Fourier transform Definition and 950 Threads

Hi friends, I was looking for signals which will have themselves as the Fourier transform. Few of them are given below. \frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}\longrightarrow e^{-\frac{\omega^2}{2}} \sum_{k=-\infty}^{\infty}\delta(t-kT)\longrightarrow...

Homework Statement This is a question from a Physics Lab i recently completed. We used a function generator to provide a signal to a spectrum analyzer that performed a Fourier transform on the signal. In this case the signal was a square wave. When viewing the Fourier transform on a log...

Homework Statement We are asked to prove that if F(\omega ) is the Fourier transform of f(x) then prove that the inverse Fourier transform of e^{i\omega \beta}F(\omega) is f(x-\beta ) Homework Equations F(\omega)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(x)e^{i\omega x}dx...

Homework Statement Given f(x,y) = DeltaFunction(y - x*tan(theta)) a) Plot function. b) Take Fourier transform. c) Plot resulting transform. Homework Equations Delta function condition non-zero condition DeltaFunction(0) = Infinity Sifting property of delta functions The...

1. Homework Statement f(t) = (sin(2t))/t Homework Equations 3. The Attempt at a Solution I know that sin(t)/t has the Fourier transform pi(w). I'm just not sure how to apply that fact to this problem. Knowing that sin(t)/t --> pi(w), I reasoned that sin(2t)/t --> 2pi(2w). I'm...

I know how to describe a square wave with Fourier analysis, but what if I'm looking for a square wave with "peaks" that are longer than the "valleys." For example, from f(x)=1 {from 0 to 2}, f(x)=-1 {from 2 to 3}, f(x)=1 {from 3 to 5}, f(x)=-1 {from 5 to 6}... and so on in a periodic fashion...

fourier transform of the gaussian (1/\sqrt{2 pi \sigma}) e ^ (^{x^2/2\sigma^2}) now the Fourier of a gaussian is said to equal another gaussian as shown by equation (4) here: http://mathworld.wolfram.com/FourierTransform.html but when i also did it using equation (1) here...

I have a problem understanding the following: I should calculate the Fourier transform of a product of three functions: \mathcal{F} \left[ f(x_{1}) g(x_{2}) h(x_{1} + x_{2}) \right] = \int dx_{1} dx_{2} f(x_{1}) g(x_{2}) h(x_{1} + x_{2}) e^{i p x_{1} + i q x_{2}} okay, and this goes over...

I've been working on this problem for around three hours, and I'm getting nowhere... I think it may be that I don't have even the most basic grasp of the material to even get a decent start on the problem, but hopefully someone here will be able to help me... Homework Statement Calculate...

I feel a bit dumb, but could someone help me see this: G(s):= \int_{-\infty}^{\infty}f(-x)e^{-2\pi isx}dx = \int_{-\infty}^{\infty}f(u)e^{-2\pi i(-s)u}du = F(-s)

Fourier Transform NMR Physics... Work Shown... Please Help! Suppose you would like to detect the NMR signal from water within an area of the brain using a 2 Tesla Magnet. Intially, the magnetization from the protons in water has a magnitude (length) represented by Mo and oriented in a direction...

Homework Statement How would you solve the one-dimensional Poisson's equation: $\nabla ^2 \phi = \frac{\rho}{\epsilon_0}$ Using Fourier Transforms? $\phi (x) = \int ^{+\infty}_{-\infty} G(k) e^{-i k x} dk$ $G(k) = \int^{+\infty}_{-\infty} \phi (x) dx$ I've been trying to understand Fourier...

Homework Statement Evaluate INT(|X(t)|^2) dt using parsevals theorem where x(t) = (sin(t)cos(10t))/(pi*t) Homework Equations parsevals theorem: int(|f(t)|^2 dt = (1/2*pi)INT(|F(W)|^2 dw The Attempt at a Solution So I've tried several attempts at this problem and this is...

Homework Statement Find the Fourier transform of f(t) = 1 / (t^2 +1) Homework Equations F(w) = Integral f(t) * e^-jwt dt The Attempt at a Solution Hi guys, so I've been having problems trying to solve Fourier transforms. It seems that slapping the e^-jwt makes it hard to...

Human population verses time, Fourier transform of that "function". Let the human population of the Earth be plotted verses time. Assume that this function is almost continuous. What would a Fourier Time Transform of that function look like? Is there a "strong" exponential component of...

Why can't Fourier transform distinguish between a clockwise and a counter clockwise rotating vector? Why does it give peaks at both + and -. If we discard the -ve frequency and use only the +ve frequency, we can just use \int f(t)coswt instead of {f(t)(coswt-isinwt)}

By taking the Fourier transform of the fundamental Helmholtz equation (\nabla^2+k^2)G(\vec{x})=-\delta(\vec{x}), one finds that G(\vec{x})=\frac{e^{ikr}}{r} and \tilde{G}(\vec{\xi})=\frac{1}{k^2-\xi^2}. However, I can't figure out how to directly confirm that this Fourier...

Homework Statement Find the Fourier transform of f(-x) Homework Equations The Attempt at a Solution The way I tried to solve is Fourier series is a sum of even and odd functions. If f(-x) is even then, f(-x)=f(x) If f(-x) is odd then, f(-x)= -f(x) Sum of even and odd...

Hello, If I've a real signal, and I do a forward Fourier transformation I receive two parts: Real and Imaginary, what's the difference between them? i need to represent the transform in a software program, which part do i represent ?

I can easily find the Fourier transform of rect(x) to be 2sinc(2\pi k) using particular conventions (irrelevant here). But when I attempt to inverse Fourier transform the sinc function, I find I have to resort to contour integration and Cauchy principal values. This is troubling to me. It...

Homework Statement T(x,t) What is the Fourier transform of \frac{1}{x}\frac{\partial T}{\partial x} F(\frac{1}{x}\frac{\partial T}{\partial x}) = \int^{\infty}_{-\infty} \frac{1}{x}\frac{\partial T}{\partial x} e^{i \theta x}dx = ?? Homework...

Hi all, I'm having a bit trouble computing the Inverse Fourier Transform of the following: \frac{\alpha}{2\pi}\exp\left(\frac{1}{2} \alpha^2 C^2(K) \tau \omega^2\right) Here, C^2(K), \alpha and \tau can be assumed to be constant. Hence, we have an integral with respect to \omega. Who...

Homework Statement Find the Fourier transform of cos(100t) The Attempt at a Solution now I know just from looking at a Fourier transform table that if the equation is in the form cos(2Pi*k*t) then the answer is just 1/2(delta(f+k) + delta(f-k)) So in this case is the answer...

Hello, I have a question about the following problem: Given a wave equation \Psi(n,t) where t is the time, and n is an integer. What is the Fourier transform? I'm trying to reproduce this paper: One-dimensional Quantum Walks by Ambainis et al...

does anyone know how to calculate (in the sense of distribution) the Fourier transform of f(x)= ln|x| that is to obtain the integral \int_{-\infty}^{\infty} dx ln|x|exp(iux)

If X_1 and X_2 are independent random variables in \mathbb{R}^n, and \rho_{X_1} and \rho_{X_2} are their probability densities, then let \rho_{X_1+X_2} be the probability density of the random variable X_1+X_2. Is it true that \hat{\rho}_{X_1+X_2}(\xi) =...

Hey, I was wondering. Since the Fourier Series coefficients can just be represented in the form of a Fourier Transform, what is the point of ever finding the Fourier coefficients and not doing the transform?

Looking at how the Fourier transform comes about from the Fourier series when the period goes to infinity, they make the following step h \left( x \right) = \frac{2}{\pi}\int^{\infty}_{0} \left( dk \right) \left[ \int^{\infty}_{0} h \left( \varsigma \right) sin \left( k \varsigma \right) sin...

Fourier Transform of sin(10t) Hi all, Can some1 explain how to get the complex Fourier transform of sin(10t) I understand how to steal it off a Fourier transform table, but i have no idea how to do it manually. Any help anyone? Cheers

Homework Statement Hi, So I'm suppose to solve the following problem: \left.\frac{d^{2}u}{dt^{2}}-4\frac{d^{3}u}{dt dx^{2}}+3\frac{d^{4}u}{dx^{4}}=0 \left.u(x,0) = f(x) \left.\frac{du}{dt}(x,0) = g(x) Homework Equations The Attempt at a Solution First I use Fourier transform on...

Hello, I'm having an issue with a given problem. Homework Statement Using Parseval's Equation find the energy of the signal z(t)=\frac{4}{4+t^{2}} Homework Equations The book solves that problem by using the tent signal CTFT and duality property (i.e ). However that properly...

Hello everyone, Finding a good query to find an answer in www search engines isn't as easy as I thought. The subject is very narrow and sophisticated. When one performs a FFT, he/she/IT ;) gets the amplitude and phase spectra. The phase spectrum ranges from -PI to PI. Then, there are of course...

Hey guys, There is something I have known and applied for a long time, that the greater the length of an interferogram the greater the resolution of the resulting frequency-domain spectrum. But I've never fully understood why, I've always waved it off as something to do with the uncertainty...

I always tended to think that we ought to use formulae which explicitly remind us that position and momentum are on equal footing in quantum theory (even though this may not be ultimately true) and write my transforms symmetrically f(x)=(2\pi)^{-3/2}\int{F(p)e^{ix\cdot p}d^3p...

let S(R) be the schwartz space, M(R) be the set of moderately decreasing functions, F be the Fourier transform Suppose F:S(R)->S(R) is an isometry, ie is satisfies ||F(g)|| = ||g|| for every g in S(R). How is it possible that there exists a unique extension G: M(R)->M(R) which is an...

I have a practice question, which is to find the Fourier Transform of cos(2^pi^t) By substitution into the FT formula, and use of eulers formula,I have managed to reduced to: INTEGRALOF ( (cos(2*pi*t) * ( cos(2*pi*F*t) - j*sin(2*pi*F*t) ) ) By plotting the frequency graph of the...

I have noticed that this result is hinted at in several books, but am having trouble proving it: f, \hat{f} \in C_c^\infty(R^n) \Rightarrow f \equiv 0. in other words, if both f and its Fourier transform are smooth, compactly supported functions on n-dimensional euclidean space then f is...

Homework Statement obtaining the Fourier transform of the hat function h(x) = 1 if modulus of x</= 1 =0 otherwise Homework Equations F(k)=1/sqrt(2*PI) *integral from -1 to 1 of exp(ikx) The Attempt at a Solution I've carried through the transform and got an answer of...

hi, i would need some info on how can implement in MATLAB or FORTRAN (g90) the Numerical evaluation of the integral \int_{-\infty}^{\infty}dxf(x)exp(iux) and the evaluation of the ivnerse Fourier transform \int_{-\infty}^{\infty}du\frac{F(u))}{\int_{-\infty}^{\infty}g(x)exp(iux)dx}...

Hi all, First a warning: my Mathematica skills, and computery-type skills in general, are not very hot. My problem is thus: I have a function which I know: \hat{f}(k) I'd like mathematica to approximate the inverse Fourier transform of this function for me and plot the result. I've...

Homework Statement F(w) is the Fourier transform of f(t). Write down the equation for F(w) in terms of f(t). A rectangular pulse has height H and total length t0 in time. Show that as a function of w, the amplitude density is propertional to sinc(wt0/2). Homework Equations F(w) =...

Homework Statement Hey guys. I need to find the Fourier transform of sin, is this right? http://img156.imageshack.us/img156/5531/scan0004r.jpg I searched the internet but all I could find is the answer with the dirac delta and I don't need that. Thanks. Homework Equations...

Homework Statement The Attempt at a Solution I'm really at a loss on this question, which is why i have achieved so little on it so far. I think i more or less understand what a Fourier transform does (transpose amplitude vs time to amplitude vs frequency, ie the Fourier transform...

I am new to wavelets. I was reading about wavelets and Fourier transforms. So the main disadvantage of Fourier Transform is that you cannot use it on a non-uniform signal. Even though you use it you have to use a window and select your region of interest. If the window is small enough you can...

To find the frequency, Why do you need to consider the signal over long period of time? For example - if you look at a sine wave from 0-360 with two cycles, isn't it enough to get the frequency? I get the second part - you need a short time window to see sudden changes in frequency.

Homework Statement Find the discrete Fourier transform X[k] = DFTn {x[n]} of the following periodic sequences x[n] = x[n - N] with period N: (a) For n = 0 . . .N - 1 we have x[n] =\delta[n]. (b) For n = 0 . . .N - 1 we have x[n] = \mu[n] -\mu[n - K] with K < N. (c) x[n] = cos( (2*pi*M*n)/N...

Hi all, How is the Fourier transform applied to non-periodic functions, such as the Rect function? Any help would be greatly appreciated, Cheers, Jamie :)

Taking a Fourier-transform of a real signal, gives me a spectrum that has symmetry. If I take the FFT of a real signal, then throw away half of the spectrum, and then do an inverse transform I get a complex-signal. I go from r(t) to rc(t) where rc(t) is a complex-signal. Now this...

Homework Statement an exponentially decaying sinusoid is defined as f (t) = a exp (-t/towel) exp (i2(pie)vt) ; t greater than or equal to 0 0 ; t less than zero Homework Equations i have to...

Confusion with Fourier Transform and Step Function...clarification needed please :) I am required to find the Fourier Transform of (without integration): s(t) = 1 for 0 < t < 4; -t/2 for 4 < t < 6. I understand that for: s(t) = t for 0 < t < 1; 1 for t > 1 that this is the same as...