FA-91491 / Digital signal filters / Open access
Q15 smoother rejects alpha = 1.0 · case 01
alpha = 32768 (pass-through) returns "bad-alpha".
ROOT CAUSE
The upper bound check is exclusive.
VERIFIED REPAIR
Accept alpha in the closed interval [0, 32768].
Unsuccessful approach: The attempted repair makes the upper bound inclusive but excludes alpha = 0 (hold).
Case contract
Input [alpha, samples]; alpha is Q15 in [0, 32768] ("bad-alpha" otherwise). Inputs are saturated to int16; y += ((x - y) * alpha + 2**14) >> 15 (round half up on the Q15 product via arithmetic shift), starting at y = 0. Return the integer outputs.
Why this case matters
Fixed-point exponential smoothers run on microcontrollers; rounding direction and saturation slips cause bias and limit cycles.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
alpha, xs = x
if not 0 <= alpha < 32768:
return 'bad-alpha'
y = 0
out = []
for v in xs:
v = max(-32768, min(32767, v))
y = y + (((v - y) * alpha + (1 << 14)) >> 15)
out.append(y)
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: input beyond range', [16384, [40000, 40000, -40000]], [16384, 24576, -4096]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: small negative tie', [16384, [-3, 0, 0]], [-1, 0, 0]]], [['regression: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 0', [24576, [-8, -38976, 37]], [-6, -24577, -6116]], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 4', [8192, [20, -2539]], [5, -631]], ['control: random smoother 6', [8192, [16294, 2, -779, 14009]], [4074, 3056, 2097, 5075]]], [['regression: random smoother 24', [32768, [-19, -16, -18365, 21638, -14929]], [-19, -16, -18365, 21638, -14929]], ['regression: random smoother 26', [32768, [-4554, -6260, -33961]], [-4554, -6260, -32768]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 7', [1000, [-8425, 39]], [-257, -248]], ['control: random smoother 8', [30000, [45, -46]], [41, -39]], ['control: random smoother 11', [24576, [12306, -35386, 1164, 8403, 12704, 26]], [9230, -22268, -4694, 5129, 10810, 2722]], ['control: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]]], [['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 13', [8192, [-8, -27561]], [-2, -6892]], ['control: random smoother 14', [24576, [-12991, -22604, 32, -18621, -30, -36, -10331]], [-9743, -19389, -4823, -15171, -3815, -981, -7993]], ['control: random smoother 15', [30000, [13636, -12457, -1, -11, 49, 0]], [12484, -10350, -875, -84, 38, 3]], ['control: random smoother 16', [8192, [-12962, 19012, 49, -24286]], [-3240, 2323, 1755, -4755]]], [['regression: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 17', [16384, [-8485, -763, -9, 25855, -11982, -28816, 38]], [-4242, -2502, -1255, 12300, 159, -14328, -7145]], ['control: random smoother 18', [24576, [-8497, -36]], [-6373, -1620]], ['control: random smoother 19', [24576, [-1436, -36936, -26, 4657, 23423, 50]], [-1077, -24845, -6231, 1935, 18051, 4550]], ['control: random smoother 20', [16384, [6, -12155, -29083, 31]], [3, -6076, -17579, -8774]]]]
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: alpha one | bad-alpha | [100, -5, 7] | Failed |
| regression: random smoother 2 | bad-alpha | [-33, 20874, -43, 6, -28] | Failed |
| repair check: alpha zero | [0, 0] | [0, 0] | Passed |
| control: negative step | [0, 0, 0] | [0, 0, 0] | Passed |
| control: input beyond range | [16384, 24576, -4096] | [16384, 24576, -4096] | Passed |
| control: bad alpha | bad-alpha | bad-alpha | Passed |
| control: small negative tie | [-1, 0, 0] | [-1, 0, 0] | Passed |
SHA-256 / 7b1a5b37f34ecde12b59a34b4fb45923edb288612e73cb811490104347b5f89f
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
alpha, xs = x
if not 0 < alpha <= 32768:
return 'bad-alpha'
y = 0
out = []
for v in xs:
v = max(-32768, min(32767, v))
y = y + (((v - y) * alpha + (1 << 14)) >> 15)
out.append(y)
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: input beyond range', [16384, [40000, 40000, -40000]], [16384, 24576, -4096]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: small negative tie', [16384, [-3, 0, 0]], [-1, 0, 0]]], [['regression: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 0', [24576, [-8, -38976, 37]], [-6, -24577, -6116]], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 4', [8192, [20, -2539]], [5, -631]], ['control: random smoother 6', [8192, [16294, 2, -779, 14009]], [4074, 3056, 2097, 5075]]], [['regression: random smoother 24', [32768, [-19, -16, -18365, 21638, -14929]], [-19, -16, -18365, 21638, -14929]], ['regression: random smoother 26', [32768, [-4554, -6260, -33961]], [-4554, -6260, -32768]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 7', [1000, [-8425, 39]], [-257, -248]], ['control: random smoother 8', [30000, [45, -46]], [41, -39]], ['control: random smoother 11', [24576, [12306, -35386, 1164, 8403, 12704, 26]], [9230, -22268, -4694, 5129, 10810, 2722]], ['control: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]]], [['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 13', [8192, [-8, -27561]], [-2, -6892]], ['control: random smoother 14', [24576, [-12991, -22604, 32, -18621, -30, -36, -10331]], [-9743, -19389, -4823, -15171, -3815, -981, -7993]], ['control: random smoother 15', [30000, [13636, -12457, -1, -11, 49, 0]], [12484, -10350, -875, -84, 38, 3]], ['control: random smoother 16', [8192, [-12962, 19012, 49, -24286]], [-3240, 2323, 1755, -4755]]], [['regression: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 17', [16384, [-8485, -763, -9, 25855, -11982, -28816, 38]], [-4242, -2502, -1255, 12300, 159, -14328, -7145]], ['control: random smoother 18', [24576, [-8497, -36]], [-6373, -1620]], ['control: random smoother 19', [24576, [-1436, -36936, -26, 4657, 23423, 50]], [-1077, -24845, -6231, 1935, 18051, 4550]], ['control: random smoother 20', [16384, [6, -12155, -29083, 31]], [3, -6076, -17579, -8774]]]]
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: alpha one | [100, -5, 7] | [100, -5, 7] | Passed |
| regression: random smoother 2 | [-33, 20874, -43, 6, -28] | [-33, 20874, -43, 6, -28] | Passed |
| repair check: alpha zero | bad-alpha | [0, 0] | Failed |
| control: negative step | [0, 0, 0] | [0, 0, 0] | Passed |
| control: input beyond range | [16384, 24576, -4096] | [16384, 24576, -4096] | Passed |
| control: bad alpha | bad-alpha | bad-alpha | Passed |
| control: small negative tie | [-1, 0, 0] | [-1, 0, 0] | Passed |
SHA-256 / c4aca503b1be2b5c93c46351a8acda9b52f46a32aef26b934db4895dd4b9dcb1
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
N = 1
observations = []
def solve(x):
alpha, xs = x
if not 0 <= alpha <= 32768:
return 'bad-alpha'
y = 0
out = []
for v in xs:
v = max(-32768, min(32767, v))
y = y + (((v - y) * alpha + (1 << 14)) >> 15)
out.append(y)
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: input beyond range', [16384, [40000, 40000, -40000]], [16384, 24576, -4096]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: small negative tie', [16384, [-3, 0, 0]], [-1, 0, 0]]], [['regression: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 0', [24576, [-8, -38976, 37]], [-6, -24577, -6116]], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 4', [8192, [20, -2539]], [5, -631]], ['control: random smoother 6', [8192, [16294, 2, -779, 14009]], [4074, 3056, 2097, 5075]]], [['regression: random smoother 24', [32768, [-19, -16, -18365, 21638, -14929]], [-19, -16, -18365, 21638, -14929]], ['regression: random smoother 26', [32768, [-4554, -6260, -33961]], [-4554, -6260, -32768]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 7', [1000, [-8425, 39]], [-257, -248]], ['control: random smoother 8', [30000, [45, -46]], [41, -39]], ['control: random smoother 11', [24576, [12306, -35386, 1164, 8403, 12704, 26]], [9230, -22268, -4694, 5129, 10810, 2722]], ['control: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]]], [['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 13', [8192, [-8, -27561]], [-2, -6892]], ['control: random smoother 14', [24576, [-12991, -22604, 32, -18621, -30, -36, -10331]], [-9743, -19389, -4823, -15171, -3815, -981, -7993]], ['control: random smoother 15', [30000, [13636, -12457, -1, -11, 49, 0]], [12484, -10350, -875, -84, 38, 3]], ['control: random smoother 16', [8192, [-12962, 19012, 49, -24286]], [-3240, 2323, 1755, -4755]]], [['regression: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['repair check: alpha zero', [0, [5, 5]], [0, 0]], ['control: random smoother 17', [16384, [-8485, -763, -9, 25855, -11982, -28816, 38]], [-4242, -2502, -1255, 12300, 159, -14328, -7145]], ['control: random smoother 18', [24576, [-8497, -36]], [-6373, -1620]], ['control: random smoother 19', [24576, [-1436, -36936, -26, 4657, 23423, 50]], [-1077, -24845, -6231, 1935, 18051, 4550]], ['control: random smoother 20', [16384, [6, -12155, -29083, 31]], [3, -6076, -17579, -8774]]]]
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: alpha one | [100, -5, 7] | [100, -5, 7] | Passed |
| regression: random smoother 2 | [-33, 20874, -43, 6, -28] | [-33, 20874, -43, 6, -28] | Passed |
| repair check: alpha zero | [0, 0] | [0, 0] | Passed |
| control: negative step | [0, 0, 0] | [0, 0, 0] | Passed |
| control: input beyond range | [16384, 24576, -4096] | [16384, 24576, -4096] | Passed |
| control: bad alpha | bad-alpha | bad-alpha | Passed |
| control: small negative tie | [-1, 0, 0] | [-1, 0, 0] | Passed |
SHA-256 / 8a28fd137a3283c70902661dad52ab3dd35e487e097b20ea210773898e37a07f
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 2026-09-29T14:51:36.386314+00:00.
Case digest / b1c053b34a845ea23b2777b9a180b06970816a8e9ed172d8effdc97f0b9bc59c