Fourier transform Definition and 951 Threads

Homework Statement Hi, I was wondering why the Fourier transform of 1 is 2\pi\delta(w) I would of though that one would be of infinite frequencies (like a square wave). Further more if g(t) = 1, for all t, g(t) = 1. Why does the Fourier transform have the argument of g(t) = 1 have...

Homework Statement Homework Equations The Attempt at a Solution For part i) I got the answer 1/((jw)^2 + 5jw +6) For part ii) I first consider input to be a unit impulse Thus, Y(w)=H(w)F(w) and F(w)=1 yI(t)=-1/2pi integrate from -infinity to infinity (e^jwt)/(w^2 - 5jw -...

Homework Statement A periodic function of period 2\pi is defined by: f(t)=\frac{t}{2} , 0<t<2\pi Show that the trigonometric Fourier series of f(t) is given by: f(t)=\frac{\pi}{2} - \sum_{n=1}^{\infty} \frac{1}{n}sin(nt) Homework Equations The Attempt at a Solution I've gotten...

Homework Statement Not really a homework question, but related none the less. I'm confused about what exactly the amplitude spectrum is. As well as the power spectrum. Homework Equations Not really taking a purely mathematical approach here, I'm using numpy for python. Specifically the fft...

First of all I apologies if I am in the wrong part of the forum for this question but here it is: How do I go about finding the Fast Fourier Transform (FFT) for a given data set? Homework Equations Ive tried using FFT Magnitude = FFT (Real numbers)^2 + FFT (imaginary numbers)^2...

Homework Statement Can someone tell me how to Fourier transform this quantity: \Sigma (x_(j+1) - x_j)^2 where the sum is from j=1 to N Homework Equations Define the Fourier transform as x_j = \Sigma A_k *exp(-iqkj) **Where i is sqrt(-1) **The Sum is from k=0 to (N-1) **q =...

Is it going from the frequency domain to the time domain? Also, is there a relationship between the Fourier series and transform? Thanks for your help!

Homework Statement find the Fourier transform of complex exponential multiplied to a unit step. given: v(t)=exp(-i*wo*t)*u(t) Homework Equations ∫(v(t)*exp(-i*w*t) dt) from -∞ to +∞ The Attempt at a Solution ∫([v(t)]*exp(-i*w*t) dt) from -∞ to +∞...

Hello, I am having a hard time finding the Fourier transform of a function like this: x(t)=4+3sin(1.5t)-4cos(2.5t) How do you do this? Thanks, Chris

Hello, I'm trying to solve the following problem: \nabla^2 p = 0 \frac{\partial p}{\partial y}(x, y_{bot}) = \frac{\partial p}{\partial y}(x, y_{top}) = 0 \frac{\partial p}{\partial x}(x_{left}, y) = \frac{\partial p}{\partial x}(x_{right}, y) = C_0 which is the laplace equation...

Homework Statement I'm supposed to take the "spatial Fourier transform" of the partial differential equation p_t = \frac{a^2}{2\tau}p_{xx} + 2g(p + xp_x) for p = p(x,t). Homework Equations Well, I guess I eventually need something like \phi(k,t) = \mathbb F(p(x,t)) =...

Homework Statement From the definition of the Fourier transform, find the Fourier transform of rect(t-5).Homework Equations G(w) = \int^{\infty}_{-\infty}g(t)e^{jwt}dtThe Attempt at a Solution So, I sketched the function which has area 1 and centre at 5, with its lower bound @ 4.5 and upper at...

This isn't really a homework problem, but I am having trouble understanding why this is true: A function with the following symmetry does not have any even harmonics in its spectrum. I understand the concept based on odd/even symmetry properties, but can anyone provide a mathematical proof?

I have about 40 tabs open on this right now and something important is slipping my grasp. I know this has been covered a million and a half times, but for some reason I cannot seem to find a straight answer (or more probably realize and understand it when I see it). When I take the Continuous...

a) let f be L-integrable on R. show that F(x) = integral (from 0 to x) f(t)dt is continuous. b) show that if F is L-integrable, then lim (as x approaches +/-∞) of F(x) = 0. i am a little stuck on part b). i am trying to use the dominated convergence theorem but i am a bit confused on what...

Hi, if we consider a constant function f(x)=1, it is well-known that its Fourier transform is the delta function, in other words: \int_{-\infty}^{+\infty}e^{-i\omega x}dx = \delta(\omega) The constant function does not tend to zero at infinity, so I was wondering: are there other...

Homework Statement I need to determine the first three terms in the Fourier series pictured in the attachment. Did I define the peace-wise functions correctly? I'm re-posting this with the tex code instead of the attached document. Homework Equations a_o=\frac{1}{2L}\int_{-L}^Lf(t)dt...

Hi, just got set a 3d Fourier transform to solve but I've never seen one before and can't find any examples online. once the integral is set up I should be fine but I'm not sure how to set it up; What is Fourier transfrom (f(k)) of following 3d function for k=(kx, ky, kz)=(1,2,3) for...

Hello, my question arises from reading the section on Smoothness/Compactness from Bracewell's "The Fourier Transform and Its Applications" page 162. I don't quite understand the following reasoning: F(\omega) = \ldots = \frac{1}{i\omega}\int_{-\infty}^{+\infty}f'(x)e^{-i\omega x}dx and...

Homework Statement Let h(t) be impulse response of unity-gain ideal lowpass filter with bandwidth of 50[Hz] and a time delay of 5[ms]. Sketch magnitude and phase of Fourier transform of h(t). The Attempt at a Solution I know that the magnitude2 of H(f) is total power gain, so perhaps by...

I have the readings from a signal in a file (floating point values) that I wish to apply the Fourier Transform to. The samples (mV) were taken every 4 milliseconds and I wish to transform them into the frequency domain. How would I apply the FT to a set of values without knowing any...

I have been playing with the FFT and graphs. The easiest example I could think of for a transform was the top hat function (ie 0,0,0,0,0...1,1,1...0,0,0,0,0). When I transform this from the time domain to the frequency domain, it returns a sinc function when I take the absolute value squared of...

my question is the following let be the Fourier transform \int_{-\infty}^{\infty}d^{4}p \frac{exp( ip*k)}{p^{2}+a^{2}} here p^{2}= p_{0}^{2}+p_{1}^{2}+p_{2}^{2}+p_{3}^{2} is the modulus of vector 'p' , here * means scalar product for the scalar product i can use the definition...

Hello, I am trying to do some self-studying in Byron & Fuller mathematical methods for classical and quantum physics. I have slightly ran aground on this one task of finding 3d Fourier transforms and I can't find the info in the book itself to free me. Google has neither been very fruitfull...

Does it make sense to take the Fourier transform of a function that blows up at some point? For example the Fourier transform of f(x)=1/x, which blows up at zero? Doesn't the integral: \int^{\infty}_{-\infty} \frac{dx}{x} e^{-ikx} not converge because of x=0? Yet for some reason analytical...

Hello. I understand that in the form of \int_{\mathbb R} f(x) \exp{2 \pi i tx} \mathrm d x the function f: \mathbb R \to \mathbb C: x \to \frac{1}{x} doesn't have a Fourier transform (because the function is not integrable). But in my analysis course, there is a theorem that states that in...

Hey guys, I have a quick question about Fourier transforms. I have been told that the Fourier transform of a function tells us the minimum components required to support that function and that a real pulse may have extra frequencies, but not too few frequencies. I don't understand why...

I am trying to compute the inverse Fourier transform numerically (using a DFT) for some complicated characteristic functions in order to compute their corresponding probability distribution functions. As a test case I thought I would invert the characteristic function for the simple exponential...

Hello, Thanks at first. If anyone can understand, then I would like to know how do I get to equation 4.15. Its a laplacian equation in which I want to apply the Fourier transform. Thanks again.

I need to deduce that \hat{\hat{f}}(x)=2\pif(-x) using the inversion formula for the Fourier transform, I was wondering if someone could explain why there's f(-x) because i just can't get started on this problem!

Does anyone know a good free library to do Fourier Transforms (FFT or DFT). I know FFTW but I'm having some problems with it. I want an alternative that do FFT in two dimensions with complex numbers. The libraries I have found doesn't fulfill this requirements. Thank you

Homework Statement Okay so i am applying a FT to an image of particles that are forming a lattice, and i need to find the average distance between the particles because its not a perfect lattice, I am getting an airy pattern and i believe that the distance to the first ring is the average...

The Fourier transform relates spacetime domain to momentum-energy (wave number - frequency) domain. For example, a generic function f(x, t) is transformed as given by photo I can't understant What does this theorem guarantee about the quantum systems?Hot to find the representation of...

Hi, Can anyone tell me if there is a convolution theorem for the Fourier transform of: \int^{t}_{0}f(t-\tau)g(\tau)d\tau I know the convolution theorem for the Fourier Transform of: \int^{\infty}_{-\infty}f(t-\tau)g(\tau)d\tau But I can't seem to find (or proove!) anything...

Find the Fourier transform of the following aperodic digital signal x[n] = 3 for -2<n<2 3. Not to surer where to start on this one any help would be great thanks

Homework Statement x(t) = t*exp(a)*exp(-a*t)*u(t-1) - exp(a)*exp(-a*t)*u(t-1) I need to find X(jw)... Homework Equations how to apply properties of Fourier transform to get an answer? Because i know that the only effective method for this.. The Attempt at a Solution For...

Hello! Can someone help me with this. Evaluate: the integral from zero to infinite of ((xcos(x)-sin(x))/x^3)cos(x/2)dx I think it has to do with Fouriers Transform but I am just stuck. Any help would be appreciated! Thank You

Hi, I have several sets of stochastic signals that oscillate about the x-axis over time. I would like to transform these signals into the frequency domain (make a periodogram) so that I can which signal has the most stable frequency. I was thinking about using taking the Fourier transform...

Got stuck in the second part, any help is appreciated, cheers.

Hi there, A quick question concerning the FFT. Let's say I explicitly know a 2D function \tilde{f}\left(\xi_1,\xi_2 \right) in the frequency domain. If I want to know the values of f\left(x_1,x_2 \right) in the time domain at some specific times, I can calculate \tilde{f} at N_jdiscrete...

Hi Could someone help me to calculate the Fourier transform of the following function: rect(x/d)exp(2ipia|x|)

Homework Statement Express the Fourier Transform of the following function ae^{2\pi iabx}f(ax-c) terms of the Fourier Transform of f . (Here a, b, c are positive constants.)Homework Equations Define the following operators acting on function f(x): T_{a}(f)(x)=f(x+a) M_{b}(f)(x)=e^{-2\pi...

Hi there! I need to calculate the Fourier transform of a continuous function in C++. To do this I need to use the Dft, but what is the relation between the Dft and the continuous Fourier transform? I mean, how can I get the continuous Fourier transform from the Dft?

I was wondering if this is correct: \phi(k-a)=\phi(k)-\phi(a) Where k=p/h (h bar that is) and a is some constant and \phi is the Fourier transform of a wave function (momentum function). I know that if I had some real formula for \phi I could just test this but the problem isn't like...

Homework Statement I have to find the Fourier transform of f(x)=\frac{\beta^2}{\beta^2+x^2} Homework Equations Fourier Transform is given by F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}f(x) dx The Attempt at a Solution I'm having trouble with the integration...

Homework Statement Let \phi (k) be the Fourier transform of the function \psi (x). Determine the Fourier transform of e^{iax} \psi (x) and discuss the physical interpretation of this result.Homework Equations (1) \tilde{f} (k) = \frac{1}{\sqrt{2 \pi}} \int{f (x) e^{-ikx} dx} (2) \psi...

[PLAIN]http://img716.imageshack.us/img716/3663/semttulont.png f(x) = 0 (|x| > 1) = x² (|x| < 1) I know that thing on integral is [F(x)]^2, but I have no clue what to do now.

Hi! I'm trying to understand how do i get the phase spectrum from a Fourier Transform. From this site http://sepwww.stanford.edu/public/docs/sep72/lin4/paper_html/node4.html#lin4_swhfactm this statement "The phase spectrum is usually calculated by taking the arctangent of the ratio...

Can someone recommend a good book or notes detailing the use of the Fourier transform with wave trains. Something short and sweet hopefully. thanks

I am learning about adv quantum and field theory and i have run across something unfamiliar mathematically. In several instances the author simpy expands the field or a wave function as a Fourier transform. that is they assume the field or wave function is simply the transform of two other...