FA-91791 / Digital signal filters / Open access

Q15 FIR shifts each product before accumulating · case 01

Many small products each truncate to zero or -1, biasing the output.

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

ROOT CAUSE

Every product is shifted down to Q15 before the sum, discarding the guard bits.

VERIFIED REPAIR

Accumulate full-precision products and round once at the end.

Unsuccessful approach: The attempted repair rounds each product before summing, which still accumulates rounding error.

Case contract

Input [coeffs, samples] as Q15 integers. Each output accumulates the full-precision products c_k x[n-k] (zero history) in a wide accumulator, then rounds once (add 2**14, arithmetic shift 15) and saturates to int16. Return {"y": outputs, "saturations": number of clipped outputs}.

Why this case matters

DSP MAC units keep guard bits and round once; per-product rounding or missing overflow accounting degrades fixed-point FIRs.

1 / The failure

Exit 1

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

N = 1
observations = []
def solve(x):
    coefs, xs = x
    hist = [0] * len(coefs)
    out = []
    sat = 0
    for v in xs:
        hist = [v] + hist[:-1]
        acc = sum((c * s) >> 15 for c, s in zip(coefs, hist))
        y = acc
        if y > 32767:
            y = 32767
            sat += 1
        elif y < -32768:
            y = -32768
            sat += 1
        out.append(y)
    return {'y': out, 'saturations': sat}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: exact positive rail', [[16384, 16384], [32767, 32767, 0]], {'y': [16384, 32767, 16384], 'saturations': 0}], ['regression: exact rail then negative overflow', [[16384, 16384, 32767], [32767, 32767, -32768, -32768]], {'y': [16384, 32767, 32766, -2], 'saturations': 0}], ['regression: random q15 fir 0', [[-32619, -22239, 2111], [244, -26104, -17179, -16]], {'y': [-243, 25820, 32767, 9993], 'saturations': 1}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}]], [['regression: random q15 fir 1', [[-2422, 8115, -26238], [81, -19840, 164, 18, -31318, 26443]], {'y': [-6, 1487, -4990, 15926, 2188, -9725], 'saturations': 0}], ['regression: random q15 fir 2', [[-1074, -846, 29231], [-7102, 287, 18053, 24830, -286, 148]], {'y': [233, 174, -6935, -1024, 15473, 22152], 'saturations': 0}], ['regression: random q15 fir 0', [[-32619, -22239, 2111], [244, -26104, -17179, -16]], {'y': [-243, 25820, 32767, 9993], 'saturations': 1}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 0}], ['control: random q15 fir 30', [[-1704], [264, 265, -20482, -11272]], {'y': [-14, -14, 1065, 586], 'saturations': 0}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}]], [['regression: random q15 fir 4', [[375, 13267], [120, 1419, -3648]], {'y': [1, 65, 533], 'saturations': 0}], ['regression: random q15 fir 5', [[21007, -2568, -1633], [-267, 32265, 7287, 49]], {'y': [-171, 20705, 2156, -2148], 'saturations': 0}], ['regression: random q15 fir 2', [[-1074, -846, 29231], [-7102, 287, 18053, 24830, -286, 148]], {'y': [233, 174, -6935, -1024, 15473, 22152], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}]], [['regression: random q15 fir 7', [[-2335, 2343], [-158, -4249, -3825, -225]], {'y': [11, 291, -31, -257], 'saturations': 0}], ['regression: random q15 fir 8', [[-249, 2187, 304], [24116, 6453, 6219, -174, -23699]], {'y': [-183, 1561, 607, 476, 226], 'saturations': 0}], ['regression: random q15 fir 5', [[21007, -2568, -1633], [-267, 32265, 7287, 49]], {'y': [-171, 20705, 2156, -2148], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 0}], ['control: random q15 fir 30', [[-1704], [264, 265, -20482, -11272]], {'y': [-14, -14, 1065, 586], 'saturations': 0}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}]], [['regression: random q15 fir 10', [[17726], [-261, 178, -17320, 220]], {'y': [-141, 96, -9369, 119], 'saturations': 0}], ['regression: random q15 fir 11', [[-1511], [227, 17400]], {'y': [-10, -802], 'saturations': 0}], ['regression: random q15 fir 8', [[-249, 2187, 304], [24116, 6453, 6219, -174, -23699]], {'y': [-183, 1561, 607, 476, 226], 'saturations': 0}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 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 fixture	Actual	Expected	Outcome
regression: exact positive rail	`{'saturations': 0, 'y': [16383, 32766, 16383]}`	`{'saturations': 0, 'y': [16384, 32767, 16384]}`	Failed
regression: exact rail then negative overflow	`{'saturations': 0, 'y': [16383, 32766, 32765, -2]}`	`{'saturations': 0, 'y': [16384, 32767, 32766, -2]}`	Failed
regression: random q15 fir 0	`{'saturations': 1, 'y': [-243, 25819, 32767, 9992]}`	`{'saturations': 1, 'y': [-243, 25820, 32767, 9993]}`	Failed
control: small products lost	`{'saturations': 0, 'y': [3, 6, 9, 9]}`	`{'saturations': 0, 'y': [3, 6, 9, 9]}`	Passed
control: negative overflow	`{'saturations': 1, 'y': [-32767, -32768]}`	`{'saturations': 1, 'y': [-32767, -32768]}`	Passed
control: positive overflow	`{'saturations': 1, 'y': [32766, 32767]}`	`{'saturations': 1, 'y': [32766, 32767]}`	Passed
control: asymmetric taps	`{'saturations': 0, 'y': [500, 250, 0, -125, 0]}`	`{'saturations': 0, 'y': [500, 250, 0, -125, 0]}`	Passed

SHA-256 / 34f29e8538f9bdf0612a5ac9041a8f76acecc66108aa6c145295d26e4fc8447c

2 / The unsuccessful fix

Exit 1

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

N = 1
observations = []
def solve(x):
    coefs, xs = x
    hist = [0] * len(coefs)
    out = []
    sat = 0
    for v in xs:
        hist = [v] + hist[:-1]
        acc = sum((c * s + (1 << 14)) >> 15 for c, s in zip(coefs, hist))
        y = acc
        if y > 32767:
            y = 32767
            sat += 1
        elif y < -32768:
            y = -32768
            sat += 1
        out.append(y)
    return {'y': out, 'saturations': sat}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: exact positive rail', [[16384, 16384], [32767, 32767, 0]], {'y': [16384, 32767, 16384], 'saturations': 0}], ['regression: exact rail then negative overflow', [[16384, 16384, 32767], [32767, 32767, -32768, -32768]], {'y': [16384, 32767, 32766, -2], 'saturations': 0}], ['regression: random q15 fir 0', [[-32619, -22239, 2111], [244, -26104, -17179, -16]], {'y': [-243, 25820, 32767, 9993], 'saturations': 1}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}]], [['regression: random q15 fir 1', [[-2422, 8115, -26238], [81, -19840, 164, 18, -31318, 26443]], {'y': [-6, 1487, -4990, 15926, 2188, -9725], 'saturations': 0}], ['regression: random q15 fir 2', [[-1074, -846, 29231], [-7102, 287, 18053, 24830, -286, 148]], {'y': [233, 174, -6935, -1024, 15473, 22152], 'saturations': 0}], ['regression: random q15 fir 0', [[-32619, -22239, 2111], [244, -26104, -17179, -16]], {'y': [-243, 25820, 32767, 9993], 'saturations': 1}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 0}], ['control: random q15 fir 30', [[-1704], [264, 265, -20482, -11272]], {'y': [-14, -14, 1065, 586], 'saturations': 0}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}]], [['regression: random q15 fir 4', [[375, 13267], [120, 1419, -3648]], {'y': [1, 65, 533], 'saturations': 0}], ['regression: random q15 fir 5', [[21007, -2568, -1633], [-267, 32265, 7287, 49]], {'y': [-171, 20705, 2156, -2148], 'saturations': 0}], ['regression: random q15 fir 2', [[-1074, -846, 29231], [-7102, 287, 18053, 24830, -286, 148]], {'y': [233, 174, -6935, -1024, 15473, 22152], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}]], [['regression: random q15 fir 7', [[-2335, 2343], [-158, -4249, -3825, -225]], {'y': [11, 291, -31, -257], 'saturations': 0}], ['regression: random q15 fir 8', [[-249, 2187, 304], [24116, 6453, 6219, -174, -23699]], {'y': [-183, 1561, 607, 476, 226], 'saturations': 0}], ['regression: random q15 fir 5', [[21007, -2568, -1633], [-267, 32265, 7287, 49]], {'y': [-171, 20705, 2156, -2148], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 0}], ['control: random q15 fir 30', [[-1704], [264, 265, -20482, -11272]], {'y': [-14, -14, 1065, 586], 'saturations': 0}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}]], [['regression: random q15 fir 10', [[17726], [-261, 178, -17320, 220]], {'y': [-141, 96, -9369, 119], 'saturations': 0}], ['regression: random q15 fir 11', [[-1511], [227, 17400]], {'y': [-10, -802], 'saturations': 0}], ['regression: random q15 fir 8', [[-249, 2187, 304], [24116, 6453, 6219, -174, -23699]], {'y': [-183, 1561, 607, 476, 226], 'saturations': 0}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 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 fixture	Actual	Expected	Outcome
regression: exact positive rail	`{'saturations': 1, 'y': [16384, 32767, 16384]}`	`{'saturations': 0, 'y': [16384, 32767, 16384]}`	Failed
regression: exact rail then negative overflow	`{'saturations': 1, 'y': [16384, 32767, 32766, -2]}`	`{'saturations': 0, 'y': [16384, 32767, 32766, -2]}`	Failed
regression: random q15 fir 0	`{'saturations': 1, 'y': [-243, 25819, 32767, 9993]}`	`{'saturations': 1, 'y': [-243, 25820, 32767, 9993]}`	Failed
control: small products lost	`{'saturations': 0, 'y': [3, 6, 9, 9]}`	`{'saturations': 0, 'y': [3, 6, 9, 9]}`	Passed
control: negative overflow	`{'saturations': 1, 'y': [-32767, -32768]}`	`{'saturations': 1, 'y': [-32767, -32768]}`	Passed
control: positive overflow	`{'saturations': 1, 'y': [32766, 32767]}`	`{'saturations': 1, 'y': [32766, 32767]}`	Passed
control: asymmetric taps	`{'saturations': 0, 'y': [500, 250, 0, -125, 0]}`	`{'saturations': 0, 'y': [500, 250, 0, -125, 0]}`	Passed

SHA-256 / 9edeb825282933bf1ad6c83e25bf66d24108a0db0c35e46e8ca4f53ec7673f4d

3 / The verified repair

Exit 0

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

N = 1
observations = []
def solve(x):
    coefs, xs = x
    hist = [0] * len(coefs)
    out = []
    sat = 0
    for v in xs:
        hist = [v] + hist[:-1]
        acc = sum(c * s for c, s in zip(coefs, hist))
        y = (acc + (1 << 14)) >> 15
        if y > 32767:
            y = 32767
            sat += 1
        elif y < -32768:
            y = -32768
            sat += 1
        out.append(y)
    return {'y': out, 'saturations': sat}
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression: exact positive rail', [[16384, 16384], [32767, 32767, 0]], {'y': [16384, 32767, 16384], 'saturations': 0}], ['regression: exact rail then negative overflow', [[16384, 16384, 32767], [32767, 32767, -32768, -32768]], {'y': [16384, 32767, 32766, -2], 'saturations': 0}], ['regression: random q15 fir 0', [[-32619, -22239, 2111], [244, -26104, -17179, -16]], {'y': [-243, 25820, 32767, 9993], 'saturations': 1}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}]], [['regression: random q15 fir 1', [[-2422, 8115, -26238], [81, -19840, 164, 18, -31318, 26443]], {'y': [-6, 1487, -4990, 15926, 2188, -9725], 'saturations': 0}], ['regression: random q15 fir 2', [[-1074, -846, 29231], [-7102, 287, 18053, 24830, -286, 148]], {'y': [233, 174, -6935, -1024, 15473, 22152], 'saturations': 0}], ['regression: random q15 fir 0', [[-32619, -22239, 2111], [244, -26104, -17179, -16]], {'y': [-243, 25820, 32767, 9993], 'saturations': 1}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 0}], ['control: random q15 fir 30', [[-1704], [264, 265, -20482, -11272]], {'y': [-14, -14, 1065, 586], 'saturations': 0}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}]], [['regression: random q15 fir 4', [[375, 13267], [120, 1419, -3648]], {'y': [1, 65, 533], 'saturations': 0}], ['regression: random q15 fir 5', [[21007, -2568, -1633], [-267, 32265, 7287, 49]], {'y': [-171, 20705, 2156, -2148], 'saturations': 0}], ['regression: random q15 fir 2', [[-1074, -846, 29231], [-7102, 287, 18053, 24830, -286, 148]], {'y': [233, 174, -6935, -1024, 15473, 22152], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}]], [['regression: random q15 fir 7', [[-2335, 2343], [-158, -4249, -3825, -225]], {'y': [11, 291, -31, -257], 'saturations': 0}], ['regression: random q15 fir 8', [[-249, 2187, 304], [24116, 6453, 6219, -174, -23699]], {'y': [-183, 1561, 607, 476, 226], 'saturations': 0}], ['regression: random q15 fir 5', [[21007, -2568, -1633], [-267, 32265, 7287, 49]], {'y': [-171, 20705, 2156, -2148], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 0}], ['control: random q15 fir 30', [[-1704], [264, 265, -20482, -11272]], {'y': [-14, -14, 1065, 586], 'saturations': 0}], ['control: small products lost', [[1000, 1000, 1000], [100, 100, 100, 100]], {'y': [3, 6, 9, 9], 'saturations': 0}], ['control: negative overflow', [[32767, 32767], [-32768, -32768]], {'y': [-32767, -32768], 'saturations': 1}]], [['regression: random q15 fir 10', [[17726], [-261, 178, -17320, 220]], {'y': [-141, 96, -9369, 119], 'saturations': 0}], ['regression: random q15 fir 11', [[-1511], [227, 17400]], {'y': [-10, -802], 'saturations': 0}], ['regression: random q15 fir 8', [[-249, 2187, 304], [24116, 6453, 6219, -174, -23699]], {'y': [-183, 1561, 607, 476, 226], 'saturations': 0}], ['control: positive overflow', [[32767, 32767], [32767, 32767]], {'y': [32766, 32767], 'saturations': 1}], ['control: asymmetric taps', [[16384, 8192, 0, -4096], [1000, 0, 0, 0, 0]], {'y': [500, 250, 0, -125, 0], 'saturations': 0}], ['control: random q15 fir 14', [[2191], [5955, -128]], {'y': [398, -9], 'saturations': 0}], ['control: random q15 fir 16', [[-1447], [-227, 108]], {'y': [10, -5], 'saturations': 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 fixture	Actual	Expected	Outcome
regression: exact positive rail	`{'saturations': 0, 'y': [16384, 32767, 16384]}`	`{'saturations': 0, 'y': [16384, 32767, 16384]}`	Passed
regression: exact rail then negative overflow	`{'saturations': 0, 'y': [16384, 32767, 32766, -2]}`	`{'saturations': 0, 'y': [16384, 32767, 32766, -2]}`	Passed
regression: random q15 fir 0	`{'saturations': 1, 'y': [-243, 25820, 32767, 9993]}`	`{'saturations': 1, 'y': [-243, 25820, 32767, 9993]}`	Passed
control: small products lost	`{'saturations': 0, 'y': [3, 6, 9, 9]}`	`{'saturations': 0, 'y': [3, 6, 9, 9]}`	Passed
control: negative overflow	`{'saturations': 1, 'y': [-32767, -32768]}`	`{'saturations': 1, 'y': [-32767, -32768]}`	Passed
control: positive overflow	`{'saturations': 1, 'y': [32766, 32767]}`	`{'saturations': 1, 'y': [32766, 32767]}`	Passed
control: asymmetric taps	`{'saturations': 0, 'y': [500, 250, 0, -125, 0]}`	`{'saturations': 0, 'y': [500, 250, 0, -125, 0]}`	Passed

SHA-256 / 828798186d7d7d4b637bef5ef65aaa4269d5d9c29f6ab9fa4ec15e85ae9afac0

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 2026-09-29T14:51:39.407242+00:00.

Case digest / a94d1b1bfcee71ba42b62a5aeb0fc3b7e24c223c6acc350fd43a656e3d5341b9