GeezerVison

We definitely need to update the documentation, but I wonder whether we shouldn't also consider what the right definition of the impulse response is for a discrete time system with non-unit sampling time. MATLAB apparently defines the impulse response for a discrete time system as being the response to an input that is 1 at the initial time and then 0 after that. This matches the description in the Wikipedia entry for impulse response.

I made this change a little while back to match what matlab does, which is a unit-area impulse for discrete-time systems: https://www.mathworks.com/help/control/ref/lti.impulse.html. This arrangement is probably what you want if you are viewing a discrete-time system as a discretized model of a continuous-time system.

Probably unit area is favored by engineering, unit magnitude by math.

Probably a subjective choice. Anybody here have further insight into such a choice?

As the sampling frequency period grows the response starts to get larger if impulse is kept at unity. Hence the same dynamical system sampled at different frequencies will give different results with the same unity impulse which is not correct. In a hand-waving argument we have to scale the energy of the impulse with frequency to have equivalent supply. Otherwise, we need to type math here which is not fun :)

You can test it yourself by generating a [1, 0, 0, 0, ...] array and supply to lsim for different sampled versions of the same system to see the effect I am attempting to describe above.

@ilayn's comment makes it clear why scaling by the sampling time is the right approach (and I was mistaken about what MATLAB does).