General procedure for the partial fraction expansion
Since X(z)/z should be rational, it takes the form
If K < L then no adjustment is required. The partial fraction expansion is straightforward.
If K ≥ L then divide until the remainder polynomial in z has the degree of L-1 or less:
The 1st part of the above expression, (cK-LzK-L + ... +c1z + c0), will eventually contribute δ functions to output sequence some of themare time-advanced such that the resulting x(n) will be noncausal. The 2nd part - the proper fraction - is expanded into the partial fractions. Suppose that we have 1 repeated pole of order m, call it z1, and all the rest are distinct, call them zm+1, zm+2,..., zL. Then let
The coefficients Aj (m of them) and Bj (L - m of them) are found as follows:
In resulting x(n) the contribution of Aj terms is several exponentials multiplied by n, (n-1), (n-2), etc., and the contribution of the Bj terms is several complex exponentials.
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