FA-91756 / Digital signal filters / Open access

DC blocker starts from a zero previous input · case 01

A constant input produces a decaying spike equal to the offset at start-up.

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

ROOT CAUSE

x[-1] is initialized to 0 instead of x[0].

VERIFIED REPAIR

Warm-start x[-1] = x[0].

Unsuccessful approach: The attempted repair warm-starts both x[-1] and y[-1] to x[0], injecting the offset through the feedback path.

Case contract

Input [R, samples]: y[n] = x[n] - x[n-1] + R y[n-1] with 0 <= R < 1 ("bad-pole"), warm-started so x[-1] = x[0] and y[-1] = 0 (no start-up step). Return exact fraction strings.

Why this case matters

DC blockers remove offsets before further processing; a cold start injects a large transient.

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):
    R, xs = Fraction(x[0]), x[1]
    if not 0 <= R < 1:
        return 'bad-pole'
    if not xs:
        return []
    xp = 0
    yp = Fraction(0)
    out = []
    for v in xs:
        y = v - xp + R * yp
        xp, yp = v, y
        out.append(str(y))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: constant input warm start', ['9/10', [5, 5, 5]], ['0', '0', '0']], ['regression: pole zero', ['0', [1, 3, 2]], ['0', '2', '-1']], ['regression: random dc blocker 0', ['3/4', [-2, -1, 1, -3]], ['0', '1', '11/4', '-31/16']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']], ['control: bad pole', ['1', [1]], 'bad-pole'], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']]], [['regression: random dc blocker 1', ['3/4', [-2, 2]], ['0', '4']], ['regression: random dc blocker 2', ['1/4', [-3, 0, 1, -3]], ['0', '3', '7/4', '-57/16']], ['regression: random dc blocker 3', ['9/10', [1, 4, -3, -3, 5, -2]], ['0', '3', '-43/10', '-387/100', '4517/1000', '-29347/10000']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']], ['control: random dc blocker 32', ['9/10', [0, 0]], ['0', '0']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']]], [['regression: random dc blocker 4', ['3/4', [1, 2, 6, 0, -2]], ['0', '1', '19/4', '-39/16', '-245/64']], ['regression: random dc blocker 5', ['9/10', [2, 5]], ['0', '3']], ['regression: random dc blocker 3', ['9/10', [1, 4, -3, -3, 5, -2]], ['0', '3', '-43/10', '-387/100', '4517/1000', '-29347/10000']], ['control: bad pole', ['1', [1]], 'bad-pole'], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']]], [['regression: random dc blocker 8', ['9/10', [-3, -3, 2, -2, 2, -2]], ['0', '0', '5', '1/2', '89/20', '1/200']], ['regression: random dc blocker 9', ['9/10', [3, 4, -2, 3]], ['0', '1', '-51/10', '41/100']], ['regression: random dc blocker 5', ['9/10', [2, 5]], ['0', '3']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']], ['control: random dc blocker 32', ['9/10', [0, 0]], ['0', '0']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']], ['control: bad pole', ['1', [1]], 'bad-pole']], [['regression: random dc blocker 11', ['1/2', [-1, 0, 6]], ['0', '1', '13/2']], ['regression: random dc blocker 12', ['3/4', [-3, 0, 6]], ['0', '3', '33/4']], ['regression: random dc blocker 8', ['9/10', [-3, -3, 2, -2, 2, -2]], ['0', '0', '5', '1/2', '89/20', '1/200']], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']]]]
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: constant input warm start	`['5', '9/2', '81/20']`	`['0', '0', '0']`	Failed
regression: pole zero	`['1', '2', '-1']`	`['0', '2', '-1']`	Failed
regression: random dc blocker 0	`['-2', '-1/2', '13/8', '-89/32']`	`['0', '1', '11/4', '-31/16']`	Failed
control: step	`['0', '4', '2', '1']`	`['0', '4', '2', '1']`	Passed
control: bad pole	`bad-pole`	`bad-pole`	Passed
control: empty	`[]`	`[]`	Passed
control: random dc blocker 7	`['0', '6', '-5', '-3/2', '9/4', '-7/8']`	`['0', '6', '-5', '-3/2', '9/4', '-7/8']`	Passed

SHA-256 / 3183eeb04210b60ebaff82a881cdcfeebc5d5bb82d0aebe0ee9ac3e307e8931b

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):
    R, xs = Fraction(x[0]), x[1]
    if not 0 <= R < 1:
        return 'bad-pole'
    if not xs:
        return []
    xp = xs[0]
    yp = Fraction(xs[0])
    out = []
    for v in xs:
        y = v - xp + R * yp
        xp, yp = v, y
        out.append(str(y))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: constant input warm start', ['9/10', [5, 5, 5]], ['0', '0', '0']], ['regression: pole zero', ['0', [1, 3, 2]], ['0', '2', '-1']], ['regression: random dc blocker 0', ['3/4', [-2, -1, 1, -3]], ['0', '1', '11/4', '-31/16']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']], ['control: bad pole', ['1', [1]], 'bad-pole'], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']]], [['regression: random dc blocker 1', ['3/4', [-2, 2]], ['0', '4']], ['regression: random dc blocker 2', ['1/4', [-3, 0, 1, -3]], ['0', '3', '7/4', '-57/16']], ['regression: random dc blocker 3', ['9/10', [1, 4, -3, -3, 5, -2]], ['0', '3', '-43/10', '-387/100', '4517/1000', '-29347/10000']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']], ['control: random dc blocker 32', ['9/10', [0, 0]], ['0', '0']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']]], [['regression: random dc blocker 4', ['3/4', [1, 2, 6, 0, -2]], ['0', '1', '19/4', '-39/16', '-245/64']], ['regression: random dc blocker 5', ['9/10', [2, 5]], ['0', '3']], ['regression: random dc blocker 3', ['9/10', [1, 4, -3, -3, 5, -2]], ['0', '3', '-43/10', '-387/100', '4517/1000', '-29347/10000']], ['control: bad pole', ['1', [1]], 'bad-pole'], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']]], [['regression: random dc blocker 8', ['9/10', [-3, -3, 2, -2, 2, -2]], ['0', '0', '5', '1/2', '89/20', '1/200']], ['regression: random dc blocker 9', ['9/10', [3, 4, -2, 3]], ['0', '1', '-51/10', '41/100']], ['regression: random dc blocker 5', ['9/10', [2, 5]], ['0', '3']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']], ['control: random dc blocker 32', ['9/10', [0, 0]], ['0', '0']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']], ['control: bad pole', ['1', [1]], 'bad-pole']], [['regression: random dc blocker 11', ['1/2', [-1, 0, 6]], ['0', '1', '13/2']], ['regression: random dc blocker 12', ['3/4', [-3, 0, 6]], ['0', '3', '33/4']], ['regression: random dc blocker 8', ['9/10', [-3, -3, 2, -2, 2, -2]], ['0', '0', '5', '1/2', '89/20', '1/200']], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']]]]
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: constant input warm start	`['9/2', '81/20', '729/200']`	`['0', '0', '0']`	Failed
regression: pole zero	`['0', '2', '-1']`	`['0', '2', '-1']`	Passed
regression: random dc blocker 0	`['-3/2', '-1/8', '61/32', '-329/128']`	`['0', '1', '11/4', '-31/16']`	Failed
control: step	`['0', '4', '2', '1']`	`['0', '4', '2', '1']`	Passed
control: bad pole	`bad-pole`	`bad-pole`	Passed
control: empty	`[]`	`[]`	Passed
control: random dc blocker 7	`['0', '6', '-5', '-3/2', '9/4', '-7/8']`	`['0', '6', '-5', '-3/2', '9/4', '-7/8']`	Passed

SHA-256 / 6c0836148a2da4c59c904ece7c50194dd007bfcdaee1f6263a12bba830154bc6

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):
    R, xs = Fraction(x[0]), x[1]
    if not 0 <= R < 1:
        return 'bad-pole'
    if not xs:
        return []
    xp = xs[0]
    yp = Fraction(0)
    out = []
    for v in xs:
        y = v - xp + R * yp
        xp, yp = v, y
        out.append(str(y))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: constant input warm start', ['9/10', [5, 5, 5]], ['0', '0', '0']], ['regression: pole zero', ['0', [1, 3, 2]], ['0', '2', '-1']], ['regression: random dc blocker 0', ['3/4', [-2, -1, 1, -3]], ['0', '1', '11/4', '-31/16']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']], ['control: bad pole', ['1', [1]], 'bad-pole'], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']]], [['regression: random dc blocker 1', ['3/4', [-2, 2]], ['0', '4']], ['regression: random dc blocker 2', ['1/4', [-3, 0, 1, -3]], ['0', '3', '7/4', '-57/16']], ['regression: random dc blocker 3', ['9/10', [1, 4, -3, -3, 5, -2]], ['0', '3', '-43/10', '-387/100', '4517/1000', '-29347/10000']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']], ['control: random dc blocker 32', ['9/10', [0, 0]], ['0', '0']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']]], [['regression: random dc blocker 4', ['3/4', [1, 2, 6, 0, -2]], ['0', '1', '19/4', '-39/16', '-245/64']], ['regression: random dc blocker 5', ['9/10', [2, 5]], ['0', '3']], ['regression: random dc blocker 3', ['9/10', [1, 4, -3, -3, 5, -2]], ['0', '3', '-43/10', '-387/100', '4517/1000', '-29347/10000']], ['control: bad pole', ['1', [1]], 'bad-pole'], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']]], [['regression: random dc blocker 8', ['9/10', [-3, -3, 2, -2, 2, -2]], ['0', '0', '5', '1/2', '89/20', '1/200']], ['regression: random dc blocker 9', ['9/10', [3, 4, -2, 3]], ['0', '1', '-51/10', '41/100']], ['regression: random dc blocker 5', ['9/10', [2, 5]], ['0', '3']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']], ['control: random dc blocker 32', ['9/10', [0, 0]], ['0', '0']], ['control: step', ['1/2', [0, 4, 4, 4]], ['0', '4', '2', '1']], ['control: bad pole', ['1', [1]], 'bad-pole']], [['regression: random dc blocker 11', ['1/2', [-1, 0, 6]], ['0', '1', '13/2']], ['regression: random dc blocker 12', ['3/4', [-3, 0, 6]], ['0', '3', '33/4']], ['regression: random dc blocker 8', ['9/10', [-3, -3, 2, -2, 2, -2]], ['0', '0', '5', '1/2', '89/20', '1/200']], ['control: empty', ['1/2', []], []], ['control: random dc blocker 7', ['1/2', [0, 6, -2, -1, 2, 0]], ['0', '6', '-5', '-3/2', '9/4', '-7/8']], ['control: random dc blocker 18', ['3/4', [0, 5, 4]], ['0', '5', '11/4']], ['control: random dc blocker 25', ['3/4', [0, 4, 6, -2, 1, -3]], ['0', '4', '5', '-17/4', '-3/16', '-265/64']]]]
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: constant input warm start	`['0', '0', '0']`	`['0', '0', '0']`	Passed
regression: pole zero	`['0', '2', '-1']`	`['0', '2', '-1']`	Passed
regression: random dc blocker 0	`['0', '1', '11/4', '-31/16']`	`['0', '1', '11/4', '-31/16']`	Passed
control: step	`['0', '4', '2', '1']`	`['0', '4', '2', '1']`	Passed
control: bad pole	`bad-pole`	`bad-pole`	Passed
control: empty	`[]`	`[]`	Passed
control: random dc blocker 7	`['0', '6', '-5', '-3/2', '9/4', '-7/8']`	`['0', '6', '-5', '-3/2', '9/4', '-7/8']`	Passed

SHA-256 / edf817a2b512b2b1601fd8f59f0dc7327db15a3fdec8e76fa5f0d23ad848bd3e

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 2026-09-29T14:51:38.910503+00:00.

Case digest / 1af3e7172b56f0b87b63da1495d60a201a7521c76ccb537c68c2d90efe064da9