linear systems - should this be viewed as an impulse response or step response

I'm trying to teach myself the relation between simple discrete ODE's and the impulse response-step response concept.

Getting back to the question: I don't expect anyone to read the whole thing but I'm confused early on which is a good thing.

Right at the beginning of the link, the author provides an example of a leaky integrator model in which the water facet is turned on and the water goes into the bucket and there is no hole in the bucket.

Given the leaky integrator model, the author then shows how, if there is no hole in the bucket, then by setting k = infinity in the model, one can model the water level for the bucket. He then provides

1. a plot of the input: the water going into the bucket and


2. a plot of the output: the water level of the bucket.

My confusion stems from the plot of the input. I think that the plot of the input, rather than being a spike as is shown at time t = 1, should instead be a step response where there are spikes all the way from zero to one ? so basically a step response ?

My thinking is that, if the water is turned out at time zero, then the spike for the input is at t = 0 and remains a spike until time t = 1 ?

The output plot seems fine to me.

If I am correct about the input plot being incorrect as is, then it kind of tells me that I understand the basic impulse response-step response concept. If I'm not, then, even though I loved the example, I'm still not quite getting it.

I asked the creator of the link in an email and he-she hasn't gotten back to me yet and I would like to understand this and don't honestly know if I'll get a response ( no pun intended ).

Answer

The plots that you're confused about are just drawn poorly. They don't show a spike at $t = 1$; instead, they are supposed to represent a short pulse covering the time period from $t=0$ to $t=1$. This is why you see the water levels in the corresponding output plots start increasing at $t=0$ and then start decaying after the water input stops at $t=1$ (except for the $k = \infty$ case of course).

This could be modeled as a linear combination of the system's step response; for this input $s(t)$, the output $y(t)$ is:

$$ y(t) = y_s(t) - y_s(t-1) $$

where $y_s(t)$ is the system's step response.

While the term impulse is thrown around in the lesson, they aren't impulses in the delta-function sense; instead, they are all just short (but finite) duration, finite-height rectangular pulses.