Convolution Definition and 364 Threads

Homework Statement Use convolution theorem to solve: \mathfrak{L} \left \{t\int_{0}^{t} \sin \tau d\tau \right \} Do not solve the integral. Homework Equations "Convolution Theorem" in textbook is stated as: \mathfrak{L}\left \{ f*g \right \}=F(s)G(s)f*g=\int_{0}^{t} \ f(\tau )g(t-\tau...

Homework Statement Calculate the convolution (\theta \ast \theta)(x) Homework Equations Convolution is defined as: (f \ast g)(x) \equiv \int_{-\infty}^{\infty} f(x - y) g(y) \ dy = \int_{-\infty}^{\infty} f(y) g(x-y) \ dy The Attempt at a Solution I know this is probably easy for...

If I have a convolution of two variables, say x * y, and I take the natural logarithm of this operation, ln(x*y), do the same properties of logarithms apply? So, does ln(x*y) = ln(x)+ln(y) ?

Hi guys, I am just having a bit of difficulty figuring out how to do convolution of impulses. Suppose I have a function consisting of impulses located at -2ω0, 0, and +2ω0 (in frequency domain) with some arbitrary amplitude A. I want to convolve this function with another function consisting of...

Homework Statement Prove the following: If \delta\in L_1(\mathbb{R}^n) and f\in L_p(\mathbb{R}^n) then the convolution \delta * f\in L_p(\mathbb{R}^n) with \lVert \delta * f\rVert_p\leq\lVert\delta\rVert_1\lVert f\rVert_p. Homework Equations We use the natural isometry (or isometric...

Hi, This is something that has appeared in a module, we've had a lab session in it but I am still not sure what it is. I don't understand the formulas given in lecture notes so I was hoping someone could explain it? Autocorrelation R1(τ) = ∫f(t)f(t+τ)dt = f V f Convolution C12(τ)=...

Hi, Suppose we have these two functions and their z-transforms are P(r,z)=\sum_{t=0}^{\infty}P(r,t)z^t and F(r,z)=\sum_{t=0}^{\infty}F(r,t)z^t. Now we are going to transform the following convolution of P and F: \sum_{t'\le{t}}F(r,t')P(0,t-t'). The result is said to be F(r,z)P(0,z). But I don't...

I solved majority of the question I just need to find the last joint density. Found the equations at part 3. Homework Statement Show P(X-Y=z ,Y=y) = P(X) = P(|Y|) I showed P(X) = P(|Y|) Homework EquationsThe Attempt at a Solution P(X=x,Y=y) = \frac{2*(2x-y)}{\sqrt{2πT^3σ^6}} *...

I have a question on the Convolution theorem for Fourier Transforms. The convolution theorem states that \mathscr{F}\{f(t) g(t)\}=\mathcal{F}\{f(t)g(t)\}=\mathcal{F}\{f(t)\}\ast \mathcal{F}\{g(t)\}-\mathscr{F}\{f(t)\} \ast \mathscr{F}\{g(t)\} \mathscr{F}\{f(t) \ast...

Hy. I have a problem about correlation depending whether it it observed as a measurement of linear fit of statistical data, and when observed as a relationship between two continuous functions. Is a result of correlation a coefficient (Pearson's product-moment coefficient) or a function...

Homework Statement Determine whether the assertions are true or false, explain. (a) If (f * g)(t) = f(t), then g(t) must be an impulse, d(t). (b) If the convolution of two functions f1(t) and f2(t) is identically zero, (f1 * f2)(t) = 0 then either f1(t) or f2(t) is identically zero...

Hello. I have a problem convolving two functions. I have attached a file with the problem in details, and will be very grateful if someone can provide me with a proper explanation. Thanks! :shy:

Homework Statement If a signal f1[n] begins in a moment N1 and ends in moment N2, and signal f2[n] begins in the moment M1, and ends in the moment M2, derive the formula which states in which moment begins and ends the signal f1[n]*f2[n] Homework Equations The Attempt at a Solution I...

Does anyone know how to take the partial derivative of a convolution integral where the derivative is taken with respect to one of the functions of the convolution integral? In the following example, the best I can come up with is: \frac{\partial}{\partial g(t)}\int...

Homework Statement Hi, I am wondering how to Laplace transform this expression f(t)=\int^{\tau}_{0} g(\tau)f'(t-\tau)d\tau or more precisely f(t)=\int^{\tau}_{0} sin(8\tau)f'(t-\tau)d\tau The f'(t-\tau) gets me confused. Homework Equations \int^{\tau}_{0}...

What is right definition? (f*g)(x)=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}f(x-\xi)g(\xi)d\xi or (f*g)(x)=\frac{1}{2\pi}\int^{\infty}_{-\infty}f(x-\xi)g(\xi)d\xior (f*g)(x)=\int^{\infty}_{-\infty}f(x-\xi)g(\xi)d\xi this is for me huge problem. For example f*\delta=f or f*2\pi\delta=f...

I'm struggling to find a function E(t) which is the energy inside a sphere with energy density \rho(t,r) where the radius r \equiv r(t) is itself a function of time. This E(t) = 8\pi \int_0^{r(t)} \rho(r,t) dr doesn't make sense, does it ? Is the thing I'm looking for some kind of...

Homework Statement h(t) = u(t) (the unit step function) x (t) = e-t The Attempt at a Solution There is only one interval where the two functions overlap, and that's from 0 to t. The integral from 0 to t of e-\tau d\tau = -e-t Doesn't look right to me... what am I doing wrong? EDIT: This is...

how would I show that y'(t) = x(t) * h'(t) and y'(t) = x'(t) * h(t) I know that in an LTI system y(t) = x(t) * h(t) = \int x(\tau) * h(t-\tau) from \infty to -\infty But how would I go about trying to prove the first two equations?

how would I show that y'(t) = x(t) * h'(t) and y'(t) = x'(t) * h(t) I know that in an LTI system y(t) = x(t) * h(t) = \int x(\tau) * h(t-\tau) from \infty to -\infty But how would I go about trying to prove the first two equations?

Homework Statement x(t) is input, h(t) is the impulse response, y(t) is output Find the system response to the input x(t) x(t): http://img10.imageshack.us/img10/5157/55570988.jpg h(t): http://img593.imageshack.us/img593/1079/52492104.jpg Homework Equations Now I know the...

Hi There The associative property of convolution is proved in literature for infinite interval. I want to prove the associative property of convolution for finite interval. I have explained the problem in the attached pdf file. Any help is appreciated. Regards Aman

Apparently, when convolving, for example: [δ(ω-π) - δ(ω+π)] * (δ(ω+50π)-δ(ω-50π)) the result is δ(ω+49π)-δ(ω-51π)-δ(ω+51π)+δ(ω-49π) where δ() is the Dirac delta function, * the convolution operator and ω the frequency variable How do we get to this? Can you help me on the intuition in...

Homework Statement Hello, I'm revising this summer for signals and systems and I came across this convolution cos(t)*u(t) Homework Equations having two signals x(t) and h(t), where x(t) is the input signal and h(t) the impulse response the output y(t) is given by y(t) = x(t)*h(t) =...

hey guys, i hope you can help. my task is to analyse data of raman spectroscopy. therefor i have to deconvolute it. that means the data must have been convoluted somewhere. is it true that the raw data which i receive is convoluted already? or is it common to convolute the data "active"...

Hi there. We know that Convolve[f,g,x,y] = f[y] if g = diracdelta. My question is, what should be g so that Convolve[f,g,x,y] = f[y1] where y1 is a parameter of the g function. I.e. Is there any function g such that, when convolved with another f, gives the evaluation of f on a given point?

I fail to understand a step made in this proof: http://en.wikipedia.org/wiki/Convolution_theorem" more specifically the last step where the integral is written as a product of 2 separate integrals (each equal to a Fourier transform): from: to: I'm quite rusty on my integration, but as far I...

finding the FT of x(t)=sin(πt) sin(50πt) : ( '*' is the convolution operator) its FT X(Ω)=(1/2π) F(sin(πt)) * F(sin(50πt)) = (1/2π) jπ(δ(Ω+π)-δ(Ω-π)) * (jπ (δ(Ω+π)-δ(Ω-π)) ) (a) from my professor's solution it next goes: = (π/2) (-δ(Ω+π)+δ(Ω-π)) * ((-δ(Ω+π)+δ(Ω-π)) )...

Hi there. I must calculate the convolution between these functions f(t)= e^{-t} H(t) g(t)=e^t H(t) H(t) the unit step Heaviside function. So I have to find: f \star g This is what I did: f \star...

Homework Statement Consider the signal x(t)=cos(4t)+cos(5t)+cos(6t), and the SLIT with impulse response: h(t)=\begin{cases} 1, & \mbox{if } |t|<T \\ 0, & \mbox{if } |t|>T \end{cases} For what value of T is the output of the system y(t) equal to Acos(4t)+Bcos(5t), when x(t) is the...

it might sound a bit hilarious.. some where i read about image processing where on the original image some operations were done (dealing with something related to convolution may be ) and say image A was obtained.. again another set of operations ( dealing with Fourier transform on the image...

Homework Statement I need to find a transfer function using the given functions which i have no problem with. But i don't know how to exactly define the Region Of Convergence of the resulting transfer function. I have h1[n] and h2[n] as to be convoluted to find the transfer function. Homework...

Homework Statement Convolve an arbitary function f(t) with comb(t) [a sum of delta functions that run from -infinity to infinity with spikes at t = nT]. Is the convolution an array of copies of f(t) or is it a set of discrete points such that f(t) is returned at every t = nT? Homework...

Homework Statement δ = dirac delta x[n] = δ[n] + 2δ[n-1] - δ[n-3] h[n] = 2δ[n+1] + 2δ[n-1] y[n] = x[n]*h[n] Homework Equations y[n] = x[n]h[n] = \sumh[k]x[n-k] The Attempt at a Solution I have graphed the x[-k] and h[n], the solution saids y[n] = h[-1]*x[n+1] +...

I am trying to understand wikipedia's definition of convolution: http://en.wikipedia.org/wiki/Convolution#Definition . I'm wondering why g(tau) is flipped in the definition.

Hello everyone, please help me to answer this question. Is this true that any LTI system can be characterized by either its impulse response or engenvalue?

Hi all, I am now doing revision for one of the statistics module. I am having some difficulty to proove the following: Given n iid Exponential distribution with rate parameter \mu, using convolution to show that the sum of them is Erlang distribution with density f(x) = \mu...

As a double major in physics an electrical engineering, I noticed that graphical convolution is used in both signal processing and quantum mechanics. In my signals course I couldn't help but notice that sometimes the professor would just convolved the function from straight integration, and...

The problem: Solve the integral equation \int\stackrel{\infty}{-\infty}exp(-abs(x-y))u(y)dy+u=f(x) for -\infty<x<\infty. The solutions say "Use the convolution theorem to find u(x)=f(x)-\frac{4}{3}\intf(t)exp(-3abs(x-t))dt." The Convolution Theorem in my book states "If the functions f(x)...

hi, i have trouble in understanding the concepts of the impulse response first of all, let's assume that we have a signal y[n] = x[n] which is time invariant and linear, hence if I understand correctly linear means that if for input a*x1[n] we have an output a*y1[n] b*x2[n] we...

Homework Statement The signal x(t) = u(t-1) - u(t-3) is the input to an LTI system with the impulse response h(t) = u(t-5) - u(t-8). the system is initially at rest. a) Compute the output y(t) of this system using convolution. b) Compute the output y(t) of this system using the Laplace...

Homework Statement i have this graph http://img858.imageshack.us/img858/1346/56954457.png and i need to find h-1[k] i don't understand, i know that the impulse response is the response for input -> δ[n], in this case it will be δ[n+1], but i don't understand how to calulate the response...

Homework Statement I have a convolution integral:H(\omega)=\int E_{L}(\omega -\omega_{T})E_{T}(\omega_{T})d\omega_{T} I would like calculate this integral at every \omega, but I have just discreet points, also first I calculated this with H=conv(E_{L},E_{T}), but I think so this is not...

Homework Statement Calculate f*f where f is the indicator function (-1,1) Homework Equations The convolution f*g of functions f and g is defined by: f*g(x)=\int^{\infty}_{-\infty} f(x-y)g(y)\ dy The Attempt at a Solution I haven't really done convolution before as I am teaching myself, so...

Hi, I have encountered with this: \delta[y-a]*\delta[y-b] where a and b are positive real numbers, and * denotes convolution. How to do this in both continuous and discrete cases? In Wikipedia, they say that: \int_{-\infty}^{\infty}\delta(\zeta-x)\delta(x-\eta)\,dx=\delta(\zeta-\eta) Can I...

Hi - I'm trying to work out the following convolution problem: I have the following integral: \int^{\infty}_{-\infty}p(x)U(x)e^{-i \omega x}dx Where p(x) is any real function which is always positive and U(x) is the step function Obviously this can easily be solved using the...

Homework Statement I have two equations: \frac{\partial}{\partial z}E_{L}\left(z,\omega \right) = i \frac{2 \mu d_{eff}(\omega+\omega_{0})^{2}}{k(\omega+\omega_{0})}\int E_{L}(z,\omega-\omega_{T})E_{T}(z,\omega_{T})d\omega_{T} \frac{\partial}{\partial z}E_{T}\left(z,\omega_{T} \right) = i...

Please help me to solve these problems in my homework. I should deliver it tomorrow.

for... p'_n = {1 Union Prime Numbers} M_n = n-th Mersenne Number (2^n - 1) T_n = n-th Triangular Number (n^2 + n)/2 x = {0,1,2,3,13} --> F_(0, 1/2, 3, 4, 7) for F_n = n-th Fibonacci Number Then... ((p'_x*p'_2x)*(M_x - (T_x - 1))) / ((T_(M_x) - T_(T_x - 1)) is in N EXPANSION ((1*1)*(0 +...

Im struggling to find proof for this suspicion I have; Given is a function f(t) and a normalised function h(t), and their convolution; f(t) * h(t) = g(t) Is it true that \int fdt = \int gdt ?