Filter Designer — Higher-Order (Butterworth/Chebyshev) Response Plotter mechanism

Model scope

The core Butterworth low-pass formulas and the standard Chebyshev Type-I pole construction are correct under the stated assumptions. The overall instrument remains NEEDS_REVIEW because Chebyshev definitions, frequency units, transformed-topology formulas, phase conventions, and optional component-synthesis assumptions are underspecified. The honest label correctly states that tolerances and op-amp non-idealities are not modeled, but it should also state that the displayed results are ideal transfer-function/prototype results and that BP/BS/HP conventions depend on explicit transformation definitions.

• The Butterworth pole formula is correct for normalized low-pass poles when k=0,...,N-1 and angular cutoff frequency ωc is used, but the specification does not state the k range or explicitly require selecting the left-half-plane poles.
• The Chebyshev pole formula is incomplete: it must define ε, the ripple convention, α=(asinh(1/ε))/N, θk=(2k−1)π/(2N), and pk=ωc[−sinh(α)sin(θk)+j cosh(α)cos(θk)].
• The specification does not provide the low-pass-to-high-pass, band-pass, or band-stop frequency transformations, so LP/HP/BP/BS outputs are not fully verifiable.
• The specification does not define whether fc means Hz or rad/s, nor whether it is the Butterworth 3 dB cutoff, Chebyshev passband edge, center frequency, or another frequency for transformed filters.
• Magnitude and phase formulas are not explicitly stated, including phase-unwrapping and zero-frequency phase conventions.
• Optional LC ladder synthesis cannot be verified without specifying the prototype normalization, source/load terminations, impedance scaling, frequency scaling, and g-value tables.

Units and conventions

• fc is interpreted as frequency in hertz and ωc=2πfc in radians per second.
• Frequency-response test values are ordinary frequency f in hertz; s=j2πf.
• Butterworth low-pass prototypes have unity DC gain and their cutoff is the −3.0102999566 dB frequency.
• Chebyshev Type-I ripple is interpreted as passband ripple Ap in decibels, with ε=sqrt(10^(Ap/10)−1).
• For Chebyshev formulas, ε>0 and the passband edge is interpreted as fc.
• All pole locations are expressed in rad/s.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "order": 2,
    "filter_type": "Butterworth",
    "topology": "LP",
    "fc_hz": 1000,
    "ripple_db": 0
  },
  "expected_outputs": {
    "canonical_transfer_function": "H(s)=(ωc^2)/(s^2+√2·ωc·s+ωc^2), ωc=2000π rad/s",
    "canonical_poles": [
      "p0=ωc·e^(j3π/4)=−ωc/√2+jωc/√2",
      "p1=ωc·e^(j5π/4)=−ωc/√2−jωc/√2"
    ],
    "magnitude_formula": "|H(j2πf)|=1/sqrt(1+(f/fc)^4)",
    "spot_checks": [
      {
        "at": {
          "f_hz": 0
        },
        "input_side": 1,
        "answer_side": 1
      },
      {
        "at": {
          "f_hz": 1000
        },
        "input_side": 0.7071067811865475,
        "answer_side": 0.7071067811865475
      }
    ]
  },
  "reference_data": {
    "standard_method": "2nd-order Butterworth low-pass prototype",
    "normalization": "H(0)=1"
  },
  "working": "ωc=2πfc=2π(1000)=2000π rad/s. For k=0, angle=π(2·0+2+1)/(2·2)=3π/4, so p0=ωc(cos 3π/4+j sin 3π/4)=−ωc/√2+jωc/√2. For k=1, angle=5π/4, so p1=−ωc/√2−jωc/√2. At f=0: |H|=1/sqrt(1+(0/1000)^4)=1. At f=fc: |H|=1/sqrt(1+1^4)=1/√2=0.7071067811865475."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

A Chebyshev Type I input with zero dB ripple makes ε=0 and the direct pole formula evaluate as asinh(1/0), which is undefined. The instrument must reject the input or explicitly handle the Butterworth limiting case.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche8/tranche8-results.json
• BOUNDARY_HAZARD_CATALOG.json