IIR high-pass pole transform uses the wrong real part

[gabrielfrasantos]

Severity: high
Domain: filters
Status: VERIFIED — read + hand-traced against a210d34 on 2026-08-10
Suggested labels: bug, numerical-correctness, filters

Summary

HighPassRoots() maps the analog prototype pole with wc / p.Real() for the real part while
correctly using |p|² for the imaginary part. The low-pass→high-pass transform requires
ωc · Re(p) / |p|². The asymmetry between the two components makes this a clear transcription slip.

Location

numerical/filters/passive/IirFilterDesign.hpp

Evidence

for (std::size_t k{ 0 }; k < order; ++k)
{
    const T mag2{ protoPoles[k].Real() * protoPoles[k].Real()
                + protoPoles[k].Imaginary() * protoPoles[k].Imaginary() };

    const ComplexT hpPole{ wc / protoPoles[k].Real(),                        // WRONG
                           -wc * protoPoles[k].Imaginary() / mag2 };         // correct

    az[k] = BilinearS2Z(hpPole, fs);
    bz[k] = { T{ 1 }, T{ 0 } };
}

Expected

The transform s → ωc/s maps a pole p to

ωc / p = ωc · conj(p) / |p|²  =  ( ωc·Re(p)/|p|² ,  −ωc·Im(p)/|p|² )

so the real part must also divide by mag2.

Trace

Butterworth prototype pole p = −0.7071 + 0.7071j, |p|² = 1:

correct:  ωc·(−0.7071)/1 = −0.7071·ωc
code:     ωc/(−0.7071)   = −1.4142·ωc

The pole is placed at twice the correct radius, so both the cutoff frequency and the Q of the
resulting section are wrong.

Suggested fix

const ComplexT hpPole{ wc * protoPoles[k].Real() / mag2,
                       -wc * protoPoles[k].Imaginary() / mag2 };

Add a test asserting the −3 dB point of a designed high-pass lands at the requested cutoff.

Notes

The band-pass/band-stop paths in the same file have separate, unverified concerns about section
storage sizing and single-cutoff parameterisation — tracked separately in the leads list.