Transfer Function & Pole-Zero Analysis:

• 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.