## Transfer Function & Pole-Zero Analysis:

• 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. ©2004-2020 McGill University. All Rights Reserved.Maintained by Gary P. Scavone.