Transform Definition and 1000 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?

Hi, I was just looking over my textbook, and it mentions a ## \Delta##-y and y-## \Delta## transformation that is helpful for dealing with circuits in these configurations. The equations can be found here...

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...

Hello, This question might seem silly, but I've tried some approaches and none of them seemed to work. Here's my problem: I need some sort of transform that maps points from any quad to an rectangle. I will be using this on a computer graphics software, so you can think of this rectangle as my...

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...

Reading Griffiths, he states that the Lorentz Transform is useful for describing where an 'event' occurs in a different inertial frame. What about describing the motion of a particle in this moving frame if I know its motion in my frame? Really, I'm looking at pickup ions in the solar wind. A...

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...

Homework Statement Say how coordinate lines of the z plane transform when applied the following transformation ##e^z=\frac{a-w}{a+w}## Homework EquationsThe Attempt at a Solution This is exactly the way the problem is stated. It is a pretty weird transformation in my opinion and I'm guessing...

Homework Statement [/B] Having a little trouble solving this fractional inverse Laplace were the den. is a irreducible repeated factor 2. The attempt at a solution tryed at first with partial fractions but that didnt got me anywhere, i know i could use tables at the 2nd fraction i got as...

Homework Statement Prove that http://www4f.wolframalpha.com/Calculate/MSP/MSP26931c1531g07285beh7000062h7f6g1ggd95eea?MSPStoreType=image/gif&s=5&w=98.&h=38. =http://www4f.wolframalpha.com/Calculate/MSP/MSP6901c153574d0bdbh20000048829f0g4d1fi1d0?MSPStoreType=image/gif&s=5&w=69.&h=35. The...

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:

qi is the cartesian coordinate, and Qi is the Generalized coordinate, why the momentum under the two coordinates have this transformation way: pi=∑Pj(∂Qj/∂qj) pi and Pi are corresponding momentum under the two coordinate respectively.

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...

Can anybody direct me towards a proof of the second shifting theorem for Laplace transforms? I'm understanding how to use it but I can't figure out where it comes from. I've been learning from Boas, which doesn't offer much in way of proof for this theorem. Are there any good resources online...

Hi, I'm just curious because I know wifi uses digital FFT to send and receive signals. (I can't really remember why) But when I imagine a signal being sent its like a squiggily wave, so what method does the reciever use to approximate the instantanious values of the signal into a mathematical...

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

Homework Statement (didn't know how to make piecewise function so I took screenshot) Homework EquationsThe Attempt at a Solution My issue here with this problem is that I have absolutely no idea where to start... I have read through the textbook numerous times, and searched all over the...

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...

When I try to apply the force transformation (the 3 vector one) to the describe following situation, I find a result that I can't make sense of. Hopefully someone can tell me what I'm doing wrong. Suppose observers A and B are in inertial frames, and B travels in the +x direction relative to A...

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...

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...

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

It's been quite a few years but I recently watched a video about how every picture can be represented by a number of overlapping constructive and destructive peaks from a Fourier (transform or series? I don't remember which). I remember that Fourier series was for periodic and transform was for...

While reading about combinatorial mathematics, I came across the Stirling transform. https://en.wikipedia.org/wiki/Stirling_transform So then, if I want to find the Stirling transform of, for instance, ##(k-1)!##, I have to compute this (using the explicit formula of the Stirling number of the...

Suppose that there is a linear relation between discrete time (n) and frequency (f), then what is the relatian between x(n) and X(f) (X(f) is DFT transform of x(n))?

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...

Hello, I am trying to find Fourier Bessel Transform (i.e. Hankel transform of order zero) for Yukuwa potential of the form f(r) = - e1*e2*exp(-kappa*r)/(r) (e1, e2 and kappa are constants). I am using the discrete sine transform routine from FFTW ( with dst routine). For this potential...

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...

Hello everybody, Sorry to ask you something that may be easy for you but I'm stuck. For example I have 2 images (size 2056x2056). One image of reference and the other is the same rotated from -90degrees. Using a program with keypoints, it gives me a transform matrix : a=2.056884522e+03...

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.

Hi guys, I am trying to find the z-tranform of the following equation: x(n) =-2n u(-n-1) Using the Z-transform definition,summation and geometric series I am getting 1/(1-2z-1) But according to my lecturer the answer is suppose to be 0.5/(1-2z-1) I have tried simplification/factorization and...

The unilateral laplace transform integrates from 0 to infinity.Then how can i take laplace transform of exp(t)u(-t)?PS:I don't want to flip it and use time scaling property.

Hi, I have an idea which when tested looks like its clearly flawed. I am hoping someone can tell me where my procedure is flawed, or point me to some other theory that has already done something similar. The first two are the laplace transform. The third line is the Fourier Transform. The...

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...

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?

Do two inertial observers in relative motion agree on their relative velocity? Velocity is distance per unit time and they don’t agree on the distance or the elapsed time. If the apparent distance in the prime system is shorter and the elapsed time is longer, then it seems that the apparent...

Homework Statement Homework Equations Laplace Trasformations The Attempt at a Solution a. done b. f(t)= t -3*t*u(t-1) + 4*u(t-1) -3*u(t-2) -2*t*(t-2) c. 1/(s^2) - (3e^-s -2e^-2s)/(s^3) + (4e^-s -3e^-2s)/s d. 1/(s-1) * (1/(s^2) - (3e^-s -2e^-2s)/(s^3) + (4e^-s -3e^-2s)/s) These are the...

Homework Statement f(t)=tcos(4t)Homework Equations tnf(t)=(-1)n dF(s)/dsn The Attempt at a Solution I don't understand why this formula is giving me the oppiste sign of the answer. If I apply the formula I get (16-s2)/(s2+16)2 Because n=1 I need to multiply by a negative but this yields...

Hi all. Sorry about creating this new threat despite existing some others on the same topic. I have a problem in understanding a very specific step in the mentioned proof. Let me take the proof given in this link as our guide. My problem is just at the ending. When it says: "The region...

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...