FA-91661 / Digital signal filters / Open access

Forward-backward filter pads with an even mirror · case 01

A ramp input gets a kink at the edges and the filtered ends droop.

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

ROOT CAUSE

The padding copies mirrored samples instead of reflecting them about the end value (2 x0 - x_i).

THE FAILURE

The padding copies mirrored samples instead of reflecting them about the end value (2 x0 - x_i).

Unsuccessful approach: The attempted repair uses odd reflection at the start only; the end is still an even mirror.

Case contract

Input [taps, samples] (integers). Zero-phase filtering: pad = min(3 (len(taps) - 1), len - 1); extend with odd reflection about each end (2 x[0] - x[i] for i = pad..1 before, 2 x[-1] - x[n-1-i] for i = 1..pad after); filter forward with zero initial state, reverse, filter again, reverse, and return the middle len samples.

Why this case matters

Forward-backward filtering is used for offline zero-phase smoothing; padding and reversal slips create edge transients and phase shift.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    b, xs = x
    n = len(xs)
    if n == 0:
        return []
    pad = min(3 * (len(b) - 1), n - 1)
    front = [xs[i] for i in range(pad, 0, -1)]
    back = [xs[n - 1 - i] for i in range(1, pad + 1)]
    ext = front + xs + back
    def fir(sig):
        return [sum(b[k] * sig[i - k] for k in range(len(b)) if i - k >= 0) for i in range(len(sig))]
    y = fir(ext)
    y = fir(y[::-1])[::-1]
    return y[pad:pad + n]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: ramp with two-tap average', [[1, 1], [1, 2, 3, 4, 5]], [4, 8, 12, 16, 20]], ['regression: step with three taps', [[1, 2, 1], [0, 0, 4, 4, 4, 4]], [0, 20, 44, 60, 64, 64]], ['regression: short signal', [[1, 1, 1], [2, 5]], [22, 21]], ['control: single sample', [[1, 2], [3]], [3]], ['control: empty', [[1, 1], []], []], ['control: random filtfilt 3', [[-1], [-4, 3, -2, 7, 9, 5, 3, -2, 1]], [-4, 3, -2, 7, 9, 5, 3, -2, 1]], ['control: random filtfilt 4', [[-1], [8, 1, 1, 8]], [8, 1, 1, 8]]], [['regression: random filtfilt 0', [[-1, -1], [-5, 3]], [-20, 12]], ['regression: random filtfilt 1', [[2, -1], [-1, 3, 2, 2, 8, 5, 5, 0, -3]], [-1, 13, 0, -10, 26, -1, 15, -4, -3]], ['regression: random filtfilt 2', [[1, 1, -2], [7, 7, 2, 7, 6, 9, -5, 8]], [0, 5, -28, 2, 26, 23, -101, 0]], ['control: random filtfilt 5', [[2, 1, -1, 3], [0]], [0]], ['control: random filtfilt 7', [[-2], [8, -5, -4, -4, -2, 9, 6, 3]], [32, -20, -16, -16, -8, 36, 24, 12]], ['control: random filtfilt 12', [[0], [-2, 4, -5, -1, 8, 4, 3, 0, 1]], [0, 0, 0, 0, 0, 0, 0, 0, 0]], ['control: random filtfilt 15', [[1], [-1, -5, 2]], [-1, -5, 2]]], [['regression: random filtfilt 6', [[1, 3, -2, -1], [-3, -1, -4]], [-2, 59, -4]], ['regression: random filtfilt 8', [[-1, -2], [5, 5, 7]], [45, 49, 63]], ['regression: random filtfilt 2', [[1, 1, -2], [7, 7, 2, 7, 6, 9, -5, 8]], [0, 5, -28, 2, 26, 23, -101, 0]], ['control: random filtfilt 16', [[-1, -1, 0], [8]], [8]], ['control: random filtfilt 17', [[-1, -2, -1], [-1]], [-1]], ['control: random filtfilt 19', [[-1], [-5, 9, -4, 7, -2]], [-5, 9, -4, 7, -2]], ['control: random filtfilt 24', [[-2], [-2, -3, 0, 4]], [-8, -12, 0, 16]]], [['regression: random filtfilt 10', [[2, 0, -2], [-3, 9, 2, 7]], [0, 104, -20, 0]], ['regression: random filtfilt 11', [[0, 3, 3, 3], [2, 0, 6]], [162, 288, 234]], ['regression: random filtfilt 8', [[-1, -2], [5, 5, 7]], [45, 49, 63]], ['control: random filtfilt 27', [[0], [0, -5, 8, -2, 9, 3]], [0, 0, 0, 0, 0, 0]], ['control: random filtfilt 33', [[3, 0], [-2, 7]], [-18, 63]], ['control: random filtfilt 36', [[-2], [2, 6, 0, 0, 4, 9, 0, -4]], [8, 24, 0, 0, 16, 36, 0, -16]], ['control: random filtfilt 39', [[3, 2, 3], [6]], [54]]], [['regression: random filtfilt 14', [[0, -1, -2, -1], [-3, 7]], [-53, -12]], ['regression: random filtfilt 18', [[2, -2], [2, -3, 5, -1, 3, -2, 3]], [0, -52, 56, -40, 36, -40, 0]], ['regression: random filtfilt 10', [[2, 0, -2], [-3, 9, 2, 7]], [0, 104, -20, 0]], ['control: random filtfilt 42', [[3], [7, -3, 8, -3, -2, 6, 1, 0, -5]], [63, -27, 72, -27, -18, 54, 9, 0, -45]], ['control: random filtfilt 43', [[2, 0], [9, -4, 9, 0]], [36, -16, 36, 0]], ['control: random filtfilt 46', [[-1, 1, 2, 2], [5]], [5]], ['control: random filtfilt 47', [[-2], [2, 2, 4, 6, 3, 6, 8]], [8, 8, 16, 24, 12, 24, 32]]]]
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: ramp with two-tap average	`[6, 8, 12, 16, 18]`	`[4, 8, 12, 16, 20]`	Failed
regression: step with three taps	`[8, 20, 44, 60, 64, 64]`	`[0, 20, 44, 60, 64, 64]`	Failed
regression: short signal	`[28, 21]`	`[22, 21]`	Failed
control: single sample	`[3]`	`[3]`	Passed
control: empty	`[]`	`[]`	Passed
control: random filtfilt 3	`[-4, 3, -2, 7, 9, 5, 3, -2, 1]`	`[-4, 3, -2, 7, 9, 5, 3, -2, 1]`	Passed
control: random filtfilt 4	`[8, 1, 1, 8]`	`[8, 1, 1, 8]`	Passed

SHA-256 / e2ba141e50dc017112a25bd73344c497da37927c1e67c5da0112948f093398f9

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(x):
    b, xs = x
    n = len(xs)
    if n == 0:
        return []
    pad = min(3 * (len(b) - 1), n - 1)
    front = [2 * xs[0] - xs[i] for i in range(pad, 0, -1)]
    back = [xs[n - 1 - i] for i in range(1, pad + 1)]
    ext = front + xs + back
    def fir(sig):
        return [sum(b[k] * sig[i - k] for k in range(len(b)) if i - k >= 0) for i in range(len(sig))]
    y = fir(ext)
    y = fir(y[::-1])[::-1]
    return y[pad:pad + n]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: ramp with two-tap average', [[1, 1], [1, 2, 3, 4, 5]], [4, 8, 12, 16, 20]], ['regression: step with three taps', [[1, 2, 1], [0, 0, 4, 4, 4, 4]], [0, 20, 44, 60, 64, 64]], ['regression: short signal', [[1, 1, 1], [2, 5]], [22, 21]], ['control: single sample', [[1, 2], [3]], [3]], ['control: empty', [[1, 1], []], []], ['control: random filtfilt 3', [[-1], [-4, 3, -2, 7, 9, 5, 3, -2, 1]], [-4, 3, -2, 7, 9, 5, 3, -2, 1]], ['control: random filtfilt 4', [[-1], [8, 1, 1, 8]], [8, 1, 1, 8]]], [['regression: random filtfilt 0', [[-1, -1], [-5, 3]], [-20, 12]], ['regression: random filtfilt 1', [[2, -1], [-1, 3, 2, 2, 8, 5, 5, 0, -3]], [-1, 13, 0, -10, 26, -1, 15, -4, -3]], ['regression: random filtfilt 2', [[1, 1, -2], [7, 7, 2, 7, 6, 9, -5, 8]], [0, 5, -28, 2, 26, 23, -101, 0]], ['control: random filtfilt 5', [[2, 1, -1, 3], [0]], [0]], ['control: random filtfilt 7', [[-2], [8, -5, -4, -4, -2, 9, 6, 3]], [32, -20, -16, -16, -8, 36, 24, 12]], ['control: random filtfilt 12', [[0], [-2, 4, -5, -1, 8, 4, 3, 0, 1]], [0, 0, 0, 0, 0, 0, 0, 0, 0]], ['control: random filtfilt 15', [[1], [-1, -5, 2]], [-1, -5, 2]]], [['regression: random filtfilt 6', [[1, 3, -2, -1], [-3, -1, -4]], [-2, 59, -4]], ['regression: random filtfilt 8', [[-1, -2], [5, 5, 7]], [45, 49, 63]], ['regression: random filtfilt 2', [[1, 1, -2], [7, 7, 2, 7, 6, 9, -5, 8]], [0, 5, -28, 2, 26, 23, -101, 0]], ['control: random filtfilt 16', [[-1, -1, 0], [8]], [8]], ['control: random filtfilt 17', [[-1, -2, -1], [-1]], [-1]], ['control: random filtfilt 19', [[-1], [-5, 9, -4, 7, -2]], [-5, 9, -4, 7, -2]], ['control: random filtfilt 24', [[-2], [-2, -3, 0, 4]], [-8, -12, 0, 16]]], [['regression: random filtfilt 10', [[2, 0, -2], [-3, 9, 2, 7]], [0, 104, -20, 0]], ['regression: random filtfilt 11', [[0, 3, 3, 3], [2, 0, 6]], [162, 288, 234]], ['regression: random filtfilt 8', [[-1, -2], [5, 5, 7]], [45, 49, 63]], ['control: random filtfilt 27', [[0], [0, -5, 8, -2, 9, 3]], [0, 0, 0, 0, 0, 0]], ['control: random filtfilt 33', [[3, 0], [-2, 7]], [-18, 63]], ['control: random filtfilt 36', [[-2], [2, 6, 0, 0, 4, 9, 0, -4]], [8, 24, 0, 0, 16, 36, 0, -16]], ['control: random filtfilt 39', [[3, 2, 3], [6]], [54]]], [['regression: random filtfilt 14', [[0, -1, -2, -1], [-3, 7]], [-53, -12]], ['regression: random filtfilt 18', [[2, -2], [2, -3, 5, -1, 3, -2, 3]], [0, -52, 56, -40, 36, -40, 0]], ['regression: random filtfilt 10', [[2, 0, -2], [-3, 9, 2, 7]], [0, 104, -20, 0]], ['control: random filtfilt 42', [[3], [7, -3, 8, -3, -2, 6, 1, 0, -5]], [63, -27, 72, -27, -18, 54, 9, 0, -45]], ['control: random filtfilt 43', [[2, 0], [9, -4, 9, 0]], [36, -16, 36, 0]], ['control: random filtfilt 46', [[-1, 1, 2, 2], [5]], [5]], ['control: random filtfilt 47', [[-2], [2, 2, 4, 6, 3, 6, 8]], [8, 8, 16, 24, 12, 24, 32]]]]
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: ramp with two-tap average	`[4, 8, 12, 16, 18]`	`[4, 8, 12, 16, 20]`	Failed
regression: step with three taps	`[0, 20, 44, 60, 64, 64]`	`[0, 20, 44, 60, 64, 64]`	Passed
regression: short signal	`[16, 15]`	`[22, 21]`	Failed
control: single sample	`[3]`	`[3]`	Passed
control: empty	`[]`	`[]`	Passed
control: random filtfilt 3	`[-4, 3, -2, 7, 9, 5, 3, -2, 1]`	`[-4, 3, -2, 7, 9, 5, 3, -2, 1]`	Passed
control: random filtfilt 4	`[8, 1, 1, 8]`	`[8, 1, 1, 8]`	Passed

SHA-256 / b607aa43eae3a5cd6b140df493602e45716f8fb27640344a0f564ea8d494b866

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 7 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.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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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 2026-09-29T14:51:38.226803+00:00.

Case digest / 2c18d7f0476d801b0113902cf2cc15c61172bae6e7f4107d784de5803b72426a