#### Real Second-Order Sections

In practice, however, signals are typically real-valued functions of time. As a result, for real filters5.1), 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

 (10.3)

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

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.

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First-Order Complex Resonators