#### 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 becomeswhere 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

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First-Order Complex Resonators