# Lecture 2: Introduction to the Z-Transform

As you recall, we talked first about differential equations, then difference equations. The methods of solving difference equations was in very many respects parallel to the methods used to solve differential equations. We then learned about the Laplace transform, which is a useful tool for solving differential equations and for doing system analysis on continuous-time systems. Our development now continues to the Z-transform. This is a transform technique used for discrete time signals and systems. As you might expect, many of the tools and techniques that we developed using Laplace transforms will transfer over to the Z-transform techniques.

The Z-transform is simply a power series representation of a discrete-time
sequence. For example, if we have the sequence
, the Z-transform
simply multiplies each coefficient in the sequence by a power of
corresponding to its index. In this example

Note that negative powers of are used for positive time indexes. This is by convention. (Comment.)

For a general causal sequence
,
the Z-transform is written as

For a general (not necessarily noncausal) sequence ,

As for continuous time systems, we will be most interested in causal signals and systems.

The inverse Z-transform has the rather strange and frightening form

where is the integral around a closed path in the complex plane, in the region of integration. (Fortunately, this is rarely done by hand, but there is some very neat theory associated with integrals around closed contours in the complex plane. If you want the complete scoop on this, you should take complex analysis.)

Notationally we will write

or

Important and useful functions have naturally been transformed and put
into tabular form.