The z-transform of a discrete-time signal x(n) is defined by the following series:

```         inf         -n
X(z) =   Sum    x(n)z
n = -inf
```

where z is a complex number. Propably the three most important properties of z-transforms are (`-z->` denoting the z-transform):

• linearity:
`ax(n) + by(n) -z-> aX(z) + bY(z)`
• delay:
`x(n-d) -z-> z-dX(z)`
• convolution:
`y(n) = h(n)*x(n) ==> Y(z) = H(z)X(z)`

The region of the complex plane in which the series above converges is called the region of convergence of the z-transform. The region of convergence is of utmost importance when considering the stability and causality of a given system. Regions of convergence are typically circular or annular. Since z-transforms can always be written as a ratio of polynomials of z-1, they are often characterized by their poles and zeros. For example, causal signals have regions of convergence which are outside the smallest circle having the origin as its center and including all the poles. If the region of convergence is a circle around the origin that does not contain any pole, then the signal is strictly anticausal. A necessary and sufficient condition for the stability of a signal is that the region of convergence of its z-transform contain the unit circle.

Z-transforms can be related to discrete-time Fourier transforms. The z-transform of a signal, evaluated on the unit circle so that z = exp(jω), is then:

```         inf
X(ω) =   Sum    x(n)exp(-jωn)
n = -inf
```

where ω is 2πf/fs, the digital or numerical frequency, fs being the sampling frequency.

Z-transforms are especially useful when analysing, designing and implementing digital filters.

Primary source: Introduction to Signal Processing, Orfanidis, Prentice-Hall Editor.

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