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

Thus, the th sample of the impulse response is given by for . Each such sample generalizes to a matrix in the multi-input, multi-output (MIMO) case ( inputs, outputs); in such a case, the input signal in Eq.(1.8) is , which is a collection of input vectors , for , each having dimension , corresponding to an impulse signal being applied to the th system input.

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.

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