Up to this point, we have ignored how to actually invert a z-transform to
find

- Inspection (look it up in a table)
- Partial fraction expansion
- Power series expansion

One can also use contour integration combined with the Cauchy Residue Theorem. See Oppenheim and Schafer for details.

**Inspection**

Basically, become familiar with the

#### Example 1

Suppose that

By now you should be able to recognize that

**Partial fraction expansion**

If

#### Example 2

Suppose that

By computing a partial fraction expansion we can decompose

where each term in the sum can be inverted by inspection.

**Power Series Expansion**

Recall that

If we know the coefficients for the Laurent series expansion of

#### Example 3

Suppose

Then

#### Example 4

Suppose

where

we can write

Thus we can infer that