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FA-91371 / Digital signal filters / Open access

Biquad stores the new output in y[n-2] · case 01

A pure second-order feedback filter never feels its a2 term correctly and the impulse response rings at the wrong rate.

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

ROOT CAUSE

The output delay update is written y2, y1 = y, y1, swapping which slot receives the new output.

VERIFIED REPAIR

Shift outputs as y2, y1 = y1, y.

Unsuccessful approach: The attempted repair chains y2 = y1 = y, which loses the old y[n-1].

Case contract

Input [[b0, b1, b2, a0, a1, a2], samples]; coefficients are rational strings, normalized by a0 ("a0-zero" if a0 == 0). y[n] = b0 x[n] + b1 x[n-1] + b2 x[n-2] - a1 y[n-1] - a2 y[n-2] with zero initial state; return outputs as exact fraction strings.

Why this case matters

The biquad is the workhorse IIR section; sign, state or normalization slips produce unstable or wrong-gain filters.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    b0, b1, b2, a0, a1, a2 = [Fraction(v) for v in x[0]]
    if a0 == 0:
        return 'a0-zero'
    b0, b1, b2, a1, a2 = b0 / a0, b1 / a0, b2 / a0, a1 / a0, a2 / a0
    x1 = x2 = y1 = y2 = Fraction(0)
    out = []
    for s in x[1]:
        y = b0 * s + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2
        x2, x1 = x1, s
        y2, y1 = y, y1
        out.append(str(y))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['regression: a0 not one', [['2', '1', '0', '2', '1', '1/2'], [2, 0, 0, 1]], ['2', '0', '-1/2', '5/4']], ['regression: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 0', [['1/3', '3/4', '1/2', '-1', '2', '1/3'], [2, 2]], ['-2/3', '-7/2']], ['regression: random biquad 1', [['1/8', '-1/4', '3/4', '2', '1/2', '1/8'], [-2, 1, 2, 2, 0, 2]], ['-1/8', '11/32', '-53/64', '223/512', '907/2048', '3019/4096']], ['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']], ['regression: random biquad 4', [['-3/8', '1/3', '-1/2', '2', '1/3', '0'], [-1, 1]], ['3/16', '-37/96']], ['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 6', [['1/8', '0', '5/4', '1', '-1/2', '5/4'], [1, 1, 1, 0, 3, 2]], ['1/8', '3/16', '21/16', '107/64', '105/128', '-183/128']], ['regression: random biquad 9', [['3/4', '1/3', '1/2', '-1', '0', '1/8'], [-1, -1, 2]], ['3/4', '13/12', '-55/96']], ['regression: random biquad 5', [['1/2', '0', '-1/2', '-1', '-1/2', '-3/8'], [0, 2, -2, -1, 1]], ['0', '-1', '3/2', '9/8', '-21/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 11', [['1/2', '3/4', '0', '1', '1/3', '5/4'], [-3, 3, -3, 0, 2]], ['-3/2', '-1/4', '65/24', '-409/144', '-1243/864']], ['regression: random biquad 13', [['1/8', '-1/4', '5/4', '2', '-1/4', '1/2'], [1, -1, 2]], ['1/16', '-23/128', '857/1024']], ['regression: random biquad 9', [['3/4', '1/3', '1/2', '-1', '0', '1/8'], [-1, -1, 2]], ['3/4', '13/12', '-55/96']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]]]
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 fixtureActualExpectedOutcome
regression: pure feedback impulse['1', '0', '0', '0']['1', '1/2', '1/4', '1/8']Failed
regression: a0 not one['2', '1/2', '-1/8', '33/32']['2', '0', '-1/2', '5/4']Failed
regression: second order feedback['1', '-1/4', '1/16', '-1/64', '1/256']['1', '0', '-1/4', '0', '1/16']Failed
control: fir only['1', '4', '8']['1', '4', '8']Passed
control: random biquad 7['0', '-9/8']['0', '-9/8']Passed
control: random biquad 8['0', '3/2']['0', '3/2']Passed
control: random biquad 30['0', '0', '1/8']['0', '0', '1/8']Passed

SHA-256 / df29591a0df73e993ac807439367cc303e6736ed6aaab6bc2394b39655fe357d

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    b0, b1, b2, a0, a1, a2 = [Fraction(v) for v in x[0]]
    if a0 == 0:
        return 'a0-zero'
    b0, b1, b2, a1, a2 = b0 / a0, b1 / a0, b2 / a0, a1 / a0, a2 / a0
    x1 = x2 = y1 = y2 = Fraction(0)
    out = []
    for s in x[1]:
        y = b0 * s + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2
        x2, x1 = x1, s
        y2 = y1 = y
        out.append(str(y))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['regression: a0 not one', [['2', '1', '0', '2', '1', '1/2'], [2, 0, 0, 1]], ['2', '0', '-1/2', '5/4']], ['regression: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 0', [['1/3', '3/4', '1/2', '-1', '2', '1/3'], [2, 2]], ['-2/3', '-7/2']], ['regression: random biquad 1', [['1/8', '-1/4', '3/4', '2', '1/2', '1/8'], [-2, 1, 2, 2, 0, 2]], ['-1/8', '11/32', '-53/64', '223/512', '907/2048', '3019/4096']], ['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']], ['regression: random biquad 4', [['-3/8', '1/3', '-1/2', '2', '1/3', '0'], [-1, 1]], ['3/16', '-37/96']], ['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 6', [['1/8', '0', '5/4', '1', '-1/2', '5/4'], [1, 1, 1, 0, 3, 2]], ['1/8', '3/16', '21/16', '107/64', '105/128', '-183/128']], ['regression: random biquad 9', [['3/4', '1/3', '1/2', '-1', '0', '1/8'], [-1, -1, 2]], ['3/4', '13/12', '-55/96']], ['regression: random biquad 5', [['1/2', '0', '-1/2', '-1', '-1/2', '-3/8'], [0, 2, -2, -1, 1]], ['0', '-1', '3/2', '9/8', '-21/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 11', [['1/2', '3/4', '0', '1', '1/3', '5/4'], [-3, 3, -3, 0, 2]], ['-3/2', '-1/4', '65/24', '-409/144', '-1243/864']], ['regression: random biquad 13', [['1/8', '-1/4', '5/4', '2', '-1/4', '1/2'], [1, -1, 2]], ['1/16', '-23/128', '857/1024']], ['regression: random biquad 9', [['3/4', '1/3', '1/2', '-1', '0', '1/8'], [-1, -1, 2]], ['3/4', '13/12', '-55/96']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]]]
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 fixtureActualExpectedOutcome
regression: pure feedback impulse['1', '1/2', '1/4', '1/8']['1', '1/2', '1/4', '1/8']Passed
regression: a0 not one['2', '-1/2', '3/8', '23/32']['2', '0', '-1/2', '5/4']Failed
regression: second order feedback['1', '-1/4', '1/16', '-1/64', '1/256']['1', '0', '-1/4', '0', '1/16']Failed
control: fir only['1', '4', '8']['1', '4', '8']Passed
control: random biquad 7['0', '-9/8']['0', '-9/8']Passed
control: random biquad 8['0', '3/2']['0', '3/2']Passed
control: random biquad 30['0', '0', '1/8']['0', '0', '1/8']Passed

SHA-256 / 4e3cc8b134cce255875155d8876c6b320c8fdf3b3a7df028912f0fdd2b71b09b

3 / The verified repair

Exit 0
"""Failure Map reference implementation. Python standard library only."""
import json
from fractions import Fraction
N = 1
observations = []
def solve(x):
    b0, b1, b2, a0, a1, a2 = [Fraction(v) for v in x[0]]
    if a0 == 0:
        return 'a0-zero'
    b0, b1, b2, a1, a2 = b0 / a0, b1 / a0, b2 / a0, a1 / a0, a2 / a0
    x1 = x2 = y1 = y2 = Fraction(0)
    out = []
    for s in x[1]:
        y = b0 * s + b1 * x1 + b2 * x2 - a1 * y1 - a2 * y2
        x2, x1 = x1, s
        y2, y1 = y1, y
        out.append(str(y))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: pure feedback impulse', [['1', '0', '0', '1', '-1/2', '0'], [1, 0, 0, 0]], ['1', '1/2', '1/4', '1/8']], ['regression: a0 not one', [['2', '1', '0', '2', '1', '1/2'], [2, 0, 0, 1]], ['2', '0', '-1/2', '5/4']], ['regression: second order feedback', [['1', '0', '0', '1', '0', '1/4'], [1, 0, 0, 0, 0]], ['1', '0', '-1/4', '0', '1/16']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 0', [['1/3', '3/4', '1/2', '-1', '2', '1/3'], [2, 2]], ['-2/3', '-7/2']], ['regression: random biquad 1', [['1/8', '-1/4', '3/4', '2', '1/2', '1/8'], [-2, 1, 2, 2, 0, 2]], ['-1/8', '11/32', '-53/64', '223/512', '907/2048', '3019/4096']], ['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 3', [['-3/8', '5/4', '1/2', '2', '5/4', '1/8'], [-2, 0, -1, 3]], ['3/8', '-95/64', '303/512', '-5999/4096']], ['regression: random biquad 4', [['-3/8', '1/3', '-1/2', '2', '1/3', '0'], [-1, 1]], ['3/16', '-37/96']], ['regression: random biquad 2', [['1/2', '5/4', '1/2', '-1', '-1/2', '-1/2'], [-2, -2, 3, -3, 1]], ['1', '3', '0', '-11/4', '25/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 6', [['1/8', '0', '5/4', '1', '-1/2', '5/4'], [1, 1, 1, 0, 3, 2]], ['1/8', '3/16', '21/16', '107/64', '105/128', '-183/128']], ['regression: random biquad 9', [['3/4', '1/3', '1/2', '-1', '0', '1/8'], [-1, -1, 2]], ['3/4', '13/12', '-55/96']], ['regression: random biquad 5', [['1/2', '0', '-1/2', '-1', '-1/2', '-3/8'], [0, 2, -2, -1, 1]], ['0', '-1', '3/2', '9/8', '-21/8']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]], [['regression: random biquad 11', [['1/2', '3/4', '0', '1', '1/3', '5/4'], [-3, 3, -3, 0, 2]], ['-3/2', '-1/4', '65/24', '-409/144', '-1243/864']], ['regression: random biquad 13', [['1/8', '-1/4', '5/4', '2', '-1/4', '1/2'], [1, -1, 2]], ['1/16', '-23/128', '857/1024']], ['regression: random biquad 9', [['3/4', '1/3', '1/2', '-1', '0', '1/8'], [-1, -1, 2]], ['3/4', '13/12', '-55/96']], ['control: fir only', [['1', '2', '1', '1', '0', '0'], [1, 2, 3]], ['1', '4', '8']], ['control: random biquad 7', [['-3/8', '0', '-1/4', '1', '2', '1/8'], [0, 3]], ['0', '-9/8']], ['control: random biquad 8', [['0', '1/2', '3/4', '-1', '3/4', '0'], [-3, 0]], ['0', '3/2']], ['control: random biquad 30', [['0', '0', '1/8', '1', '0', '-1/2'], [1, -2, 1]], ['0', '0', '1/8']]]]
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 fixtureActualExpectedOutcome
regression: pure feedback impulse['1', '1/2', '1/4', '1/8']['1', '1/2', '1/4', '1/8']Passed
regression: a0 not one['2', '0', '-1/2', '5/4']['2', '0', '-1/2', '5/4']Passed
regression: second order feedback['1', '0', '-1/4', '0', '1/16']['1', '0', '-1/4', '0', '1/16']Passed
control: fir only['1', '4', '8']['1', '4', '8']Passed
control: random biquad 7['0', '-9/8']['0', '-9/8']Passed
control: random biquad 8['0', '3/2']['0', '3/2']Passed
control: random biquad 30['0', '0', '1/8']['0', '0', '1/8']Passed

SHA-256 / ff4bd3aa0bc92f8a982beb30440c2da6fafecce3ac0c372067d90994a8eb446e

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 2026-09-29T14:51:35.432707+00:00.

Case digest / b092e0c3298f3c5c327108797c26463dddac45cf4b2585034f161466e12e5318