Polar Form of the Frequency Response

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

(from Eq.(7.2)) in polar form gives

which implies

This states explicitly that the output magnitude spectrum equals the
input magnitude spectrum *times* the filter amplitude response,
and the output phase equals the input phase *plus* the filter
phase at each frequency
.

Equation (7.3) 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):

Going the other way from polar to rectangular (using Euler's formula),

Application of these formulas to some basic example filters are carried out in Appendix B. Some useful trig identities are summarized in Appendix A. A matlab listing for computing the frequency response of any IIR filter is given in §7.5.1 below.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University