Given the pole-zero plot of the transfer function , we can qualitatively learn the system's behavior as frequency changes from 0 to infinity (i.e., the Bode plot if in log scale). We first review the representation of complex values in the complex plane (e.g., the s-plane).

**Vector representation of complex values**Any complex number can be written in two forms

where the real and imaginary parts and and the magnitude and angle of are related by

This complex can be represented in the complex plane by a point with coordinates or the corresponding vector from the origin to . This vector can also be expressed by its length and angle between the vector and the real positive axis of the complex plane.**Vector representation of complex sum and difference**The sum and difference of two complex numbers and can also be easily obtained graphically as the vector sum and vector difference.

**Vector representation of complex product**If a complex number is given as the product of two other complex numbers and , then we have

i.e.,

**Vector representation of complex ratio**If a complex number is given as the ratio between two other complex numbers and , then we have

i.e.,

where

i.e.,

Given the pole-zero plot of a system's rational transfer function , the magnitude and phase angle of the corresponding Fourier transform , i.e., its Bode plot, can be obtained qualitatively by observing the magnitude and angle changes of the zero and pole vectors, as shown below for the first and second order systems.

- First order system
- Second order system
- Root locus of Second Order System
- Frequency Response for Small
- Higher Order Systems