Convolution of the step function, three times

[nightshade123]

Homework Statement

u(t) * u(t) * u(t)

* indicates convolution

Homework Equations

i know u(t) * u(t) = t u(t)

The Attempt at a Solution

so (t u(t)) * u(t) = [tex]\int[/tex] [tex]\tau[/tex] d [tex]\tau[/tex] limits of integration are 0 to t
so the answer is 1/2 t^2 u(t)?

 

[Redbelly98]

Looks good to me

 

[Mene]

I would like to clarify that the convolution of a step function, u(t), with itself is not simply t*u(t), but rather a piecewise function. The convolution of u(t) with itself is given by the following equation:

u(t) * u(t) = { 0, t < 0
{ t, 0 <= t < 1
{ 1, t >= 1

Therefore, the convolution of u(t) * u(t) * u(t) would be:

u(t) * u(t) * u(t) = { 0, t < 0
{ t^2/2, 0 <= t < 1
{ t, 1 <= t < 2
{ 1, t >= 2

This can be derived by convolving u(t) * u(t) with u(t) using the convolution integral:

u(t) * u(t) * u(t) = \int_{-\infty}^\infty u(\tau) u(t-\tau) u(t-\tau) d\tau

= \int_0^t 0 d\tau + \int_t^1 \tau d\tau + \int_1^t 1 d\tau + \int_t^\infty 1 d\tau

= t^2/2, 0 <= t < 1
t - 1/2, 1 <= t < 2
1, t >= 2

= { 0, t < 0
{ t^2/2, 0 <= t < 1
{ t, 1 <= t < 2
{ 1, t >= 2

Therefore, the final answer is correct, but it is important to note that it is a piecewise function.