An important tool for inverting the z transform and converting among digital filter implementation structures is the partial fraction expansion (PFE). The term ``partial fraction expansion'' refers to the expansion of a rational transfer function into a sum of first and/or second-order terms. The case of first-order terms is the simplest and most fundamental:
and . (The case is addressed in the next section.) The denominator coefficients are called the poles of the transfer function, and each numerator is called the residue of pole . Equation (6.7) is general only if the poles are distinct. (Repeated poles are addressed in §6.8.5 below.) Both the poles and their residues may be complex. The poles may be found by factoring the polynomial into first-order terms,7.4e.g., using the roots function in matlab. The residue corresponding to pole may be found analytically as
Note that in Eq. (6.8), there is always a pole-zero cancellation at . That is, the term is always cancelled by an identical term in the denominator of , which must exist because has a pole at . The residue is simply the coefficient of the one-pole term in the partial fraction expansion of at . The transfer function is , in the limit, as .