FA-91666 / Digital signal filters / Open access

Forward-backward filter runs the second pass forward · case 01

The result has twice the group delay instead of zero phase.

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

ROOT CAUSE

The second pass filters y without reversing it first.

VERIFIED REPAIR

Reverse, filter and reverse back.

Unsuccessful approach: The attempted repair reverses before the second pass but not after it, returning a time-reversed signal.

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 = [2 * xs[0] - xs[i] for i in range(pad, 0, -1)]
    back = [2 * xs[-1] - 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)
    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 4', [[-1], [8, 1, 1, 8]], [8, 1, 1, 8]], ['control: random filtfilt 5', [[2, 1, -1, 3], [0]], [0]]], [['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: short signal', [[1, 1, 1], [2, 5]], [22, 21]], ['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 16', [[-1, -1, 0], [8]], [8]], ['control: random filtfilt 17', [[-1, -2, -1], [-1]], [-1]], ['control: random filtfilt 27', [[0], [0, -5, 8, -2, 9, 3]], [0, 0, 0, 0, 0, 0]]], [['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 1', [[2, -1], [-1, 3, 2, 2, 8, 5, 5, 0, -3]], [-1, 13, 0, -10, 26, -1, 15, -4, -3]], ['control: random filtfilt 39', [[3, 2, 3], [6]], [54]], ['control: random filtfilt 46', [[-1, 1, 2, 2], [5]], [5]], ['control: single sample', [[1, 2], [3]], [3]], ['control: empty', [[1, 1], []], []]], [['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]], ['repair check: 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]], ['control: random filtfilt 5', [[2, 1, -1, 3], [0]], [0]], ['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 16', [[-1, -1, 0], [8]], [8]]], [['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]], ['repair check: random filtfilt 7', [[-2], [8, -5, -4, -4, -2, 9, 6, 3]], [32, -20, -16, -16, -8, 36, 24, 12]], ['control: random filtfilt 17', [[-1, -2, -1], [-1]], [-1]], ['control: random filtfilt 27', [[0], [0, -5, 8, -2, 9, 3]], [0, 0, 0, 0, 0, 0]], ['control: random filtfilt 39', [[3, 2, 3], [6]], [54]], ['control: random filtfilt 46', [[-1, 1, 2, 2], [5]], [5]]]]
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	`[0, 4, 8, 12, 16]`	`[4, 8, 12, 16, 20]`	Failed
regression: step with three taps	`[-44, -20, 0, 20, 44, 60]`	`[0, 20, 44, 60, 64, 64]`	Failed
regression: short signal	`[0, 6]`	`[22, 21]`	Failed
control: single sample	`[3]`	`[3]`	Passed
control: empty	`[]`	`[]`	Passed
control: random filtfilt 4	`[8, 1, 1, 8]`	`[8, 1, 1, 8]`	Passed
control: random filtfilt 5	`[0]`	`[0]`	Passed

SHA-256 / 1a1f5f41fb3b5c16e306987ebc18abbce6202a17705e8c8a0a08ba6655c3e56e

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 = [2 * xs[-1] - 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])
    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 4', [[-1], [8, 1, 1, 8]], [8, 1, 1, 8]], ['control: random filtfilt 5', [[2, 1, -1, 3], [0]], [0]]], [['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: short signal', [[1, 1, 1], [2, 5]], [22, 21]], ['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 16', [[-1, -1, 0], [8]], [8]], ['control: random filtfilt 17', [[-1, -2, -1], [-1]], [-1]], ['control: random filtfilt 27', [[0], [0, -5, 8, -2, 9, 3]], [0, 0, 0, 0, 0, 0]]], [['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 1', [[2, -1], [-1, 3, 2, 2, 8, 5, 5, 0, -3]], [-1, 13, 0, -10, 26, -1, 15, -4, -3]], ['control: random filtfilt 39', [[3, 2, 3], [6]], [54]], ['control: random filtfilt 46', [[-1, 1, 2, 2], [5]], [5]], ['control: single sample', [[1, 2], [3]], [3]], ['control: empty', [[1, 1], []], []]], [['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]], ['repair check: 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]], ['control: random filtfilt 5', [[2, 1, -1, 3], [0]], [0]], ['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 16', [[-1, -1, 0], [8]], [8]]], [['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]], ['repair check: random filtfilt 7', [[-2], [8, -5, -4, -4, -2, 9, 6, 3]], [32, -20, -16, -16, -8, 36, 24, 12]], ['control: random filtfilt 17', [[-1, -2, -1], [-1]], [-1]], ['control: random filtfilt 27', [[0], [0, -5, 8, -2, 9, 3]], [0, 0, 0, 0, 0, 0]], ['control: random filtfilt 39', [[3, 2, 3], [6]], [54]], ['control: random filtfilt 46', [[-1, 1, 2, 2], [5]], [5]]]]
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	`[20, 16, 12, 8, 4]`	`[4, 8, 12, 16, 20]`	Failed
regression: step with three taps	`[64, 64, 60, 44, 20, 0]`	`[0, 20, 44, 60, 64, 64]`	Failed
regression: short signal	`[21, 22]`	`[22, 21]`	Failed
control: single sample	`[3]`	`[3]`	Passed
control: empty	`[]`	`[]`	Passed
control: random filtfilt 4	`[8, 1, 1, 8]`	`[8, 1, 1, 8]`	Passed
control: random filtfilt 5	`[0]`	`[0]`	Passed

SHA-256 / 1a950d97e8d4b08ea710ede08231dfaf2aaa538860292df690148f45975ccea7

3 / The verified repair

Exit 0

"""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 = [2 * xs[-1] - 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 4', [[-1], [8, 1, 1, 8]], [8, 1, 1, 8]], ['control: random filtfilt 5', [[2, 1, -1, 3], [0]], [0]]], [['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: short signal', [[1, 1, 1], [2, 5]], [22, 21]], ['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 16', [[-1, -1, 0], [8]], [8]], ['control: random filtfilt 17', [[-1, -2, -1], [-1]], [-1]], ['control: random filtfilt 27', [[0], [0, -5, 8, -2, 9, 3]], [0, 0, 0, 0, 0, 0]]], [['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 1', [[2, -1], [-1, 3, 2, 2, 8, 5, 5, 0, -3]], [-1, 13, 0, -10, 26, -1, 15, -4, -3]], ['control: random filtfilt 39', [[3, 2, 3], [6]], [54]], ['control: random filtfilt 46', [[-1, 1, 2, 2], [5]], [5]], ['control: single sample', [[1, 2], [3]], [3]], ['control: empty', [[1, 1], []], []]], [['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]], ['repair check: 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]], ['control: random filtfilt 5', [[2, 1, -1, 3], [0]], [0]], ['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 16', [[-1, -1, 0], [8]], [8]]], [['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]], ['repair check: random filtfilt 7', [[-2], [8, -5, -4, -4, -2, 9, 6, 3]], [32, -20, -16, -16, -8, 36, 24, 12]], ['control: random filtfilt 17', [[-1, -2, -1], [-1]], [-1]], ['control: random filtfilt 27', [[0], [0, -5, 8, -2, 9, 3]], [0, 0, 0, 0, 0, 0]], ['control: random filtfilt 39', [[3, 2, 3], [6]], [54]], ['control: random filtfilt 46', [[-1, 1, 2, 2], [5]], [5]]]]
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, 20]`	`[4, 8, 12, 16, 20]`	Passed
regression: step with three taps	`[0, 20, 44, 60, 64, 64]`	`[0, 20, 44, 60, 64, 64]`	Passed
regression: short signal	`[22, 21]`	`[22, 21]`	Passed
control: single sample	`[3]`	`[3]`	Passed
control: empty	`[]`	`[]`	Passed
control: random filtfilt 4	`[8, 1, 1, 8]`	`[8, 1, 1, 8]`	Passed
control: random filtfilt 5	`[0]`	`[0]`	Passed

SHA-256 / b43d25acbcdb90519d18da9417760bfaf0bb760cef72e0f1bd749f05422ef514

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 2026-09-29T14:51:38.185200+00:00.

Case digest / 01be6b40a7bcb94160b878850ba9b73d107e5bbf125c8e6ce2149bb0976e5d41