What is Transform: Definition and 1000 Discussions
In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable
t
{\displaystyle t}
(often time) to a function of a complex variable
s
{\displaystyle s}
(complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms linear differential equations into algebraic equations and convolution into multiplication.For suitable functions f, the Laplace transform is the integral
Homework Statement
Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teacher is against using it..)
y'' + 2y' + 2y = 2 ; y(0)= y'(0) = 0
Homework Equations
Lf'' = ((s^2)*F) - s*f(0) - f'(0)
Lf' = sF - f(0)
Lf = F(s)
The...
Homework Statement
Use laplace transforms to find following initial value problem -- there is no credit for partial fractions. (i assume my teach is against using it..)
y'' - 4y' + 3y = 0 ; y(0)=2 y'(0) = 8
Homework Equations
Lf'' = ((s^2)*F) - s*f(0) - f'(0)
Lf' = sF -...
I read about how MRI works briefly, by flipping the water molecules using a magnetic field to the correct state then send the radio wave to these atoms and have it bounces back to be received by receiver coils and apply Fourier Transform to figure out the imaging. My question is, how does...
3d Fourier transform of function which has only radial dependence ##f(r)##. Many authors in that case define
\vec{k} \cdot \vec{r}=|\vec{k}||\vec{r}|\cos\theta
where ##\theta## is angle in spherical polar coordinates.
So
\frac{1}{(2\pi)^3}\int\int_{V}\int e^{-i \vec{k} \cdot...
I'm learning digital signal processing in my engineer class, but I'm more interested in apply these things into Astrophysics, so i know a little bit about for what is useful the Fourier Transform, so i thought why not use this in Analyzing the sun spectra! But what do you think!? Is it useful...
Homework Statement
So I know 1/(s-a)=e^(a1), but why is say, 2/((s+4)^2) equal to 2xe^-4x? Do I just simply add an X if the numeration is a constant other than 1?
Homework Statement
Find a function ##u## such that
##\int_{-\infty}^\infty u(x-y)e^{-|y|}dy=e^{-x^4}##.
Homework Equations
Not really sure how to approach this but here's a few of the formulas I tried to use.
Fourier transform of convolution
##\mathscr{F} (f*g)(x) \to \hat f(\xi ) \hat g(\xi...
In a book the Fourier transform is defined like this. Let g(t) be a nonperiodic deterministic signal... and then the integrals are presented.
So, I understand that the signal must be deterministic and not random. But why it has to be nonperiodic (aperiodic).
The sin function is periodic and we...
Homework Statement
Let
##f(t)=
\begin{cases}
\sin t , \; \; 0 \le t < \pi \\
0 , \; \; \; \; \; \text{else.}
\end{cases}##
Use Laplace transform to solve the initial value problem
##x'(t)+x(t)=f(t), \; \; \; x(0)=0.##
Homework Equations
Some useful Laplace transforms...
Homework Statement
Find the Fourier transform F(w) of the function f(x) = [e-2x (x>0), 0 (x ≤ 0)]. Plot approximate curves using CAS by replacing infinite limit with finite limit.
Homework Equations
F(w) = 1/√(2π)*∫ f(x)*e-iwxdx, with limits of integration (-∞,∞).
The Attempt at a Solution
I...
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...
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...
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...
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...
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...