FA-91771 / Digital signal filters / Open access

Impulse truncation counts quiet samples cumulatively · case 01

A response that dips below eps, rebounds and dips again is truncated while still ringing.

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

ROOT CAUSE

The quiet counter is never reset by a loud sample, so non-consecutive quiet samples add up.

VERIFIED REPAIR

Reset the counter whenever a sample is not quiet.

Unsuccessful approach: The attempted repair only decrements the counter on a loud sample.

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 abs(acc) < eps:
            quiet += 1
            if quiet >= hold:
                break
        else:
            pass
    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: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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']], ['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['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: 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: 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: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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: bad a0', [['1'], ['0', '1'], '1/8', 5, 1], 'bad-a0'], ['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']]], [['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['regression: 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: 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: 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']], ['control: 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: 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']], ['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']]], [['regression: 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']], ['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['regression: 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']], ['control: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['control: random impulse 5', [['1'], ['-1', '1/3'], '1/8', 9, 2], ['-1', '-1/3', '-1/9', '-1/27']], ['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']]], [['regression: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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']], ['regression: 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']], ['control: 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 9', [['1'], ['-1', '1/4'], '1/8', 9, 3], ['-1', '-1/4', '-1/16', '-1/64', '-1/256']], ['control: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: random impulse 11', [['-1', '0'], ['2', '-1/2', '-1/2'], '1/10', 6, 2], ['-1/2', '-1/8', '-5/32', '-9/128', '-29/512']]]]
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: noisy tail resets hold	`['1', '0', '-1/4', '0', '1/16']`	`['1', '0', '-1/4', '0', '1/16', '0']`	Failed
regression: sparse echo hold 4	`['1', '0', '0', '0', '1/2', '0']`	`['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']`	Failed
regression: sparse echo hold 3	`['1', '0', '0', '1/2', '0']`	`['1', '0', '0', '1/2', '0', '0', '1/4']`	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: a0 not one	`['1', '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']`	Passed

SHA-256 / 1e41cb1bb75cf78eba902ba7c5fc4d6358e7db811776eb618ccebe33033d1ef4

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 = max(quiet - 1, 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: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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']], ['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['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: 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: 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: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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: bad a0', [['1'], ['0', '1'], '1/8', 5, 1], 'bad-a0'], ['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']]], [['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['regression: 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: 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: 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']], ['control: 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: 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']], ['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']]], [['regression: 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']], ['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['regression: 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']], ['control: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['control: random impulse 5', [['1'], ['-1', '1/3'], '1/8', 9, 2], ['-1', '-1/3', '-1/9', '-1/27']], ['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']]], [['regression: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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']], ['regression: 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']], ['control: 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 9', [['1'], ['-1', '1/4'], '1/8', 9, 3], ['-1', '-1/4', '-1/16', '-1/64', '-1/256']], ['control: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: random impulse 11', [['-1', '0'], ['2', '-1/2', '-1/2'], '1/10', 6, 2], ['-1/2', '-1/8', '-5/32', '-9/128', '-29/512']]]]
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: noisy tail resets hold	`['1', '0', '-1/4', '0', '1/16', '0']`	`['1', '0', '-1/4', '0', '1/16', '0']`	Passed
regression: sparse echo hold 4	`['1', '0', '0', '0', '1/2', '0', '0']`	`['1', '0', '0', '0', '1/2', '0', '0', '0', '1/4']`	Failed
regression: sparse echo hold 3	`['1', '0', '0', '1/2', '0', '0']`	`['1', '0', '0', '1/2', '0', '0', '1/4']`	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: a0 not one	`['1', '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']`	Passed

SHA-256 / af17930c16e9fa84e130ca1e0e5dac1180e0851bf263eb168dca35e6bd7cda1a

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: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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']], ['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['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: 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: 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: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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: bad a0', [['1'], ['0', '1'], '1/8', 5, 1], 'bad-a0'], ['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']]], [['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['regression: 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: 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: 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']], ['control: 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: 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']], ['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']]], [['regression: 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']], ['regression: sparse echo hold 3', [['1'], ['1', '0', '0', '-1/2'], '1/3', 20, 3], ['1', '0', '0', '1/2', '0', '0', '1/4']], ['regression: 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']], ['control: random impulse 4', [['2'], ['-1', '1/4'], '1/10', 9, 1], ['-2', '-1/2', '-1/8', '-1/32']], ['control: random impulse 5', [['1'], ['-1', '1/3'], '1/8', 9, 2], ['-1', '-1/3', '-1/9', '-1/27']], ['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']]], [['regression: noisy tail resets hold', [['1'], ['1', '0', '1/4'], '1/10', 12, 3], ['1', '0', '-1/4', '0', '1/16', '0']], ['regression: 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']], ['regression: 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']], ['control: 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 9', [['1'], ['-1', '1/4'], '1/8', 9, 3], ['-1', '-1/4', '-1/16', '-1/64', '-1/256']], ['control: random impulse 10', [['-1'], ['-1', '-1/4', '0'], '1/16', 12, 1], ['1', '-1/4', '1/16', '-1/64']], ['control: random impulse 11', [['-1', '0'], ['2', '-1/2', '-1/2'], '1/10', 6, 2], ['-1/2', '-1/8', '-5/32', '-9/128', '-29/512']]]]
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: noisy tail resets hold	`['1', '0', '-1/4', '0', '1/16', '0']`	`['1', '0', '-1/4', '0', '1/16', '0']`	Passed
regression: 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
regression: sparse echo hold 3	`['1', '0', '0', '1/2', '0', '0', '1/4']`	`['1', '0', '0', '1/2', '0', '0', '1/4']`	Passed
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: a0 not one	`['1', '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']`	Passed

SHA-256 / f2893cb37c24aa95897f1857023ea324816ea464a6a1394089ea2523c8dd0ea1

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 2026-09-29T14:51:38.963512+00:00.

Case digest / bc0618f3b5af579bac1fc54d916be2a7de5c7a869b2a3c798deae8d91ee90205