When the complex-valued frequency response is expressed in polar form, the amplitude response and phase response explicitly appear:
Writing the basic frequency response description
Equation (7.2) gives the frequency response in polar form. For completeness, recall the transformations between polar and rectangular forms (i.e., for converting real and imaginary parts to magnitude and angle and vice versa). The real part of a complex number is written as re, and the imaginary part of is denoted im.
Going the other way from polar to rectangular (using Euler's formula),
Application of these formulas to some basic example filters will be carried out in §10.1. A matlab listing for computing the frequency response of any IIR filter is given in §7.5.1 below.