Convolution has the form
(f\star g)(t) = \int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau
However, I for my own purposes I have invented a similar but different type of "convolution" which has the form
(f\star g)(t) = \int_0^{\infty}f(\tau)g(t/\tau)d\tau
So instead of shifting the function g(t)...
1. Homework Statement
Hi there, I've been trying to gain some intuition on how the convolution sum works, but as I dig deeper I am realizing that there is an issue with my intuition of signals and systems, in particular the unit impulse response.
My issue is trying to understand how a unit...
1. Homework Statement
Hello everyone,
In the following problem I have to find the unknown impulse response g1(t) given the input and output signals, as shown below:
(the answer is already there, at the moment I am trying to understand how to get there).
2. Homework Equations
I have...
1. Homework Statement
Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ##
Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...
Given a convolution:
\begin{equation}
\begin{split}
g(x) * h(x) &\doteq \int_{-\infty}^{\infty} g(z) h(x-z) dz
\end{split}
\end{equation}
Do ##z## and ##x## have to be independent? For example, can one set ##x=z+y## such that:
\begin{equation}
\begin{split}
\int_{-\infty}^{\infty} g(z)...
Hi all! I'm new to Mathematica.
I have written a code for performing a convolution integral (as follows) but it seems to be giving out error messages:
My code is:
a[x_?NumericQ] := PDF[NormalDistribution[40, 2], x]
b[k_?NumericQ, x_?NumericQ] := 0.0026*Sin[1.27*k/x]^2
c[k_?NumericQ...
Prelude
Consider the convolution h(t) of two function f(t) and g(t):
$$h(t) = f(t) \ast g(t)=\int_0^t f(t-\tau) g(\tau) d \tau$$
then we know that by the properties of convolution
$$\frac{d h(t)}{d t} = \frac{d f(t)}{d t} \ast g(t) = f(t) \ast \frac{d g(t)}{d t}$$
Intermezzo
We also know that...
Dear "General Math" Community,
my goal is to calculate the following integral $$\mathcal{I} = \int_{-\infty }^{+\infty }\frac{f\left ( \mathbf{\vec{x}} \right )}{\left | \mathbf{\vec{c}}- \mathbf{\vec{x}} \right |}d^{3}x $$ in the particular case in which f\left ( \mathbf{\vec{x}} \right...
1. Homework Statement
Calculate the convolution of ##sinc(at)## and ##sinc(bt),## where ##a## and ##b## are positive real numbers and ##a>b.##
2. Homework Equations
Convolution integral
3. The Attempt at a Solution
The fact that ##a>b## tells us that the graph of ##sinc(at)## is ##a-b##...
1. Homework Statement
Differential equation: ##Ay''+By'+Cy=f(t)## with ##y_{0}=y'_{0}=0##
Write the solution as a convolution (##a \neq b##). Let ##f(t)= n## for ##t_{0} < t < t_{0}+\frac{1}{n}##. Find y and then let ##n \rightarrow \infty##.
Then solve the differential equation with...
1. Homework Statement
The impulse response ##h(t)## of a linear time invariant system is real valued. Where * denotes the complex conjugate, the transfer function satisfies:
$$H(- \nu) = H^* (- \nu)$$
Use this result to show that for such a system, given the input ##f(t)= \sin(2 \pi \nu...
1. Homework Statement
Either by using the properties of convolution or directly from the definition, show that:
If
$$F(t)=\int^t_{-\infty} f (\tau) d \tau$$
then
$$(F * g) (t) = \int^t_{-\infty} (f * g) (\tau) d \tau$$
2. Homework Equations
The convolution of ##f## with ##g## is given...
1. Homework Statement
y'' + 3y' + 2y = r(t),
r(t) = u(t - 1) - u(t - 2),
y(0) = y'(0) = 0.
I need to solve this by convolution, which I know is commutative. The problem is that my calculation gives (f * g) =/= (g * f). Could someone please tell me where my mistake is?
2. Homework...
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...
1. Homework Statement
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...
I greatly appreciate this chance to submit a query.
I have the following integral: $$\int_{1}^t 2sin(t-\tau)e^{-2(t-1)} d\tau$$
and it has been suggested to me that if I sketch the two constituent functions and multiply them, I can read the answer off the paper. So here are my sketches: go...
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...
1. Homework Statement
An electrical network has the unit-impulse response :h(t)=3t⋅e-4t .If a unit voltage step is applied to the network, use the convolution integral to work out the value of the output after 0.25 seconds.
2. Homework Equations
Convolution integral: y(t)=f(t)*h(t)
Unit step...
Hey guys so I have this Calc 3 project and the end is throwing me for a loop. I've done the encoding part, and i've coded the standard iterative methods, but I don't see how the two correlate so I can use the iterative methods to decode a "y stream" with the inputs specified...
Hi ☺️ i have to do a convolution with a periodic signal and a dirac impulse:
x(t)=sen(πt)(u(t)−u(t−2))
h(t)=u(t−1)−u(t−3)
The first is a periodic graph that intersect axis x in points 0 , 1 and 2 (ecc)
The se ing is a rectangle ( Dirac impulse ) that intersect AxiS x in points 1 and 3.
For...