z-Transform

The discrete $z$-transform is defined as

$${\mathcal Z}[a]=\sum_{n=0}^{N-1} a_nz^{kn}.$$ (1)

The Discrete Fourier Transform is a special case of the $z$-transform with

$$z\equiv e^{-2\pi i/N}.$$ (2)

A $z$-transform with

$$z\equiv e^{-2\pi i\alpha/N}$$ (3)

for $\alpha\not=\pm 1$ is called a Fractional Fourier Transform.

See also Discrete Fourier Transform, Fractional Fourier Transform

References

Arndt, J. ``The $z$-Transform (ZT).'' Ch. 3 in ``Remarks on FFT Algorithms.'' http://www.jjj.de/fxt/.