d wave How does the finite Fourier transform work exactly? The transform of f(x) is \widetilde{f}(\lambda_{n}) =\int^{L}_{0} f(x) X_{n} dx If I had a 3d wave equation pde and I applied Finite Fourier transform on the pde for z(x,y,t)=X(x)Y(y)T(t)...

Hi all, as a physics student, I seldom use Fourier transform but from my understanding, given a periodic function you can decompose the function into sine function with different frequencies. Also, to get a ultra short pulse in time domain, this would require mixing many frequencies. I would...

Homework Statement \psi (x) = Ne^{ \frac{-|x|}{a}+ \frac{ixp_o}{/hbar}} Compute Fourier transform defined by ##\phi (p) = \frac{1}{ \sqrt{2 \pi \hbar}} \int \psi (x) e^{ \frac{-ipx} {\hbar}} dx## to obtain ## \phi (x) ## Homework Equations Fourier transform = ##g(x)= \frac {1}{2 \pi} \int...

Hi, I have the following question: A signal x(t) which is band-limited to 10kHz is sampled with a sampling frequency of 20kHz. The DFT (Discrete Fourier Transform) of N= 1000 samples of x(n) is then computed. To what analogue frequency does the index k=120 respond to? I'm trying to...

Hi all, I want to calculate \int_0^{\infty}e^{-a t^2}\cos(2xt)dt=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{\frac{-x^2}{a}}. The answer is known from the literature, but I don't know how to do it step by step. Any one has a clue? Thanks. Jo

I took f(t) = SIN(10*t) +SIN(5*t) and got this f(0) = 0 f(1) = -1.5 f(2) = 0.4 f(3) = -0.3 now I tried to do the DFT Fs = 4Hz N = 4 samples 3 f[r] = Ʃ x[k]ε^(-j(2πkr/4) k=0 f[r] = 0 -1.5ε^(-j(2πr/4) + 0.4ε^(-j(2π(2)r/4) -0.3ε^(-j(2π(3)r/4) f[0] = 0 - 1.5 +...

Hi there, I am trying to get some practice with Fourier Transforms, there is a long way to go. For example, let me consider the function $$ \gamma (t) = \int_{-\infty}^{t} C(t-\tau) \sigma(\tau) \mathrm{d}{\tau}$$ Defining the Fourier Transform as $$ \gamma(\omega) = \frac{1}{2 \pi}...

I was looking through some examples which applied the duality principle while studying for an up and coming exam when it hit me that the transform applied 4 times gives you back the same function. So is there some theory that uses this? perhaps some sort of operator? I thought it...

Homework Statement Compute the Fourier transform of a function of norm f(\norm{x}). Homework Equations \mathbb{F}{\frac{1}{1+\norm{x}} The Attempt at a Solution Attempt at using Cauchy theorem and the contour integral with the contour [(-R,R),(R,R+ip),(R+ip,-R+ip),(-R+ip,-R)] does...

Checked around a buch and could not find any help. But I needed help with: Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant image/data i.e. for every tick on the axis, how do I know the spatial length? More detailed...

Hello everyone: I have some question using the FFT in MATLAB for data interpolating. I don't know what the relation between the normal Fourier series and the real, image number. For example, given a set of measurement data, I can use the curve fitting toolbox to fit a curve. The general...

Homework Statement Hi, this is not a homework question per se, but something I'm wondering. Let C be a circulant n x n matrix, let x, b, be vectors such that C x = b. We would like to find a solution x. One way is to use the DFT: According to section 5, In Linear Equations, in the wikipedia...

I have been studying Fourier Optics and I have a basic conceptual question. I understand the mathematics of how to perform Fourier Transforms however the part of this topic I seem to have missed is why the action of a lens on light is the same as performing a Fourier Transform on the functional...

Hi all, I'm a complete novice when it comes to describing images in frequency space and i understand that it is a way of representing images as being composed of a series of sinusoids. So a horizontal striped pattern with a single spatial frequency would have a magnitude image in frequency...

Homework Statement I have am doing a two dimensional discrete Fourier transform on an image (using MATLAB). What are the units associated with each pixel of the image in the frequency domain? Homework Equations The Attempt at a Solution I thought that the frequency should be...

Homework Statement Find the inverse Fourier transform of F(jω) = cos(4ω + pi/3)Homework Equations δ(t) <--> 1 δ(t - to) <--> exp(-j*ωo*t) cos(x) = 1/2 (exp(jx) + exp(-jx))The Attempt at a Solution So first I turned the given equation into its complex form using Euler's Formula. F(jω) = 1/2...

can you use Fourier transform to find a moving average on a data set? so, you do a Fourier transform on your one dimensional data set. next remove high order harmonics from FT result. do reverse Fourier transform on new FT result. And, vola! smoothed out data set.

Hi, I was wondering what would the Fourier transform of a signal like below give: s(t) = sin(2πt*10) ; t in [0s,5s] = sin(2πt*20) ; t in [5s,10s] I certainly did not expect it to give me 2 sharp peaks at frequencies 10Hz and 20Hz - because I understand that the addition of...

Hallo, I really don't understand Fourier transform. Do somebody know a good book for beginners? Something like Fourier transform for dummies or so? I need it just for physics. So it don't have to be to mathematical. ^^ THX

Fourier Transform on the "connected part" of QFT transition prob. Homework Statement Calculate ⟨0|T[ϕ(x₁)ϕ(x₂)ϕ(x₃)ϕ(x₄)]|0⟩ up to order λ from the generating functional Z[J] of λϕ⁴-theory. Using the connected part, derive the T-matrixelement for the reaction a(p₁) + a(p₂) → a(p₃) +...

Homework Statement In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that: F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv) where F_{l,v}(g(l)) is the Fourier transform of g(l) and...

Hello, Consider I have a linear time-invariant (LTI) system, with ##x(t)##, ##y(t)##, and ##h(t)##, as input, output, and impulse response functions, respectively. I have two choices to write the convolution integral to get ##y(t)##: $$ 1)\ \ \ y(t) = \int_{0}^{t} h(t-t')x(t')dt' $$ and...

Homework Statement Find the Fourier transform of (1/p)e^{[(-pi*x^2)/p^2]} for any p > 0 Homework Equations The Fourier transform of e^{-pi*x^2} is e^{-pi*u^2}. The scaling property is given to be f(px) ----> (1/p)f(u/p) The Attempt at a Solution Using the information above, I got...