- #1
- 2,076
- 140
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:
$$y(t) = g(t) * x(t) = \int_0^t g(t - x) x(x) \space dx = \int_0^t u(t-x)u(x) \space dx - \int_0^t u(t-x)u(x-2) \space dx$$
The integrals should reduce to this I think:
$$\int_0^t u(t-x) \space dx - \int_2^t u(t-x) \space dx$$
How do I finish this integration? I thought about setting ##v = t - x##, but I am unsure.
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:
$$y(t) = g(t) * x(t) = \int_0^t g(t - x) x(x) \space dx = \int_0^t u(t-x)u(x) \space dx - \int_0^t u(t-x)u(x-2) \space dx$$
The integrals should reduce to this I think:
$$\int_0^t u(t-x) \space dx - \int_2^t u(t-x) \space dx$$
How do I finish this integration? I thought about setting ##v = t - x##, but I am unsure.