The z-transform (ZT) is a generalization of the discrete-time Fourier transform (DTFT) for discrete-time signals, but the ZT applies to a broader class of signals than the DTFT. The two-sided or bilateral z-transform (ZT) of sequence x[n] is defined as
![image0.jpg](https://www.dummies.com/wp-content/uploads/405286.image0.jpg)
The ZT operator transforms the sequence x[n] to X(z), a function of the continuous complex variable z. The relationship between a sequence and its transform is denoted as
![image1.jpg](https://www.dummies.com/wp-content/uploads/405287.image1.jpg)
You can establish the connection between the discrete-time Fourier transform (DTFT) and the ZT by first writing
![image2.jpg](https://www.dummies.com/wp-content/uploads/405288.image2.jpg)
The special case of r = 1 evaluates X(z) over the unit circle —
![image3.jpg](https://www.dummies.com/wp-content/uploads/405289.image3.jpg)
and is represented as
![image4.jpg](https://www.dummies.com/wp-content/uploads/405290.image4.jpg)
the DTFT of x[n]. This result holds as long as the DTFT is absolutely summable (read: impulse functions not allowed).
The view that
![image5.jpg](https://www.dummies.com/wp-content/uploads/405291.image5.jpg)
sampled around the unit circle in the z-plane
![image6.jpg](https://www.dummies.com/wp-content/uploads/405292.image6.jpg)
shows that the DTFT has period 2π because
![image7.jpg](https://www.dummies.com/wp-content/uploads/405293.image7.jpg)
![image8.jpg](https://www.dummies.com/wp-content/uploads/405294.image8.jpg)
![image9.jpg](https://www.dummies.com/wp-content/uploads/405295.image9.jpg)