I'm currently studying what a convolution operation is via the book Deep Learning (Goodfellow et al., 2015) and had a question regarding a specific part where the book explains the convolution operation.
More specifically, on page 322 of Chapter 9: Convolution Networks, they attempt to explain the convolution operation with a specific example:
Suppose we are tracking the location of a spaceship with a laser sensor. Our laser sensor provides a single output $x(t)$, the position of the spaceship at time $t$.
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We want to make more recent measurements more relevant, and so we'll perform a weighted average on the measurements with a weighting function $w(a)$, where $a$ is the age of the measurement. Applying this new function at every moment, we obtain the following estimate:
$$ s(t) = \int x(a)w(t - a)da$$
More specifically, we want $w$ to be a valid PDF that is $0$ for all negative arguments, otherwise we would be looking into the future.
I vaguely grasp what the convolution equation means: As the authors said, at a specific time point $t$, we're taking all previous measurements (i.e. all $a$'s) and weighting them. However, what I don't understand is the last sentence that says something about looking into the future.
My interpretation is that a negative argument (i.e. cases where $t \lt a$) means a point in time where the age of the measurement is larger than the point in time we're at. How does this mean that we're "looking into the future?"
Furthermore, it seems that the range of $a$ is within $(-\infty, \infty)$. Does this even make sense? How would we have measurements from time periods from $-\infty$?