{"id":4823,"date":"2025-03-31T17:01:41","date_gmt":"2025-03-31T08:01:41","guid":{"rendered":"https:\/\/www.ktech.biz\/jp\/?p=4823"},"modified":"2025-05-06T16:43:46","modified_gmt":"2025-05-06T07:43:46","slug":"13-fft","status":"publish","type":"post","link":"https:\/\/www.ktech.biz\/jp\/tutorial\/13-fft\/","title":{"rendered":"13. \u9ad8\u901f\u30d5\u30fc\u30ea\u30a8\u5909\u63db (FFT)"},"content":{"rendered":"\n

13.1 \u6982\u8981<\/h3>\n

\u6ce8 – \u30d5\u30fc\u30ea\u30a8\u5909\u63db\/\u9006\u5909\u63db, DFT\/IDFT \u306a\u3069\u306e\u5b9a\u7fa9\u306b\u304a\u3044\u3066, \u6b63\u898f\u5316\u4fc2\u6570 (1\/N \u306a\u3069) \u304c\u6587\u732e\u306b\u3088\u3063\u3066\u7570\u306a\u308b\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u306e\u3067\u9069\u5b9c\u8aad\u307f\u66ff\u3048\u3066\u304f\u3060\u3055\u3044.<\/p>\n

13.1.1 \u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/h4>\n

\\(f(x)\\) \u304c\u8907\u7d20\u95a2\u6570\u306e\u3068\u304d, \u6b21\u3092\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u3068\u3044\u3044\u307e\u3059.
\n\\[
\nF(u) = (1\/2\\pi)\\int_{-\\infty}^{\\infty}f(x)e^{-iux}dx
\n\\]\n\u3053\u306e\u9006\u5909\u63db\u3092\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db\u3068\u3044\u3044\u307e\u3059.
\n\\[
\nf(x) = \\int_{-\\infty}^{\\infty}F(u)e^{iux}du
\n\\]\n

13.1.2 \u96e2\u6563\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/h4>\n

\u6709\u9650\u500b\u306e\u5165\u529b\u95a2\u6570\u5024(\u96e2\u6563\u5024)\u3092\u4f7f\u7528\u3057\u3066\u6570\u5024\u7684\u306b\u30d5\u30fc\u30ea\u30a8\u5909\u63db (\u306e\u96e2\u6563\u8fd1\u4f3c) \u3092\u6c42\u3081\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059.<\/p>\n

\u7b49\u9593\u9694\u306b\u3068\u3063\u305f \\(N\\) \u500b\u306e\u5206\u70b9 \\(x_j = 2{\\pi}j\/N (j = 0 \\sim N – 1)\\) \u306b\u304a\u3051\u308b\u95a2\u6570\u5024 \\(f(x_j)\\) \u3092\u4f7f\u7528\u3059\u308b\u3053\u3068\u306b\u3057, \\(f_j\\) \u3068\u8868\u3057\u307e\u3059.<\/p>\n

\\(f(x)\\) \u304c\u5468\u671f \\(2\\pi\\) \u306e\u5468\u671f\u95a2\u6570\u3067\u3042\u308b\u3068\u4eee\u5b9a\u3057, \u7a4d\u5206\u533a\u9593\u3092 \\([0, 2\\pi]\\) \u3068\u3057\u3066 \\(F(u)\\) \u306e\u7a4d\u5206\u3092\u8907\u5408\u53f0\u5f62\u516c\u5f0f\u306b\u3088\u308a\u6c42\u3081\u308b\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\nF(u) = (1\/N)\\sum_{j=0}^{N-1}f_je^{-iux_j}
\n\\]\n\\(u_k = 2{\\pi}k, F_k = F(u_k)\\) \u3068\u8868\u3059\u3068\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\nF_k = (1\/N)\\sum_{j=0}^{N-1}f_je^{-i2{\\pi}jk\/N} \\space (k = 0 \\sim N – 1)
\n\\]\n\u3053\u308c\u3092\u96e2\u6563\u30d5\u30fc\u30ea\u30a8\u5909\u63db (DFT) \u3068\u3044\u3044\u307e\u3059. \u3053\u308c\u306f\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u306e\u8fd1\u4f3c\u306b\u306a\u3063\u3066\u3044\u307e\u3059.<\/p>\n

\u540c\u69d8\u306b\u96e2\u6563\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db (IDFT) \u306f\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059.
\n\\[
\nf_j = \\sum_{k=0}^{N-1}F_ke^{i2{\\pi}kj\/N} \\space (j = 0 \\sim N – 1)
\n\\]\n

13.1.3 \u5b9f\u6570\u306e\u96e2\u6563\u30d5\u30fc\u30ea\u30a8\u5909\u63db<\/h4>\n

\u5165\u529b\u95a2\u6570\u5024\u304c\u5b9f\u6570\u306e\u5834\u5408\u306b\u306f \\(F_{N-k} = \\overline{F_k}\\) \u3068\u306a\u308b\u3053\u3068\u304b\u3089 \\(k = 0 \\sim N\/2\\) \u306b\u3064\u3044\u3066\u3060\u3051\u6c42\u3081\u308c\u3070\u3088\u3044\u3053\u3068\u306b\u306a\u308a\u307e\u3059 (\\(\\overline{F_k}\\) \u306f \\(F_k\\) \u306e\u5171\u5f79\u8907\u7d20\u6570\u3092\u8868\u3057\u307e\u3059). \u307e\u305f, \\(F_0\\) \u3068 \\(F_{N\/2}\\) \u306f\u5b9f\u6570\u306b\u306a\u308a\u307e\u3059.<\/p>\n

\u901a\u5e38\u306f, \\(a_k = 2Re(F_k), b_k = -2Im(F_k)\\) \u3068\u7f6e\u304d\u63db\u3048\u3066, \u6b21\u306e\u3088\u3046\u306b\u5b9f\u6570\u6f14\u7b97\u3067\u8868\u3057\u307e\u3059 (\\(Re\\) \u306f\u5b9f\u6570\u90e8, \\(Im\\) \u306f\u865a\u6570\u90e8\u3092\u8868\u3057\u307e\u3059).
\n\\[
\n\\begin{align}
\n& a_0 = (1\/N)\\sum_{j=0}^{N-1}f_j \\\\
\n& a_k = (2\/N)\\sum_{j=0}^{N-1}f_jcos(2{\\pi}jk\/N) \\space (k = 1 \\sim N2) \\\\
\n& a_{N\/2} = (1\/N)\\sum_{j=0}^{N-1}(-1)^jf_j \\\\
\n& b_k = (2\/N)\\sum_{j=0}^{N-1}f_jsin(2{\\pi}jk\/N) \\space (k = 1 \\sim N2) \\\\
\n\\end{align}
\n\\]\n\u305f\u3060\u3057, \\(N2 = N\/2 – 1\\) (\\(N\\) \u304c\u5076\u6570\u306e\u5834\u5408), \\(N2 = (N – 1)\/2\\) (\\(N\\) \u304c\u5947\u6570\u306e\u5834\u5408).<\/p>\n

IDFT \u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059.
\n\\[
\n\\begin{align}
\n& N \u304c\u5076\u6570\u306e\u5834\u5408:
\n f_j = a_0 + \\sum_{k=1}^{N\/2}(a_kcos(2{\\pi}kj\/N) + b_ksin(2{\\pi}kj\/N)) + (-1)^ka_N\/2 \\space (j = 0 \\sim N – 1) \\\\
\n& N \u304c\u5947\u6570\u306e\u5834\u5408:
\n f_j = a_0 + \\sum_{k=1}^{(N-1)\/2}(a_kcos(2{\\pi}kj\/N) + b_ksin(2{\\pi}kj\/N)) \\space (j = 0 \\sim N – 1) \\\\
\n\\end{align}
\n\\]\n

13.2 \u9ad8\u901f\u30d5\u30fc\u30ea\u30a8\u5909\u63db (FFT)<\/h3>\n

DFT \u304a\u3088\u3073 IDFT \u3092\u5c11\u306a\u3044\u6f14\u7b97\u56de\u6570\u3067\u6c42\u3081\u308b\u6570\u5024\u8a08\u7b97\u6cd5\u3092\u9ad8\u901f\u30d5\u30fc\u30ea\u30a8\u5909\u63db (FFT) \u3068\u3044\u3044\u307e\u3059.<\/p>\n

DFT \u3092\u5b9a\u7fa9\u3069\u304a\u308a\u8a08\u7b97\u3059\u308b\u306a\u3089\u3070, \\(\\sim\\) \u306e\u4e2d\u3067 \\(N\\) \u56de\u306e\u8a08\u7b97\u3092, \\(N\\) \u500b\u306e \\(k\\) \u306b\u3064\u3044\u3066\u884c\u3046\u3068\u5168\u4f53\u3067 \\(N^2\\) \u56de\u306e\u30aa\u30fc\u30c0\u30fc\u306e\u8a08\u7b97\u304c\u5fc5\u8981\u306b\u306a\u308a\u307e\u3059.<\/p>\n

\u3057\u304b\u3057, \\(N\\) \u304c\u5c0f\u3055\u306a\u7d20\u6570\u306e\u3079\u304d\u4e57\u306e\u7a4d\u3067\u8868\u3055\u308c\u308b\u5834\u5408\u306b\u306f FFT \u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u4f7f\u3048\u3070\u305a\u3063\u3068\u5c11\u306a\u3044\u8a08\u7b97\u91cf\u3067\u6e08\u307f\u307e\u3059. \u4f8b\u3048\u3070, \\(N = 2^n\\) \u306e\u3088\u3046\u306b 2 \u306e\u3079\u304d\u4e57\u3067\u8868\u3055\u308c\u308c\u3070 \\(Nlog_2 N\\) \u56de\u306e\u30aa\u30fc\u30c0\u30fc\u3067\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059 (\u4f8b: \\(N = 1024\\) \u3067\u3042\u308c\u3070, \u901a\u5e38\u7d04100\u4e07\u56de\u5fc5\u8981\u306a\u306e\u304c\u7d041\u4e07\u56de\u3067\u3088\u304f\u306a\u308a, 100 \u500d\u901f\u304f\u306a\u308a\u307e\u3059).<\/p>\n

13.3 XLPack \u3092\u4f7f\u3063\u305f\u9ad8\u901f\u30d5\u30fc\u30ea\u30a8\u5909\u63db (FFT) \u306e\u4f7f\u3044\u65b9<\/h3>\n

VBA \u30b5\u30d6\u30eb\u30fc\u30c1\u30f3 Rfft1f<\/strong>, Rfft1b<\/strong>, Rfft1i<\/strong> \u3042\u308b\u3044\u306f\u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u95a2\u6570 WRfft1f<\/strong>, WRfft1b<\/strong> \u3092\u4f7f\u3044 1 \u6b21\u5143\u306e\u5b9f\u30d5\u30fc\u30ea\u30a8\u5909\u63db\u304a\u3088\u3073\u5b9f\u30d5\u30fc\u30ea\u30a8\u9006\u5909\u63db\u3092\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.<\/p>\n

\u4f8b\u984c<\/h5>\n

2.5kHz \u306e\u642c\u9001\u6ce2\u3092 (100Hz \u306e\u6b63\u5f26\u6ce2 + 200Hz \u306e\u6b63\u5f26\u6ce2) \u3067 AM \u5909\u8abf\u3059\u308b\u3068\u6b21\u306e\u3088\u3046\u306a\u6ce2\u5f62\u306b\u306a\u308a\u307e\u3059. <\/p>\n

\"\"<\/p>\n

\u3053\u306e\u30c7\u30fc\u30bf\u3092\u4f7f\u3063\u3066 DFT \u304a\u3088\u3073 IDFT \u3092\u6c42\u3081\u3066\u307f\u307e\u3059.<\/p>\n

13.3.1 \u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u95a2\u6570\u3092\u4f7f\u7528\u3057\u305f\u6c42\u3081\u65b9<\/h4>\n

1\/20000 \u79d2\u3054\u3068\u306b (20 kHz \u3067) \u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3057\u305f 1000 \u500b\u306e\u30c7\u30fc\u30bf\u3092\u7528\u610f\u3057, DFT \u3092\u6c42\u3081\u307e\u3059 (\u5909\u63db\u7d50\u679c\u306e\u683c\u7d0d\u5185\u5bb9\u306b\u3064\u3044\u3066\u306f\u30ea\u30d5\u30a1\u30ec\u30f3\u30b9\u30de\u30cb\u30e5\u30a2\u30eb\u3092\u53c2\u7167). \u305d\u3057\u3066, \u3055\u3089\u306b\u305d\u306e IDFT \u3092\u6c42\u3081\u3066\u5909\u63db\u304c\u6b63\u3057\u304f\u884c\u308f\u308c\u3066\u3044\u308b\u3053\u3068\u3092\u78ba\u8a8d\u3057\u307e\u3059.<\/p>\n

DFT \u3092\u6c42\u3081\u308b\u305f\u3081\u306b\u306f, WRfft1f<\/strong> \u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u95a2\u6570\u3092\u4f7f\u3044\u307e\u3059.<\/p>\n

\"\"<\/p>\n

DFT \u304c\u6c42\u3081\u3089\u308c\u305f\u3089, \u305d\u306e\u5185\u5bb9\u306b\u5bfe\u3057\u3066 WRfft1b<\/strong> \u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u95a2\u6570\u306b\u3088\u308a IDFT \u3092\u9069\u7528\u3057\u307e\u3059.<\/p>\n

\"\"<\/p>\n

\u4e0b\u306e\u3088\u3046\u306b, \u5143\u306e\u30c7\u30fc\u30bf\u304c\u5fa9\u5143\u3055\u308c\u305f\u306e\u304c\u78ba\u8a8d\u3067\u304d\u307e\u3059.<\/p>\n

\"\"<\/p>\n

13.3.2 \u30d1\u30ef\u30fc\u30b9\u30da\u30af\u30c8\u30eb<\/h4>\n

FFT \u306e\u5fdc\u7528\u306e\u3072\u3068\u3064\u306b\u30d1\u30ef\u30fc\u30b9\u30da\u30af\u30c8\u30eb\u5206\u6790\u304c\u3042\u308a\u307e\u3059. \u3053\u308c\u306f, \u30b9\u30da\u30af\u30c8\u30e9\u30e0\u30a2\u30ca\u30e9\u30a4\u30b6\u3092\u4f7f\u3063\u3066\u5165\u529b\u4fe1\u53f7\u306e\u5468\u6ce2\u6570\u5206\u5e03\u3092\u89b3\u6e2c\u3059\u308b\u306e\u306b\u76f8\u5f53\u3057\u307e\u3059.<\/p>\n

\\(N\\) \u500b\u306e\u30c7\u30fc\u30bf \\(f_j\\) \u306e DFT \u51fa\u529b\u304b\u3089, \u5468\u6ce2\u6570 \\(\\omega_k\\) \u306b\u304a\u3051\u308b\u30d1\u30ef\u30fc \\(P_k\\) \u3092\u6b21\u306e\u3088\u3046\u306b\u3057\u3066\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059. \u305f\u3060\u3057, \\(\\omega_k = (k\/N)\\omega_s\\) (\\(\\omega_s\\) \u306f\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u5468\u6ce2\u6570) \u3067\u3059.
\n\\[
\n\\begin{align}
\n& P_0 = P(f_0) = N |F_0|^2 = N a_k^2 \\\\
\n& P_k = P(f_k) = N (|F_k|^2 + |F_{N-k}|^2) = (N\/2)(a_k^2 + b_k^2) \\space (k = 1 \\sim N\/2 – 1) \\\\
\n& P_N\/2 = P(f_N\/2) = N |F_N\/2|^2 = N a_{N\/2}^2 \\space (N \u304c\u5076\u6570\u306e\u5834\u5408) \\\\
\n\\end{align}
\n\\]\n\u5168\u30a8\u30cd\u30eb\u30ae\u30fc \\(E\\) \u306f\u3053\u308c\u3089\u306e\u7dcf\u548c\u306b\u3088\u308a\u6c42\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059.
\n\\[
\nE = \\sum_{k=0}^{N\/2} P_k
\n\\]\n\u3055\u3066, \u4e0a\u306e\u4f8b\u984c\u3067\u4f7f\u7528\u3057\u305f\u30c7\u30fc\u30bf\u306f 20kHz \u3067\u30b5\u30f3\u30d7\u30ea\u30f3\u30b0\u3057\u305f AM \u5909\u8abf\u6ce2\u306e\u30c7\u30fc\u30bf\u3067\u3057\u305f. \u3053\u306e DFT \u7d50\u679c\u3088\u308a\u30d1\u30ef\u30fc\u30b9\u30da\u30af\u30c8\u30eb\u3092\u6c42\u3081\u3066\u307f\u307e\u3059.<\/p>\n

\u30ef\u30fc\u30af\u30b7\u30fc\u30c8\u4e0a\u3067\u8a08\u7b97\u3059\u308b\u3068\u8907\u96d1\u306b\u306a\u308b\u306e\u3067, \u30d1\u30ef\u30fc\u3092\u6c42\u3081\u308b\u6b21\u306e VBA \u95a2\u6570\u3092\u7528\u610f\u3057\u307e\u3059.<\/p>\n

\n
Function PsdR(N As Long, VR As Variant) As Variant\r\n    Dim R() As Double, P() As Double, I As Long\r\n    ReDim R(N - 1), P(N \/ 2, 0)\r\n    For I = 1 To N\r\n        R(I - 1) = VR(I)\r\n    Next\r\n    P(0, 0) = N * R(0) ^ 2\r\n    For I = 1 To N - 3 Step 2\r\n        P((I + 1) \/ 2, 0) = (N \/ 2) * (R(I) ^ 2 + R(I + 1) ^ 2)\r\n    Next\r\n    P(N \/ 2, 0) = N * R(N - 1) ^ 2\r\n    PsdR = P\r\nEnd Function<\/code><\/pre>\n<\/div>\n
\u3053\u306e\u95a2\u6570\u3092\u4f7f\u3063\u3066\u30d1\u30ef\u30fc\u30b9\u30da\u30af\u30c8\u30eb\u3092\u6c42\u3081\u307e\u3059.<\/p>\n
\"\"<\/p>\n
DFT \u7d50\u679c\u3092\u5165\u529b\u306b\u6307\u5b9a\u3057\u307e\u3059.<\/p>\n
\"\"<\/p>\n
\u30d1\u30ef\u30fc\u30b9\u30da\u30af\u30c8\u30eb\u304c\u6c42\u3081\u3089\u308c\u307e\u3057\u305f. \u3053\u308c\u304b\u3089\u4f5c\u6210\u3055\u308c\u305f\u5468\u6ce2\u6570\u5206\u5e03\u306e\u30b0\u30e9\u30d5\u306f\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3057\u305f (\u30d4\u30ea\u30aa\u30c9\u30b0\u30e9\u30e0\u3068\u3044\u3044\u307e\u3059).<\/p>\n
\"\"<\/p>\n
2.5kHz \u306e\u642c\u9001\u6ce2\u306e\u4e0a\u4e0b \u00b1100Hz \u3068 \u00b1200Hz \u306e\u3068\u3053\u308d\u306b\u4fe1\u53f7\u304c\u3042\u308b AM \u5909\u8abf\u6ce2\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u304c\u78ba\u304b\u3081\u3089\u308c\u307e\u3057\u305f.<\/p>\n","protected":false},"excerpt":{"rendered":"
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