As derived in Book II [#!JOSFP!#, Appendix G], the impulse response of the state-space model can be summarized in the single-input, single-output (SISO) case as
In our force-driven-mass example, we have , , and . For a position output we have while for a velocity output we would set . Choosing simply feeds the whole state vector to the output, which allows us to look at both simultaneously:
Thus, when the input force is a unit pulse, which corresponds physically to imparting momentum at time 0 (because the time-integral of force is momentum and the physical area under a unit sample is the sampling interval ), we see that the velocity after time 0 is a constant , or , as expected from conservation of momentum. If the velocity is constant, then the position must grow linearly, as we see that it does: . The finite difference approximation to the time-derivative of now gives , for , which is consistent.