The z-transform is extensively used to evaluate the properties of discrete-time systems such as digital filters. In particular, it is convenient for determining the stability of a system.
The numerator and denominator of a system's transfer function are polynomials in z. The roots of these polynomials can be determined by factorization.
Roots of the numerator polynomial indicate values of z at which the transfer function evaluates to zero. These are called zeros.
Roots of the denominator polynomial indicate values of z at which the transfer function evaluates to infinity. These are called poles.
The zeros and poles of a transfer function can be plotted in the z-plane. Their locations with respect to the unit circle indicate radian frequencies at which the system's magnitude response has local minima (near zeros) or maxima (near poles).
In Matlab, the functions roots and zplane can be used to determine and plot the poles and zeros of a system.
When the coefficients of a transfer function are all real, complex roots are given by complex-conjugate pairs.
For a system to have a stable frequency response, all of its poles must lie within the unit circle in the z-plane.