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FA-91786 / Digital signal filters / Open access

Impulse truncation normalizes the denominator before the numerator · case 01

With a0 = 2 the numerator is never scaled and the response is twice too large.

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

ROOT CAUSE

a is divided by a0 first, so the numerator is then divided by the new a[0] = 1.

THE FAILURE

a is divided by a0 first, so the numerator is then divided by the new a[0] = 1.

Unsuccessful approach: The attempted repair normalizes b but leaves a unnormalized, so the recursion still uses raw feedback coefficients.

Case contract

Input [b, a, eps, maxlen, hold]; normalize by a0 ("bad-a0"), compute the impulse response exactly and stop after hold consecutive samples with |h| < eps, counted only once n >= len(b) - 1 (any non-quiet sample resets the count), or at maxlen. Return the kept samples as fraction strings.

Why this case matters

FIR approximations of IIR filters truncate the impulse response; wrong stop rules cut real energy or run forever.

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):
    b = [Fraction(v) for v in x[0]]
    a = [Fraction(v) for v in x[1]]
    eps, maxlen, hold = Fraction(x[2]), x[3], x[4]
    if not a or a[0] == 0:
        return 'bad-a0'
    a = [v / a[0] for v in a]
    b = [v / a[0] for v in b]
    h = []
    quiet = 0
    for n in range(maxlen):
        acc = b[n] if n < len(b) else Fraction(0)
        for k in range(1, len(a)):
            if n - k >= 0:
                acc -= a[k] * h[n - k]
        h.append(acc)
        if n >= len(b) - 1 and abs(acc) < eps:
            quiet += 1
            if quiet >= hold:
                break
        else:
            quiet = 0
    return [str(v) for v in h]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: a0 not one', [['2', '1'], ['2', '-1'], '1/16', 10, 2], ['1', '1', '1/2', '1/4', '1/8', '1/16', '1/32', '1/64']], ['regression: random impulse 0', [['0', '1', '1'], ['2', '1/2', '1/3'], '1/16', 9, 3], ['0', '1/2', '3/8', '-17/96', '-7/384', '157/4608', '-101/18432']], ['regression: random impulse 1', [['1', '1/2', '2'], ['-1', '1/3'], '1/16', 7, 1], ['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']], ['control: negative alternating tail', [['1'], ['1', '1/2'], '1/8', 12, 2], ['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']], ['control: exact epsilon boundary', [['1'], ['1', '-1/2'], '1/8', 12, 1], ['1', '1/2', '1/4', '1/8', '1/16']], ['control: zero taps inside fir part', [['1', '0', '0', '1'], ['1', '-1/2'], '1/100', 12, 1], ['1', '1/2', '1/4', '9/8', '9/16', '9/32', '9/64', '9/128', '9/256', '9/512', '9/1024']], ['control: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']]], [['regression: random impulse 3', [['-1', '-1', '1/2'], ['-1', '-1/4'], '1/8', 12, 3], ['1', '3/4', '-11/16', '11/64', '-11/256', '11/1024', '-11/4096']], ['regression: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['regression: random impulse 1', [['1', '1/2', '2'], ['-1', '1/3'], '1/16', 7, 1], ['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']], ['control: bad a0', [['1'], ['0', '1'], '1/8', 5, 1], 'bad-a0'], ['control: sparse echo hold 4', [['1'], ['1', '0', '0', '0', '-1/2'], '1/3', 20, 4], ['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']], ['control: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['control: sparse negative echo hold 3', [['2'], ['1', '0', '0', '1/2'], '1/2', 20, 3], ['2', '0', '0', '-1', '0', '0', '1/2', '0', '0', '-1/4']]], [['regression: random impulse 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']], ['regression: random impulse 9', [['1'], ['-1', '1/4'], '1/8', 9, 3], ['-1', '-1/4', '-1/16', '-1/64', '-1/256']], ['regression: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['control: gap inside numerator', [['1', '0', '1'], ['1'], '1/2', 8, 1], ['1', '0', '1', '0']], ['control: gap inside numerator with pole', [['1', '0', '0', '2'], ['1', '1/4'], '1/8', 10, 1], ['1', '-1/4', '1/16', '127/64', '-127/256', '127/1024']], ['control: two gaps in numerator', [['1', '0', '0', '-1'], ['1'], '1/2', 8, 2], ['1', '0', '0', '-1', '0', '0']], ['control: random impulse 2', [['0', '1/2', '-1'], ['1', '1/2'], '1/4', 8, 3], ['0', '1/2', '-5/4', '5/8', '-5/16', '5/32', '-5/64', '5/128']]], [['regression: random impulse 11', [['-1', '0'], ['2', '-1/2', '-1/2'], '1/10', 6, 2], ['-1/2', '-1/8', '-5/32', '-9/128', '-29/512']], ['regression: random impulse 13', [['2', '0', '2'], ['-1', '-1/4'], '1/4', 11, 3], ['-2', '1/2', '-17/8', '17/32', '-17/128', '17/512', '-17/2048']], ['regression: random impulse 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']], ['control: random impulse 6', [['-1', '1', '-1'], ['1', '1/2', '1/2'], '1/16', 9, 3], ['-1', '3/2', '-5/4', '-1/8', '11/16', '-9/32', '-13/64', '31/128', '-5/256']], ['control: random impulse 7', [['-1', '1', '-1'], ['1', '1/2'], '1/16', 11, 3], ['-1', '3/2', '-7/4', '7/8', '-7/16', '7/32', '-7/64', '7/128', '-7/256', '7/512']], ['control: random impulse 12', [['1/2', '-1'], ['1', '1/2'], '1/4', 11, 1], ['1/2', '-5/4', '5/8', '-5/16', '5/32']], ['control: random impulse 14', [['1'], ['1', '-1/4', '1/4'], '1/10', 8, 3], ['1', '1/4', '-3/16', '-7/64', '5/256', '33/1024', '13/4096']]], [['regression: random impulse 17', [['1/2', '1'], ['2', '1/2'], '1/8', 12, 3], ['1/4', '7/16', '-7/64', '7/256', '-7/1024']], ['regression: random impulse 18', [['1', '-1', '0'], ['2', '1/4', '0'], '1/16', 5, 2], ['1/2', '-9/16', '9/128', '-9/1024', '9/8192']], ['regression: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: random impulse 15', [['2'], ['1', '0'], '1/8', 11, 1], ['2', '0']], ['control: random impulse 19', [['1/2', '-1', '0'], ['1', '1/3', '0'], '1/8', 9, 2], ['1/2', '-7/6', '7/18', '-7/54', '7/162', '-7/486']], ['control: random impulse 21', [['-1'], ['1', '1/4'], '1/16', 12, 1], ['-1', '1/4', '-1/16', '1/64']], ['control: random impulse 22', [['0'], ['-1', '0', '1/4'], '1/16', 11, 1], ['0']]]]
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: a0 not one['2', '2', '1', '1/2', '1/4', '1/8', '1/16', '1/32', '1/64']['1', '1', '1/2', '1/4', '1/8', '1/16', '1/32', '1/64']Failed
regression: random impulse 0['0', '1', '3/4', '-17/48', '-7/192', '157/2304', '-101/9216', '-953/110592', '587/147456']['0', '1/2', '3/8', '-17/96', '-7/384', '157/4608', '-101/18432']Failed
regression: random impulse 1['1', '5/6', '41/18', '41/54', '41/162', '41/486', '41/1458']['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']Failed
control: negative alternating tail['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']Passed
control: exact epsilon boundary['1', '1/2', '1/4', '1/8', '1/16']['1', '1/2', '1/4', '1/8', '1/16']Passed
control: zero taps inside fir part['1', '1/2', '1/4', '9/8', '9/16', '9/32', '9/64', '9/128', '9/256', '9/512', '9/1024']['1', '1/2', '1/4', '9/8', '9/16', '9/32', '9/64', '9/128', '9/256', '9/512', '9/1024']Passed
control: noisy tail resets hold['1', '0', '-1/4', '0', '1/16', '0']['1', '0', '-1/4', '0', '1/16', '0']Passed

SHA-256 / 224dd0026439c55b49459513ba62efdc99c080b8dc32d15193f6fdea2d04935f

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):
    b = [Fraction(v) for v in x[0]]
    a = [Fraction(v) for v in x[1]]
    eps, maxlen, hold = Fraction(x[2]), x[3], x[4]
    if not a or a[0] == 0:
        return 'bad-a0'
    b = [v / a[0] for v in b]
    h = []
    quiet = 0
    for n in range(maxlen):
        acc = b[n] if n < len(b) else Fraction(0)
        for k in range(1, len(a)):
            if n - k >= 0:
                acc -= a[k] * h[n - k]
        h.append(acc)
        if n >= len(b) - 1 and abs(acc) < eps:
            quiet += 1
            if quiet >= hold:
                break
        else:
            quiet = 0
    return [str(v) for v in h]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: a0 not one', [['2', '1'], ['2', '-1'], '1/16', 10, 2], ['1', '1', '1/2', '1/4', '1/8', '1/16', '1/32', '1/64']], ['regression: random impulse 0', [['0', '1', '1'], ['2', '1/2', '1/3'], '1/16', 9, 3], ['0', '1/2', '3/8', '-17/96', '-7/384', '157/4608', '-101/18432']], ['regression: random impulse 1', [['1', '1/2', '2'], ['-1', '1/3'], '1/16', 7, 1], ['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']], ['control: negative alternating tail', [['1'], ['1', '1/2'], '1/8', 12, 2], ['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']], ['control: exact epsilon boundary', [['1'], ['1', '-1/2'], '1/8', 12, 1], ['1', '1/2', '1/4', '1/8', '1/16']], ['control: zero taps inside fir part', [['1', '0', '0', '1'], ['1', '-1/2'], '1/100', 12, 1], ['1', '1/2', '1/4', '9/8', '9/16', '9/32', '9/64', '9/128', '9/256', '9/512', '9/1024']], ['control: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']]], [['regression: random impulse 3', [['-1', '-1', '1/2'], ['-1', '-1/4'], '1/8', 12, 3], ['1', '3/4', '-11/16', '11/64', '-11/256', '11/1024', '-11/4096']], ['regression: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['regression: random impulse 1', [['1', '1/2', '2'], ['-1', '1/3'], '1/16', 7, 1], ['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']], ['control: bad a0', [['1'], ['0', '1'], '1/8', 5, 1], 'bad-a0'], ['control: sparse echo hold 4', [['1'], ['1', '0', '0', '0', '-1/2'], '1/3', 20, 4], ['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']], ['control: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['control: sparse negative echo hold 3', [['2'], ['1', '0', '0', '1/2'], '1/2', 20, 3], ['2', '0', '0', '-1', '0', '0', '1/2', '0', '0', '-1/4']]], [['regression: random impulse 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']], ['regression: random impulse 9', [['1'], ['-1', '1/4'], '1/8', 9, 3], ['-1', '-1/4', '-1/16', '-1/64', '-1/256']], ['regression: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['control: gap inside numerator', [['1', '0', '1'], ['1'], '1/2', 8, 1], ['1', '0', '1', '0']], ['control: gap inside numerator with pole', [['1', '0', '0', '2'], ['1', '1/4'], '1/8', 10, 1], ['1', '-1/4', '1/16', '127/64', '-127/256', '127/1024']], ['control: two gaps in numerator', [['1', '0', '0', '-1'], ['1'], '1/2', 8, 2], ['1', '0', '0', '-1', '0', '0']], ['control: random impulse 2', [['0', '1/2', '-1'], ['1', '1/2'], '1/4', 8, 3], ['0', '1/2', '-5/4', '5/8', '-5/16', '5/32', '-5/64', '5/128']]], [['regression: random impulse 11', [['-1', '0'], ['2', '-1/2', '-1/2'], '1/10', 6, 2], ['-1/2', '-1/8', '-5/32', '-9/128', '-29/512']], ['regression: random impulse 13', [['2', '0', '2'], ['-1', '-1/4'], '1/4', 11, 3], ['-2', '1/2', '-17/8', '17/32', '-17/128', '17/512', '-17/2048']], ['regression: random impulse 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']], ['control: random impulse 6', [['-1', '1', '-1'], ['1', '1/2', '1/2'], '1/16', 9, 3], ['-1', '3/2', '-5/4', '-1/8', '11/16', '-9/32', '-13/64', '31/128', '-5/256']], ['control: random impulse 7', [['-1', '1', '-1'], ['1', '1/2'], '1/16', 11, 3], ['-1', '3/2', '-7/4', '7/8', '-7/16', '7/32', '-7/64', '7/128', '-7/256', '7/512']], ['control: random impulse 12', [['1/2', '-1'], ['1', '1/2'], '1/4', 11, 1], ['1/2', '-5/4', '5/8', '-5/16', '5/32']], ['control: random impulse 14', [['1'], ['1', '-1/4', '1/4'], '1/10', 8, 3], ['1', '1/4', '-3/16', '-7/64', '5/256', '33/1024', '13/4096']]], [['regression: random impulse 17', [['1/2', '1'], ['2', '1/2'], '1/8', 12, 3], ['1/4', '7/16', '-7/64', '7/256', '-7/1024']], ['regression: random impulse 18', [['1', '-1', '0'], ['2', '1/4', '0'], '1/16', 5, 2], ['1/2', '-9/16', '9/128', '-9/1024', '9/8192']], ['regression: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: random impulse 15', [['2'], ['1', '0'], '1/8', 11, 1], ['2', '0']], ['control: random impulse 19', [['1/2', '-1', '0'], ['1', '1/3', '0'], '1/8', 9, 2], ['1/2', '-7/6', '7/18', '-7/54', '7/162', '-7/486']], ['control: random impulse 21', [['-1'], ['1', '1/4'], '1/16', 12, 1], ['-1', '1/4', '-1/16', '1/64']], ['control: random impulse 22', [['0'], ['-1', '0', '1/4'], '1/16', 11, 1], ['0']]]]
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: a0 not one['1', '3/2', '3/2', '3/2', '3/2', '3/2', '3/2', '3/2', '3/2', '3/2']['1', '1', '1/2', '1/4', '1/8', '1/16', '1/32', '1/64']Failed
regression: random impulse 0['0', '1/2', '1/4', '-7/24', '1/16', '19/288', '-31/576', '17/3456', '107/6912']['0', '1/2', '3/8', '-17/96', '-7/384', '157/4608', '-101/18432']Failed
regression: random impulse 1['-1', '-1/6', '-35/18', '35/54', '-35/162', '35/486', '-35/1458']['-1', '-5/6', '-41/18', '-41/54', '-41/162', '-41/486', '-41/1458']Failed
control: negative alternating tail['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']Passed
control: exact epsilon boundary['1', '1/2', '1/4', '1/8', '1/16']['1', '1/2', '1/4', '1/8', '1/16']Passed
control: zero taps inside fir part['1', '1/2', '1/4', '9/8', '9/16', '9/32', '9/64', '9/128', '9/256', '9/512', '9/1024']['1', '1/2', '1/4', '9/8', '9/16', '9/32', '9/64', '9/128', '9/256', '9/512', '9/1024']Passed
control: noisy tail resets hold['1', '0', '-1/4', '0', '1/16', '0']['1', '0', '-1/4', '0', '1/16', '0']Passed

SHA-256 / 67906a8c60af4e71496b4bc2f271bfd96747f2db9cec3837efdf703892dc5800

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.

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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 2026-09-29T14:51:39.410769+00:00.

Case digest / 96f9df9486153019aaa8ad120bdb9c11c6c3b9d761104db3bcffd21b54cbf27b