Cascading identical biquad filters degrades the corner response — each stage needs a unique Q derived from pole spacing

Stacking two identical second-order Butterworth filters does not produce a fourth-order Butterworth: at the corner frequency, each stage already attenuates by -3 dB, so the cascaded result is -6 dB at the nominal cutoff, and the rolloff shape changes. To maintain a Butterworth response in higher-order designs, each cascaded biquad stage requires a different Q value calculated from the pole positions in the s-plane. Poles are evenly spaced at angles of π/n around the unit circle; for a given pole angle θ, Q = 1/(2cos(θ)). All stages share the same corner frequency but have different Q values.

Examples

4-pole Butterworth lowpass from two biquads: pole angles π/8 and 3π/8 give Q values 0.541196 and 1.306563 (rounded 0.54 and 1.31). A naive approach sets both to Q = 0.707 — the result is a softer rolloff and a corner sitting -6 dB rather than -3 dB.

Assessment

Why does cascading two identical Butterworth biquads not produce a higher-order Butterworth filter? Describe what the resulting filter actually does at the corner compared to a properly calculated 4-pole design.