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
I was looking through some examples which applied the duality principle while studying for an up and coming exam when it hit me that the transform applied 4 times gives you back the same function.
So is there some theory that uses this? perhaps some sort of operator?
I thought it...
Homework Statement
Find the inverse Laplace transform of F(s)=(2s-3)/(s^2-4).
Homework Equations
I don't want to find the answer by looking at the Table.
F(s)=2s/(s^2-4)-3/(s^2-4)
The Attempt at a Solution
The answer is f(t)=2 cosh 2t - (3/2) sinh 2t.
Homework Statement
Compute the Fourier transform of a function of norm f(\norm{x}).
Homework Equations
\mathbb{F}{\frac{1}{1+\norm{x}}
The Attempt at a Solution
Attempt at using Cauchy theorem and the contour integral with the contour [(-R,R),(R,R+ip),(R+ip,-R+ip),(-R+ip,-R)] does...
Homework Statement
The problem is tough to type out correctly. Pasting problem statement image
http://postimg.org/image/a0r92a0wl/
http://postimg.org/image/a0r92a0wl/
The Attempt at a Solution
I just need to know how to proceed with the problem. Not the answer. This is the scan...
I attached the problem as a word document. I'm stuck trying to determine the laplace transform for t-tU(t-1). I know I'm supposed to work with 1/s^2(s+2) and solve for A, B,C. I got B=1/2, A=-1/4, and C=1/4 when 1=(As+B)(s+2)+Cs^2. The answer to the problem is
y= 1/4 + 1/2t +1/4 e^-2t -[1/4...
Why do we say that t'=t for Galilean transformation, when the low velocity limit of the Lorentz transformation is t'=t+vx/c2?
If x is really big, then doesn't time cease to be absolute, no matter how small v/c is?
Checked around a buch and could not find any help. But I needed help with:
Understanding that if I get the Inverse FT of K-space data, what is the scaling on the X-space (object space) resultant image/data i.e. for every tick on the axis, how do I know the spatial length?
More detailed...
what is the inverse laplace transform of (2s)(1/(s-2))?
could i use the identity ∫f(T)g(t-T)dT=F(s)G(s)?
i was hesitant so i figured i'd just ask before i continue..
Hello everyone:
I have some question using the FFT in MATLAB for data interpolating. I don't know what the relation between the normal Fourier series and the real, image number.
For example, given a set of measurement data, I can use the curve fitting toolbox to fit a curve.
The general...
Homework Statement
Hi, this is not a homework question per se, but something I'm wondering. Let C be a circulant n x n matrix, let x, b, be vectors such that
C x = b.
We would like to find a solution x. One way is to use the DFT: According to section 5, In Linear Equations, in the wikipedia...
I have been studying Fourier Optics and I have a basic conceptual question. I understand the mathematics of how to perform Fourier Transforms however the part of this topic I seem to have missed is why the action of a lens on light is the same as performing a Fourier Transform on the functional...
In the attachment that I added I highlighted the portion I am questioning.
I will define L[f(t)](s) to be the laplace transform of the function f(t).
f(t) = e^t
L[f(t)](s) = 1/(s-1). The laplace transform is defined for all values s≠1.
L[f(t)](2) = 1.
Question: "What do they mean by...
Homework Statement
y(t) solves the following IVP
y''(t) + 2y'(t) + 10y(t) = r(t)
y(0) = 2
y'(0) = 3
r(t) =
0 if t < 0
t if 0 ≤ t ≤ 1
0 if t > 1
Demonstrate that the laplace transform of y(t) is
Y(s) = \frac{2s+7}{s^{2}+2s+7} + \frac{e^{-s}}{s(s^{2}+2s+7)} +...
Hi all,
I'm a complete novice when it comes to describing images in frequency space and i understand that it is a way of representing images as being composed of a series of sinusoids. So a horizontal striped pattern with a single spatial frequency would have a magnitude image in frequency...
Hi!
I am new here, thought to join as I am trying to learn Relativity, in this case Special Relativity. I have solved a bunch of problems already but ...
The Lorentz Transform formulation I am dealing with is a 4x4 matrix. I understand the invariance of the spacetime interval and have...
Homework Statement
I have am doing a two dimensional discrete Fourier transform on an image (using MATLAB). What are the units associated with each pixel of the image in the frequency domain?
Homework Equations
The Attempt at a Solution
I thought that the frequency should be...
is it also possible to transform any these kinds summation to any product notation:
1. infinite - convergent
2. infinite - divergent
3. finite (but preserves the "description" of the sequence)
For example, I could describe the number 6, from the summation of i from i=0 until 3.
Could I...
Hey guys!
if anyone can help me I guess it is you! :)
I'm trying to find the Fourier Series demonstration to continuous and periodic functions.
I don't understand why people keep using X(jw) and X[e^jw] and even sometimes X(w) and X(f)
If anyone can help me I'm really not understanding that...
Homework Statement
Find the inverse Fourier transform of F(jω) = cos(4ω + pi/3)Homework Equations
δ(t) <--> 1
δ(t - to) <--> exp(-j*ωo*t)
cos(x) = 1/2 (exp(jx) + exp(-jx))The Attempt at a Solution
So first I turned the given equation into its complex form using Euler's Formula.
F(jω) = 1/2...
can you use Fourier transform to find a moving average on a data set?
so, you do a Fourier transform on your one dimensional data set.
next remove high order harmonics from FT result.
do reverse Fourier transform on new FT result.
And, vola! smoothed out data set.
Also just working on another question, especially stuck with the last part.
It's basically definitions.
This is what I've got so far, correct me if I'm wrong.
a) k components, k components.
b) R^n to R^n
c) R^rank(T)
d)R^nullity(T)
e) Completely unsure (need help with this)
I have posted a question on here before regarding the generation of a number sequence. I followed up that question with an answer. However, as I have developed my code more I need to use an equation instead of a lookup table.
Note: I'm using MATLAB
Given a matrix A of size r x c where r >=...
Homework Statement
Find the following integral:
Homework Equations
\int \frac{e^{x}}{\sqrt{(1+e^{2x})(1-e^{4x})}}dx
The Attempt at a Solution
I changed the integral to: \int \frac{e^{x}}{(1+e^{2x})\sqrt{(1-e^{2x})}}dx
The let u=e^x
The integral becomes: \int...
Hi,
I was wondering what would the Fourier transform of a signal like below give:
s(t) = sin(2πt*10) ; t in [0s,5s]
= sin(2πt*20) ; t in [5s,10s]
I certainly did not expect it to give me 2 sharp peaks at frequencies 10Hz and 20Hz - because I understand that the addition of...
When applying Kirchhoff's transformation to heat conduction PDE with temperature dependent thermophysical properties (k,ρ , Cp) , one obtains a transformed energy variable
u=∫Cp(τ) dτ and a term for a thermal diffusivity (α=k/ρ*Cp), thus reducing the nonlinerarity of the equation. When...
Function f(t)=\frac{t-i}{t+i} for t\in \mathbb{R} maps real ax into complex circle. Show that for any hermitian operator H operator U:=(H-iI)(H+iI)^{-1} is unitary (where H+iI is reversible)
If I understand correctly U is unitary when U=U^{T} right?
So I tried to show that U is unitary like...
Homework Statement
(6-t)heaviside(t-2)
This is just one term of the real problem I'm working, but it will serve to help me figure this out.
Homework Equations
The Attempt at a Solution
http://www.wolframalpha.com/input/?i=laplace+transform+%7B%286-t%29heaviside%28t-2%29%7D...
Homework Statement
Hi, for a project for school, I need to implement the Discrete Haar Wavelet Transform to compress an audio signal. This would be fine and dandy, but I do not really understand how to use the the DHWT. Could anyone direct me towards some resources that would be very helpful...
Homework Statement
Use the Laplace transform approach to find the renewal function for a renewal process with interrenewal p.d.f. as follows:
g(x) = (c^2)xe^(-cx) , x > 0
The Attempt at a Solution
M*(s) = G*(s)/(1-G*(s)) where M*(s) and G*(s) denote laplace transforms
I have that G*(s) =...
Google seems to provide not much information on this. In essence, I am asking about the eigenfunctions of the Laplace transform when λ=1? Anyone have any insights on this rather unusual problem?
BiP
Homework Statement
Evaluate the laplace transform of {t2e7tsinh(3t)}
Homework Equations
Laplace transform of {tnf(t)}=(-1)ndn/ds2 * F(s)
The Attempt at a Solution
I've replaced it with (-1)2d2L{e7tsinh(3t)}
I'm not sure how to proceed, though, as I don't really see how to take...
Hallo,
I really don't understand Fourier transform.
Do somebody know a good book for beginners?
Something like Fourier transform for dummies or so?
I need it just for physics.
So it don't have to be to mathematical. ^^
THX
Fourier Transform on the "connected part" of QFT transition prob.
Homework Statement
Calculate ⟨0|T[ϕ(x₁)ϕ(x₂)ϕ(x₃)ϕ(x₄)]|0⟩ up to order λ from the generating functional Z[J] of λϕ⁴-theory.
Using the connected part, derive the T-matrixelement for the reaction a(p₁) + a(p₂) → a(p₃) +...
Hey all,
Learning the Laplace transform and I get the point that it is a transformation but I would like to know what are some of the merits of the Laplace transform or more general why perform transformations in the first place. Any examples would be helpful.
Homework Statement
In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that:
F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv)
where F_{l,v}(g(l)) is the Fourier transform of g(l) and...
Hello,
Consider I have a linear time-invariant (LTI) system, with ##x(t)##, ##y(t)##, and ##h(t)##, as input, output, and impulse response functions, respectively.
I have two choices to write the convolution integral to get ##y(t)##:
$$ 1)\ \ \ y(t) = \int_{0}^{t} h(t-t')x(t')dt' $$
and...
Homework Statement
Homework Equations
The Attempt at a Solution
First, I found all the impedance values.
(starting from the left)
Z1 = 1 / (j * 16E5 * 15E-9) = -j41.67
Z2 = 80
Z3 = 30
Z4 = j * 16E5 * 60E-6 = j96
I then combined Z1 and Z2.
Z1 || Z2 = (-j41.67 * 80) /...
I've been wondering whether the Laplace transform is injective. Suppose I have that
\int^{∞}_{0}e^{-st}f(t)dt = \int^{∞}_{0}e^{-st}g(t)dt for all s for which both integrals converge. Then is it true that f(t) = g(t) ? If so, any hints on how I might prove it?
Thanks!
BiP
1. The limit as b approaches infinity always shows up as undefined on my calc so I don't know what to put for that section of the work.
2. What pat of the work is supposed to need L'hopital's rule? The integration?
Homework Statement
Find the La Place transform of cos(x)*(u(x-\pi))
Homework Equations
L{u(t-a)}(s)=(e^(-as))/s
The Attempt at a Solution
I don't think I can just multiply this by the La Place transform of cos (x), which is s/(s^2) ?
Homework Statement
Find the inverse transform of the function
F(s) = log\frac{s-2}{s+2}
Homework Equations
L(\frac{f(t)}{t}) = \int^{∞}_{s}F(x)dx
f(t) = tL^{-1}(\int^{∞}_{s}F(x)dx)
The Attempt at a Solution
I missed the lecture on this and while I was able to figure out...
Homework Statement
I want to eliminate spurious peaks of Hilbert transform for finding Glottal closure in LP residual. I have 4 step :
Homework Equations
1-down-sample.
2-Hilbert Transform.
3-Identify Peaks in Hilbert Transform.
4-consider this hypothesis that time gap between two...
Homework Statement
L-1{\frac{s}{s^2+4s+5}}
Homework Equations
\frac{s-a}{(s-a)^2+k^2}
\frac{k}{(s-a)^2+k^2}
The Attempt at a Solution
I completed the square for the denominator and got:
L-1{\frac{s}{(s+2)^2+1}}
(a= -2, k=1)
But how do I get rid of the s in the numerator? Or do I have...
Homework Statement
Find the Fourier transform of (1/p)e^{[(-pi*x^2)/p^2]} for any p > 0
Homework Equations
The Fourier transform of e^{-pi*x^2} is e^{-pi*u^2}.
The scaling property is given to be f(px) ----> (1/p)f(u/p)
The Attempt at a Solution
Using the information above, I got...
Find the DTFT of:
h[n]=(-1)^{n}\frac{sin(\frac{\pi}{2}n}{sin(\pi n}
useful properties:
x[n]y[n] --> X[Ω]*Y[Ω]
\frac{sin(\frac{\pi}{2}n}{sin(\pi n} --> rect[\frac{2Ω}{\pi}
I have no clue how to deal with the (-1)[itex]^{n}[\itex] the DTFT of that doesn't converge. . .
any help...