Fourier transform Definition and 951 Threads

Homework Statement Use the Fourier transform to compute \int_{-\infty}^\infty \frac{(x^2+2)^2}{(x^4+4)^2}dx Homework Equations The Plancherel Theorem ##||f||^2=\frac{1}{2\pi}||\hat f ||^2## for all ##f \in L^2##. We also have a table with the Fourier transform of some function, the ones of...

I'm having a hard time understand this theorem in our book: The Plancherel Theorem The Fourier transform, defined originally on ##L^1\cap L^2## extends uniquely to a map from ##L^2## from ##L^2## to itself that satisfies ##\langle \hat f, \hat g \rangle = 2\pi \langle f,g\rangle## and ##||\hat...

Homework Statement The complex amplitudes of a monochromatic wave of wavelength ##\lambda## in the z=0 and z=d planes are f(x,y) and g(x,y), redprctively. Assume ##d=10^4 \lambda##, use harmonic analysis to determine g(x,y) in the following cases: (a) f(x,y)=1 ... (d) ##f(x,y)=cos^2(\pi y / 2...

Considering two functions of ##t##, ##f\left(t\right) = e^{3t}## and ##g\left(t\right) = e^{7t}##, which are to be convolved analytically will result to ##f\left(t\right) \ast g\left(t\right) = \frac{1}{4}\left(e^{7t} - e^{3t}\right)##. According to a Convolution Theorem, the convolution of two...

If I have a wave function given to me in momentum space, bounded by constants, and I have to find the wave function in position space, when taking the Fourier transform, what will be my bounds in position space?

I have been very briefly introduced to Fourier transformations but the topic was not explained especially well (or I just didn't understand it!) We were shown the graphs with equations below and then their Fourier transformation (RHS). I understand the one for cos(2pist) but NOT the sin(2pist)...

Hi I am trying to program excel to take the DFT of a signal, then bring it back to the time domain after a low pass filter. I have a code that can handle simple data for example t = [ 0, 1, 2, 3] y = [2, 3, -1, 4] So I think everything is great and so I plug in my real signal and things go off...

I have been studying Fourier transforms lately. Specifically, I have been studying the form of the formula that uses the square root of 2π in the definition. Now here is the problem: In some sources, I see the forward and inverse transforms defined as such: F(k) = [1/(√2π)] ∫∞-∞ f(x)eikx dx...

Hello everyone, The question that I have may not be fully relevant to the title, but I thought that could be the best point to start the main question! I'm working on 2-D data which are images. For some reason, I have converted my data to a 1-D vector, and then transformed them to the...

There are many waves and oscillations books out there that also include Fourier analysis but very few give the subject a thorough treatment, they just pass it in a few pages. If anybody has any sources(particularly books) that have Fourier analysis and particularly Fourier Transforms, I would...

With Dirac Comb is defined as follow: $$III(t)=\sum_{n=-\infty}^\infty\delta(t-nT)$$ Fourier Transform from t domain to frequency domain can be obtained by: $$F(f)=\int_{-\infty}^{\infty}f(t)\cdot e^{-i2\pi ft}dt$$ I wonder why directly apply the above equation does not work for the Dirac Comb...

I have the following problem:

i have read many of the answers and explanations about the similarities and differences between laplace and Fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of laplace. Some signals have Fourier but laplace is not defined , for instance cosine or sine...

Hi, I have a general function u(x,y,z,t). Then, (1) what would be the space-time Fourier transform of G⊗(∂nu/∂tn) and (2) would the relation G⊗(∂nu/∂tn) = ∂n(G⊗u)/∂tn hold true? Here, note that the symbol ⊗ represents convolution and G is a function of (x,y,z) only (i.e. it does not depend on...

Hi everyone, in the course of trying to solve a rather complicated statistics problem, I stumbled upon a few difficult integrals. The most difficult looks like: I(k,a,b,c) = \int_{-\infty}^{\infty} dx\, \frac{e^{i k x} e^{-\frac{x^2}{2}} x}{(a + 2 i x)(b+2 i x)(c+2 i x)} where a,b,c are...

Hi In making Fourier Transform of a function, why is it said that large space (r) corresponds to low wavenumber(k)?

A convolution can be expressed in terms of Fourier Transform as thus, ##\mathcal{F}\left\{f \ast g\right\} = \mathcal{F}\left\{f\right\} \cdot \mathcal{F}\left\{g\right\}##. Considering this equation: ##g\left(x, y\right) = h\left(x, y\right) \ast f\left(x, y\right)## Are these steps valid...

I need a good book on the Fourier transform, which I know almost noting about. Some online sources were suggesting Bracewell's "The Fourier Transform & Its Applications." I gave it shot, but it's competely unreadable. On page 1 he throws out an internal expression and says "There, that's the...

I constructed my code of the Angular Spectrum Method. However, as the distance between the object and the plane of interest increases, the diffraction pattern never disappears; there is still some sort of a diffraction pattern, and I am expecting that it disappears as distance increases. Here...

The Fourier Transform transforms a function of space into a function of frequency. Considering a function ##f\left(x, y\right)##, the Fourier Transform of such a function is ##\mathcal{F}\left\{f\left(x, y\right)\right\} = F\left(p, q\right)##, where ##p## and ##q## are the spatial frequencies...

This is my first time attempting to use the MKL's FFT functions, and I'm having trouble understanding them. The available examples (https://software.intel.com/en-us/node/522290) provide little insight into the reasons for doing things, and looking into the commands themselves doesn't provide...

Hi everyone! How do I transform Momentum to Position in spherical coordinates?Thinker301

I was told to do a Fourier transform of function by using a Filon's method. I have found the code but I don't know how to include any function to the subroutine. I would be grateful for any example of how to use this code. SUBROUTINE FILONC ( DT, DOM, NMAX, C, CHAT ) C...

Hello guys. I need an easy explanation regarding Laplace Transform and Fourier Transform. I know it is quite a mathematics question but I need an explanation in which it has something to do with engineering. I already search a bit about them but still cannot find and explanation that easy enough...

We know that in the Fourier transform formula ,there are mainly two terms function f(t) and complex exponential term ( function). But I am confused that what should i call Fourier transform formula as a correlation or convolution formula? So can anybody help regarding it?

(1) For a real function, g(x), the Fourier integral transform is defined by g(x) = \int_{0}^{\infty} A(\omega )cos(2\pi \omega x)d\omega - \int_{0}^{\infty} B(\omega )sin(2\pi \omega x)d\omega where A(\omega ) = 2 \int_{-\infty}^{\infty} g(x)cos(2\pi \omega x)dx and B(\omega ) = 2...

I am working on a project which is based on importance of phase only reconstruction of a signal obtained from fft. Now ,I have detected vehicles from the Video of Traffic on road taken using stationary camera ( Please download the 1.47 MB video for testing MATLAB Code by ( step1) click on the...

We know that Fourier series is used for periodic sinusoidal signals and Fourier transform is used for aperiodic sinusoidal signals. But i want to know that Is there any relation present between Fourier Series and Fourier transform ? Also,Can we derive mathematical formula of Fourier...

Are the results of the Angular Spectrum Method and the Fourier Transform of a Fresnel Diffraction be different, or the same? Given the same distance between the input and output plane, and the same aperture.

Homework Statement given a continuous-time signal g(t) . Its Fourier transform is G(f) ( see definition in picture / "i" is the imaginary number) . It is required to find the Fourier transform of the shifted-time-reversed signal g(a-t) where a is a real constant . That is , find the Fourier...

Hello! (Wave) I want to write a version of FastFourierTransform(fft) for the case that $N$ is a power of $3$, seperating the input-vector into $3$ subvectors, solving the problem recursively at them and combining the solutions of the subproblems. I have tried the following: We assume that...

Homework Statement f'(p) is the Fourier transform of f(x). Show that the Fourier transform of e^(ip0x)f(x) is f'(p - p0). (using f'(p) for transform) Homework Equations f(x) = 1/√(2pi) ∫e^(ipx) f'(p) dp (intergral from -∞ to ∞) f'(p) = 1/√(2pi) ∫e^(-ipx) f(x) dx (also from -∞ to ∞) The...

When using the Quantum Fourier transform to find the period of the function f(x)\equiv a^x\mod N why is it that the input register is 2n qubits in size and the output register is n qubits?

Homework Statement Hi all, I'm currently reviewing for a final and would like some help understanding a certain part of this particular problem: Determine the retarded Green's Function for the D'Alembertian operator ##D = \partial_s^2 - \Delta##, where ##\Delta \equiv \nabla \cdot \nabla## ...

I am fond of Fourier series & Fourier transform. In Fourier domain, we can come to know what frequency components are present and the contribution of each component in forming the given signal.But every approach has some advantages and disadvantages.Here, I want to know what are the limitations/...

I'm currently working through Nielsen & Chuang's section on the circuit design for implementing the QFT. I'm confused as to why swap gates are used in the model to swap the order of qubits. Heres what I'm looking at http://www.johnboccio.com/research/quantum/notes/QC10th.pdf page 247 figure...

I was wondering what the physical insight is of integrating a product of two functions. When we do that for a Fourier transform, we decompose a function into its constituent frequencies, and that's because the exponential with an imaginary x in the transform can be seen as a weighting function...

Suppose 3 1D Signals x(t), y1(t) and y2(t) are given as x(t)=sin(40*pi*t); y1(t)=.5*sin(40*pi*t) and y2(t)=x(t)+y1(t). Left side =Right side Here,values of x(t)and y1(t)i.e.(Right side) are given and we get y2(t)(Left side).The plots of x(t),y1(t) and y2(t) are plotted below...

I am beginer in image processing. Any signal whether it is 1D,2D or any multidimensional signal can be represented using combination of number of sine and cosine waves.Similerly any image can be termed as a sinusoidal function. Fourier series and transform plays vital role in image processing...

I am engineering student and studying signal processing. The term Fourier transform comes in the discussion several times. There are many transforms like Laplace transform,Z transform,Wavelet transform.But as per my view ,Fourier transform is mostly used compared to others in general. My...

Hi, I have a Fourier problem that i do not know if it is valid to do the calculations like this. The Fourier transform looks like this ## \hat{v}(x,\omega) = \frac{\hat{F}(\omega)}{4(EI)^{\frac{1}{4}}i \omega^{\frac{3}{2}}(\rho A)^{\frac{3}{4}}}\left[ e^{-i\left[\omega^2 \frac{\rho A}{EI}...

Homework Statement Derive the FT for a full-wave rectified sine wave, i.e., |sin(wt)| Homework Equations $$1/(√2π)\int_{a}^{b} |Sin[wt]| {e}^{-i w t}dt$$ The Attempt at a Solution I'm not entirely sure how to start doing this problem. What I tried doing was noticing that both of these...

I am studying Fourier Transform and it's inverse. We get phase and magnitude of a signal from it's Fourier transform and reconstruct it back from both together(magnitude of signal +phase of signal) My question is that is it possible to reconstruct given signal back using it's phase only or...

We know that Fourier Transform F(W) of function f(t) is summation from -infinity to +infinity product of f(t) and exp^{-j w t}Here, what does the exponential term mean?

Hi. Here is one example from my book. Calculate Fourier transform of signal: Here is solution: We can write x(n) as: , where x1(n) is u(n+N)-u(n-N-1). We can write: (we used that cos(n)=(1/2)*(exp(j*n)+exp(-j*n)). Using properties of Fourier transform of discrete signal: , Fourier...

Hi. I'm familiar with Fourier series but I have some hard times in learning Fourier transform. Why we use it? What's purpose of Fourier transform? Here is one signal and plot of Fourier transform of that signal: What this graph tells us? Thanks in advance.

Hi I am currently trying to find the water absorption lines by using a Michelson interferometer, as a detector I am using an ocean spectrometer. The data obtained is thus the spectrum's "received" by the spectrometer. Am I right to assume that in order to find the absorption peak/peaks I should...

Homework Statement derive the Fourier sine and cosine transforms of $$f(x) = e^{-cx}$$ by using $$e^{iax}=cos(ax)+isin(ax)$$ and computing the integral $$\int_0 ^{\infty} e^{-cx}e^{iax}dx$$.Homework EquationsThe Attempt at a Solution i'm completely clueless, all i did was evaluate what they...

Homework Statement $$u_{xx} + u_{yy} = 0 : x < 0, -\infty < y < \infty$$ Homework Equations We can use Fourier Transform, which is defined over some function ##f(x)## as ##F(f(x)) = 1/ 2\pi \int_{-\infty}^{\infty} f(x) \exp (i \omega x) dx##. The Attempt at a Solution Using the Fourier...

Textbook says, Fourier transform expresses a function in time domain as a function in frequency domain. Basically, Fourier transform gives two different expressions in terms of t domain and f domain but they represent the same signal. It also says Hilbert transform is a different type of...