1. The Z-Transform of a given discrete signal, x(n), is given by:
where z is a complex variable.

2. In most cases the signal will be causal giving rise to the One- Sided Z-Transform:

3. The Z-Transform is derived from the Laplace Transform of a train of unit pulses.
The Laplace Transform of the unit pulse is unity, the delayed pulse an exponential term:

4. L The Laplace Transform of a sampled function is:
Note that the Laplace Transform of a sampled function is in the form of an infinite series and involves factors of the form of powers of e-st.

5. We define a new transform called the Z-Transform such that:

6. Example:
For the unit impulse, d(t), x(n)=1 at n=0 only, therefore:-
Likewise,

7. For the unit step function:
Note that the result is an infinite geometric series and the following identity holds:

8. so that:
Another useful identity exists for finite geometric series.

12. Digital filters are characterized by polynomials (FIR filters) or rational fractions (IIR filters).
It is preferred to represent DSP filter function using negative powers of z. Any positive power rational fraction can be converted to one with negative powers of z.

13. Suppose we have:
we can multiply numerator and denominator by z-2.

14. Consider the FIR filter:
Since z represents a delay:

15. so that the Z transform ofis
which becomes

16. Find the system function H(z) of a FIR filter whose impulse response is:
The solution is:

17. Suppose that
The inverse Z transform is: