I'm trying to understand the basics of a convolution and have troubles with the following task:
$$ x_1 = \cos(2 \pi t ) \cdot u(t) $$ $$ x_2 = u(t-0,5)$$
The task is to compute the convolution $$x_1 \ast x_2 $$
So I tried to compute the integral $$ \int_{-\infty}^{+\infty} = x_{1}(\tau)\cdot x_{2}(t-\tau) \, d\tau $$
Since the cosine starts at $0$ and the unit step is mirrored in the convolution, I start the integral from $0$ instead of $-\infty$.
$$ \int_0^{+\infty} = \cos(2\pi\tau)\cdot u(t-\tau-0.5) \, d\tau $$
Now this is where I'm a bit confused. What about the upper bound of the integral?
Does someone know how I can continue from here?