#### Real Second-Order Sections

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

where is one of the poles, and is its corresponding residue. This is a special case of the

*biquad section*discussed in §B.1.6.

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

Figures 3.25 and 3.26 (p. ) illustrate filter realizations consisting of one first-order and two second-order filter sections in parallel.

**Next Section:**

Implementation of Repeated Poles

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First-Order Complex Resonators