Thus, the th ``sample'' of the impulse response is given by for . Each such ``sample'' is a matrix, in general.
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.
Zero-Input Response of State Space Models