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
Transform the line element of special relativity from the usual (t, x, y, z) coordinates rectangular coordinates to new coordinates (t', x', y', z') related by
t =\left (\frac{c}{g} + \frac{x'}{c} \right )sinh\left (\frac{gt'}{c} \right)
x =\left (\frac{c}{g} +...
In relativistic limit the spin and the angular momentum are not of conservation because of spin-orbit interaction.Then the symmetry SU(2) is broken because vector spin does not commute with the interaction Hamintonian.The SO(3) symmetry is also broken for the same reason.So I do not understand...
Homework Statement
Laplace Transform teat
Homework Equations
The Attempt at a Solution
http://img521.imageshack.us/img521/6449/homeworkhelp.jpg
Not sure where I am going wrong. I feel like I did integration by parts incorrectly, because the anser I have boxed is not the...
Homework Statement
Let's say that I have a set called M, which is a subset of real numbers. Let's say that I want to create a sequence {s_1, s_2, ..., s_3} with all of the members of M, which would be ordered in an ascending (increasing) order. For example, if M = {4, 5, 1, 3, 2}, then s_0 = 1...
"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...
I am trying to find a real life inverse linear transform which can be used to motivate students. Does anyone have an example or two? I am looking for an example which will have a real impact. Thanks in advance.
I am trying to find a real life inverse linear transform which can be used to motivate students. Does anyone have an example or two? I am looking for an example which will have a real impact. Thanks in advance.
Homework Statement
Show that the normal coordinates for the equation we derived in Problem Set 4, Problem 2 are given by the discrete Fourier transform of an infinite series and the eigenfrequencies corresponding to each k.
http://www.ph.utexas.edu/~asimha/PHY315/Solutions-4.pdf
The...
Homework Statement
Semi-infnite bar (0 < x < ∞) with unit thermal conductivity is insulated at x = 0, and is constantly heated at x = 1 over such a narrow interval that the
heating may be represented by a delta function:
∂U/∂t = ∂2U/∂t2 + δ(x-1)
U(x; t) is the temperature. Assume...
Homework Statement
f(t) is a piecewise function:
{0 0<= t< 1
{t*exp(2t) t = >1Homework Equations
F(s)= L{f(t)}The Attempt at a Solution
F(s)= L{t*exp(2t)}
for this problem I just took the Laplace Transformer directly from the table which is: n!/ (s-a)^(n+1)
and after plucking in the...
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} )...
Hello,
I am confused as to how to transform nonlinear ODEs to linear ones by change of variables. Usually its pretty straight forward and I can do it, but this particular problem has me stumped and I don't know where to begin.
Homework Equations
Thank you guys!
I want to perform the inverse of
\frac s { [(s+α)^2-β^2](s^2+ω^2)}
I know the conventional way is
\frac s { [(s+α)^2-β^2](s^2+ω^2)}= \frac{As+B}{[(s+α)^2-β^2]}+\frac{Ds+E}{(s^2+ω^2)}
s= (As+B)(s^2+ω^2)+(Ds+E)[(s+α)^2-β^2]
\Rightarrow\; A+D=0,\; B+E+2\alpha...
Homework Statement
Find f(t) for:
2\int_{0}^{t}f'(u)sin(9(t-u)du +5cos(9t), t\geq0
The Attempt at a Solution
F(s)=2\frac{9(sF(s)-f(0))}{s^2+81}+\frac{5s}{s^2+81}
At this point i don't know what to do with f(0) since there are no initial conditions.
What do I do with it?
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...
now say we have cos^2(3t), how would you go about computing it with the 3t?
i can manage cos^2(t) but I'm not sure how to take it that one step further
in the link below is what I've managed so far.. SOLVEDI worked it out.
If anyone's interested in the future, Just start it off as cos^2(t)...
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...
I am trying to do laplace transform of y(t)=e^{-at}\cos (bt)\;. The answer should be:
\frac {s+a}{(s+a)^2+b^2}
But here is my work, I can't get rid of the -ve sign. I must be too blind to see the obvious, please help:
\overline{y(s)}=\int^∞_0 e^{-(s+a)t}\cos (bt) dt\;=\; \frac 1 b...
Homework Statement
d'^2 (y)/dt + 4 (dy/dt) + 4y = -7(e^(-3t)). Here I need to forced response of this differential equation using laplace transform technique.
Homework Equations
The Attempt at a Solution
I understand the part of converting each term to each laplace,
d^2y/dt to...
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...
Hi, my name is Tim, I am new to this forum. Here's the description:
I don't understand how they've done it. I realize I can plot this, but what is the logic.
As a graph, I drew ln(-x) first, then added 3, so move left by 3 but that was not correct
I can see they've added 3 first, move...
Homework Statement
Let u1 = [1 1]^T and u2 = [0 -1]^T. Find a 2 x 2 real matrix A so that the function T_A is a map from ℝ^2 to ℝ^2, given by multiplication by A,
T_A := Av,
sends T_A(u1) = v1 and T_A(u2) = v2 where v1 = [cosθ sinθ]^T and v2 = [-sinθ cosθ]^T. Explain/justify your...
Show the diffusion equation is invariant to a linear transformation in the temperature field
$$
\overline{T} = \alpha T + \beta
$$
Since $\overline{T} = \alpha T + \beta$, the partial derivatives are
\begin{alignat*}{3}
\overline{T}_t & = & \alpha T_t\\
\overline{T}_{xx} & = & \alpha T_{xx}...
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 ∞
Dear All,
I have encountered the following situation:
Given a node that is connected through n resistors to n ports.
One if the resistors is a negative resistance and equals to minus of all other resistors in parallel.
\frac{1}{R_n} =- \sum_{i=1}^{n-1} \frac{1}{R_i}
Using the well known...
Homework Statement
Hi
I have a set of five coupled ODE, and I would like to find a solution to the first variable X in the set (the rest I call Y, Z, V, W). The equations are of the form
\frac{dX}{dt} = A + BY - CX
This isn't homework, but something I been working with for some time. OK, so...
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...
maple issue -- inverse laplace transform equation from a basic series RLC circuit
Pretty simple for some of you I know, but I have a laplace transform equation from a basic series RLC circuit
((sV-v(0))/L+si(0))/(s^2+sR/L+1/LC) = I(s)
I want to take the inverse laplace of it, I am given all...
I want to use Matlab and Fourier transforms to solve linear systems and am attempting to implement a very simple linear system (with the idea of implimenting a much more complex one later) that I can't seem to get working. The system will take the derivative with respect to time of the input as...
I'm inverting this:
Y = s2 + 15s + 17 / [(s+1)(s2 + 13s - 4)]
I'm using PF expansion,
A/(s+1) + Bs + C/(s2 + 13s - 4), I however keep on getting wrong answers, seeing how Runge-Kutta and Taylor approximation disagrees with my final equation.
My final equation is...
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...
Homework Statement
Hi
I am trying to solve the following system of ODE's by Laplace transforming:
x' = 1 + 21y - 6x \\
y' = 6x-53y
with the initial conditions x(0)=y(0)=0. Laplace transforming gives me (X and Y denote the Laplace transformed variables)
sX = 1 + 21y-6x \\
sY = 6x-53y
From...
Wavelet transform for images...cannot interpret it correctly...
Hello Forum,
The discrete wavelet transform and the continuous wavelet transform are two different beasts.
When we take the DWT of an image, we get a bunch of subimages that are different in size. Each subimage is like a...
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...
H is a contravariant transformation matrix, M is a covariant transformation matrix, G is the metric tensor and G-1 is its inverse. Consider an oblique coordinates system with angle between the axes = α
I have G = 1/sin2α{(1 -cosα),(-cosα 1)} <- 2 x 2 matrix
I compute H = G*M where M =...
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...
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...
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...
Hi guys :) I'm looking to get a jump start on my uni course and have been going through some topics on my own before classes start in the Fall, I've reached the point of Laplace Transform and Fourier maths - and it's tough!
I have a small question on an inverse laplace transform equation in...