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 \ 1 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
       1 b  
 ) =



ZLW K
F D UU
S R LQ
VWD Q
LQ G H

b
 1;
ZbNM ]M
1 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 1

H U
G LF
V R
D


b
}NM
 1;
HbL 1 ]M
1 M 

ZD \ IR U P LQ
D W H G  U H TX LU
I 1  H J  1
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 1  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 1
    

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 1 LV D S R ZH U R I   LH  1
 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 1
 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  =

3

u
N


 )1  =


1

1 b


HbL
M 
1 M HYHQ

}NM
1

]M


N

b
}NM
 1;
HbL 1 ]M
1 M 
3

1 b

M 

1 M RGG

HbL

}NM
1

]M


b
b
 ; bL }NC
 ; bL }NC
1 ]C 
1
H
H
]C
1 C 
1 C 
1

1

b
b
 ; bL 1}NC
 bL }N ; bL 1}NC
H 1
H
]C 
H
]C
1 C 
1
C 

1

1



}N





HbL 1 

„


)1  = 

1 

 t


N

)1  = 


u

N





„




HbL



)1   = 

1 

}N t
1

)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
7  = 

 ] ] c c c ]1 b
7 

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 1  V K R ZQ  7K X V ZH K D YH
„



3 1 b

C 
1  C  U

 % 
#

1 

b

;

1

U

„



D Q G D OV R


)1  = 

1 



HbL

HbL

N

}NC
1  ]
C 


1

b

C




;


1 

HbL





1 

}N





HbL 1 


N

}NC

3 1 b

C 
1  C  U

}NU
 bL }N % 
1  ] &
U $  H 1  #


)1   = 

1 

}NC
1  ]
C

„

HbL 1  ]C

1

b

U




;



HbL


)1  = 

1 

b
;
}NC

HbL 1  ]C
1  C 
1

u
 t
)1   = 

N
1 



}NU
&
1  ]
U$

N


1
3 1 b
bL }NC
 3C b
 H 1  ]C  
C 
1  C  U
1  C  U


1 


„



1

b

U




;


)1  =

1


)1   = 

1 
„

w




N

}N


 HbL 1


„„

}NC
1  ]
C

 b

b
L 1}N ; bL 1}NU
 ]U
H
U  1  H
U 
1

}NU
HbL 1  ]






N

}N
HbL 1 

„


)1  = 

1 

3 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  =