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FA-91496 / Digital signal filters / Open access

Windowed sinc centres even-length filters on a sample · case 01

Even-length designs are not symmetric and lose linear phase.

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

ROOT CAUSE

The centre is N // 2 instead of (N - 1) / 2, which is a half-integer for even N.

THE FAILURE

The centre is N // 2 instead of (N - 1) / 2, which is a half-integer for even N.

Unsuccessful approach: The attempted repair uses (N - 1) // 2, still an integer centre for even N.

Case contract

Input [N, fc, window]; fc is the cutoff as a fraction of Nyquist in (0, 1], window hamming (0.54 - 0.46 cos(2 pi n/(N-1))), hann (0.5 - 0.5 cos(...)) or rect (N = 1 uses 1). Taps h[n] = fc sinc(fc (n - (N-1)/2)) w[n], normalized to unit DC gain and rounded to 8 decimals; "bad-spec" for invalid input (Hann needs N != 2).

Why this case matters

Windowed-sinc design is the default FIR recipe; centre or normalization slips yield non-linear-phase or wrong-gain filters.

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):
    N, fc_s, win = x
    fc = float(Fraction(fc_s))
    if N < 1 or not 0 < fc <= 1:
        return 'bad-spec'
    c = N // 2
    taps = []
    for n in range(N):
        t = n - c
        s = 1.0 if t == 0 else math.sin(math.pi * fc * t) / (math.pi * fc * t)
        if N == 1 or win == 'rect':
            w = 1.0
        elif win == 'hann':
            w = 0.5 - 0.5 * math.cos(2 * math.pi * n / (N - 1))
        else:
            w = 0.54 - 0.46 * math.cos(2 * math.pi * n / (N - 1))
        taps.append(fc * s * w)
    tot = sum(taps)
    return [round(v / tot, 8) + 0.0 for v in taps]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: N=2 fc=1/2 hamming', [2, '1/2', 'hamming'], [0.5, 0.5]], ['regression: N=2 fc=1/2 rect', [2, '1/2', 'rect'], [0.5, 0.5]], ['regression: N=2 fc=1/4 hamming', [2, '1/4', 'hamming'], [0.5, 0.5]], ['control: N=1 fc=1/2 hamming', [1, '1/2', 'hamming'], [1.0]], ['control: N=1 fc=1/2 hann', [1, '1/2', 'hann'], [1.0]], ['control: N=1 fc=1/2 rect', [1, '1/2', 'rect'], [1.0]], ['control: N=1 fc=1/4 hamming', [1, '1/4', 'hamming'], [1.0]]], [['regression: N=2 fc=1/4 rect', [2, '1/4', 'rect'], [0.5, 0.5]], ['regression: N=2 fc=3/5 hamming', [2, '3/5', 'hamming'], [0.5, 0.5]], ['regression: N=2 fc=1/4 hamming', [2, '1/4', 'hamming'], [0.5, 0.5]], ['control: N=1 fc=1/4 hann', [1, '1/4', 'hann'], [1.0]], ['control: N=1 fc=1/4 rect', [1, '1/4', 'rect'], [1.0]], ['control: N=1 fc=3/5 hamming', [1, '3/5', 'hamming'], [1.0]], ['control: N=1 fc=3/5 hann', [1, '3/5', 'hann'], [1.0]]], [['regression: N=2 fc=1 hamming', [2, '1', 'hamming'], [0.5, 0.5]], ['regression: N=2 fc=1 rect', [2, '1', 'rect'], [0.5, 0.5]], ['regression: N=2 fc=3/5 hamming', [2, '3/5', 'hamming'], [0.5, 0.5]], ['control: N=1 fc=3/5 rect', [1, '3/5', 'rect'], [1.0]], ['control: N=1 fc=1 hamming', [1, '1', 'hamming'], [1.0]], ['control: N=1 fc=1 hann', [1, '1', 'hann'], [1.0]], ['control: N=1 fc=1 rect', [1, '1', 'rect'], [1.0]]], [['regression: N=4 fc=1/2 hann', [4, '1/2', 'hann'], [0.0, 0.5, 0.5, 0.0]], ['regression: N=4 fc=1/2 rect', [4, '1/2', 'rect'], [0.125, 0.375, 0.375, 0.125]], ['regression: N=2 fc=1 hamming', [2, '1', 'hamming'], [0.5, 0.5]], ['control: N=5 fc=1/2 hamming', [5, '1/2', 'hamming'], [0.0, 0.20371237, 0.59257526, 0.20371237, 0.0]], ['control: N=5 fc=1/2 hann', [5, '1/2', 'hann'], [0.0, 0.19449226, 0.61101547, 0.19449226, 0.0]], ['control: N=5 fc=1/2 rect', [5, '1/2', 'rect'], [0.0, 0.28004958, 0.43990085, 0.28004958, 0.0]], ['control: N=5 fc=1/4 hamming', [5, '1/4', 'hamming'], [0.02455383, 0.23438946, 0.4821134, 0.23438946, 0.02455383]]], [['regression: N=4 fc=1/4 hann', [4, '1/4', 'hann'], [0.0, 0.5, 0.5, 0.0]], ['regression: N=4 fc=1/4 rect', [4, '1/4', 'rect'], [0.22295145, 0.27704855, 0.27704855, 0.22295145]], ['regression: N=4 fc=1/2 hamming', [4, '1/2', 'hamming'], [0.0167364, 0.4832636, 0.4832636, 0.0167364]], ['control: N=5 fc=1/4 hann', [5, '1/4', 'hann'], [0.0, 0.23688591, 0.52622818, 0.23688591, 0.0]], ['control: N=5 fc=1/4 rect', [5, '1/4', 'rect'], [0.15626896, 0.22099768, 0.24546671, 0.22099768, 0.15626896]], ['control: N=5 fc=3/5 hamming', [5, '3/5', 'hamming'], [-0.00820621, 0.17925211, 0.65790821, 0.17925211, -0.00820621]], ['control: N=5 fc=3/5 hann', [5, '3/5', 'hann'], [0.0, 0.16767497, 0.66465005, 0.16767497, 0.0]]]]
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: N=2 fc=1/2 hamming[0.38898453, 0.61101547][0.5, 0.5]Failed
regression: N=2 fc=1/2 rect[0.38898453, 0.61101547][0.5, 0.5]Failed
regression: N=2 fc=1/4 hamming[0.47377182, 0.52622818][0.5, 0.5]Failed
control: N=1 fc=1/2 hamming[1.0][1.0]Passed
control: N=1 fc=1/2 hann[1.0][1.0]Passed
control: N=1 fc=1/2 rect[1.0][1.0]Passed
control: N=1 fc=1/4 hamming[1.0][1.0]Passed

SHA-256 / 58c987dbc579dea5bd8dabf9ee83019035a3d5d93882d04381a7a7827a1643ee

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):
    N, fc_s, win = x
    fc = float(Fraction(fc_s))
    if N < 1 or not 0 < fc <= 1:
        return 'bad-spec'
    c = (N - 1) // 2
    taps = []
    for n in range(N):
        t = n - c
        s = 1.0 if t == 0 else math.sin(math.pi * fc * t) / (math.pi * fc * t)
        if N == 1 or win == 'rect':
            w = 1.0
        elif win == 'hann':
            w = 0.5 - 0.5 * math.cos(2 * math.pi * n / (N - 1))
        else:
            w = 0.54 - 0.46 * math.cos(2 * math.pi * n / (N - 1))
        taps.append(fc * s * w)
    tot = sum(taps)
    return [round(v / tot, 8) + 0.0 for v in taps]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: N=2 fc=1/2 hamming', [2, '1/2', 'hamming'], [0.5, 0.5]], ['regression: N=2 fc=1/2 rect', [2, '1/2', 'rect'], [0.5, 0.5]], ['regression: N=2 fc=1/4 hamming', [2, '1/4', 'hamming'], [0.5, 0.5]], ['control: N=1 fc=1/2 hamming', [1, '1/2', 'hamming'], [1.0]], ['control: N=1 fc=1/2 hann', [1, '1/2', 'hann'], [1.0]], ['control: N=1 fc=1/2 rect', [1, '1/2', 'rect'], [1.0]], ['control: N=1 fc=1/4 hamming', [1, '1/4', 'hamming'], [1.0]]], [['regression: N=2 fc=1/4 rect', [2, '1/4', 'rect'], [0.5, 0.5]], ['regression: N=2 fc=3/5 hamming', [2, '3/5', 'hamming'], [0.5, 0.5]], ['regression: N=2 fc=1/4 hamming', [2, '1/4', 'hamming'], [0.5, 0.5]], ['control: N=1 fc=1/4 hann', [1, '1/4', 'hann'], [1.0]], ['control: N=1 fc=1/4 rect', [1, '1/4', 'rect'], [1.0]], ['control: N=1 fc=3/5 hamming', [1, '3/5', 'hamming'], [1.0]], ['control: N=1 fc=3/5 hann', [1, '3/5', 'hann'], [1.0]]], [['regression: N=2 fc=1 hamming', [2, '1', 'hamming'], [0.5, 0.5]], ['regression: N=2 fc=1 rect', [2, '1', 'rect'], [0.5, 0.5]], ['regression: N=2 fc=3/5 hamming', [2, '3/5', 'hamming'], [0.5, 0.5]], ['control: N=1 fc=3/5 rect', [1, '3/5', 'rect'], [1.0]], ['control: N=1 fc=1 hamming', [1, '1', 'hamming'], [1.0]], ['control: N=1 fc=1 hann', [1, '1', 'hann'], [1.0]], ['control: N=1 fc=1 rect', [1, '1', 'rect'], [1.0]]], [['regression: N=4 fc=1/2 hann', [4, '1/2', 'hann'], [0.0, 0.5, 0.5, 0.0]], ['regression: N=4 fc=1/2 rect', [4, '1/2', 'rect'], [0.125, 0.375, 0.375, 0.125]], ['regression: N=2 fc=1 hamming', [2, '1', 'hamming'], [0.5, 0.5]], ['control: N=5 fc=1/2 hamming', [5, '1/2', 'hamming'], [0.0, 0.20371237, 0.59257526, 0.20371237, 0.0]], ['control: N=5 fc=1/2 hann', [5, '1/2', 'hann'], [0.0, 0.19449226, 0.61101547, 0.19449226, 0.0]], ['control: N=5 fc=1/2 rect', [5, '1/2', 'rect'], [0.0, 0.28004958, 0.43990085, 0.28004958, 0.0]], ['control: N=5 fc=1/4 hamming', [5, '1/4', 'hamming'], [0.02455383, 0.23438946, 0.4821134, 0.23438946, 0.02455383]]], [['regression: N=4 fc=1/4 hann', [4, '1/4', 'hann'], [0.0, 0.5, 0.5, 0.0]], ['regression: N=4 fc=1/4 rect', [4, '1/4', 'rect'], [0.22295145, 0.27704855, 0.27704855, 0.22295145]], ['regression: N=4 fc=1/2 hamming', [4, '1/2', 'hamming'], [0.0167364, 0.4832636, 0.4832636, 0.0167364]], ['control: N=5 fc=1/4 hann', [5, '1/4', 'hann'], [0.0, 0.23688591, 0.52622818, 0.23688591, 0.0]], ['control: N=5 fc=1/4 rect', [5, '1/4', 'rect'], [0.15626896, 0.22099768, 0.24546671, 0.22099768, 0.15626896]], ['control: N=5 fc=3/5 hamming', [5, '3/5', 'hamming'], [-0.00820621, 0.17925211, 0.65790821, 0.17925211, -0.00820621]], ['control: N=5 fc=3/5 hann', [5, '3/5', 'hann'], [0.0, 0.16767497, 0.66465005, 0.16767497, 0.0]]]]
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: N=2 fc=1/2 hamming[0.61101547, 0.38898453][0.5, 0.5]Failed
regression: N=2 fc=1/2 rect[0.61101547, 0.38898453][0.5, 0.5]Failed
regression: N=2 fc=1/4 hamming[0.52622818, 0.47377182][0.5, 0.5]Failed
control: N=1 fc=1/2 hamming[1.0][1.0]Passed
control: N=1 fc=1/2 hann[1.0][1.0]Passed
control: N=1 fc=1/2 rect[1.0][1.0]Passed
control: N=1 fc=1/4 hamming[1.0][1.0]Passed

SHA-256 / ef40bc078c8342b26c2d0da4bd49ae9d95b4116cd942c7ba75507fbfcfda5727

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 2026-09-29T14:51:36.487487+00:00.

Case digest / a4956a71b857421acd874a643a3b4f672d0b7746c18f3d0bd44f9e3a98590b2b