2.10 Inverse z-transform

Inverse $z$ -transform

Up to this point, we have ignored how to actually invert a z-transform to find $x\left[n\right]$ from $X\left(z\right)$ . Doing so is very different from inverting a DTFT. We will consider three main techniques:

1. Inspection (look it up in a table)
2. Partial fraction expansion
3. 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 $z$ -transform pairs listed in tables, and “reverse engineer”

Partial fraction expansion

If $X\left(z\right)$ is rational, break it up into a sum of elementary forms, each of which can be inverted by inspection.

Power series expansion

Recall that

If we know the coefficients for the Laurent series expansion of $X\left(z\right)$ , then these coefficients give us the inverse $z$ -transform.