Convolution Operation
In its most general form, convolution is an operation on two functions of a real valued argument. To motivate the definition of convolution, we start with examples of two functions we might use. 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 $t$. Both $x$ and $t$ are real-valued, i.e., we can get a different reading from the laser sensor at any instant in time.Now suppose that our laser sensor is somewhat noisy. To obtain a less noisy estimate of the spaceship’s position, we would like to average together several measurements. Of course, more recent measurements are more relevant, so we will want this to be a weighted average that gives more weight to recent measurements. We can do this with a weighting function $w(a)$, where $a$ is the age of a measurement.If we apply such a weighted average operation at every moment, we obtain a new function providing a smoothed estimate ...