In practice, however, signals are typically real-valued functions of
time. As a result, for real filters (§5.1),
it is typically more efficient computationally to combine
complex-conjugate one-pole sections together to form real second-order
sections (two poles and one zero each, in general). This process was
discussed in §6.8.1, and the resulting transfer function of
each second-order section becomes
When the two poles of a real second-order section are complex, they form a complex-conjugate pair, i.e., they are located at in the plane, where is the modulus of either pole, and is the angle of either pole. In this case, the ``resonance-tuning coefficient'' in Eq. (9.3) can be expressed as
which is often more convenient for real-time control of resonance tuning and/or bandwidth. A more detailed derivation appears in §B.1.3.
Figures 3.25 and 3.26 (p. ) illustrate filter realizations consisting of one first-order and two second-order filter sections in parallel.