5 16.4 FourierMD? Fouriers 1 p/ ?l ￥f ?￥ FourierMD ),( +∞?∞ ò ? ? ? << = ;,0 ,0, )(  ? δxA xf ó fx ax () e || = ?  ;a > 0 ? fx ax () e= ? 2  ; a > 0 ? ? ? ? < ≥ = ? ;0,0 ,0,e )( 2 x x xf x ? ? ? ? > ≤ = ;||,0 ,||,cos )( 0 δ δω x xxA xf  0 0 ≠ω ^è ? 0 ω π δ = b 3  1 () () ix f fxe dx ω ω +∞ ? ?∞ = ∫  0 ix Aedx δ ω? = ∫  )1( ωδ ω i e i A ? ? b  2 () () ix f fxe dx ω ω +∞ ? ?∞ = ∫  0 () () 0 ai x ai x ede ωω +∞ ?+ ? ?∞ =+ ∫∫ d 11 ai aiω ω =+ +? = 22 2 ω+a a b   () () ix f fxe dx ω ω +∞ ? ?∞ = ∫  2 ax i x ed ω +∞ ?? ?∞ == ∫ 2 cos ax exω +∞ ? ?∞ ∫ dx 2 0 2cos t tt ed aa ω +∞ ? = ∫  ?¨ è ￥2T  2 2 a e a ω π ?? ? ?? ?? == a e a 4 2 ω π ? b   () () ix f fxe dx ω ω +∞ ? ?∞ = ∫  (2 ) 0 ix edx ω +∞ ?+ == ∫ ωi+2 1 b   () () ix f fxe dx ω ω +∞ ? ?∞ = ∫   0 cos ix A xe dx δ ω δ ω ? ? ∫  0 cos cosA xx δ δ ωω ? = ∫ dx ′?1 f ?s1  00 [cos( ) cos( ) ] 2 A x xdx δ δ ωω ωω ? =?++ ∫  " 00 sin( ) sin( ) () () A ω ωδ ωωδ ωω ωω ? ??+ + ? ? ? ? b 2 p   ￥??MD???MDb fx ax () e= ? ),0[ +∞∈x a > 0 3 ??MD 0 () ()sinf fx xdxωω +∞ = ∫  0 sin ax exdx 22 ω ω +a ω +∞ ? = = ∫ ??MD 1  0 () ()cosf fx xdxωω +∞ = ∫  0 cos ax exdx 22 ω+a a b ω +∞ ? = = ∫ 3 !  ? ? ? < ≥ = ? ,0,0 ,0,e )( 1 x x xf x ? ? ? ? ? ≤≤ = ,,0 , 2 0,sin )( 2  ? π xx xf  p )( 21 xff ? b 3 : ()Fx= 2 12 21 1 0 () () sin() ( )f fx f fx tfxtdt π ?=?= ? ∫  I n [0, ] 2 t π ∈  ? H0x≤ 1 ()fxt 0? =  ?[ () 0Fx=  ? 2 x π > H  ?[ () 1 () xt fxt e ?? ?= 22 0 1 () sin() (1 ) 2 xt x Fx e e tdt e e π π ?? == ∫ + ? 0 2 x π <≤ H () 1 , () 0, xt ex fxt t x t ?? ? > ?= ? ≤ ?  ?[ 0 1 ( ) sin( ) (sin cos ) 2 x xt x F xe e tdt x xe==? ∫ +b ? ^   ? ? ? ? ? ? ? ? ? >+ ≤<+? ≤ =? ? ? . 2 ),1( 2 1 , 2 0),cos(sin 2 1 ,0,0 )( 2 21 π π π xee xexx x xff x x 2 5 16.5 y ? FourierMD 1 a ü ? ? FourierMD Xj xn i nj N n N () ()e= ? = ? ∑ 2 0 1 π V[ A? FourierMD ∫ ∞+ ∞? ? = dxxff xiω ω e)()( ?  ￥ ? ?í ?? T￥w<b 3 L ! 0ω>  2 2? () ()e x i ffxd ω π π ω ?+∞ ?∞ = ∫ 2( ) 2 ()e nx i n fnx x ω π π ? +∞ ? =?∞ ≈ ?? ∑  | Px? 1 2 x N ω π ? = : 2 e i N W π ? = 5 1? ? H b? ^k kN n n WW + = 1 0 ? () ( ) ) N n nk f WfkNnxω ?+∞ ==?∞ =+ ∑∑ x? : () (( ) ) k x nfkNnx +∞ =?∞ =+? ∑ x ?[  ? () ( )X jfjω= 1 0 () N jn n Wxn ? = = ∑ 1 2 0 ()e nj N i N n xn π ? ? = = ∑ b 2 ￡ ü??1" T kj N kn i N n N jn i N , 2 1 0 2 ee 1 δ ππ = ∑ ? = ? b 3 A ? Hjk= 1 22 0 1 ee nk nk N ii NN n N ππ ? ? = 1= ∑ b / ? I n jk≠ ?^ ! jk< b? ? 1≠ξ ^Z? ￥B?? H  μ  7 1= N x 0 1 0 = ∑ ? = N n n ξ () 2 e1 kj i N π ξ ? =≠5 2( ) e Nkjiπ ξ ? 1= = b? ^ 1 0 N n n ξ ? = = ∑ () 11 22 2 00 1 ee e nj nk nk j NN ii i NN N nn ππ π ? ?? ? == 0= = ∑∑ b 3 !  /o3Np= q qp, N∈ ))(( NqpO + Q ?￥ FourierMD 3 ?Eb 3 7 2 e i N W π ? = 5 k1? ? H bL !  kN n n WW + = 101 0 ,0,1,1,0,1,jjqj j p j q=+ = ? = ?""1, 101 0 , 0,1, , 1, 0,1, , 1nnpn n q n p=+ = ? = ?""b Xj xn i nj N n N () ()e= ? = ? ∑ 2 0 1 π 1 0 () N jn n x nW ? = = ∑ 10 01 11 () 10 00 () pq jnpn nn xnp n W ?? + == =+ ∑∑ 01 01 11 10 00 () pq jn n j p nn WxnpnW ?? == =+ ∑∑ 0 b %? 9 ? 31j 10 1 1 10 0 () q njp n xnp n W ? = + ∑ 1q? QeE  1 0n = ?31Se E?M]￥  ^M]￥í3×ˉ9 ? ? μN ?? T 3 QeEb p?31 0 j 10 1 1 10 0 () q njp n xnp n W ? = + ∑ (1qq? ) 0 n (1)pN? QeE ?[ 9 31 QeEb (1)( 1) ( )qq p N O p qN?+ ? = + ) 4 N = 2 3  8[ 21?￥ FFT￥9 ? @?b 3 : 2 8 4 ee i i W π π ? ? ==5 48 1, 1WW=?=b V¤9 ? T 7 0 () () , 0,1, ,7. jn n Xj xnW j = == ∑ " [ (0) ( 1) (4)] [ (1) ( 1) (5)] jj j xxWx=+? + +?x 2 {[ (2) ( 1) (6)] [ (3) ( 1) (7)]} jjjj Wx x Wx x++?++?b . 9 ? @? ?B? 1 1 () () ( 4), (4) [() (4)], 0,1,2,3 i x i xi xi xi Wxi xi i = ++ += ?+ = 4 ?=? 211 2 1 () () ( 2), (2) [() (2)], 0,1,4,5 i xi xi xi xi W xi xi i . = ++ += ? + = ? ?? 22 22 22 22 () () ( 1), (4) () (1), 0,2, ( ) ( 3) ( 4), (4) (3) (4),1,3 Xi x i x i Xi xi xi i Xi x i x i Xi x i x i i . = ++ += ? + = =+++ += +? + = 5 9 ? L 5  ? =￥·?/I???è09 ?  L=9 ? ? ?¨C?￥ ?DY¨ èq  ? MATLABaMathematicaaMaple?  ?  N = 32 64 128,, ò 3? L ??  {(  )}xk k N = ? 0 1 ó ¨ FFT9 ? {( ￥ ? ? FourierMD?  {(  )}xk k N = ? 0 1 )}Xj j N = ? 0 1 ? T ? {| ￥mié?s ?nm  {( )}xk ( )|}Xj ? !? 0 0 >δ | {| ? ?@( )|}Xj 0 |)(| δ<jX ￥ ? ???1 , é? ? ? Fourier IMD|¤?￥ ? D {( 1? )}xk ? ?M 0 δ ￥′×ˉ?s?]￥ 0 δ  IMD ?¤?￥ ? ￥?Yb 3 ÷??1 GVODUJPOFY /  U/ YSBOEO /  SBOEO ZGGU Y /  [BCT Z  QMPU U Y   U [ P   EFMUBJOQVU  h { ?μ  GPSJ/ JG[ J  EFMUB Z J   FOE FOE [SFBM JGGU Z  QMPU U Y   U [ P  ?2Ts[ N=128 1 èb'?? ? ^ ?á 3￥ o+p1 e S ?  oop1MDa￥ ￥ ? b | 0 5δ = | {| ? ?@( )|}Xj 0 |)(| δ<jX ￥ ? ???1 ,é? ? ? 6 Fourier IMDb o+p1e S ?  oop1? ,aMD¤?￥ ?  D {( 1?+?×? )}xk | 0 50δ = ]") ?a¤?￥ ? D {( 1?μtlμb )}xk | 0 100δ = ]") ?a¤?￥ ? D {( 1?μ b Vn? ??vb )}xk ?? ? ÷?]2T?μ ?μsb ? ?  N = 32 64 128,, ò á 3 ? L ??  ? {(  {( )}xk k N = ? 0 1 )}yk k N = ? 0 1 ó ¨°¤ZE9 ? ? {( ￥  {(  {( )}xk )}yk )}zk k N = ? 0 1 ? ?¨ ? ?FourierMD￥ ±X¨ FFT9 ? {(  )}zk ? 2? N 1? ? ?E ?¨￥ HWb 3 ÷??1 GVODUJPOUFY /  YSBOEO /  SBOEO ZSBOEO /  SBOEO UJD ? eV [DPOW Y Z  7 GPSJ/ [ J   GPSKJ [ J  [ J  Y K  Z JK   FOE GPSKJ / [ J  [ J  Y K  Z / JK   FOE FOE  UUPD9 H UJD YGGU Y /  ZGGU Z /  [JGGU Y Z  UUPD U<U U> s 9 ? ?? HWD P¨￥9 ?? ?μ1??9 ?9 H  ￥K l?ê?v ???￥9 ? ' P?N=128 ?? HW+?1 0b7 O?? ?¨} ?3 d3? ?? ¤ FourierMD?¨ =? f ? ?? y¨ FFT9 ? ?1 y¤b à ¨ FFT9 ?[ T () ()! ? + + = ∑ 1 21 21 0 nn n m x n ? () ()! ? = ∑ 1 2 2 0 nn n m x n ￥e iD sin 2 2 x ￥ Taylor) ?￥M?[1?b 3 ÷??1 GVODUJPO<[ NBYFSSPS>FY N  [e NBYFSSPSKvμN ¨ ? MFO N  B[FSPT MFO  $e T" ? B   GPSJ N  B J  B J  J  J  FOE 8 C[FSPT MFO  e T" ? C   GPSJ N  C J  C J  J  J  FOE D[FSPT MFO  e" ? D   GPSJMFO D J  D J  J  J  FOE  YGGU B MFO 'PVSJFSM D  ZGGU C MFO 'PVSJFSM D  [Y Z [JGGU [ 'PVSJFS IMD NBYFSSPS GPSJMFO FBCT [ J D J  JGFNBYFSSPS NBYFSSPSF FOE FOE 9 9 ?2Tμs N μ      F  F  F  F ?" N￥ 9Fμá ?h b  10