{"id":7712,"date":"2009-02-22T01:07:00","date_gmt":"2009-02-22T01:07:00","guid":{"rendered":"http:\/\/melotopia.net\/b\/?p=7712"},"modified":"2009-02-22T01:07:00","modified_gmt":"2009-02-22T01:07:00","slug":"%ed%91%b8%eb%a6%ac%ec%97%90-%eb%b3%80%ed%99%98fourier-transformation-5","status":"publish","type":"post","link":"http:\/\/melotopia.net\/b\/?p=7712","title":{"rendered":"\ud478\ub9ac\uc5d0 \ubcc0\ud658(Fourier Transformation) #5"},"content":{"rendered":"
\n

\n *\uc5ec\uae30\uc5d0 \uc124\uba85\ub41c FFT\ub294 \ub2e8\uc21c\ud788 \uac1c\ub150\ub9cc\uc744 \uc124\uba85\ud558\uace0 \uc788\ub2e4. \uc2e4\uc81c\uc801\uc778 \uc54c\uace0\ub9ac\uc998\uc740 \uc88b\uc740 \ucc45\uacfc \uc778\ud130\ub137 \uac80\uc0c9\uc744 \ud65c\uc6a9\ud558\uc5ec\uc11c \uc815\ud655\ud55c \ub0b4\uc6a9\uc744 \uc774\ud574\ud55c \ud6c4\uc5d0 \uc0ac\uc6a9\ud558\uc5ec\uc57c \ud55c\ub2e4.
\n \n<\/p>\n

\n \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc740 \ud604\ub300 \uacf5\ud559\uc5d0\uc11c \uc544\uc8fc\uc544\uc8fc\uc544\uc8fc \ub2e4\uc591\ud55c \ubc94\uc704\uc5d0\uc11c \uc751\uc6a9\ub418\ub294\ub370, \uadf8\uac83\uc740 \uc2e4\uc81c \uc138\uc0c1\uc744 \ubd84\uc11d\ud558\ub294 \uc2e4\uc9c8\uc801\uc774\uba74\uc11c \uac15\ub825\ud55c \ub3c4\uad6c\uc774\uae30 \ub54c\ubb38\uc774\ub2e4. \ud558\uc9c0\ub9cc \uc55e\uc11c \uc598\uae30\ud588\ub4ef, \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc740 \uc801\ubd84\uc5d0 \uad00\ud55c \ubb38\uc81c\uc774\uace0 \uc801\ubd84\uc744 \uc798 \ud558\ub294 \uac83\uc740 \uc544\uc8fc\uc544\uc8fc\uc544\uc8fc \uc5b4\ub824\uc6b4 \ubb38\uc81c\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774\uac83\uc740 \ub300\ub2e8\ud788 \ud798\ub4e0 \uc77c\uc774 \ub41c\ub2e4. \uc801\ubd84\uc740 \ub204\uad6c\uc5d0\uac8c\ub098 \uc5b4\ub824\uc6b4 \uc77c\uc778\ub370, \ud558\ubb3c\uba70 \uc218\uc2dd\uc744 \uc774\ud574\uc870\ucc28 \ubabb\ud558\ub294 \ucef4\ud4e8\ud130\uac00 \uc5b4\ub5bb\uac8c \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc744 \ud560 \uc218 \uc788\uc744\uae4c? \uadf8\ub798\uc11c, \uc5f0\uc18d\uccb4\uc5d0 \uad00\ub828\ub41c \uc801\ubd84 \ubb38\uc81c\ub97c \ucef4\ud4e8\ud130\ub3c4 \uc798 \ud560 \uc904 \uc544\ub294 \ub2e8\uc21c\ud55c \ub367\uc148\uc73c\ub85c \ubc14\uafd4\uc11c \ube44\uad50\uc801 \uc815\ud655\ud55c \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc744 \ud558\ub294 \uac83\uc774 DFT\uac00 \ub41c\ub2e4. \ud558\uc9c0\ub9cc DFT\ub294 N\uac1c\uc758 \uad6c\uac04\uc73c\ub85c \ub098\ub234\uc744 \ub54c N*N\uac1c\uc758 \uacc4\uc0b0\uc744 \ud558\ub294 \ubc29\ubc95\uc774\ub2e4. \uc880 \ub354 \ud68d\uae30\uc801\uc778 \ubc29\ubc95\uc774 \ub4f1\uc7a5\ud558\ub294\ub370, \uadf8\uac83\uc774 \ube60\ub978 \ud478\ub9ac\uc5d0 \ubcc0\ud658(Fast Fourier Transform=FFT)\uc774\ub2e4. \ube60\ub978 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc740 \uc2e4\uc81c\ub85c N*N\ubc88 \uac78\ub9b4 \uacc4\uc0b0\uc744 N*logN\ubc88 \uc815\ub3c4\ub85c \uc904\uc5ec\uc8fc\uac8c \ub41c\ub2e4.<\/p>\n

\uac00\uc7a5 \uae30\ubcf8\uc801\uc73c\ub85c, \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc5d0\uc11c \uc0ac\uc6a9\ud558\ub294 \ud568\uc218\ub4e4\uc774 \uc9c0\uc218\ud568\uc218\ub77c\ub294 \uac83\uc744 \uc0dd\uac01\ud574\ubcf4\uc790.
\n
\n $e^{a}*e^{b}=e^{a+b}$
\n
\n \uc798 \ubcf4\uc2dc\ub77c, \uc9c0\uc218 \ud568\uc218\uc640 \ub85c\uadf8 \ud568\uc218\uc758 \uac00\uc7a5 \ud070 \ud2b9\uc9d5\uc740 \ub367\uc148\uacfc \uacf1\uc148\uc744 \uc11c\ub85c \ubc14\uafd4 \uc4f8 \uc218 \uc788\ub2e4\ub294 \uac83\uc774\ub2e4. \uc9c0\uc218\ud568\uc218\uc5d0\uc11c\ub294 \uacf1\uc148\uc774 \ub367\uc148\uc73c\ub85c \ubc14\ub00c\uace0, \ub85c\uadf8 \ud568\uc218\uc5d0\uc11c\ub294 \ub367\uc148\uc774 \uacf1\uc148\uc73c\ub85c \ubc14\ub010\ub2e4. \uc544\ubb34\ud2bc \uac00\uc7a5 \uc911\uc694\ud55c \ud2b9\uc131\uc740 \uc9c0\uc218 \ubc11\uc5d0 \uc788\ub358 \uacf1\uc148\uc774 \uc9c0\uc218 \uc704\uc5d0\uc11c \ub367\uc148\uc73c\ub85c \ubcc0\uc2e0\ud588\ub2e4\ub294 \uc0ac\uc2e4.<\/p>\n

\uadf8\ub7fc, DFT\uc5d0 \ub098\uc624\ub294 \uc218\uc2dd\uc744 \uc798 \uc0b4\ud3b4\ubcf4\uc790.
\n
\n $a_1=f(n2\\pi\/N)*e^{ik(n2\\pi\/N)$
\n
\n \uc800\uac78 $a_1$\uc774\ub77c\uace0 \ubd80\ub974\uc790.
\n
\n \uadf8\ub7f0\ub370 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc5d0\uc11c \ub098\uc640\uc57c \ud558\ub294 \uac83\uc740 \ud568\uc218\uc774\ubbc0\ub85c k\uac00 \ubcc0\ud558\uac8c \ub418\uba74 \uacc4\uc0b0\uc744 \uc0c8\ub85c \ud574\uc57c \ud55c\ub2e4.\n <\/p>\n

\ud558\uc9c0\ub9cc – \uac04\uacfc\ud574\uc11c\ub294 \uc548\ub418\ub294 \uc0ac\uc2e4\uc774 \uc788\ub2e4. \uc800\uae30\uc5d0 \ub354\ud558\uae30 \uc804\uc5d0 \uacf1\ud558\ub294 \uc22b\uc790\ub294 \ubc14\ub85c \ud06c\uae30\uac00 1\uc778 \uc9c0\uc218\ud568\uc218\ub77c\ub294 \uac83. \uc544\ubb34\ub9ac \uacf1\ud574\ub3c4 \ud06c\uae30\ub294 \ubcc0\ud558\uc9c0 \uc54a\uc73c\uba70, \uadf8 \uc704\uc0c1\uac01\ub9cc \uacc4\uc18d \ub3cc\uc544\uac04\ub2e4. \uac8c\ub2e4\uac00, \ub9cc\uc57d \uad6c\uac04\uc744 \uc815\uc218\uc778 N\ub4f1\ubd84 \ud574 \ub193\ub294\ub2e4\uba74 k\uac00 \ucee4\uc9c0\uba74\uc11c \uacc4\uc0b0\ud574\uc57c \ud560 \uacc4\uc218\ub294 \uc0c8\ub85c \uacc4\uc0b0\ud560 \ud544\uc694 \uc5c6\uc774, $e^{i(n2\\pi\/N)$\uc744 \uacc4\uc18d \uacf1\ud574\uac00\uba74\uc11c \uacc4\uc0b0\ud558\uba74 \ub41c\ub2e4.
\n
\n $e^{i(n2\\pi\/N)$\ub97c 2\ubc88 \uba3c\uc800 \uacf1\ud558\uace0 3\ubc88 \uacf1\ud558\ub4e0, 3\ubc88 \uba3c\uc800 \uacf1\ud558\uace0 2\ubc88 \uacf1\ud558\ub4e0, 5\ubc88 \uacf1\ud588\ub2e4\ub294 \uc0ac\uc2e4\uc740 \ubcc0\ud568\uc774 \uc5c6\ub2e4. \ubc14\ub85c \uc774\ub7f0 \uc131\uc9c8\uc744 \uc774\uc6a9\ud558\ub294 \uac83\uc774\ub2e4.<\/p>\n

\uc6b0\uc120, \uadc0\ucc2e\uc73c\ub2c8\uae4c $e^{i(n2\\pi\/N)$\ub97c w\ub77c\uace0 \uc4f0\uace0, $w(k)=w^k$\ub77c\uace0 \ud558\uc790.<\/p>\n

\uac00\ub839, \uc810\uc774 4\uac1c \uc788\ub2e4\uace0 \ud574 \ubcf4\uc790. 0, 1, 2, 3\uc774\ub77c\ub294 \ubc88\ud638\ub97c \ubd99\uc5ec\ubcf4\uc790. \uadf8\ub7fc 4\uac1c\uc758 \uc810\uc740 x(0), x(1), x(2), x(3)\uc774 \ub41c\ub2e4.
\n
\n \uc774 \uc810 4\uac1c\uc5d0 \ub300\ud55c \uc774\uc0b0 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4
\n
\n $X(0)=x(0)w(0)+x(1)w(1*0)+x(2)w(1*0)+x(3)w(1*0)$
\n
\n $X(1)=x(0)w(0*1)+x(1)w(1*1)+x(2)w(2*1)+x(3)w(3*1)$
\n
\n $X(2)=x(0)w(0*2)+x(1)w(1*2)+x(2)w(2*2)+x(3)w(3*2)$
\n
\n $X(3)=x(0)w(0*3)+x(1)w(1*3)+x(2)w(2*3)+x(3)w(3*3)$<\/p>\n

$w(k+N)=w(k)$\uc778 \uc131\uc9c8\uc744 \uc774\uc6a9\ud574\uc11c \uc815\ub9ac\ud574\uc8fc\uba74 (\uc774 \uc131\uc9c8\uc774 \uac00\uc7a5 \uc911\uc694\ud558\ub2e4. \uc774\ub7ec\ud55c \uc131\uc9c8\uc774 \uc5c6\ub2e4\uba74, FFT\ub294 \ubd88\uac00\ub2a5\ud558\uace0 DFT\uc804\uccb4\ub97c \uacc4\uc0b0\ud574\uc57c\ub9cc \ud55c\ub2e4. \uacf1\uc148\uc758 \uad50\ud658\ubc95\uce59\uacfc \uc720\ud55c\uad70\ub860\uc758 \uc2b9\ub9ac.)
\n
\n $X(0)=x(0)w(0)+x(1)w(0)+x(2)w(0)+x(3)w(0)$
\n
\n $X(1)=x(0)w(0)+x(1)w(1)+x(2)w(2)+x(3)w(3)$
\n
\n $X(2)=x(0)w(0)+x(1)w(2)+x(2)w(0)+x(3)w(2)$
\n
\n $X(3)=x(0)w(0)+x(1)w(3)+x(2)w(2)+x(3)w(1)$<\/p>\n

\uc774 \uc218\uc2dd\uc744 \ub6ab\uc5b4\uc9c0\uac8c \uccd0\ub2e4\ubcf4\uba74, \uc774\ubbf8 \uacc4\uc0b0\ud55c \ud56d\uc774 \uc5ec\ub7ec\ubc88 \ub4f1\uc7a5\ud558\uace0 \uc788\ub294 \uac83\uc744 \ud655\uc778\ud560 \uc218 \uc788\ub2e4.
\n
\n \ub2e4\uc2dc\ub9d0\ud574\uc11c, DFT\ub97c \uacc4\uc0b0\ud560 \ub54c \uc800\uac78 \uc804\ubd80 \uacc4\uc0b0\ud55c\ub2e4\uba74 \ubcf4\ub2e4\uc2dc\ud53c N*N\ubc88\uc758 \uacc4\uc0b0\uc744 \uc804\ubd80 \ub2e4 \ud574\uc57c \ud55c\ub2e4. \ud558\uc9c0\ub9cc, \ubc18\ubcf5\ub418\ub294 \ubd80\ubd84\uc744 \uc798 \uc54c\uace0 \uc788\ub2e4\uac00, \ub2e4\uc74c\ubc88\uc5d0 \uacc4\uc0b0\ud560 \ub54c \uc368\uba39\ub294\ub2e4\uba74 \uacc4\uc0b0 \ube44\uc6a9\uc774 \uc808\uc57d\ub41c\ub2e4. FFT\ub294 \uc218\uc2dd\uc744 \ub6ab\uc5b4\uc9c0\uac8c \uccd0\ub2e4\ubcf8 \uc0ac\ub78c\uc758 \uc2b9\ub9ac\ub77c\uace0 \ud560 \uc218 \uc788\uaca0\ub2e4.
\n
\n \ud558\uc9c0\ub9cc, \uc544\ubb34\ub9ac \ucef4\ud4e8\ud130\ub77c\ub3c4 \uc800 \uc22b\uc790\ub4e4\uc744 \ub2e4 \uc678\uc6b0\uace0 \uc788\ub2e4\ubcf4\uba74 \uba38\ub9ac\uac00 \uc544\ud504\uaca0\uc9c0. (\uc751?)<\/p>\n

\uc815\ud655\ud558\uac8c\ub294, \uc9dd\uc218\ud56d\uacfc \ud640\uc218\ud56d\uc73c\ub85c \ub098\ub204\uace0, \uadf8 \uc9dd\uc218\ud56d\uacfc \ud640\uc218\ud56d\uc73c\ub85c \ub098\ub208 \uac83\uc744 \ub2e4\uc2dc \uac01\uac01 \uc9dd\uc218\ud56d\uacfc \ud640\uc218\ud56d\uc73c\ub85c \ub098\ub204\uace0, \uc774 \uac83\uc744 \ubc18\ubcf5\ud574\uc11c 2\ud56d \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc774 \ub420 \ub54c\uae4c\uc9c0 \uacc4\uc18d\ud55c\ub2e4. \uadf8 \ub2e4\uc74c\uc5d0, \uc870\uac01\ub09c \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc744 \uc801\ub2f9\ud788 \ub354\ud574\uc11c \uc804\uccb4 \ud478\ub9ac\uc5d0 \ubcc0\ud658\uc744 \uc644\uc131\ud558\uac8c \ub41c\ub2e4.
\n
\n $x(0)w(0)+x(2)w(0)$\uacfc $x(0)w(0)+x(2)w(2)$\uc744 \uacc4\uc0b0\ud558\uace0, $x(1)w(0)+x(3)w(0)$\uacfc $x(1)w(1)+x(3)w(3)$\uc744 \uacc4\uc0b0\ud55c\ub2e4.
\n
\n $E1=x(0)w(0)+x(2)w(0)$
\n
\n $E2=x(0)w(0)+x(2)w(2)$
\n
\n $E3=x(1)w(0)+x(3)w(0)$
\n
\n $E4=x(1)w(1)+x(3)w(3)$
\n
\n \ub77c\uace0 \ud574\ubcf4\uc790.
\n
\n $X(0)=E1+E3$
\n
\n $X(1)=E2+E4$
\n
\n $X(2)=E1+E3*w(2)$
\n
\n $X(3)=E2+E4*w(2)$
\n
\n \uc774\ub807\uac8c \ub41c\ub2e4.
\n
\n \ucc98\uc74c\uc758 DFT\uc5d0\uc11c 16\ubc88\uc758 \uacf1\uc148\uacfc 12\ubc88\uc758 \ub367\uc148\uc774 \uc788\uc5c8\ub2e4\uba74, \ub098\uc758 4\uc810 FFT\uc5d0\uc11c\ub294 10\ubc88\uc758 \uacf1\uc148\uacfc 8\ubc88\uc758 \ub367\uc148\uc774 \uc788\uac8c \ub418\uc5c8\ub2e4. \ud655\uc2e4\ud788 \uc904\uc5b4\ub4e4\uc5c8\ub2e4. (\uc65c NlogN\ubc88\uc774 \uc544\ub2c8\ub0d0\uace0\ub294 \ubb3b\uc9c0\ub3c4 \ub530\uc9c0\uc9c0\ub3c4 \ub9d0\uc790. -_-; \uc22b\uc790\uac00 \uc791\uc544\uc11c \uadf8\ub807\ub2e4.)<\/p>\n

\uc22b\uc790\uac00 \uc801\uc744 \ub54c\ub294 \uadf8\ub2e4\uc9c0 \ucc28\uc774\uac00 \uc5c6\uc9c0\ub9cc, \ub9cc\uc57d \uc774 \uc22b\uc790\uac00 100\ub9cc\uac1c \uc815\ub3c4 \ub41c\ub2e4\uba74 \uadf8\ub550 \ubb34\uc2dc\ud560 \uc218 \uc5c6\ub294 \uc815\ub3c4\ub85c \ucc28\uc774\uac00 \ub098\uac8c \ub41c\ub2e4.<\/p>\n

\uc790. \uadf8\ub7f0\ub370, \uc774\ub7f0 \uac83\uc744 \ub3c4\ub300\uccb4 \uc5b4\ub5bb\uac8c \uc774\ud574\ud558\uba74 \uc88b\uc744\uae4c.
\n
\n gnuplot\uc758 \ub3c4\uc6c0\uc744 \ubc1b\uc544\uc11c \ub2e4\uc74c \uadf8\ub9bc\uc744 \uadf8\ub824\ubcf4\uc558\ub2e4.
\n
\n \uadf8\ub9bc\uc774 \uc880 \ud750\ub9bf\ud558\uc9c0\ub9cc, \ub300\ucda9 \ubcf4\uc790. \uc5b4\ub290\ucabd\uc774 x\ucd95\uc774\uace0 y\ucd95\uc778\uc9c0\ub294 \uc911\uc694\ud558\uc9c0 \uc54a\ub2e4.
\n <\/p>\n

\n\"\"\n<\/div>\n

\n \uc5ec\uae30\uc11c\ub294 sin\ud568\uc218\ub97c \uacc4\uc0b0\ud588\uc9c0\ub9cc, \ub300\ucda9 exp(ikx)\ud568\uc218\ub77c\uace0 \uc0dd\uac01\ud558\ub3c4\ub85d \ud55c\ub2e4. \ubcf5\uc18c\uc218 \uc0bc\uac01\ud568\uc218\ub294 \uadf8\ub9bc\uc73c\ub85c \uadf8\ub9b4 \ubc29\ubc95\uc774 \uc5c6\ub2e4.
\n
\n x=1\uc778 \uc120\uc744 \ub530\ub77c\uac08 \ub54c \uc0dd\uae30\ub294 \ud568\uc218\ub97c \uc54c\uace0 \uc788\uc744 \ub54c, x=2\uc778 \uc120\uc744 \ub530\ub77c\uac08 \ub54c \uc0dd\uae30\ub294 \ud568\uc218\ub294 \uc5b4\ub5bb\uac8c \uacc4\uc0b0\ud560\uae4c? \ub2f9\uc5f0\ud788 \uc81c\uacf1\ud558\uba74 \ub41c\ub2e4. (\uc0ac\uc778\uacf5\uc2dd\uc5d0\ub3c4 \ub098\uc624\uc9c0\ub9cc, \uc81c\uacf1\ud558\uba74 \ud568\uc218 \uc548\uc758 \uacc4\uc218\uac00 2\ubc30\uac00 \ub41c \ub2e4\ub978 \uc0ac\uc778\ud568\uc218\ub85c \ubcc0\uc2e0\ud55c\ub2e4.) \uc81c\uacf1\uc774\ub77c\ub294 \uac74 \uc790\uae30 \uc790\uc2e0\uc744 \uacf1\ud558\ub294 \uac83\uc774\ub2e4. x=3\uc778 \uc120\uc744 \ub530\ub77c\uac08 \ub54c \uc0dd\uae30\ub294 \ud568\uc218\ub294, \ub2e4\uc2dc \uc790\uae30 \uc790\uc2e0\uc744 \ud55c\ubc88 \ub354 \uacf1\ud558\uba74 \uc54c \uc218 \uc788\uac8c \ub41c\ub2e4. \uc798 \ubcf4\uba74 \uc54c\uaca0\uc9c0\ub9cc, x\uac00 1\uc778 \uc120\uc744 \ub530\ub77c\uc11c, y\uac00 -2\uc5d0\uc11c 2\uae4c\uc9c0 \ubcc0\ud560 \ub54c \uc0dd\uae30\ub294 \ud568\uc218\ub294 x\uac00 2\uc778 \uc120\uc744 \ub530\ub77c\uc11c y\uac00 0\uc5d0\uc11c 2\uae4c\uc9c0 \ubcc0\ud560\ub54c \uc0dd\uae30\ub294 \ud568\uc218\uc640 \ub611\uac19\uc774 \uc0dd\uacbc\ub2e4. \uc7a1\uc544 \ub2f9\uaca8\uc11c \ub298\ub824\uc8fc\uae30\ub9cc \ud558\uba74 \ub611\uac19\ub2e4.<\/p>\n

\uc989, \uc0bc\uac01\ud568\uc218\ub294 \uc81c\uacf1\ud574\uc11c \uc0dd\uae34 \ud568\uc218\uc758 \uc77c\ubd80\uac00 \uc6d0\ub798\uc758 \ud568\uc218\uc640 \uac19\ub2e4. (\ud504\ub799\ud0c8…\uadf8\ub7f0\uac74 \ub530\uc9c0\uc9c0 \ub9d0\uc790. \ud504\ub799\ud0c8 \uc544\ub2c8\ub2e4.)<\/p>\n

\uc800\ub7f0\uac78 \ud568\uc218 f(x)\uc5d0 \uacf1\ud574\uc11c \uc801\ubd84\ud55c\ub2e4\uace0 \ud558\uba74, \uc0bc\uac01\ud568\uc218\uc758 \ud2b9\uc9d5\uc744 \uc798 \uc774\uc6a9\ud574 \uc904 \uc218 \uc788\uac8c \ub418\ub294 \uac83\uc774\ub2e4.<\/p>\n

\uc774\ud574 \uc548\uac00\ub294 \ubd80\ubd84\uc740 \ub313\uae00\ub85c. -_-;
\n <\/p>\n

\n
\n
\n\"\uc2e0\uace0\"
\n<\/a>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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