• Hmm... This looks a lot like my course notes... Perhaps you want to write this somewhere, otherwise one might think that you are pretending that you wrote this yourself. --Mboutin 12:26, 23 September 2009 (UTC)
                                               Inverse Z-transform


$x[n] = \frac{1}{2 \prod j} \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 \frac{f^n (X_0) (X-X_0)^{n}}{n!} \$ , near $X_0$

$\frac{1}{(1-x)} = \sum_{n=0}^\infty X^n \$ , geometric series where |X|< 1

Computing equivalent to complex integration formula's

1) Write X(z) as a power series.

$X(z) = \sum_{n=-\infty}^{\infty} \ C_n z^n \$ , series must converge for all z's on the ROC of X(z)

2) Observe that

$X(z) = \sum_{n=-\infty}^{\infty} \ x[n] z^{-n} \$

i.e.,

$X(z) = \sum_{n=-\infty}^{\infty} \ x[-n] z^n \$

3) By comparison

$X[-n] = \ C_n \$

or

$X[n] = \ C_ {-n}$

Example 1:

$X(z) = \frac{1}{(1-z)} \$

Two possible ROC

Case 1: |z|<1

$X(z) = \sum_{n=0}^\infty z^n \$

   $= \sum_{k=-\infty}^{0} \ z^{-k} \$
$= \sum_{n=-\infty}^{\infty} \ u(-k) z^{-k} \$


so, x[n]=u[-n]

Consistent as having inside a circle as ROC.

Case 2: |z|>1

$X(z) = \frac{1}{(1-z)} \$

   $= \frac{1}{z(\frac{1}{z}-1)} \$
$= \frac {-1}{z} {\frac{1}{1-\frac{1}{z}}} \$    ,observe $\ |\frac {1}{z}|< 1 \$

Now by using the geometric series formula, the series can be formed as

      $= \frac {-1}{z} \sum_{n=0}^\infty (\frac {1}{z})^{n} \$

      $= - \sum_{n=0}^\infty z^{-n-1} \$


Let k=(n+1)

      $= - \sum_{k=1}^\infty z^{-k} \$

      $= \sum_{k=1}^\infty -u(k-1) z^{-k} \$


By coparison with the Z- transform formula

$x[n]= -u[n-1] \$

Consistent as having outside of circle as the ROC.

## Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood