FFTの振幅が合わない原因と補正方法：Hann窓・片側スペクトル・ゼロパディングをPythonで検証

import numpy as np
import matplotlib.pyplot as plt
from scipy.signal.windows import hann


def amplitude_spectrum(x, fs, window=None, nfft=None):
    x = np.asarray(x)
    if x.ndim != 1 or x.size < 2 or np.iscomplexobj(x):
        raise ValueError("x must be a real one-dimensional record")
    x = x.astype(float)
    n = len(x)
    if not np.isfinite(fs) or fs <= 0 or not np.all(np.isfinite(x)):
        raise ValueError("finite samples and positive fs are required")
    m = n if nfft is None else int(nfft)
    if m < n or (nfft is not None and m != nfft):
        raise ValueError("nfft must be an integer >= len(x)")
    w = np.ones(n) if window is None else np.asarray(window, dtype=float)
    if w.shape != x.shape or not np.all(np.isfinite(w)) or w.sum() <= 0:
        raise ValueError("window must match x and have a positive sum")
    z = np.fft.rfft(x * w, n=m, norm="backward")
    a = np.abs(z) / w.sum()
    if m % 2 == 0:
        a[1:-1] *= 2
    else:
        a[1:] *= 2
    f = np.fft.rfftfreq(m, d=1 / fs)
    return f, a


fs, amplitude = 1024.0, 2.0
for n in (1024, 1023):
    k = 37
    t = np.arange(n) / fs
    x = amplitude * np.cos(2 * np.pi * (k * fs / n) * t + 0.3)
    for w in (np.ones(n), hann(n, sym=False)):
        for m in (n, 4 * n):
            f, a = amplitude_spectrum(x, fs, w, m)
            np.testing.assert_allclose(a[k * (m // n)], amplitude, atol=1e-10)
    # DC is not doubled, for either record length.
    _, dc = amplitude_spectrum(np.full(n, 1.25), fs)
    np.testing.assert_allclose(dc[0], 1.25, atol=1e-12)
    print(f"N={n}: rectangular/Hann and padded amplitude checks passed")

# An even FFT has a Nyquist bin; an odd FFT has an ordinary final bin.
n = 1024
_, a = amplitude_spectrum(0.7 * (-1.0) ** np.arange(n), fs)
np.testing.assert_allclose(a[-1], 0.7, atol=1e-12)
n = 1023
k = n // 2
_, a = amplitude_spectrum(0.7 * np.cos(2 * np.pi * k * np.arange(n) / n), fs)
np.testing.assert_allclose(a[-1], 0.7, atol=1e-10)

# Off-bin sinusoid: compare native and padded frequency grids.
n, fs, f0, amplitude = 256, 256.0, 10.5, 2.0
t = np.arange(n) / fs
x = amplitude * np.cos(2 * np.pi * f0 * t + 0.3)
w = hann(n, sym=False)
fig, ax = plt.subplots(figsize=(8, 4.5))
for m, label, marker in ((n, "256-point FFT", "o"), (8 * n, "2048-point FFT", None)):
    f, a = amplitude_spectrum(x, fs, w, m)
    ax.plot(f, a, marker=marker, label=label)
    band = (f > 5) & (f < 16)
    idx = np.flatnonzero(band)[np.argmax(a[band])]
    print(f"M={m}: peak={a[idx]:.6f} at {f[idx]:.3f} Hz")
ax.axhline(amplitude, color="black", linestyle="--", label="True peak amplitude = 2")
ax.set(xlim=(6, 15), ylim=(0, 2.2), xlabel="Frequency [Hz]", ylabel="Peak amplitude",
       title="Same 1-second record, different FFT grids (periodic Hann)")
ax.grid(alpha=0.3)
ax.legend()
fig.tight_layout()
fig.savefig("amplitude_padding.png", dpi=170)
plt.close(fig)