The fast fourier transform (fft)

7KH )DVW )RXULHU 7UDQVIRUP ))7 &R K D V WR WR R E W $ OW H U Q G LQ J W K H G LV F U H W H )R X U LH U W U D Q V IR U P LV Q R W S D U W LF X OD U O\ G Lr F X OW  D OO R Q H G R LV W R IR U P W K H P D W U L[ )1  F R Q MX J D W H LW D Q G G LYLG H E \ LQ R U G H U D LQ W K H P D W U L[ )1b X V H G W R IR U P ) = IR U D J LYH Q LQ S X W YH F W R U = D W LYH O\ ZH F D Q V LP S O\ IR U P  IR U N        b    ) = ZLW K F D UU S R LQ VWD Q LQ G H b  1; ZbNM ]M M  N R X W H YH U V W R U LQ J W K H P D W U L[ ( LW K \LQ J R X W W K H V X P P D W LR Q V MX V W LQ W P X OW LS OLF D W LR Q V  )R U OD U J H YD OX H G D U G LQ D S S OLF D W LR Q V  W K LV F D Q E H H G  LW LV R YH U R Q H P LOOLR Q IR U H U G LF V R D  b }NM  1; HbL ]M M  ZD \ IR U P LQ D W H G  U H TX LU I  H J  YH U \ OD U J H Q     R YH U   J W K H S U R G X F W )1b  = R U H V D W R W D O R I  q R D W LQ J         R U     D U H X P E H U R I P X OW LS OLF D W LR Q V P LOOLR Q IR U      7K H &RROH\  7XNH\ DOJRULWKP X V X D OO\ F D OOH G W K H )D V W )R X U LH U 7U D Q V IR U P  R U MX V W ))7 H Q D E OH V R Q H W R V X E V W D Q W LD OO\ U H G X F H W K H Q X P E H U R I P X OW LS OLF D  W LR Q V U H TX LU H G  S U R YLG H G LV D S R ZH U R I   LH   Q IR U V R P H S R V LW LYH LQ W H J H U Q ,W U H V X OW V LQ D K X J H LQ F U H D V H LQ F R P S X W D W LR Q D O H r F LH Q F \ IR U OD U J H Q D Q G LV U H V S R Q V LE OH IR U P D NLQ J W K H ))7 W K H ZLG H O\ X V H G D Q D O\W LF D O W R R O W K D W LW LV LQ F X U U H Q W V F LH Q F H D Q G W H F K Q R OR J \ $ V V X P LQ J  Q ZH K D YH  X V LQ J W K H V X E V F U LS W R Q ) K H U H W R LQ G LF D W H W K H G LP H Q V LR Q R I W K H W U D Q V IR U P X Q G H U G LV F X V V LR Q U D W K H U W K D Q W K H IR U P R I W K H W U D Q V IR U P D V LQ R X U VHFWLRQ RQ FRQYROXWLRQ HTXDWLRQV t )1  = u N   )1  = 1 b  HbL M  M HYHQ }NM ]M  N b }NM  1; HbL ]M M  b  M   M RGG HbL }NM ]M b b  ; bL }NC  ; bL }NC ]C  H H ]C C  C  1 b b  ; bL 1}NC  bL }N ; bL 1}NC H H ]C  H ]C C  C  1  }N   HbL  „  )1  =    t  N )1  =  u N   „   HbL   )1   =   }N t )1  =  N u N  + H U H =  LV W K H YH F W R U R I G LP H Q V LR Q 1 F R Q V LV W LQ J R I W K H H YH Q LQ G H [H G F R P  S R Q H Q W V R I = D Q G =  LV W K H YH F W R U R I W K H V D P H G LP H Q V LR Q F R Q V LV W LQ J R I W K H R G G LQ G H [H G F R P S R Q H Q W V R I = =  ] ] c c c ]1 b  =   ] ] c c c ]1 b  7K LV S U R F H V V F D Q E H U H S H D W H G IR U H D F K R I W K H )R X U LH U W U D Q V IR U P V R I G LP H Q  V LR Q  V K R ZQ  7K X V ZH K D YH „  1 b  C   C  U  %  #   b ; U „   D Q G D OV R  )1  =    HbL HbL N }NC  ] C   b C   ;   HbL    }N   HbL   N }NC 1 b  C   C  U }NU  bL }N %   ] & U $  H  #  )1   =   }NC  ] C „ HbL  ]C b U   ;  HbL  )1  =   b ; }NC  HbL  ]C  C  u  t )1   =  N   }NU &  ] U$ N 1 b bL }NC  3C b  H  ]C   C   C  U  C  U    „  b U   ;  )1  =  )1   =   „ w   N }N   HbL  „„ }NC  ] C  b  b L 1}N ; bL 1}NU  ]U H U   H U  }NU HbL  ]    N }N HbL  „  )1  =   X W W LQ J D OO R I W K LV W R J H W K H U ZH Q R Z K D YH v HbL  )1  =    )1  =   N    }N HbL  }N  HbL  N + H U H W K H V X S H U V F U LS W V LQ G LF D W H W K H OR ZH V W R I W K H LQ G LF H V M R I W K H ]M LQ F OX G H G LQ W K H ZK LF K 1  )1  LV D S S OLH G  7K X V =  LQ F OX ]  LQ F OX G H V ]M V X F K W K D W ZK H Q M LV G LYLG R Q  R UG G H H G „ „  N  )1   =    )1  =   H U G LJ LW V LQ W K  G LP H Q V LR Q D V ]M V X F K W K D W E \ WK H UH P N  N H E LQ D U \ H [S D Q V LR Q O V X E YH F W R U R I = W R M LV G LYLV LE OH E \   D LQ G H U LV   D Q G V R R Z OH W X V D Q D O\] H W K H Q X P E H U R I P X OW LS OLF D W LR Q V LQ YR OYH G  ,Q W K H p U V W VWH S  }N  )1   =  N  HbL   )1  =  N   )1  = N D WR WD O E H Q R WH F R P S X W WR R E WD R E V H U YD R I P X OW LS OLF D W LR Q V G  K R ZH YH U  W K D W R Q H G ? P R G X OR   W K LQ D Q LQ W H J H U LQ W K H W LR Q  LW G R H V Q R W D oH F ZLW K W K W K H U LJ K D W LV  LI UD Q J H  W WK H Q X }N H ID F W R U HbL  D U H U H TX LU H G  ,W V K R X OG W K D Q G V LG H W K H VXEVFULSW N V K R X OG E H N y  W K H Q ZH V K R X OG V X E W U D F W  W R  b    7K H OD W W H U LV MX V W D Q P E H U R I P X OW LS OLF D W LR Q V  $ W W K H V H F R Q G V W D J H ZH K D YH W K H H TX D W LR Q V P D U NH G  e D E R YH  + H U H ZH K D YH  P X OW LS OLF D W LR Q V  H YH Q W K R X J K LW V H H P V D W p U V W J OD Q F H W K D W ZH V K R X OG  }N K D YH   7K H U H D V R Q LV W K D W W K H Q X P E H U V HbL  U H S H D W W K H P V H OYH V ZK H Q N H [F H H G V  b   W K H U H D U H U H D OO\ R Q O\  R I W K H V H Q X P E H U V E H F D X V H LI N   NA W K H Q }N HbL  }NA HbL } HbL  }NA HbL   ,I ZH U H S H D W W K H D E R YH S U R F H V V  H [S U H V V LQ J W K H F R P S R Q H Q W  )1  = N R I W K H R U LJ LQ D O W U D Q V IR U P LQ W H U P V R I F R P S R Q H Q W V R I V X F F H V V LYH O\ OR ZH U R U G H U W U D Q V IR U P V  LH   )1 P  = N  ZH p Q G W K D W W K H Q X P E H U R I P X OW LS OLF D W LR Q V U H TX LU H G LV P1  )LQ D OO\ ZK H Q ZH U H D F K P Q W K H W U D Q V IR U P LV H [S U H V V H G LQ W H U P V R I W K H F R P S R Q H Q W V R I W K H R U LJ LQ D O YH F W R U = ZLW K Q1 OR J  1 P X OW LS OLF D W LR Q V LQ YR OYH G D OW R J H W K H U  )R U OD U J H YD OX H V R I W K LV F D Q E H D K X J H U H G X F W LR Q LQ W K H Q X P E H U R I U H TX LU H G P X OW LS OLF D W LR Q V D V F R P S D U H G ZLW K W K H  U H TX LU H G IR U ? E U X W H IR U F H F R P S X W D W LR Q R I W K H R U LJ LQ D O S U R G X F W )1  =  )R U H [D P S OH  LI ZH W D NH Q   1        W K H E U X W H IR U F H D S S U R D F K U H TX LU H V    S UR F H VV G H VF WK D Q D K X Q G          P X OW LS OLF D W LR Q V ZK H U H D V  W D NLQ J D G YD Q W D J H R I W K H U LE H G D E R YH  W K H Q X P E H U U H TX LU H G LV   d           P R UH U H G IR OG U H G X F W LR Q  )LQ D OO\ ZH V K R X OG Q R W H )1  =  D V W K X V F R Q V W U X F W H G  F D Q E H P X OW LS OLH G E \  Q R Q H F D Q  D W W K H H Q G R I W K H S U R F H V V W R R E W D LQ )1  =  R U  V LQ F H G LYLG H E \  D W H D F K V W D J H R I W K H U H G X F W LR Q S U R F H V V  &R G LQ J W K H G LV F U H W H )R X U LH U W U D Q V IR U P LQ W K H ))7 P R G H U H TX LU H V F D U H IX O D W W H Q W LR Q W R LQ G H [LQ J  D YR LG D Q F H R I U H G X Q G D Q W P X OW LS OLF D W LR Q  H W F  : H K D YH Q R W S U R YLG H G H Q R X J K G H W D LO LQ W K H D E R YH H [S R V LW LR Q W R LQ G LF D W H H [D F W O\ K R Z W K LV ZR X OG E H G R Q H LQ S U D F W LF H  H YH U W K H OH V V LW LV LP S R U W D Q W W R NQ R Z D E R X W W K H &R R OH \7X NH \ ))7 D OJ R U LW K P E H F D X V H W K H U H G X F W LR Q LQ W K H Q X P E H U R I U H TX LU H G P X OW LS OLF D W LR Q V ZK LF K LW S H U P LW V  W K H P D W K H P D W LF D O H V V H Q F H R I W K H ))7 LV D W W K H F R U H R I W K H YH U \ OD U J H Q X P E H U R I V X F F H V V IX O D S S OLF D W LR Q V R I W K H  G LV F U H W H )R X U LH U W U D Q V IR U P LQ P D Q \ G LYH U V H D U H D V R I V F LH Q F H D Q G W H F K Q R OR J \