# The Two-Sided z-Transform

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

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

You can establish the connection between the discrete-time Fourier transform (DTFT) and the ZT by first writing

The special case of *r* = 1 evaluates *X*(*z*) over the unit circle —

and is represented as

the DTFT of *x*[*n*]. This result holds as long as the DTFT is absolutely summable (read: impulse functions not allowed).

The view that

sampled around the unit circle in the *z-*plane

shows that the DTFT has period 2π because