Thus, the impulse response of every proper LTI filter (with distinct
poles) can be interpreted as a linear combination of sampled
exponentials. Recall that a uniformly sampled exponential is the
same thing as a geometric sequence. Thus, is a linear
combination of geometric sequences. The term ratio of the
th geometric sequence is just the th pole, , and the
coefficient of the th sequence is just the th residue,
In the improper case, discussed in the next section, we
additionally obtain an FIR part in the z transform to be inverted:
The FIR part (a finite-order polynomial in ) is also easily
inverted by inspection.
The case of repeated poles is addressed in §6.8.5 below.