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<math> = \sum_{poles a_i ( X(Z) Z ^ (n-1))} Residue ( X(Z) Z ^ (n-1)) \ </math> | <math> = \sum_{poles a_i ( X(Z) Z ^ (n-1))} Residue ( X(Z) Z ^ (n-1)) \ </math> | ||
<math> = \sum_{poles a_i ( X(Z) Z ^ (n-1))} Coefficient of degree (-1) term on the power series expansion of ( X(Z) Z ^ (n-1)) about a_i \ </math> | <math> = \sum_{poles a_i ( X(Z) Z ^ (n-1))} Coefficient of degree (-1) term on the power series expansion of ( X(Z) Z ^ (n-1)) about a_i \ </math> | ||
| + | |||
| + | So inverting X(Z) involves power series. | ||
| + | |||
| + | <math>f(X)= \sum_{n=0}^\infty k^2</math> | ||
Revision as of 04:37, 23 September 2009
Inverse Z-transform
$ x[n] = \oint_C {X(Z)}{Z ^ (n-1)} , dZ \ $
where C is a closed counterwise countour inside the ROC of the Z- transform and around the origin.
$ = \sum_{poles a_i ( X(Z) Z ^ (n-1))} Residue ( X(Z) Z ^ (n-1)) \ $ $ = \sum_{poles a_i ( X(Z) Z ^ (n-1))} Coefficient of degree (-1) term on the power series expansion of ( X(Z) Z ^ (n-1)) about a_i \ $
So inverting X(Z) involves power series.
$ f(X)= \sum_{n=0}^\infty k^2 $
