FA-91776 / Digital signal filters / Open access

Impulse truncation treats any negative sample as quiet · case 01

An alternating response is cut at its first large negative sample.

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

ROOT CAUSE

The test is h < eps instead of |h| < eps.

VERIFIED REPAIR

Compare the magnitude |h| with eps.

Unsuccessful approach: The attempted repair uses |h| <= eps, which also stops on samples exactly equal to eps.

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'
    b = [v / a[0] for v in b]
    a = [v / a[0] for v in a]
    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 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: negative alternating tail', [['1'], ['1', '1/2'], '1/8', 12, 2], ['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']], ['regression: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['repair check: 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: 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']]], [['regression: 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']], ['regression: two gaps in numerator', [['1', '0', '0', '-1'], ['1'], '1/2', 8, 2], ['1', '0', '0', '-1', '0', '0']], ['repair check: 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']], ['control: gap inside numerator', [['1', '0', '1'], ['1'], '1/2', 8, 1], ['1', '0', '1', '0']], ['control: 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']], ['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 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']]], [['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']], ['regression: 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 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: 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']], ['control: random impulse 15', [['2'], ['1', '0'], '1/8', 11, 1], ['2', '0']], ['control: random impulse 16', [['2', '-1', '1'], ['-1', '-1/2'], '1/16', 4, 3], ['-2', '2', '-2', '1']], ['control: 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 5', [['1'], ['-1', '1/3'], '1/8', 9, 2], ['-1', '-1/3', '-1/9', '-1/27']], ['regression: 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']], ['repair check: random impulse 29', [['1/2'], ['2', '1/4', '-1/2'], '1/4', 7, 3], ['1/4', '-1/32', '17/256', '-33/2048']], ['control: random impulse 18', [['1', '-1', '0'], ['2', '1/4', '0'], '1/16', 5, 2], ['1/2', '-9/16', '9/128', '-9/1024', '9/8192']], ['control: random impulse 20', [['1', '0'], ['2', '1/3'], '1/8', 6, 1], ['1/2', '-1/12']], ['control: random impulse 22', [['0'], ['-1', '0', '1/4'], '1/16', 11, 1], ['0']], ['control: random impulse 23', [['1'], ['1', '0', '0'], '1/4', 10, 3], ['1', '0', '0', '0']]], [['regression: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['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']], ['repair check: random impulse 40', [['-1'], ['2', '1/2', '1/3'], '1/8', 11, 3], ['-1/2', '1/8', '5/96', '-13/384', '-1/4608']], ['control: random impulse 26', [['1/2', '0'], ['-1', '-1/4', '1/2'], '1/16', 9, 2], ['-1/2', '1/8', '-9/32', '17/128', '-89/512', '225/2048', '-937/8192', '2737/32768', '-10233/131072']], ['control: random impulse 27', [['2', '-1'], ['1', '1/2', '-1/2'], '1/8', 9, 3], ['2', '-2', '2', '-2', '2', '-2', '2', '-2', '2']], ['control: random impulse 28', [['2', '1', '1/2'], ['2', '1/2'], '1/8', 5, 2], ['1', '1/4', '3/16', '-3/64', '3/256']], ['control: random impulse 30', [['0'], ['1', '1/2'], '1/8', 8, 3], ['0', '0', '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 fixture	Actual	Expected	Outcome
regression: negative alternating tail	`['1', '-1/2', '1/4', '-1/8', '1/16']`	`['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']`	Failed
regression: noisy tail resets hold	`['1', '0', '-1/4', '0']`	`['1', '0', '-1/4', '0', '1/16', '0']`	Failed
repair check: 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: bad a0	`bad-a0`	`bad-a0`	Passed
control: sparse echo hold 4	`['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']`	`['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']`	Passed
control: sparse echo hold 3	`['1', '0', '0', '1/2', '0', '0', '1/4']`	`['1', '0', '0', '1/2', '0', '0', '1/4']`	Passed

SHA-256 / f2b48315ab49ac04b210a7fb0f81881332abebceac521258299d05d3e57102e4

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]
    a = [v / a[0] for v in a]
    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: negative alternating tail', [['1'], ['1', '1/2'], '1/8', 12, 2], ['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']], ['regression: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['repair check: 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: 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']]], [['regression: 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']], ['regression: two gaps in numerator', [['1', '0', '0', '-1'], ['1'], '1/2', 8, 2], ['1', '0', '0', '-1', '0', '0']], ['repair check: 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']], ['control: gap inside numerator', [['1', '0', '1'], ['1'], '1/2', 8, 1], ['1', '0', '1', '0']], ['control: 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']], ['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 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']]], [['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']], ['regression: 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 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: 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']], ['control: random impulse 15', [['2'], ['1', '0'], '1/8', 11, 1], ['2', '0']], ['control: random impulse 16', [['2', '-1', '1'], ['-1', '-1/2'], '1/16', 4, 3], ['-2', '2', '-2', '1']], ['control: 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 5', [['1'], ['-1', '1/3'], '1/8', 9, 2], ['-1', '-1/3', '-1/9', '-1/27']], ['regression: 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']], ['repair check: random impulse 29', [['1/2'], ['2', '1/4', '-1/2'], '1/4', 7, 3], ['1/4', '-1/32', '17/256', '-33/2048']], ['control: random impulse 18', [['1', '-1', '0'], ['2', '1/4', '0'], '1/16', 5, 2], ['1/2', '-9/16', '9/128', '-9/1024', '9/8192']], ['control: random impulse 20', [['1', '0'], ['2', '1/3'], '1/8', 6, 1], ['1/2', '-1/12']], ['control: random impulse 22', [['0'], ['-1', '0', '1/4'], '1/16', 11, 1], ['0']], ['control: random impulse 23', [['1'], ['1', '0', '0'], '1/4', 10, 3], ['1', '0', '0', '0']]], [['regression: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['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']], ['repair check: random impulse 40', [['-1'], ['2', '1/2', '1/3'], '1/8', 11, 3], ['-1/2', '1/8', '5/96', '-13/384', '-1/4608']], ['control: random impulse 26', [['1/2', '0'], ['-1', '-1/4', '1/2'], '1/16', 9, 2], ['-1/2', '1/8', '-9/32', '17/128', '-89/512', '225/2048', '-937/8192', '2737/32768', '-10233/131072']], ['control: random impulse 27', [['2', '-1'], ['1', '1/2', '-1/2'], '1/8', 9, 3], ['2', '-2', '2', '-2', '2', '-2', '2', '-2', '2']], ['control: random impulse 28', [['2', '1', '1/2'], ['2', '1/2'], '1/8', 5, 2], ['1', '1/4', '3/16', '-3/64', '3/256']], ['control: random impulse 30', [['0'], ['1', '1/2'], '1/8', 8, 3], ['0', '0', '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 fixture	Actual	Expected	Outcome
regression: negative alternating tail	`['1', '-1/2', '1/4', '-1/8', '1/16']`	`['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']`	Failed
regression: noisy tail resets hold	`['1', '0', '-1/4', '0', '1/16', '0']`	`['1', '0', '-1/4', '0', '1/16', '0']`	Passed
repair check: exact epsilon boundary	`['1', '1/2', '1/4', '1/8']`	`['1', '1/2', '1/4', '1/8', '1/16']`	Failed
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: bad a0	`bad-a0`	`bad-a0`	Passed
control: sparse echo hold 4	`['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']`	`['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']`	Passed
control: sparse echo hold 3	`['1', '0', '0', '1/2', '0', '0', '1/4']`	`['1', '0', '0', '1/2', '0', '0', '1/4']`	Passed

SHA-256 / 4bb1981da777ce776386e5b5da8683945de049e0a40cd0d46d413d51e13f4f9c

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):
    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]
    a = [v / a[0] for v in a]
    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: negative alternating tail', [['1'], ['1', '1/2'], '1/8', 12, 2], ['1', '-1/2', '1/4', '-1/8', '1/16', '-1/32']], ['regression: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['repair check: 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: 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']]], [['regression: 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']], ['regression: two gaps in numerator', [['1', '0', '0', '-1'], ['1'], '1/2', 8, 2], ['1', '0', '0', '-1', '0', '0']], ['repair check: 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']], ['control: gap inside numerator', [['1', '0', '1'], ['1'], '1/2', 8, 1], ['1', '0', '1', '0']], ['control: 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']], ['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 8', [['1/2', '2', '-1'], ['-1', '1/4'], '1/8', 5, 1], ['-1/2', '-17/8', '15/32', '15/128']]], [['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']], ['regression: 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 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: 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']], ['control: random impulse 15', [['2'], ['1', '0'], '1/8', 11, 1], ['2', '0']], ['control: random impulse 16', [['2', '-1', '1'], ['-1', '-1/2'], '1/16', 4, 3], ['-2', '2', '-2', '1']], ['control: 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 5', [['1'], ['-1', '1/3'], '1/8', 9, 2], ['-1', '-1/3', '-1/9', '-1/27']], ['regression: 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']], ['repair check: random impulse 29', [['1/2'], ['2', '1/4', '-1/2'], '1/4', 7, 3], ['1/4', '-1/32', '17/256', '-33/2048']], ['control: random impulse 18', [['1', '-1', '0'], ['2', '1/4', '0'], '1/16', 5, 2], ['1/2', '-9/16', '9/128', '-9/1024', '9/8192']], ['control: random impulse 20', [['1', '0'], ['2', '1/3'], '1/8', 6, 1], ['1/2', '-1/12']], ['control: random impulse 22', [['0'], ['-1', '0', '1/4'], '1/16', 11, 1], ['0']], ['control: random impulse 23', [['1'], ['1', '0', '0'], '1/4', 10, 3], ['1', '0', '0', '0']]], [['regression: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['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']], ['repair check: random impulse 40', [['-1'], ['2', '1/2', '1/3'], '1/8', 11, 3], ['-1/2', '1/8', '5/96', '-13/384', '-1/4608']], ['control: random impulse 26', [['1/2', '0'], ['-1', '-1/4', '1/2'], '1/16', 9, 2], ['-1/2', '1/8', '-9/32', '17/128', '-89/512', '225/2048', '-937/8192', '2737/32768', '-10233/131072']], ['control: random impulse 27', [['2', '-1'], ['1', '1/2', '-1/2'], '1/8', 9, 3], ['2', '-2', '2', '-2', '2', '-2', '2', '-2', '2']], ['control: random impulse 28', [['2', '1', '1/2'], ['2', '1/2'], '1/8', 5, 2], ['1', '1/4', '3/16', '-3/64', '3/256']], ['control: random impulse 30', [['0'], ['1', '1/2'], '1/8', 8, 3], ['0', '0', '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 fixture	Actual	Expected	Outcome
regression: 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
regression: noisy tail resets hold	`['1', '0', '-1/4', '0', '1/16', '0']`	`['1', '0', '-1/4', '0', '1/16', '0']`	Passed
repair check: 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: bad a0	`bad-a0`	`bad-a0`	Passed
control: sparse echo hold 4	`['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']`	`['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']`	Passed
control: sparse echo hold 3	`['1', '0', '0', '1/2', '0', '0', '1/4']`	`['1', '0', '0', '1/2', '0', '0', '1/4']`	Passed

SHA-256 / 993d54fa647be23c780557e20e9361bb7e21425cb67ed734dfd4fb0c1c52fbde

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 2026-09-29T14:51:39.090554+00:00.

Case digest / 1c8ae55cbc2e28a9c6ce4c20d3068f28669fd556b7a6373a9e58d7061c840762