Fourier transform Definition and 950 Threads

Homework Statement I'm trying to derive the result on slide 1 of this link: http://www.physics.ucf.edu/~schellin/teaching/phz3113/lec13-3.pdf Unfortunately, I'm not sure how to integrate the Fourier transform when my u(x,t) function is undefined. Could someone help me get the...

The DCT of an even function is comprised of just cosine coefficients, correct? I'm playing around in MATLAB and I came up with a simple even function 1.0000 0.7500 0.5000 0.2500 0 0.2500 0.5000 0.7500 1.0000 0.7500 0.5000 0.2500 0 0 0 0...

I know the result: \widehat{H(f)}=i\textrm{sgn}\hspace{1mm}(k)\hat{f} I thought I could use fft, and ifft to compute the transform easily, is there a MATLAB command for sgn? Mat

Homework Statement See figure attached. Homework Equations The Attempt at a Solution See pdf attached for my attempt at the solution. I'm a little confused as to how to draw the phase spectrum for y(t). Would it simply be a line equation of, -\frac{\pi}{6000}f \pm...

Homework Statement Is the Fourier transform of a even/odd function also even/odd ? Homework Equations The Attempt at a Solution So far this result seems to be true. I can't find a confirmation however... Thanks ahead. Daniel.

Homework Statement Hi there! I'm just trying to figure out the Fourier transform of the hyperbolic secant function... I already know the outcome: 4\sum\ ((-1)^n*(1+2n))/(ω^2*(2n+1)^2) But sadly, I cannot figure out how to work round to it! :( maybe one of you could help me... Homework...

Why does a discrete Fourier transform seems to produce two peaks for a single sine wave? It seems to be the case that the spectrum ends halfway through the transform and then reappears as a mirror image; why is that? And what is the use of this mirror image? If I want to recover the frequency...

Homework Statement Find the cosine Fourier transform of the function f(t)=e-at Homework Equations The Attempt at a Solution F(w)=(2/π)0.5∫dt e-atcos(wt) The integral is from 0 to +∞ Using euler's formula I got the result F(w)=(2/π)0.5( eit(w-a)/i(w-a) - e-it(w+a)/i(w+a)...

Trying not to get too confused with this but I'm not clear about switching from coordinate representation to momentum representation and back by changing basis thru the Fourier transform. My concern is: why do we need to change basis? One would naively think that being in a Hilbert space where...

Homework Statement a) Find the Fourier transform of the function f(x) defined as: f(x) = 1-3|x| , |x|<2 and 0 for |x|>2 b) Find the values of the inverse Fourier transform of the function F(k) obtained in a) Homework Equations F(k) = \frac{1}{\sqrt{2π}}\int f(t) eikx dx f(x) =...

I've been trying to figure out why it's standard to use complex discrete Fourier transforms instead of just the real version. It's discussed a bit here. http://dsp.stackexchange.com/questions/1406/real-discrete-fourier-transform As far as I can tell there's a hypothetical efficiency...

In calculating some basic Fourier transform I seem stumble on the proble that I don't know how to take the limit in infinity of an exponentialfunction with imaginary exponent. In the attached example it just seems to give zero but I don't know what asserts this property. I would have thought...

Homework Statement A function f(x) has the following series expansion: ##f(x)=\sum _{n=0}^\infty \frac{c_n x^n}{n!}##. Write down the function ##g(y)=\sum _{n=0}^\infty c_n y^n## under a closed form in function of f(x). Homework Equations Not sure at all. The Attempt at a Solution...

Homework Statement I am looking at finding the Fourier transform of: f(t)=\exp \left[ \frac{-(t-m)^2}{2 \sigma^2}\right] Homework Equations \hat{f}(t)=\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} f(t) e^{-i \omega t} dt The Attempt at a Solution I did it a little differently that my...

I was going to post this in the learning material section but i didnt have access to it for some reason. but i guess i can post it here. its homework after all. so i have noticed that there is almost nothing learning material on fourer transform on the web. like how to transform a function to...

Homework Statement For a physics problem I must take the inverse Fourier transform of 2 functions. Namely I must compute the integral ##\frac{1}{\sqrt{2\pi}}\int_{-\infty} ^\infty [A\cos (ckt)+B\sin (ckt)]e^{ikx}dk##.Homework Equations Already given. i is the complex number. t is greater or...

Evaluating a "Fourier Transform" Integral Homework Statement Evaluate I = ∫[0,∞] e-ktw2 cos(wx) dw in the following way: Determine ∂I/∂x, then integrate by parts. Homework Equations Possibly? The Attempt at a Solution Since integral limits do not depend on x, the partial with respect...

Hi All, I'm just trying to practice graphing signals in frequency domain and I came across a stiuation I wasn't familiar with. If the exp() has a constant*t in it I'm not sure how to graph it, I remember that just cos it like a double sided exp(jwt) but with half the magnitude. I've attached a...

I have calculated a k-space function to be f(k) = \frac{1}{2k} I want to Fourier transform this to find f(x), I have found many different Fourier transform equations...can I use this one? f(x) = \frac{1}{\sqrt{2π}}\int\frac{1}{2k}e-ikxdk Limits fo integration -Infinity to Infinity...

Homework Statement Find the Fourier Transform of: f(t)=\frac{cos(\alpha t)}{t^2+\beta^2} Homework Equations F(\omega)=\frac{1}{2\pi}\int^{∞}_{-∞}\frac{cos(\alpha t)exp(i \omega t)}{t^2+\beta^2} The Attempt at a Solution I start with: cos(\alpha t)=\frac{exp(i \alpha t)+exp(-i...

Homework Statement Ok I know Fourier transform pair for u(t) is pi*del(w)+1/(j*w) Am I right to say the transform pair of u(t)-u(t-1) is [pi*del(w)+1/(j*w)]-[pi*del(w-1)+1/(j*(w-1)] If not what is it? thanks

Say you have some function that is periodic in a parameter k. The discrete Fourier transform from a sampling may be found in the usual way, giving the frequency spectrum in k. But what if I want to find the frequency spectrum in 1/k ? I'm not really sure what this is called, and so I've had a...

Homework Statement I'm trying to Solve for an impulse response h(t) Given the excitation signal x(t) and the output signal y(t) x(t) = 4rect(t/2) y(t) = 10[(1-e-(t+1))u(t+1) - (1-e-(t-1))u(t-1)] h(t) = ? y(t) = h(t)*x(t) --> '*' meaning convolution! I am unsure how to take the Fourier...

Is there a name for a transformation using the orthonormal base s_k(x)=\lceil \sin kx \rceil,\: c_k(x) = \lceil \cos kx \rceil \quad ? So basically a Fourier transform or Fourier series using periodic rectangles. What are the properties? Is there some kind of convolution theorem?

I am having trouble with this homework problem, I know how to get started but I just don't know how to carry through the completion of the problem: Question: Given the Fourier transform of an aperiodic signal X(ω) = 2*sin(3(ω-2π))/ω-2π (a)find its inverse Fourier transform x(t) using...

"Sketch the form of the Fourier transform" - is this right? Question ~ sketch the "form of the Fourier transform" for the function: f(k) = sin^2(ka/2) / (ka/2)^2So I'm thinking it will look like a cos [or sin] graph (shifted so that its 'above' *f(k)=0*) and that there will be some sort...

Homework Statement S(t) = S(0)e^{-i \pi f_{o}t} e^{-t/T^{*}_{2}}, 0 \leq t < \infty S(t) = 0, t < 0 Show that the spectrum G(f) corresponding to this signal is given by: G(f) = S(0) { \frac{T^{*}_{2}}{ 1 + [2 \pi (f- f_{o} )T^{*}_{2}]^{2}} + \frac{i2 \pi (f- f_{o} )...

Homework Statement I've been stuck on this for a while: Find the Fourier transform of f(t)=sin(\omega0t+\phi) Homework Equations I know that I have to use F(ω)=\intf(x)e^-iωt dt (between - and + infinity) to solve this The Attempt at a Solution So far I have...

I really need your help - i can't work out how to do a FFT in excel. The main problem is I don't have a constant sampling rate - I recorded the time and then the corresponding magnitude of the wave. I have followed everything oneline but I can't seem to get anything to work as I can't fill the...

Homework Statement Any wavepacket can be obtained by the superposition of an infinite number of plane waves using the so-called Fourier integral or Fourier transform f(x,t) = \frac{1}{\sqrt{2\pi}} _{-\infty}\int^\infty A(k)e^{i(kx-\omega t)} dk Find at t=0 the representation of the...

Show G(k)=\sqrt{2π}g1(k)g2(k) Given that G(k) is the Fourier transform of F(x), g1(k) is Fourier trans of f1(x), g2(k) is Fourier trans of f2(X) and F(x)=^{+∞}_{-∞}∫dyf1(y)f2(x-y) SO FAR G(k)=1/\sqrt{2π}^{+∞}_{-∞}∫F(x)e-ikxdx <-def'n of Fourier transform...

Hi, I am taking a random process class and I came across a problem that has stumped me. I believe I know the end result but I would like to know how it is solved. I have been out of college for a while and I am a little rusty with integration. Homework Statement What I need is to find out...

how to get the Fourier transform of (1+at^2)^-n ? n is a natural number such that (n>1) and a is any positive number. i.e. ∫((1+at^2)^-n)*exp(-jωt)dt; limits of integration goes from -∞ to ∞

Homework Statement Find the Fourier transform of x(t) = e-t sin(t), t >=0. We're barely 3 weeks into my signals course, and my professor has already introduced the Fourier transform. I barely understand what it means, but I just want to get through this problem set.Homework Equations I...

Hi all, I have a seemingly simple problem which is I'd like to efficiently evaluate the following sums: Y_k = \sum_{j=0}^{n-1} c_j e^{\frac{i j k \alpha}{n}} for k=0...n-1. Now if \alpha = 2\pi, then this reduces to a standard DFT and I can use a standard FFT library to compute the...

This is for an assignment, (not sure if its in the right section) but anyway I'm considering the system response to H(w) = 10/(jw + 10) when the input is x(t) = 2 + 2*cos(50*t + pi/2) so I know that Y(w) = X(w).H(w) but I'm not sure what to do about the '2 + ' in the input. I know that...

Homework Statement The argument of the kernel of the Fourier transform has a different sign for the forward and inverse transform. For a general function, show how the original function isn’t recovered upon inverse transformation if the sign of the argument is the same for both the forward and...

Hi, I was wondering if anyone might know what the analytic Fourier transform of a Super-Gaussian is? cheers

Since I lack the understand of real world applications of Fourier Transform in the real world I decided to buy a signals and systems book (Lathi) do some Fourier Transform problems and them do the same problem in Matlab. The question in the book wants me to find the Fourier Transform of...

Homework Statement Determine Fourier Transform of f(t) = cos^2 ω_p t ... for |t|<T also, for |t|>T, f(x) = 0, although i don't think you need to do anything with that. The Attempt at a Solution okay so: f(t) = cos^2 ω_p t ... for |t|<T becomes f(t) =...

Homework Statement I am using the time differentiation property to find the Fourier transform of the following function: Homework Equations f(t)=2r(t)-2r(t-1)-2u(t-2) The Attempt at a Solution f'(t)=2u(t)-2u(t-1)-2δ(t-2) f''(t)=2δ(t)-2δ(t-1)-?? Can somebody explain what the...

hi I know the Fourier transform of a lorentzian function is a lorentzian but i was wondering if the Fourier transform of the second derivation of a lorentzian function is also a second derivative of a lorentzian function Thanks

Hi all, I've been trying to solve the following I = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{x}{(x^2+y^2+d^2)^{\frac{5}{2}}} e^{-i(kx+\ell y)} \ dx \ dy where d,k,\ell are constants. I haven't been able to put this into a tractable analytic form and I figured I'd consult all...

Hey Physics Forums, Grading an assignment, the current topic is continuous Fourier Transforms. They're trying to prove the convenient property: \mathcal{F} \left[ \frac{d^n}{dx^n} f(x) \right] = (i \omega)^n \mathcal{F} \left[ f(x) \right] So there's a simple way to get it: Let f(x) be...

Homework Statement This is copied from a book: $$\eqalign{ & {\rm{Time Differentitation}} \cr & {\rm{Given that: }}F(\omega ) = F\left[ {f(t)} \right] \cr & F\left[ {f'(t)} \right] = jwF(\omega ) \cr & {\rm{Proof:}} \cr & f(t) = {F^{ - 1}}\left[ {F\left( \omega \right)}...

Hi, I am having a little trouble with the physical meaning of a Fourier transform. I will try to pose a concrete example. Mathematically, the Fourier transform of an exponential decay results in a Lorentzian function. Let's say I have a population that decays exponentially with time. Now, if...

Hi guys~ I have got a few things about some Fourier transform Q/A that i wanted to check...so here you go: 1) Find the Fourier sine and cosine transform of f(x)=x 0<x<3 ok, for the sine, i get -9/n∏ but i get zero for cosine part, is it wrong? and the second one: find the Fourier transform...

I originally asked this in the Calculus & Analysis forum. But perhaps this is better suited as a question in Abstract algebra. For the set of all Dirac delta functions that have a difference for an argument, we have the property that: \int_{ - \infty }^\infty {{\rm{\delta (x -...

Hi there, I have a little problem in wave optics: I have a wave function \psi_{ap} that depends on some geometric parameters, but that has no units itself (as one would expect). But unfortunately when I calculate the Fourier transform of this wave function the Fourier transform has a unit...

If you take the absolute value of the FFT output, does that give you the amplitude? I am asking because I have seen example where that is taken as the amplitude, and examples were the absolute value is multiplied by either SQRT2 or by 2 to get the magnitude. So my question is what is...