Convolution Definition and 364 Threads

Homework Statement [/B] Suppose we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{C} whose Fourier coefficients are known. Parseval's theorem tells us that: \sum_{n = -\infty}^{\infty}|\widehat{f(n)}|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|f(x)|^{2}dx, where...

I have a set of data which are probably convolutions of a Cauchy distribution with some other distribution. I am looking for some model for this other distribution so that a tractable analytic formula results. I know that the convolution Cauchy with Cauchy is again Cauchy, but I want the other...

Hi Two questions: 1) I saw this definition of expectation value: E[g(X)] = integral wrt x from -inf to inf of g(x)*f(x)*dx for some function g(x) of a random variable X and its density function f(x). Can this be used to derive why convolution gives the density of a random variable sum? 2) In...

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...

If f and g are continuous functions on the right half-line, [0,∞], then f✶g, the convolution of f and g, is defined by f✶g(x) = ∫[0,x] f(t)g(x-t)dt. I would like to know if f✶g is a differentiable function of x. If, for example, g(t) = 1 for t ≥ 0 then f✶g(x) = ∫[0,x]f(t)dt has a derivative...

Homework Statement Homework Equations f*f = integral from -inf to inf of f(t)f(x-t) dt = integral from -inf to inf of f(x-t)f(t)dt The Attempt at a Solution My question concerns some issues with part a. I break the problem up into two cases. Case 1: When rectangle is sliding into the...

Physically or mathematically, what does the Convolution integral compute?

I'm working on a software application that is difficult to describe. I have a fixed printer that prints a binary image (black or white). The ultimate goal is to print columns of 1" diameter circular dots, slightly interlaced for efficiency. The caveat is that the printing substrate is not...

Homework Statement Compute the convolution ##(f*h)(t)## where $$f(t) = \left\{\begin{matrix}1, \ \ for \ \ |t|<1 \\ 0, \ \ \ \ otherwise \end{matrix}\right.$$ and $$h(t) = \left\{\begin{matrix}2|t|-1, \ \ for \ \ |t|<1/2 \\ 0, \ \ \ \ otherwise \end{matrix}\right.$$ Homework Equations...

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)##...

I need some help to make sure my reasoning is correct. Bear with me please. I have a time distribution for a process and I want to construct a distribution for the time it takes to perform two processes. So I would define ##\tau = t + t## This would create a new distribution with is a...

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 Find ##x[n] \ast h[n]## when ##x[n] = 3 u[2-n]## and ##h[n] = 4\left( \frac{1}{2} \right)^{n+2}u[n+4]## where ##u[n-k]## is the unit step function. Homework Equations None really The Attempt at a Solution So I know this is probably simple but I am confused. So the...

So I have to solve an initial value problem involving the Laplace Transformation method. I have all the terms in Y(t) besides one term, I cannot figure how to change it from frequency domain back into time domain.Not sure how to type in Latex, so i uploaded a picture, using the whiteboard...

Homework Statement I am trying to figure out following problem. Let A ⊂ R. Then we can define the characteristic function: \begin{align} \chi_A : R → \{0, 1\}, x = \begin{cases} 1 & \text{if } x \in A \\ 0 & \text{else } \end{cases} \end{align} Let a be bigger than 0. I am...

Homework Statement Use the convolution property to obtain the inverse Laplace transform of F(s)= e-3s * ((3s+15)/s2+25) Homework EquationsThe Attempt at a Solution = (3*(s/s2+52) + 15*(1/s2+52)) *e-3s Using table of Laplace: 3*(s/s2+52) = 3*cos(5*t) = T7 15*(1/s2+52) = 15/5*sin(5*t) =T18 e-3s...

< Mentor Note -- Poster has been reminded that they need to show their work on schoolwork questions > Does anybody know how to solve this exercise? Derive the response of an undamped single-degree-offreedom system to force f(t)=F_0*cos(w_n*t)*u(t) with null initial conditions, w_n=(k/m)^1/2...

Hi there, I've recently been looking into applications of cross-correlation in the context of signal processing. I've noticed that the mathematical operations that yield the cross correlation between two signals is very similar to the operations in calculating the convolution of a signal and...

Some textbooks like (Numerical recipes the art of scientific computing) derive the DFT as a Riemann sum of the CTFT. With this in mind it would be natural then to approximate the identity ##y(t)=x*h=\mathcal{F}^{-1}\big\{XH\big\}## with the mathlab code y=ifft(fft(x).*fft(h)) which roughly...

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...

Homework Statement Find a function ##u## such that ##\int_{-\infty}^\infty u(x-y)e^{-|y|}dy=e^{-x^4}##. Homework Equations Not really sure how to approach this but here's a few of the formulas I tried to use. Fourier transform of convolution ##\mathscr{F} (f*g)(x) \to \hat f(\xi ) \hat g(\xi...

Hi everyone, I'm having some trouble understanding how convolutions are applied to systems. Suppose I'm given the impulse response of a system as ##g(t) = u(t)## and I'm also given the system input ##x(t) = u(t) - u(t-2)##. The system output should then be given by: $$y(t) = g(t) * x(t) =...

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 Use MATLAB to find the convolution between a) ##f(t) = u(t) -u(t -3)## and ##g(t) = u(t) -u(t -1)## Homework Equations The Attempt at a Solution t = -10: 0.1: 10; f = heavisde(t) -heaviside(t -3); g = heaviside(t) -heaviside(t -1); t = -20: 0.1: 20; c = conv(f, g); plot(t...

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 as...

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...

Hi, I have a general function u(x,y,z,t). Then, (1) what would be the space-time Fourier transform of G⊗(∂nu/∂tn) and (2) would the relation G⊗(∂nu/∂tn) = ∂n(G⊗u)/∂tn hold true? Here, note that the symbol ⊗ represents convolution and G is a function of (x,y,z) only (i.e. it does not depend on...

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...

We know that in the Fourier transform formula ,there are mainly two terms function f(t) and complex exponential term ( function). But I am confused that what should i call Fourier transform formula as a correlation or convolution formula? So can anybody help regarding it?

Hi all. Sorry about creating this new threat despite existing some others on the same topic. I have a problem in understanding a very specific step in the mentioned proof. Let me take the proof given in this link as our guide. My problem is just at the ending. When it says: "The region...

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...

Let ##G## be a group and let ##R## be the set of reals. Consider the set ## R(G) = \{ f : G \rightarrow R \, | f(a) \neq 0 ## for finitely many ## a \in G \} ##. For ## f, g \in R(G) ##, define ## (f+g)(a) = f(a) + g(a) ## and ## (f * g)(a) = \sum_{b \in G} f(b)g(b^{-1}a) ##. Prove that ##...

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...

Homework Statement Recall that we have defined the Gaussian ##f_s## by ##f_s (t) = \sqrt{s}e^{-st^2}## and shown that ##\hat{f_s}(\lambda) = \frac{1}{\sqrt{2}}e^{\frac{-\lambda^2}{4s}}##. Show that ##f_3 \ast f_6 (t) = \sqrt{\pi}f_{1/2}(t) = \sqrt{\pi/2}e^{-t^{2}/2}## The Attempt at a...

Hi, Can anyone please help me in convoluting atomic data. Along x-axis I have wavelenght and on Y axis I have scattered intensity. I want to convolute with FWHM of 20mm. Thanks Kartik

Hi, does anyone help me please to convolute atomic data. On x-axis I have wavelength and on Y axis scattered intensity data. Thanks Kartik

Homework Statement Mainly concerned with part (a). Here's the answer: I understand where the answers inside the bracket came from, but I don't understand how they got their bounds (-infinity to 3, 3 to 5, and 5 to infinity) Homework Equations x is the impulse function here so y(t) =...

Homework Statement I have found the following exercise just solved, but I haven't understood some steps... f(t)=\begin {cases}1, t\in [-1,1] \\ 0, t\not \in [-1,1]\end{cases} We have to do the convolution f*f=\int^{+\infty}_{-\infty}f(\tau)f(\tau)f(t- \tau) d\tau=\int^{+1}_{-1}f(t- \tau)...

Homework Statement The question is in the attached image . My problem starts when dealing with the limits of integration . I need an analytic procedure of solving such problems without involving graphical method . The equations of the graphs of h(t) and x(t) are easily derived . Homework...

Homework Statement According to my notes, if we have a sinusoidal aperture/transmission function of the form a(x)=1+sin(wx) and a 'top-hat' aperture function given by b(x)=1, -0.5d≤x≤0.5d, b(x)=0 otherwise, then their convolution should give a finite sinusoidal aperture function, i.e sinusoidal...

Hi! Can someone show me how to get the range of tau for case (4) in the picture? Because when I draw it, the overlapped range I get is from t-T/2 to T, but in the solution, it says t-T to T/2.

Homework Statement Have 2 iid random variables following the distribution f(x) = \frac{\lambda}{2}e^{-\lambda |x|}, x \in\mathbb{R} I'm asked to solve for E[X_1 + X_2 | X_1 < X_2] Homework EquationsThe Attempt at a Solution So what I'm trying to do is create a new random variable Z = X_1 +...

Hi. If you have a LTI system with an impulse response function ##h(t)## taking in an input ##x(t)##, why does its output ##y(t)## become ? I realize ##y(t)## not only depends on the instantaneous input ##x(t)##, but also on the lingering effects of previous inputs ##x(\tau)## with ##\tau <...

I'm trying to teach myself a bit of the content of signals and systems before the term starts using BP Lathi's "Linear Systems and Signals". I'm on convolutions now, and while I understand how to do them, I don't think I fully understand what exactly they are. So, for a given unit impulse...

Homework Statement With the Fourier transform of f(x) defined as F(k)=1/√(2π)∫-∞∞dxf(x)e-ikx and a convolution of g(x) and h(x) defined by f(x)=[g*h](x)=∫-∞∞h(x-y)g(y)dy show that the Fourier transform of f(x) equals √(2π)H(k)G(k). Homework Equations In problem The Attempt at a Solution So I...

So this is a probability question, and I am asked to find P(0.6 < Y <= 2.2) where Y = X1 + X2 X1~U(0,1) and X2~exp(2). X1 and X2 are both independent random variables. Our professor worked it out, but I do not understand his explanation. So he starts by using the convolution:$$f_y (t) =...

I'm asked to transform y(t) = x(t)*x(t) (where * is the convolution product) and x(t)= sinc(t)cos(2π10t) ( sinc(t)= sin(πt)/(πt) ).The attempt at a solution Clearly everything is simple if you know X(f), because y(t)=InverseFourier{ X(f)2 }. The problem is that I can't find X(f). By the way...

Homework Statement x(t) = cos(3πt) h(t) = e-2tu(t) Find y(t) = x(t) * h(t) (ie convolution) Homework Equations Y(s) = X(s)H(s) and then take inverse laplace tranform of Y(s) The Attempt at a Solution L(x(t)) = \frac{s}{s^2+9π^2} L(h(t)) = \frac{1}{s+2} I then try to find the partial...

Let f(x)=1+cos 2\pix and let fk=f*...*f (k-times convolution) what is the value of lim fk(1/2) when k tends to infinity Should use something about the Fourier Analysis, Could someone help me how to solve this problem?