In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions (f and g) that produces a third function (
f
∗
g
{\displaystyle f*g}
) that expresses how the shape of one is modified by the other. The term convolution refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reversed and shifted. The integral is evaluated for all values of shift, producing the convolution function.
Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation (
f
⋆
g
{\displaystyle f\star g}
) only in that either f(x) or g(x) is reflected about the y-axis; thus it is a cross-correlation of f(x) and g(−x), or f(−x) and g(x). For complex-valued functions, the cross-correlation operator is the adjoint of the convolution operator.
Convolution has applications that include probability, statistics, acoustics, spectroscopy, signal processing and image processing, engineering, physics, computer vision and differential equations.The convolution can be defined for functions on Euclidean space and other groups. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. (See row 18 at DTFT § Properties.) A discrete convolution can be defined for functions on the set of integers.
Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, and in the design and implementation of finite impulse response filters in signal processing.Computing the inverse of the convolution operation is known as deconvolution.
Hello,
First of all, I checked several other threads mentioning duality, but could not find a satisfying answer, and I don't want to revive years old posts on the subject; if this is bad practice, please notify me (my apologies if that is the case).
For the past few days, I have had a lot of...
##x[n] = (\frac{1}{2})^{-2} u[n-4]##
##h[n] = 4^{n} u[2-n]##
So I plotted x[k] and h[n-k] in picture
but x[n] is 0 for n < 4, therefore ##y[n]## only has value for n >= 4. Therefore my sum is like that:
##y[n]=\sum_{k=4}^{\infty} 4^{n-k} (-\frac{1}{2})^k##
##y[n]=-4^{n}...
I'd like to plot the normalized convolution of a Gaussian with a Lorentzian (see the definitions in terms of full width half maximum (fwhm) in the attached image). Here is my attempt, but the print statements with np.trapz() do not return 1 in both cases, but rather ##\approx##0.2. I'd also like...
Hi all.
I would like to know about "binning window".
This paper I'm reading says like this.
Why do "convolving the data with the ##b(t)## before the sampling" and "binning into time bins with a width ##δt##" have the same meaning???
I know I'm addicted to post to PF 😅
But this forum is so...
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)...
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...
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).
Homework Equations
[/B]
I have...
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...
Homework Statement
Calculate the convolution of ##sinc(at)## and ##sinc(bt),## where ##a## and ##b## are positive real numbers and ##a>b.##
Homework Equations
Convolution integral
The Attempt at a Solution
The fact that ##a>b## tells us that the graph of ##sinc(at)## is ##a-b## times more...
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...
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 t)##...
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
$$(F * g) (t) = \int^t_{-\infty} (f * g) (\tau) d \tau$$
Homework Equations
The convolution of ##f## with ##g## is given by...
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?
Homework Equations...
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...
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...
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...
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.
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...