FA-91741 / Digital signal filters / Open access

Feedback comb stores the input in its delay line · case 01

The impulse response has a single echo instead of a decaying train.

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

ROOT CAUSE

The ring buffer stores x[n] instead of y[n], making it a feedforward comb.

THE FAILURE

The ring buffer stores x[n] instead of y[n], making it a feedforward comb.

Unsuccessful approach: The attempted repair stores g y[n], applying the gain twice per trip.

Case contract

Input [D, g, samples]: y[n] = x[n] + g y[n-D] using a D-slot ring buffer with zero initial state; "bad-delay" if D < 1, "unstable" if |g| >= 1. Return exact fraction strings.

Why this case matters

Feedback combs build resonators and reverberators; ring-buffer indexing slips change the delay and the pitch.

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):
    D, g, xs = x[0], Fraction(x[1]), x[2]
    if D < 1:
        return 'bad-delay'
    if abs(g) >= 1:
        return 'unstable'
    buf = [Fraction(0)] * D
    pos = 0
    out = []
    for v in xs:
        delayed = buf[pos]
        y = v + g * delayed
        buf[pos] = v
        pos = (pos + 1) % D
        out.append(str(y))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: impulse delay 2', [2, '1/2', [1, 0, 0, 0, 0, 0]], ['1', '0', '1/2', '0', '1/4', '0']], ['regression: delay 1', [1, '-1/2', [1, 0, 0]], ['1', '-1/2', '1/4']], ['regression: random comb 0', [2, '1/2', [-1, 3, 3, 0, 3, 2, -3, -1, -2]], ['-1', '3', '5/2', '3/2', '17/4', '11/4', '-7/8', '3/8', '-39/16']], ['control: unit gain rejected', [3, '1', [1]], 'unstable'], ['control: negative unit gain', [3, '-1', [1]], 'unstable'], ['control: bad delay', [0, '1/2', [1]], 'bad-delay'], ['control: random comb 6', [4, '-2/3', [1, 0, -3, 3]], ['1', '0', '-3', '3']]], [['regression: random comb 1', [4, '1/2', [-3, -2, 2, -3, 2, -1, -3, -2, -3]], ['-3', '-2', '2', '-3', '1/2', '-2', '-2', '-7/2', '-11/4']], ['regression: random comb 5', [2, '-2/3', [-3, -3, -1, -3, -1]], ['-3', '-3', '1', '-1', '-5/3']], ['regression: random comb 0', [2, '1/2', [-1, 3, 3, 0, 3, 2, -3, -1, -2]], ['-1', '3', '5/2', '3/2', '17/4', '11/4', '-7/8', '3/8', '-39/16']], ['control: random comb 8', [4, '-2/3', [1, 3, -2, -1]], ['1', '3', '-2', '-1']], ['control: random comb 14', [4, '-2/3', [0, 0, -3, -3, -3]], ['0', '0', '-3', '-3', '-3']], ['control: random comb 17', [4, '1/3', [2, -2, 1, -1]], ['2', '-2', '1', '-1']], ['control: random comb 20', [4, '-1/2', [3, 2, 3, 2]], ['3', '2', '3', '2']]], [['regression: random comb 9', [3, '1/2', [-2, 0, -3, 1, -3, 0, 3, 1]], ['-2', '0', '-3', '0', '-3', '-3/2', '3', '-1/2']], ['regression: random comb 11', [1, '3/4', [0, 1, 2, 0, -2, 1, 0]], ['0', '1', '11/4', '33/16', '-29/64', '169/256', '507/1024']], ['repair check: random comb 2', [4, '-2/3', [2, 2, -1, 3, -1, 2, 1]], ['2', '2', '-1', '3', '-7/3', '2/3', '5/3']], ['control: random comb 31', [4, '-1/2', [-2, -2, 0, 2]], ['-2', '-2', '0', '2']], ['control: random comb 34', [4, '-2/3', [3, -3, -3, -3]], ['3', '-3', '-3', '-3']], ['control: unit gain rejected', [3, '1', [1]], 'unstable'], ['control: negative unit gain', [3, '-1', [1]], 'unstable']], [['regression: random comb 22', [3, '-2/3', [2, 1, -3, 2, 2, 0, 2, 1, 3]], ['2', '1', '-3', '2/3', '4/3', '2', '14/9', '1/9', '5/3']], ['regression: random comb 23', [3, '1/2', [2, 3, -1, 3, 0, -3, 0, 2, -1]], ['2', '3', '-1', '4', '3/2', '-7/2', '2', '11/4', '-11/4']], ['repair check: random comb 4', [3, '1/3', [-3, 3, 1, -1]], ['-3', '3', '1', '-2']], ['control: bad delay', [0, '1/2', [1]], 'bad-delay'], ['control: random comb 6', [4, '-2/3', [1, 0, -3, 3]], ['1', '0', '-3', '3']], ['control: random comb 8', [4, '-2/3', [1, 3, -2, -1]], ['1', '3', '-2', '-1']], ['control: random comb 14', [4, '-2/3', [0, 0, -3, -3, -3]], ['0', '0', '-3', '-3', '-3']]], [['regression: random comb 25', [1, '1/3', [0, -2, 2, 3, 0, 2, -2, -3]], ['0', '-2', '4/3', '31/9', '31/27', '193/81', '-293/243', '-2480/729']], ['regression: random comb 29', [1, '1/2', [-3, 0, -3, -3, 2]], ['-3', '-3/2', '-15/4', '-39/8', '-7/16']], ['regression: random comb 7', [4, '-2/3', [1, 3, 3, -1, -1, 2, 3, -2, 2]], ['1', '3', '3', '-1', '-5/3', '0', '1', '-4/3', '28/9']], ['control: random comb 17', [4, '1/3', [2, -2, 1, -1]], ['2', '-2', '1', '-1']], ['control: random comb 20', [4, '-1/2', [3, 2, 3, 2]], ['3', '2', '3', '2']], ['control: random comb 31', [4, '-1/2', [-2, -2, 0, 2]], ['-2', '-2', '0', '2']], ['control: random comb 34', [4, '-2/3', [3, -3, -3, -3]], ['3', '-3', '-3', '-3']]]]
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 delay 2	`['1', '0', '1/2', '0', '0', '0']`	`['1', '0', '1/2', '0', '1/4', '0']`	Failed
regression: delay 1	`['1', '-1/2', '0']`	`['1', '-1/2', '1/4']`	Failed
regression: random comb 0	`['-1', '3', '5/2', '3/2', '9/2', '2', '-3/2', '0', '-7/2']`	`['-1', '3', '5/2', '3/2', '17/4', '11/4', '-7/8', '3/8', '-39/16']`	Failed
control: unit gain rejected	`unstable`	`unstable`	Passed
control: negative unit gain	`unstable`	`unstable`	Passed
control: bad delay	`bad-delay`	`bad-delay`	Passed
control: random comb 6	`['1', '0', '-3', '3']`	`['1', '0', '-3', '3']`	Passed

SHA-256 / f1e1f2bbbe04f71b4946dca1bea21502756befa531783fbead97bc78e28cea61

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):
    D, g, xs = x[0], Fraction(x[1]), x[2]
    if D < 1:
        return 'bad-delay'
    if abs(g) >= 1:
        return 'unstable'
    buf = [Fraction(0)] * D
    pos = 0
    out = []
    for v in xs:
        delayed = buf[pos]
        y = v + g * delayed
        buf[pos] = g * y
        pos = (pos + 1) % D
        out.append(str(y))
    return out
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: impulse delay 2', [2, '1/2', [1, 0, 0, 0, 0, 0]], ['1', '0', '1/2', '0', '1/4', '0']], ['regression: delay 1', [1, '-1/2', [1, 0, 0]], ['1', '-1/2', '1/4']], ['regression: random comb 0', [2, '1/2', [-1, 3, 3, 0, 3, 2, -3, -1, -2]], ['-1', '3', '5/2', '3/2', '17/4', '11/4', '-7/8', '3/8', '-39/16']], ['control: unit gain rejected', [3, '1', [1]], 'unstable'], ['control: negative unit gain', [3, '-1', [1]], 'unstable'], ['control: bad delay', [0, '1/2', [1]], 'bad-delay'], ['control: random comb 6', [4, '-2/3', [1, 0, -3, 3]], ['1', '0', '-3', '3']]], [['regression: random comb 1', [4, '1/2', [-3, -2, 2, -3, 2, -1, -3, -2, -3]], ['-3', '-2', '2', '-3', '1/2', '-2', '-2', '-7/2', '-11/4']], ['regression: random comb 5', [2, '-2/3', [-3, -3, -1, -3, -1]], ['-3', '-3', '1', '-1', '-5/3']], ['regression: random comb 0', [2, '1/2', [-1, 3, 3, 0, 3, 2, -3, -1, -2]], ['-1', '3', '5/2', '3/2', '17/4', '11/4', '-7/8', '3/8', '-39/16']], ['control: random comb 8', [4, '-2/3', [1, 3, -2, -1]], ['1', '3', '-2', '-1']], ['control: random comb 14', [4, '-2/3', [0, 0, -3, -3, -3]], ['0', '0', '-3', '-3', '-3']], ['control: random comb 17', [4, '1/3', [2, -2, 1, -1]], ['2', '-2', '1', '-1']], ['control: random comb 20', [4, '-1/2', [3, 2, 3, 2]], ['3', '2', '3', '2']]], [['regression: random comb 9', [3, '1/2', [-2, 0, -3, 1, -3, 0, 3, 1]], ['-2', '0', '-3', '0', '-3', '-3/2', '3', '-1/2']], ['regression: random comb 11', [1, '3/4', [0, 1, 2, 0, -2, 1, 0]], ['0', '1', '11/4', '33/16', '-29/64', '169/256', '507/1024']], ['repair check: random comb 2', [4, '-2/3', [2, 2, -1, 3, -1, 2, 1]], ['2', '2', '-1', '3', '-7/3', '2/3', '5/3']], ['control: random comb 31', [4, '-1/2', [-2, -2, 0, 2]], ['-2', '-2', '0', '2']], ['control: random comb 34', [4, '-2/3', [3, -3, -3, -3]], ['3', '-3', '-3', '-3']], ['control: unit gain rejected', [3, '1', [1]], 'unstable'], ['control: negative unit gain', [3, '-1', [1]], 'unstable']], [['regression: random comb 22', [3, '-2/3', [2, 1, -3, 2, 2, 0, 2, 1, 3]], ['2', '1', '-3', '2/3', '4/3', '2', '14/9', '1/9', '5/3']], ['regression: random comb 23', [3, '1/2', [2, 3, -1, 3, 0, -3, 0, 2, -1]], ['2', '3', '-1', '4', '3/2', '-7/2', '2', '11/4', '-11/4']], ['repair check: random comb 4', [3, '1/3', [-3, 3, 1, -1]], ['-3', '3', '1', '-2']], ['control: bad delay', [0, '1/2', [1]], 'bad-delay'], ['control: random comb 6', [4, '-2/3', [1, 0, -3, 3]], ['1', '0', '-3', '3']], ['control: random comb 8', [4, '-2/3', [1, 3, -2, -1]], ['1', '3', '-2', '-1']], ['control: random comb 14', [4, '-2/3', [0, 0, -3, -3, -3]], ['0', '0', '-3', '-3', '-3']]], [['regression: random comb 25', [1, '1/3', [0, -2, 2, 3, 0, 2, -2, -3]], ['0', '-2', '4/3', '31/9', '31/27', '193/81', '-293/243', '-2480/729']], ['regression: random comb 29', [1, '1/2', [-3, 0, -3, -3, 2]], ['-3', '-3/2', '-15/4', '-39/8', '-7/16']], ['regression: random comb 7', [4, '-2/3', [1, 3, 3, -1, -1, 2, 3, -2, 2]], ['1', '3', '3', '-1', '-5/3', '0', '1', '-4/3', '28/9']], ['control: random comb 17', [4, '1/3', [2, -2, 1, -1]], ['2', '-2', '1', '-1']], ['control: random comb 20', [4, '-1/2', [3, 2, 3, 2]], ['3', '2', '3', '2']], ['control: random comb 31', [4, '-1/2', [-2, -2, 0, 2]], ['-2', '-2', '0', '2']], ['control: random comb 34', [4, '-2/3', [3, -3, -3, -3]], ['3', '-3', '-3', '-3']]]]
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 delay 2	`['1', '0', '1/4', '0', '1/16', '0']`	`['1', '0', '1/2', '0', '1/4', '0']`	Failed
regression: delay 1	`['1', '1/4', '1/16']`	`['1', '-1/2', '1/4']`	Failed
regression: random comb 0	`['-1', '3', '11/4', '3/4', '59/16', '35/16', '-133/64', '-29/64', '-645/256']`	`['-1', '3', '5/2', '3/2', '17/4', '11/4', '-7/8', '3/8', '-39/16']`	Failed
control: unit gain rejected	`unstable`	`unstable`	Passed
control: negative unit gain	`unstable`	`unstable`	Passed
control: bad delay	`bad-delay`	`bad-delay`	Passed
control: random comb 6	`['1', '0', '-3', '3']`	`['1', '0', '-3', '3']`	Passed

SHA-256 / 2fd233917187c67daeddf0b848cc103b980a39b015611d8df414f2e5db8e39db

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 2026-09-29T14:51:38.847483+00:00.

Case digest / 94b32a62d0e42aadd9a7b11f917215c504896435c4b349d8d8b3e93851c1e5ed