FA-91531 / Digital signal filters / Open access

Frequency response only reports exact zeros · case 01

A notch that evaluates to 1e-16 due to rounding is reported as about -320 dB with a random phase.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The zero test compares the magnitude with == 0.

THE FAILURE

The zero test compares the magnitude with == 0.

Unsuccessful approach: The attempted repair uses 1e-3, which also hides legitimate -70 dB attenuation.

Case contract

Input [b, a, freqs]; coefficients and frequencies are numbers or rational strings, frequencies are fractions of Nyquist. For w = pi f evaluate H = B(e^{jw})/A(e^{jw}) with z^-k = e^{-jwk}; return per frequency [20 log10 |H| rounded to 4, phase in degrees rounded to 3], "pole" if |A| < 1e-12, "zero" if |H| < 1e-9.

Why this case matters

Magnitude and phase plots are how filters are verified; unit or sign slips misreport cutoff and delay.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import cmath
from fractions import Fraction
N = 1
observations = []
def solve(x):
    b, a, fs = x
    b = [float(Fraction(v)) for v in b]
    a = [float(Fraction(v)) for v in a]
    out = []
    for f in fs:
        w = math.pi * float(Fraction(f))
        B = sum(c * cmath.exp(-1j * w * k) for k, c in enumerate(b))
        A = sum(c * cmath.exp(-1j * w * k) for k, c in enumerate(a))
        if abs(A) < 1e-12:
            out.append('pole')
            continue
        H = B / A
        mag = abs(H)
        if mag == 0:
            out.append('zero')
            continue
        out.append([round(20 * math.log10(mag), 4) + 0.0, round(math.degrees(cmath.phase(H)), 3) + 0.0])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: first difference', [[1, -1], [1], ['0', '1/2', '1']], ['zero', [3.0103, 45.0], [6.0206, 0.0]]], ['control: one pole', [[1], [1, '-1/2'], ['0', '1/4', '1/2']], [[6.0206, 0.0], [2.6529, -28.675], [-0.9691, -26.565]]], ['control: pole on unit circle', [[1], [1, -1], ['0']], ['pole']], ['control: pure delay phase', [[0, 1], [1], ['1/4', '1/2']], [[0.0, -45.0], [0.0, -90.0]]]], [['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: random response 0', [['-1/4', 1], [1, 0], ['1/3', '1/8', '1/8']], [[-0.9018, -73.898], [-2.2144, -29.591], [-2.2144, -29.591]]], ['control: random response 1', [[0, -1, 1, 0], [1, '1/3', '1/3'], ['1/2', '3/4', '1/8']], [[5.563, 161.565], [7.5974, 60.22], [-12.1798, -110.508]]], ['control: random response 2', [['1/2', '-1/4', 0, 2], [1, '1/2', '1/2'], ['1/2', '1/2', '1/2']], [[10.2633, 122.471], [10.2633, 122.471], [10.2633, 122.471]]], ['control: random response 3', [[0, 0, -1], [1], ['1/2', '1/8', '1']], [[0.0, 0.0], [0.0, 135.0], [0.0, -180.0]]]], [['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: random response 4', [[-1], [1, '1/2'], ['3/4', '1', '0']], [[2.6529, -151.325], [6.0206, -180.0], [-3.5218, 180.0]]], ['control: random response 5', [[1], [1, '1/2', '1/2'], ['1/4', '1', '1/8']], [[-4.0835, 32.236], [0.0, 0.0], [-5.5545, 16.706]]], ['control: random response 6', [[1, -1], [1, '1/3', '1/3'], ['1/8', '1', '1/2']], [[-12.1798, 91.992], [6.0206, 0.0], [5.563, 71.565]]], ['control: random response 7', [[2, -1], [1, '1/2'], ['3/4', '1/3', '1/3']], [[11.5896, 43.314], [2.3408, 49.107], [2.3408, 49.107]]]], [['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: random response 8', [[1, 2, -1, 1], [1, '-1/4', 0], ['1', '1/4', '0']], [[7.6042, -180.0], [7.6969, -45.419], [12.0412, 0.0]]], ['control: random response 9', [[1], [1, '1/2'], ['1', '1/4', '1']], [[6.0206, 0.0], [-2.9161, 14.639], [6.0206, 0.0]]], ['control: random response 10', [[-1, 1], [1], ['1/3', '1', '1/3']], [[0.0, -120.0], [6.0206, -180.0], [0.0, -120.0]]], ['control: random response 11', [[1, 2, 0], [1], ['1/2', '0', '1/8']], [[6.9897, -63.435], [9.5424, 0.0], [9.393, -15.043]]]], [['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: random response 12', [['-1/4'], [1, '1/2', 0], ['1/8', '0', '0']], [[-15.4136, -172.543], [-15.563, 180.0], [-15.563, 180.0]]], ['control: random response 13', [[-1, 2], [1, 0, '1/3'], ['1/2', '1/2', '1/8']], [[10.5115, -116.565], [10.5115, -116.565], [-0.8391, -31.277]]], ['control: random response 14', [[2, '1/2', -1], [1], ['1/8', '1/3', '0']], [[5.2445, 16.379], [8.893, 8.948], [3.5218, 0.0]]], ['control: random response 15', [[0, -1, '1/2', 0], [1], ['0', '1/8', '3/4']], [[-6.0206, 180.0], [-4.8662, 177.076], [2.9161, 59.639]]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression: two-tap average at nyquist	`[[-318.2398, -90.0]]`	`['zero']`	Failed
regression: deep notch	`[[-318.2398, -90.0], [-24.0378, -88.2]]`	`['zero', [-24.0378, -88.2]]`	Failed
repair check: small magnitude	`[[-80.0, 0.0]]`	`[[-80.0, 0.0]]`	Passed
control: first difference	`['zero', [3.0103, 45.0], [6.0206, 0.0]]`	`['zero', [3.0103, 45.0], [6.0206, 0.0]]`	Passed
control: one pole	`[[6.0206, 0.0], [2.6529, -28.675], [-0.9691, -26.565]]`	`[[6.0206, 0.0], [2.6529, -28.675], [-0.9691, -26.565]]`	Passed
control: pole on unit circle	`['pole']`	`['pole']`	Passed
control: pure delay phase	`[[0.0, -45.0], [0.0, -90.0]]`	`[[0.0, -45.0], [0.0, -90.0]]`	Passed

SHA-256 / 111d5585a425921831f6c6f056b206bcf80515b4fbea5251f1bd8283dda9389c

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import cmath
from fractions import Fraction
N = 1
observations = []
def solve(x):
    b, a, fs = x
    b = [float(Fraction(v)) for v in b]
    a = [float(Fraction(v)) for v in a]
    out = []
    for f in fs:
        w = math.pi * float(Fraction(f))
        B = sum(c * cmath.exp(-1j * w * k) for k, c in enumerate(b))
        A = sum(c * cmath.exp(-1j * w * k) for k, c in enumerate(a))
        if abs(A) < 1e-12:
            out.append('pole')
            continue
        H = B / A
        mag = abs(H)
        if mag < 1e-3:
            out.append('zero')
            continue
        out.append([round(20 * math.log10(mag), 4) + 0.0, round(math.degrees(cmath.phase(H)), 3) + 0.0])
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: first difference', [[1, -1], [1], ['0', '1/2', '1']], ['zero', [3.0103, 45.0], [6.0206, 0.0]]], ['control: one pole', [[1], [1, '-1/2'], ['0', '1/4', '1/2']], [[6.0206, 0.0], [2.6529, -28.675], [-0.9691, -26.565]]], ['control: pole on unit circle', [[1], [1, -1], ['0']], ['pole']], ['control: pure delay phase', [[0, 1], [1], ['1/4', '1/2']], [[0.0, -45.0], [0.0, -90.0]]]], [['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: random response 0', [['-1/4', 1], [1, 0], ['1/3', '1/8', '1/8']], [[-0.9018, -73.898], [-2.2144, -29.591], [-2.2144, -29.591]]], ['control: random response 1', [[0, -1, 1, 0], [1, '1/3', '1/3'], ['1/2', '3/4', '1/8']], [[5.563, 161.565], [7.5974, 60.22], [-12.1798, -110.508]]], ['control: random response 2', [['1/2', '-1/4', 0, 2], [1, '1/2', '1/2'], ['1/2', '1/2', '1/2']], [[10.2633, 122.471], [10.2633, 122.471], [10.2633, 122.471]]], ['control: random response 3', [[0, 0, -1], [1], ['1/2', '1/8', '1']], [[0.0, 0.0], [0.0, 135.0], [0.0, -180.0]]]], [['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: random response 4', [[-1], [1, '1/2'], ['3/4', '1', '0']], [[2.6529, -151.325], [6.0206, -180.0], [-3.5218, 180.0]]], ['control: random response 5', [[1], [1, '1/2', '1/2'], ['1/4', '1', '1/8']], [[-4.0835, 32.236], [0.0, 0.0], [-5.5545, 16.706]]], ['control: random response 6', [[1, -1], [1, '1/3', '1/3'], ['1/8', '1', '1/2']], [[-12.1798, 91.992], [6.0206, 0.0], [5.563, 71.565]]], ['control: random response 7', [[2, -1], [1, '1/2'], ['3/4', '1/3', '1/3']], [[11.5896, 43.314], [2.3408, 49.107], [2.3408, 49.107]]]], [['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: random response 8', [[1, 2, -1, 1], [1, '-1/4', 0], ['1', '1/4', '0']], [[7.6042, -180.0], [7.6969, -45.419], [12.0412, 0.0]]], ['control: random response 9', [[1], [1, '1/2'], ['1', '1/4', '1']], [[6.0206, 0.0], [-2.9161, 14.639], [6.0206, 0.0]]], ['control: random response 10', [[-1, 1], [1], ['1/3', '1', '1/3']], [[0.0, -120.0], [6.0206, -180.0], [0.0, -120.0]]], ['control: random response 11', [[1, 2, 0], [1], ['1/2', '0', '1/8']], [[6.9897, -63.435], [9.5424, 0.0], [9.393, -15.043]]]], [['regression: two-tap average at nyquist', [[1, 1], [1], ['1']], ['zero']], ['regression: deep notch', [[1, 0, 1], [1], ['1/2', '0.49']], ['zero', [-24.0378, -88.2]]], ['repair check: small magnitude', [[1, '-0.9999'], [1], ['0']], [[-80.0, 0.0]]], ['control: random response 12', [['-1/4'], [1, '1/2', 0], ['1/8', '0', '0']], [[-15.4136, -172.543], [-15.563, 180.0], [-15.563, 180.0]]], ['control: random response 13', [[-1, 2], [1, 0, '1/3'], ['1/2', '1/2', '1/8']], [[10.5115, -116.565], [10.5115, -116.565], [-0.8391, -31.277]]], ['control: random response 14', [[2, '1/2', -1], [1], ['1/8', '1/3', '0']], [[5.2445, 16.379], [8.893, 8.948], [3.5218, 0.0]]], ['control: random response 15', [[0, -1, '1/2', 0], [1], ['0', '1/8', '3/4']], [[-6.0206, 180.0], [-4.8662, 177.076], [2.9161, 59.639]]]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression: two-tap average at nyquist	`['zero']`	`['zero']`	Passed
regression: deep notch	`['zero', [-24.0378, -88.2]]`	`['zero', [-24.0378, -88.2]]`	Passed
repair check: small magnitude	`['zero']`	`[[-80.0, 0.0]]`	Failed
control: first difference	`['zero', [3.0103, 45.0], [6.0206, 0.0]]`	`['zero', [3.0103, 45.0], [6.0206, 0.0]]`	Passed
control: one pole	`[[6.0206, 0.0], [2.6529, -28.675], [-0.9691, -26.565]]`	`[[6.0206, 0.0], [2.6529, -28.675], [-0.9691, -26.565]]`	Passed
control: pole on unit circle	`['pole']`	`['pole']`	Passed
control: pure delay phase	`[[0.0, -45.0], [0.0, -90.0]]`	`[[0.0, -45.0], [0.0, -90.0]]`	Passed

SHA-256 / 381616e0be187ba536b07352d18cea3c21cadb6c9263eda7e367159eaf98538c

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 7 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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Verification & scope

A deterministic bounded teaching model with a stipulated toy contract; exact rational arithmetic or fixed-decimal rounding keeps outputs strict JSON. It is not a production DSP library and claims no standards conformance. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:51:36.930701+00:00.

Case digest / 47537a0271164dd0c8733db3273c42a64a8d8e62811fccfe742b7eccb0d77a51