The Fast Fourier Transform (fft)
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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
}NU bL }N % 1 ] & U $ H 1 #
)1 = 1
1
}N HbL 1
N
}NC
3 1 b C 1 C U
}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 = 1
)1 = 1
N
}N HbL 1
}N HbL 1
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 1 G H H G
N
)1 = 1
)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
1 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 )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 1 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 1 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 1 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 1 W K H Q ZH V K R X OG V X E W U D F W 1 W R 1 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 1 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 1 7K H U H D V R Q LV W K D W W K H Q X P E H U V HbL 1 U H S H D W W K H P V H OYH V ZK H Q N H [F H H G V 1 b W K H U H D U H U H D OO\ R Q O\ 1 R I W K H V H Q X P E H U V E H F D X V H LI N 1 NA W K H Q }N
HbL 1
}NA
HbL } HbL 1
}NA
HbL 1
,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 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 1 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 1 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
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
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 \
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
}NU bL }N % 1 ] & U $ H 1 #
)1 = 1
1
}N HbL 1
N
}NC
3 1 b C 1 C U
}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 = 1
)1 = 1
N
}N HbL 1
}N HbL 1
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 1 G H H G
N
)1 = 1
)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
1 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 )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 1 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 1 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 1 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 1 W K H Q ZH V K R X OG V X E W U D F W 1 W R 1 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 1 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 1 7K H U H D V R Q LV W K D W W K H Q X P E H U V HbL 1 U H S H D W W K H P V H OYH V ZK H Q N H [F H H G V 1 b W K H U H D U H U H D OO\ R Q O\ 1 R I W K H V H Q X P E H U V E H F D X V H LI N 1 NA W K H Q }N
HbL 1
}NA
HbL } HbL 1
}NA
HbL 1
,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 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 1 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 1 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 1 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 1 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 1 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 \
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