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
##A\dot{x} + By = 0##
##C\dot{y} + Dx = 0##Homework Equations
##\int u'v = uv - \int uv'##
The Attempt at a Solution
This is a system of linear DE:
##A\dot{x} + By = 0##
##C\dot{y} + Dx = 0##
Where the constants A-D are non-zero and x and y are functions of time.
This is...
In thermodynamics we use a variation of the Legandere Legendre transform to move from one description of the system to another ( depending on what is the control variable...), but I don't understand why choose to use the Legandere Legendre transform over writing x in terms of s=dy/dx and back...
A tad embarrassed to ask, but I've been going in circles for a while! Maybe i'll rubber duck myself out of it.
If f(t) = f(t+T) then we can find the Fourier transform of f(t) through a sequence of delta functions located at the harmonics of the fundamental frequency modulated by the Fourier...
I am using a Tascam recorder to record an environmental nuisance noise that is occurring in my home. I then use Virtins Multi Instrument Software, which includes an oscilloscope, band pass filter, and a spectrum analyser.
Noise source is probably machinery at a legal marijuana grow op. That...
As the Heaviside function is a function of t - 4, that means all other terms must also be functions of t - 4. The sine function is, but the exponential isn't. However with a little manipulation, we get
$\displaystyle \begin{align*} f\left( t\right) &= \mathrm{H}\,\left( t - 4 \right) \,\sin{...
I am a little familiar with Fourier Analysis, but I don't know where to get tools to get the answer to this question:
Consider a discrete signal A[0..N-1], consisting of N samples. Suppose we Fourier transform it and get a series of harmonics.
Now, consider the discrete signal A[1..N], that is...
It's not entirely obvious what to do with this question, as the denominator does not easily factorise. However, if we realize that $\displaystyle \begin{align*} s^4 + 40\,000 = \left( s^2 \right) ^2 + 200^2 \end{align*}$ it's possible to do a sneaky completion of the square...
$\displaystyle...
Dear all,
In my quantum mechanics book it is stated that the Fourier transform of the Coulomb potential
$$\frac{e^2}{4\pi\epsilon_0 r}$$
results in
$$\frac{e^2}{\epsilon_0 q^2}$$
Where ##r## is the distance between the electrons and ##q## is the difference in wave vectors.
What confuses me...
Hello.
I am reviewing the use of the Laplace Transform to do circuit analysis and I am slightly confused about the transform of a constant voltage source.
For example, let's say we have a constant voltage source V1(t) applied to a circuit for a long time - let's say it reaches steady state. We...
I know the result:
\widehat{\mathscr{H}(f)}(k)=-i\sgn (k)\hat{f}(k)
I want to use this to compute the Hilbert transform. I have written code for Fourier transform,inverse Fourier transform and that the Hilbert transform. My code is the following:
function y=ft(x,f,k)
n=length(k); %See now long...
Suppose that a parameter y= 123.
That parameter is somehow "perturbed" and its instantaneous value is:
y(t)= 123 +
sin(t - 50°) * 9 +
sin(t * 3 + 10°) * 3 +
sin(t * 20 + 60°) * 4
Suppose that I don't know the above formula, but I can calculate y(t) for any t. Hence I decide to use the...
Homework Statement
Homework Equations
The equation describing the balance will be f(n+1)=f(n)+R/12*Dm-Cf
with f(n)=initial deposit
R=Annual Rate
Dm=Each mouth Deposit 150
Cf= each month fee
The Attempt at a Solution
Can someone shed some lights on it?
Thanks[/B]
Homework Statement
FIGURE 4(a) represents a system to measure acceleration (i.e. an accelerometer). It shows a piezoelectric crystal that is connected to an amplifier and display via a length of coaxial cable2.A piezoelectric current is produced when the crystal is distorted by an applied...
I think this is probably a very basic question: why does the Fourier transform of a wavefunction describing position probabilities gives us a function describing momentum probabilities ?
Is there a fairly simple explanation for this ? What leads us to this relation ?
Hi guys, I have been trying to solve the Helmholtz equation with no luck at all; I'm following the procedure found in "Engineering Optics with MATLAB" by Poon and Kim, it goes something like this:
Homework Statement
Homework Equations
Let's start with Helmholtz eq. for the complex amplitude ##...
Hello everyone.
I'm trying to better understand structured illumination microscopy and in the literature, I keep coming across bits of text like this.
Source: http://www.optics.rochester.edu/workgroups/fienup/PUBLICATIONS/SAS_JOSAA09_PhShiftEstSupRes.pdf
From Fourier analysis, if I take the...
Hi members,
Laplace transform using differential equations.(see attached PDF file)
My question d/ds(s^2 y- s Y(0)-Y'(0).)...
Y(t)=sin(sqrt(t)) Y(o)=0
Now Y'= cos(sqrt(t)/2sqrt(t) Y'(0)=infinity
d/ds (Y'(0)=?? can it be treated as a constant or can we change limit and differentiation??I...
Homework Statement
find the laplace transform of (e^-s) / [ (s)(s-3) ]
since there's (e^-s) which can be found in L { f(t-a) H(t-a) } = (e^-(as)) F(s) , so , i found a = 1 , then i found F(s) = 1/ [ (s)(s-3) ] , formula :
i have attached the working below , is it correct ? btw , the...
In an image processing paper, it was explained that a 2D Gabor filter is constructed in the Fourier domain using the following formula:
$$ H(u,v)=H_R(u,v) + i \cdot H_I(u,v)$$
where HR(u,v) and HI(u,v) are the real and imaginary components, respectively. It also mentions that the real and...
Mod note: Moved from a Homework section
can i use the Laplace transform to solve a nonhomogeneous equation if
i have these Initial condition s(x) and s(-x)
I'm trying to find the distribution of a random variable ##T## supported on ##[t_1, t_2]## subject to ## \mathbb{E}[V(t', T)] = K, \forall t' \in [t_1, t_2]##. In integral form, this is : $$ \int_{t_1}^{t_2} V(t', t).f(t) \, dt = K,\forall t' \in [t_1, t_2], $$ which is just an exotic integral...
Homework Statement
Homework EquationsThe Attempt at a Solution
First write ##\phi(x,t)## as its transform
##\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \! e^{ipx} \widetilde{\phi}(p,t) \, \mathrm{d}p##
which I then plug into the PDE in the question to get...
Homework Statement
Using:
\mathcal{L}\big\{t^n\big\}=\frac{n!}{s^{n+1}}\text{for all s>0}
Give a formula for the Laplace transform of an arbitrary nth degree polynomial
p(t)=a_0+a_1t^1+a_2t^2+...+a_nt^n
Homework Equations
\mathcal{L}=\lim_{b\rightarrow\infty}\int_{0}^{b}p(t)e^{-st}dt
The...
Homework Statement
Given that r(t) = L^-1 (Inverse laplace) *H(S) and by making the link between the time-domain and frequency-domain responses of a network, explain in detail why the ideal “brick-wall” lowpass filter is not realisable in practice. [/B]Homework EquationsThe Attempt at...
Homework Statement Homework EquationsThe Attempt at a Solution
So we want sine in terms of the exponentials when we take the Fourier transform F(k)=\int_{-\infty}^{\infty}f(x)e^{-ikx}dx where f(x)=\sin(3\pi x/L). Let a=3pi/L. Then \sin(ax)=\frac{e^{iax}-e^{-iax}}{2i}.
(Is this correct?)
Then we...
Hi there,
I've recently been doing some studying into time-frequency analysis. I've covered some of the basic materials regarding the Short-Time Fourier Transform (STFT) along with the concepts of temporal and frequency resolution (along with the uncertainty principle of course).
I've now...
Homework Statement
Given the Laplace transform
$$F_L(s) = \frac{1}{(s+2)(s^2+4)},$$
by using the complex inversion formula compute the inverse Laplace transform, ##f(t),## for the following regions of convergence:
(i) ##Re(s)<-2;##
(ii) ##-2<Re(s)<0;##
(iii) ##Re(s)>0.##
Homework Equations...
Homework Statement
Show that the Hilbert transform of ##\frac{\sin(at)}{at}## is given by
$$\frac{\sin^2(at/2)}{at/2}.$$
Homework Equations
The analytic signal of a function is given by ##f_a(t) = 2 \int^\infty_0 F(\nu) \exp(j2 \pi \nu t) \ d\nu,## where ##F(\nu)## is the Fourier transform...
Homework Statement
Im trying to understand the Legendre transform from Lagrange to Hamiltonian but I don't get it. This pdf was good but when compared to wolfram alphas example they're slightly different even when accounting for variables. I think one of them is wrong. I trust wolfram over the...
In lectures, I have learned that F(k)= \int_{-\infty}^{\infty} e^{-ikx}f(x)dx where F(k) is the Fourier transform of f(x) and the inverse Fourier transform is f(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx}f(k)dk .
But on the same chapter in the lecture notes, there is an example solving...
Homework Statement
For a real, band-limited function ##m(t)## and ##\nu_v > \nu_m,## show that the Hilbert transform of
$$h(t) = m(t) cos(2\pi \nu_c t)$$
is
$$\hat{h}(t) = m(t) sin(2 \pi \nu_c t),$$
and therefore the envelope of ##h(t)## is ##|m(t)|.##
Homework Equations
Analytic signal...
The FT decomposes images into its individual frequency components
In its absolute crudest form, would the sum of these two images (R) give the L image?
Suppose a PDE for a function of that depends on position, ##\mathbf{x}## and time, ##t##, for example the wave equation $$\nabla^{2}u(\mathbf{x},t)=\frac{1}{v^{2}}\frac{\partial^{2}}{\partial t^{2}}u(\mathbf{x},t)$$ If I wanted to solve such an equation via a Fourier transform, can I Fourier...
Evening All
I have had a go at a laplace transform and got stuck.
$$\frac{d^2v}{dt^2}+\frac R L \d v t+\frac 1{LC}v=\frac 1{LC}V_0$$
$$R=12 \Omega, L=0.16H, C=10^{-4}F, V_0=6V, v(0)=0, v'(0)=0$$
so subbing these in i get
$$\mathscr L \left[ \frac {d^2v}{dt^2}+75\d v t+62500 v...
Quote: "The Fourier transform is a generalization of the complexFourier series in the limit as http://mathworld.wolfram.com/images/equations/FourierTransform/Inline1.gif. Replace the discrete http://mathworld.wolfram.com/images/equations/FourierTransform/Inline2.gif with the continuous while...
Fourier Transform of Piecewise linear spline wavelet is defined by 1-|t|, 0<t<1; 0, otherwise, is (sinc(w/2))^2. Can anyone please show me the steps. Thanks
Homework Statement
a(x)=f-Nd(x) + f-(N-1)d(x) +...+ f(N-1)d(x) + fNdHomework Equations
fd(x) = (1/a for |x-d| < a and 0 otherwise)
Fourier transform of function g(x) is g~(p) = 1/root(2pi) ∫ dx e-ipx g(x)
The Attempt at a Solution
[/B]
I have found the general Fourier transform for the...
Hi, I have a FORTRAN code with an array called Chi that I want to run an inverse FT on. I have defined two spaces X and K which each consist of 3 vectors running across my physical verse and inverse space.
My code (If it works??) is extremely slow and inefficient (see below). What is the best...
Homework Statement
For a periodic sawtooth function ##f_p (t) = t## of period ##T## defined over the interval ##[0, T]##, calculate the Fourier transform of a function made up of only a single period of ##f_p (t),## i.e.
$$f(t)=\left\{\begin{matrix}f_p (t) \ \ 0<t<T\\0 \ \ elsewhere...
This question is a little basic but.. how are signals stored in a Fourier Transform function f(t)?
In my PDE class we were always given a base function to put in terms of sin and cos. But when taking a bunch of samples, all I end up with is a table/array over some time T. How might I use this...
Homework Statement
A process can be represented by the first order equation
(4δy(t)/δt) + y(t) = 3u(t)
Assume the initial state is steady (y = 0 at t = –0).
(a) Determine the transfer function of this process in the s domain.
(b) If the input is a ramp change in u(t) = 4t, determine the...
Homework Statement
Find the Fourier transform of H(x-a)e^{-bx}, where H(x) is the Heaviside function.
Homework Equations
\mathcal{F}[f(t)]=\frac{1}{2 \pi} \int_{- \infty}^{\infty} f(t) \cdot e^{-i \omega t} dt
Convolution theory equations that might be relevant:
\mathcal{F}[f(t) \cdot...
I have a function f(x,y) which i have defined in this way:
a vector x and a vector y
meshgrid[x,y]
z= f(meshgrid[x,y]).
how do i do a 2-d Fourier transform of f(x,y)?
the transform must be done without using operations like fft, and must be done using summations written in the code.
I have a function of 2 variables [f(x,y)] where if there was an ellipse in the x-y plane, all values of the function are 1 inside the ellipse and 0 outside. I can plot this function as a surface in 3d where it looks like an elevated ellipse hovering over an elliptical hole in a sheet.
My...
Hello! (Wave)
I want to calculate the Fourier transform of $g(x)=|x|$.
I got so far that $\hat{g}(\omega)=2 \left[ \frac{x \sin{(x \omega)}}{\omega}\right]_{x=0}^{+\infty}-2 \int_0^{+\infty} \frac{\sin{(x \omega)}}{\omega} dx$
Is it right so far?
How can we calculate $\lim_{x \to +\infty}...
Homework Statement
A certain function ##v(x)## has Fourier transform ##V(\nu)##. The plot of the function is shown in the figure attached below.
For each of these functions give their Fourier transform in terms of ##V(\nu)##. And also state if the FT is Hermitian/anti-Hermitian, even/odd...