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FA-91596 / Digital signal filters / Open access

Coefficient quantizer rounds ties to even · case 01

0.125 at two fractional bits quantizes to code 0 instead of 1 (half away from zero).

Verified by executionVariant 1 · 6 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Python round() on the scaled value uses banker rounding.

THE FAILURE

Python round() on the scaled value uses banker rounding.

Unsuccessful approach: The attempted repair uses floor(v + 1/2), which rounds negative ties toward positive infinity.

Case contract

Input [coeffs, f, B, mode]: quantize each decimal coefficient to a B-bit two's-complement code with f fractional bits; "nearest" rounds half away from zero, "truncate" floors. Codes saturate to [-2^(B-1), 2^(B-1)-1]. Return {"codes", "saturated": count of clipped codes, "max_error": max |code/2^f - c| after saturation as a fraction string}.

Why this case matters

Coefficient quantization decides whether a fixed-point filter still meets spec; rounding and saturation slips shift poles.

1 / The failure

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
from fractions import Fraction
N = 1
observations = []
def solve(x):
    coeffs, f, B, mode = x
    scale = 2 ** f
    hi, lo = 2 ** (B - 1) - 1, -2 ** (B - 1)
    codes = []
    sat = 0
    err = Fraction(0)
    for s in coeffs:
        c = Fraction(s)
        v = c * scale
        if mode == 'nearest':
            q = round(v)
        else:
            q = math.floor(v)
        if q > hi or q < lo:
            sat += 1
            q = max(lo, min(hi, q))
        codes.append(q)
        err = max(err, abs(Fraction(q, scale) - c))
    return {'codes': codes, 'saturated': sat, 'max_error': str(err)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: negative truncate', [['-0.3', '0.3'], 3, 8, 'truncate'], {'codes': [-3, 2], 'saturated': 0, 'max_error': '3/40'}], ['control: positive overflow', [['1', '0.99'], 7, 8, 'nearest'], {'codes': [127, 127], 'saturated': 1, 'max_error': '1/128'}], ['control: negative full scale', [['-1'], 7, 8, 'nearest'], {'codes': [-128], 'saturated': 0, 'max_error': '0'}], ['control: negative overflow', [['-1.5'], 7, 8, 'truncate'], {'codes': [-128], 'saturated': 1, 'max_error': '1/2'}], ['control: exact grid truncate', [['0.5', '-0.25', '0.75'], 2, 4, 'truncate'], {'codes': [2, -1, 3], 'saturated': 0, 'max_error': '0'}]], [['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: exact grid', [['0.5', '-0.25'], 2, 4, 'nearest'], {'codes': [2, -1], 'saturated': 0, 'max_error': '0'}], ['control: exact grid truncate', [['0.5', '-0.25', '0.75'], 2, 4, 'truncate'], {'codes': [2, -1, 3], 'saturated': 0, 'max_error': '0'}], ['control: integer multiple truncate', [['-0.5', '1.25'], 3, 6, 'truncate'], {'codes': [-4, 10], 'saturated': 0, 'max_error': '0'}], ['control: random quantize 0', [['-1.79', '1.59'], 3, 5, 'truncate'], {'codes': [-15, 12], 'saturated': 0, 'max_error': '9/100'}], ['control: negative full scale', [['-1'], 7, 8, 'nearest'], {'codes': [-128], 'saturated': 0, 'max_error': '0'}]], [['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: random quantize 1', [['-1.919', '-2.19', '0.764', '-2.046'], 4, 7, 'truncate'], {'codes': [-31, -36, 12, -33], 'saturated': 0, 'max_error': '3/50'}], ['control: random quantize 2', [['-0.09', '-0.09', '0.397'], 4, 7, 'nearest'], {'codes': [-1, -1, 6], 'saturated': 0, 'max_error': '11/400'}], ['control: random quantize 3', [['-1.8'], 3, 5, 'nearest'], {'codes': [-14], 'saturated': 0, 'max_error': '1/20'}], ['control: random quantize 4', [['-1.49', '-1.91'], 2, 3, 'nearest'], {'codes': [-4, -4], 'saturated': 2, 'max_error': '91/100'}], ['control: negative overflow', [['-1.5'], 7, 8, 'truncate'], {'codes': [-128], 'saturated': 1, 'max_error': '1/2'}]], [['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: random quantize 5', [['1.92'], 4, 6, 'truncate'], {'codes': [30], 'saturated': 0, 'max_error': '9/200'}], ['control: random quantize 6', [['2.04', '0.49'], 3, 6, 'nearest'], {'codes': [16, 4], 'saturated': 0, 'max_error': '1/25'}], ['control: random quantize 7', [['-2.111', '2.022', '-1.492'], 3, 6, 'truncate'], {'codes': [-17, 16, -12], 'saturated': 0, 'max_error': '11/500'}], ['control: random quantize 8', [['0.08', '-2.194', '-0.29'], 4, 6, 'nearest'], {'codes': [1, -32, -5], 'saturated': 1, 'max_error': '97/500'}], ['control: exact grid', [['0.5', '-0.25'], 2, 4, 'nearest'], {'codes': [2, -1], 'saturated': 0, 'max_error': '0'}]], [['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: random quantize 9', [['-0.92', '1.79'], 4, 5, 'truncate'], {'codes': [-15, 15], 'saturated': 1, 'max_error': '341/400'}], ['control: random quantize 10', [['-1.232', '-0.29', '-0.82', '0.594'], 4, 5, 'truncate'], {'codes': [-16, -5, -14, 9], 'saturated': 1, 'max_error': '29/125'}], ['control: random quantize 11', [['-0.158', '1.311', '1.13', '-0.89'], 6, 8, 'truncate'], {'codes': [-11, 83, 72, -57], 'saturated': 0, 'max_error': '113/8000'}], ['control: random quantize 12', [['-0.52', '-2.037', '1.37'], 3, 6, 'nearest'], {'codes': [-4, -16, 11], 'saturated': 0, 'max_error': '37/1000'}], ['control: exact grid truncate', [['0.5', '-0.25', '0.75'], 2, 4, 'truncate'], {'codes': [2, -1, 3], 'saturated': 0, 'max_error': '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: tie cases nearest{'codes': [0, 2, 0], 'max_error': '1/8', 'saturated': 0}{'codes': [1, 2, -1], 'max_error': '1/8', 'saturated': 0}Failed
control: negative truncate{'codes': [-3, 2], 'max_error': '3/40', 'saturated': 0}{'codes': [-3, 2], 'max_error': '3/40', 'saturated': 0}Passed
control: positive overflow{'codes': [127, 127], 'max_error': '1/128', 'saturated': 1}{'codes': [127, 127], 'max_error': '1/128', 'saturated': 1}Passed
control: negative full scale{'codes': [-128], 'max_error': '0', 'saturated': 0}{'codes': [-128], 'max_error': '0', 'saturated': 0}Passed
control: negative overflow{'codes': [-128], 'max_error': '1/2', 'saturated': 1}{'codes': [-128], 'max_error': '1/2', 'saturated': 1}Passed
control: exact grid truncate{'codes': [2, -1, 3], 'max_error': '0', 'saturated': 0}{'codes': [2, -1, 3], 'max_error': '0', 'saturated': 0}Passed

SHA-256 / 57d74ee6425f22c313546f1009b50947eaf08398d06d6190af04684421cd9d3a

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
from fractions import Fraction
N = 1
observations = []
def solve(x):
    coeffs, f, B, mode = x
    scale = 2 ** f
    hi, lo = 2 ** (B - 1) - 1, -2 ** (B - 1)
    codes = []
    sat = 0
    err = Fraction(0)
    for s in coeffs:
        c = Fraction(s)
        v = c * scale
        if mode == 'nearest':
            q = math.floor(v + Fraction(1, 2))
        else:
            q = math.floor(v)
        if q > hi or q < lo:
            sat += 1
            q = max(lo, min(hi, q))
        codes.append(q)
        err = max(err, abs(Fraction(q, scale) - c))
    return {'codes': codes, 'saturated': sat, 'max_error': str(err)}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: negative truncate', [['-0.3', '0.3'], 3, 8, 'truncate'], {'codes': [-3, 2], 'saturated': 0, 'max_error': '3/40'}], ['control: positive overflow', [['1', '0.99'], 7, 8, 'nearest'], {'codes': [127, 127], 'saturated': 1, 'max_error': '1/128'}], ['control: negative full scale', [['-1'], 7, 8, 'nearest'], {'codes': [-128], 'saturated': 0, 'max_error': '0'}], ['control: negative overflow', [['-1.5'], 7, 8, 'truncate'], {'codes': [-128], 'saturated': 1, 'max_error': '1/2'}], ['control: exact grid truncate', [['0.5', '-0.25', '0.75'], 2, 4, 'truncate'], {'codes': [2, -1, 3], 'saturated': 0, 'max_error': '0'}]], [['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: exact grid', [['0.5', '-0.25'], 2, 4, 'nearest'], {'codes': [2, -1], 'saturated': 0, 'max_error': '0'}], ['control: exact grid truncate', [['0.5', '-0.25', '0.75'], 2, 4, 'truncate'], {'codes': [2, -1, 3], 'saturated': 0, 'max_error': '0'}], ['control: integer multiple truncate', [['-0.5', '1.25'], 3, 6, 'truncate'], {'codes': [-4, 10], 'saturated': 0, 'max_error': '0'}], ['control: random quantize 0', [['-1.79', '1.59'], 3, 5, 'truncate'], {'codes': [-15, 12], 'saturated': 0, 'max_error': '9/100'}], ['control: negative full scale', [['-1'], 7, 8, 'nearest'], {'codes': [-128], 'saturated': 0, 'max_error': '0'}]], [['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: random quantize 1', [['-1.919', '-2.19', '0.764', '-2.046'], 4, 7, 'truncate'], {'codes': [-31, -36, 12, -33], 'saturated': 0, 'max_error': '3/50'}], ['control: random quantize 2', [['-0.09', '-0.09', '0.397'], 4, 7, 'nearest'], {'codes': [-1, -1, 6], 'saturated': 0, 'max_error': '11/400'}], ['control: random quantize 3', [['-1.8'], 3, 5, 'nearest'], {'codes': [-14], 'saturated': 0, 'max_error': '1/20'}], ['control: random quantize 4', [['-1.49', '-1.91'], 2, 3, 'nearest'], {'codes': [-4, -4], 'saturated': 2, 'max_error': '91/100'}], ['control: negative overflow', [['-1.5'], 7, 8, 'truncate'], {'codes': [-128], 'saturated': 1, 'max_error': '1/2'}]], [['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: random quantize 5', [['1.92'], 4, 6, 'truncate'], {'codes': [30], 'saturated': 0, 'max_error': '9/200'}], ['control: random quantize 6', [['2.04', '0.49'], 3, 6, 'nearest'], {'codes': [16, 4], 'saturated': 0, 'max_error': '1/25'}], ['control: random quantize 7', [['-2.111', '2.022', '-1.492'], 3, 6, 'truncate'], {'codes': [-17, 16, -12], 'saturated': 0, 'max_error': '11/500'}], ['control: random quantize 8', [['0.08', '-2.194', '-0.29'], 4, 6, 'nearest'], {'codes': [1, -32, -5], 'saturated': 1, 'max_error': '97/500'}], ['control: exact grid', [['0.5', '-0.25'], 2, 4, 'nearest'], {'codes': [2, -1], 'saturated': 0, 'max_error': '0'}]], [['regression: tie cases nearest', [['0.125', '0.375', '-0.125'], 2, 8, 'nearest'], {'codes': [1, 2, -1], 'saturated': 0, 'max_error': '1/8'}], ['control: random quantize 9', [['-0.92', '1.79'], 4, 5, 'truncate'], {'codes': [-15, 15], 'saturated': 1, 'max_error': '341/400'}], ['control: random quantize 10', [['-1.232', '-0.29', '-0.82', '0.594'], 4, 5, 'truncate'], {'codes': [-16, -5, -14, 9], 'saturated': 1, 'max_error': '29/125'}], ['control: random quantize 11', [['-0.158', '1.311', '1.13', '-0.89'], 6, 8, 'truncate'], {'codes': [-11, 83, 72, -57], 'saturated': 0, 'max_error': '113/8000'}], ['control: random quantize 12', [['-0.52', '-2.037', '1.37'], 3, 6, 'nearest'], {'codes': [-4, -16, 11], 'saturated': 0, 'max_error': '37/1000'}], ['control: exact grid truncate', [['0.5', '-0.25', '0.75'], 2, 4, 'truncate'], {'codes': [2, -1, 3], 'saturated': 0, 'max_error': '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: tie cases nearest{'codes': [1, 2, 0], 'max_error': '1/8', 'saturated': 0}{'codes': [1, 2, -1], 'max_error': '1/8', 'saturated': 0}Failed
control: negative truncate{'codes': [-3, 2], 'max_error': '3/40', 'saturated': 0}{'codes': [-3, 2], 'max_error': '3/40', 'saturated': 0}Passed
control: positive overflow{'codes': [127, 127], 'max_error': '1/128', 'saturated': 1}{'codes': [127, 127], 'max_error': '1/128', 'saturated': 1}Passed
control: negative full scale{'codes': [-128], 'max_error': '0', 'saturated': 0}{'codes': [-128], 'max_error': '0', 'saturated': 0}Passed
control: negative overflow{'codes': [-128], 'max_error': '1/2', 'saturated': 1}{'codes': [-128], 'max_error': '1/2', 'saturated': 1}Passed
control: exact grid truncate{'codes': [2, -1, 3], 'max_error': '0', 'saturated': 0}{'codes': [2, -1, 3], 'max_error': '0', 'saturated': 0}Passed

SHA-256 / 3b40465180b296fbdc0a2b854f35efff77e9e4510604d2ba6df88797850d461d

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

This mechanism has 6 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 2026-09-29T14:51:37.528413+00:00.

Case digest / c9924116ead0d0f86709f8484d9ffcdfe9ae169a3aa8366ac5d567fbe0a95e1d