Source: https://guthlabs.ai/docs/instruments/filter-designer-higher-order-butterworth-chebyshev-response-plotter/mechanism/

instrument mechanism / verified-clean

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

The inspectable method record for Filter Designer — Higher-Order (Butterworth/Chebyshev) Response Plotter, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## 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

[01**Overview**](<https://guthlabs.ai/docs/instruments/filter-designer-higher-order-butterworth-chebyshev-response-plotter/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/filter-designer-higher-order-butterworth-chebyshev-response-plotter/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/filter-designer-higher-order-butterworth-chebyshev-response-plotter/qualification/>)

## 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`