FA-91541 / Digital signal filters / Open access

Polyphase decimator splits the filter into contiguous blocks · case 01

Long filters produce outputs that do not match filter-then-downsample.

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

ROOT CAUSE

Branch p takes h[pD:(p+1)D] instead of every D-th tap starting at p.

VERIFIED REPAIR

Use the interleaved split h[p::D].

Unsuccessful approach: The attempted repair interleaves from the end, h[D-1-p::D], which pairs taps with the wrong phases.

Case contract

Input [h, D, samples] (integers). Output y[m] = sum_k h[k] x[mD - k] for m = 0 .. ceil(len/D) - 1 with x = 0 outside the signal, computed through the polyphase split E_p[j] = h[jD + p].

Why this case matters

Polyphase decimators avoid computing discarded outputs; commutator and phase-split slips alias the output.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(x):
    h, D, xs = x
    L = len(xs)
    M = (L + D - 1) // D
    phases = [h[p * D:(p + 1) * D] for p in range(D)]
    out = []
    for m in range(M):
        acc = 0
        for p in range(D):
            for j, c in enumerate(phases[p]):
                idx = (m - j) * D - p
                if 0 <= idx < L:
                    acc += c * xs[idx]
        out.append(acc)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: impulse through D=2', [[1, 2, 3], 2, [1, 0, 0, 0, 0]], [1, 3, 0]], ['regression: length divisible', [[1, 1], 2, [1, 2, 3, 4]], [1, 5]], ['repair check: first sample matters', [[1], 3, [7, 1, 1, 1]], [7, 1]], ['regression: long filter D=3', [[1, 2, 3, 4, 5], 3, [1, 0, 0, 2, 0, 0, 3]], [1, 6, 11]], ['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]]], [['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: long filter D=3', [[1, 2, 3, 4, 5], 3, [1, 0, 0, 2, 0, 0, 3]], [1, 6, 11]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]]], [['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]], ['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]]], [['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]], ['regression: random polyphase 8', [[-3, 0, 2, 1, 2], 4, [-2, 0, -1, -1, -4, -2, 1]], [6, 6]], ['repair check: random polyphase 3', [[-3], 4, [5, -1, 2, 1, -2]], [-15, 6]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]]], [['regression: random polyphase 10', [[3, -1, 1, 0, -2, 2], 4, [3, -3, 0, -3, 2, 1]], [9, 3]], ['regression: random polyphase 11', [[-2, 1, 4, -2, -3], 4, [-1, -4, -2, 5, 4, 5, 3, 1, 0]], [2, 0, -9]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]], ['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]]]]
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: impulse through D=2	`[1, 2, 0]`	`[1, 3, 0]`	Failed
regression: length divisible	`[1, 4]`	`[1, 5]`	Failed
repair check: first sample matters	`[7, 1]`	`[7, 1]`	Passed
regression: long filter D=3	`[1, 4, 10]`	`[1, 6, 11]`	Failed
regression: random polyphase 0	`[6, 5, -10, 5]`	`[6, 11, -4, -3]`	Failed
regression: random polyphase 1	`[-6, 14]`	`[-6, 4]`	Failed
regression: random polyphase 2	`[-8, -8]`	`[-8, 5]`	Failed

SHA-256 / c686e775874a4b0584348e0f37804d4b7e689a8706c53aa1be6b92d97fbcb84e

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(x):
    h, D, xs = x
    L = len(xs)
    M = (L + D - 1) // D
    phases = [h[D - 1 - p::D] for p in range(D)]
    out = []
    for m in range(M):
        acc = 0
        for p in range(D):
            for j, c in enumerate(phases[p]):
                idx = (m - j) * D - p
                if 0 <= idx < L:
                    acc += c * xs[idx]
        out.append(acc)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: impulse through D=2', [[1, 2, 3], 2, [1, 0, 0, 0, 0]], [1, 3, 0]], ['regression: length divisible', [[1, 1], 2, [1, 2, 3, 4]], [1, 5]], ['repair check: first sample matters', [[1], 3, [7, 1, 1, 1]], [7, 1]], ['regression: long filter D=3', [[1, 2, 3, 4, 5], 3, [1, 0, 0, 2, 0, 0, 3]], [1, 6, 11]], ['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]]], [['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: long filter D=3', [[1, 2, 3, 4, 5], 3, [1, 0, 0, 2, 0, 0, 3]], [1, 6, 11]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]]], [['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]], ['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]]], [['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]], ['regression: random polyphase 8', [[-3, 0, 2, 1, 2], 4, [-2, 0, -1, -1, -4, -2, 1]], [6, 6]], ['repair check: random polyphase 3', [[-3], 4, [5, -1, 2, 1, -2]], [-15, 6]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]]], [['regression: random polyphase 10', [[3, -1, 1, 0, -2, 2], 4, [3, -3, 0, -3, 2, 1]], [9, 3]], ['regression: random polyphase 11', [[-2, 1, 4, -2, -3], 4, [-1, -4, -2, 5, 4, 5, 3, 1, 0]], [2, 0, -9]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]], ['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]]]]
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: impulse through D=2	`[2, 0, 0]`	`[1, 3, 0]`	Failed
regression: length divisible	`[1, 5]`	`[1, 5]`	Passed
repair check: first sample matters	`[0, 1]`	`[7, 1]`	Failed
regression: long filter D=3	`[3, 6, 9]`	`[1, 6, 11]`	Failed
regression: random polyphase 0	`[-3, -10, -1, 9]`	`[6, 11, -4, -3]`	Failed
regression: random polyphase 1	`[-3, 5]`	`[-6, 4]`	Failed
regression: random polyphase 2	`[4, -8]`	`[-8, 5]`	Failed

SHA-256 / 6aec40d2d5227978dc03451cbc0d11656ae648d41bb042099fa6536a7d3bcfa5

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(x):
    h, D, xs = x
    L = len(xs)
    M = (L + D - 1) // D
    phases = [h[p::D] for p in range(D)]
    out = []
    for m in range(M):
        acc = 0
        for p in range(D):
            for j, c in enumerate(phases[p]):
                idx = (m - j) * D - p
                if 0 <= idx < L:
                    acc += c * xs[idx]
        out.append(acc)
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: impulse through D=2', [[1, 2, 3], 2, [1, 0, 0, 0, 0]], [1, 3, 0]], ['regression: length divisible', [[1, 1], 2, [1, 2, 3, 4]], [1, 5]], ['repair check: first sample matters', [[1], 3, [7, 1, 1, 1]], [7, 1]], ['regression: long filter D=3', [[1, 2, 3, 4, 5], 3, [1, 0, 0, 2, 0, 0, 3]], [1, 6, 11]], ['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]]], [['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: long filter D=3', [[1, 2, 3, 4, 5], 3, [1, 0, 0, 2, 0, 0, 3]], [1, 6, 11]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]]], [['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 0', [[2, -1], 2, [3, -3, 4, -2, -3, 5, 1]], [6, 11, -4, -3]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]], ['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]]], [['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]], ['regression: random polyphase 8', [[-3, 0, 2, 1, 2], 4, [-2, 0, -1, -1, -4, -2, 1]], [6, 6]], ['repair check: random polyphase 3', [[-3], 4, [5, -1, 2, 1, -2]], [-15, 6]], ['regression: random polyphase 1', [[-2, 4, -1, -1, 1, -3], 4, [3, 0, 1, 0, -1, -2]], [-6, 4]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]]], [['regression: random polyphase 10', [[3, -1, 1, 0, -2, 2], 4, [3, -3, 0, -3, 2, 1]], [9, 3]], ['regression: random polyphase 11', [[-2, 1, 4, -2, -3], 4, [-1, -4, -2, 5, 4, 5, 3, 1, 0]], [2, 0, -9]], ['regression: random polyphase 5', [[3, 1, -1, 0, -1], 2, [-3, 5, 4, 1]], [-9, 20]], ['regression: random polyphase 2', [[4, 2, 3, -2, 4], 4, [-2, -4, 3, 0, -1, -2, 3, -1]], [-8, 5]], ['regression: random polyphase 4', [[0, -1, 1, -1, 4], 3, [3, 2, 5, 3, -1, 0, -3, 2, -1, -2]], [0, -6, 16, 6]], ['regression: random polyphase 6', [[1, -2, 3, 2, -1], 4, [4, 3, 5, -3, -4, -2]], [4, 19]], ['regression: random polyphase 7', [[1, -1, 0], 4, [4, 5, -2, -1, 0]], [4, 1]]]]
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: impulse through D=2	`[1, 3, 0]`	`[1, 3, 0]`	Passed
regression: length divisible	`[1, 5]`	`[1, 5]`	Passed
repair check: first sample matters	`[7, 1]`	`[7, 1]`	Passed
regression: long filter D=3	`[1, 6, 11]`	`[1, 6, 11]`	Passed
regression: random polyphase 0	`[6, 11, -4, -3]`	`[6, 11, -4, -3]`	Passed
regression: random polyphase 1	`[-6, 4]`	`[-6, 4]`	Passed
regression: random polyphase 2	`[-8, 5]`	`[-8, 5]`	Passed

SHA-256 / cb4c842736f512d8e42ac1d08b70ab1a6ef4558a0e39ad5ffe4b4de7e7273c31

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 2026-09-29T14:51:36.919714+00:00.

Case digest / fc460f22f6fc6fc41ecf14895a2fbb17143156a39dc034753192bb785d0115d2