The fast fourier transform (fft)

Trang 1

7KH )DVW )RXULHU 7UDQVIRUP ))7

&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 LV Q R W S D U W LF X OD U O\ G Lr F X OW D OO R Q H

K D V W R 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

W R R E W D LQ W K H P D W U L[ )b

1 X V H G W R IR U P ) = IR U D J LYH Q LQ S X W YH F W R U =

$ OW H U Q D W LYH O\ ZH F D Q V LP S O\ IR U P IR U N 1 b 

) = N

1

1b ;

M

ZbNM]M

1

1b ;

M

HbL}NM1 ]M

ZLW K R X W H YH U V W R U LQ J W K H P D W U L[ ( LW K H U ZD \ IR U P LQ J W K H S U R G X F W )b

1 = R U

F D U U \LQ J R X W W K H V X P P D W LR Q V MX V W LQ G LF D W H G U H TX LU H V D W R W D O R I 1 q R D W LQ J

S R LQ W P X OW LS OLF D W LR Q V )R U OD U J H YD OX H V R I 1 H J 1 R U D U H

V W D Q G D U G LQ D S S OLF D W LR Q V W K LV F D Q E H D YH U \ OD U J H Q X P E H U R I P X OW LS OLF D W LR Q V

LQ G H H G LW LV R YH U R Q H P LOOLR Q IR U 1 R YH U 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 = uN 1 )1 = N 1 1b;

M

HbL }NM

1 ]M

1

31b

M HbL }NM

1 ]M

1

31b

M HbL }NM

1 ]M

M RGG

Trang 2

1

b

;

C

HbL }NC

1 ]C 1

1

b

;

C

HbL }NC

1 ]C

1

b

;

C

HbL}NC1 ]C 1 HbL }N

1

b

;

C

HbL}NC1 ]C

1 )1 =

N

HbL}N1

1 )1 =

N

t

)1 = uN HbL }N

1

t

)1 = uN

+ 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 ]1b 7 = ]] c c c ]1b 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

1 )1 =

N

1 

1

b

;

C

HbL}NC1 ]C

1 

3 1

b

C HbL}NC1 ]C

1 

3 1

b

C HbL}NC1 ]C

C U 

%

1 

1

b

;

U HbL}NU1 ]U

& HbL}N1

%

1 

1

b

;

U HbL}NU1 ]U

&

1 )1 =

N

HbL}N1

1 )1 =

N

D Q G D OV R

1 

t

)1 = uN 1 

1

b

;

C

HbL}NC1 ]C

Trang 3

1 

3 1

b

C HbL}NC1 ]C

1 

3 1

b

C HbL}NC1 ]C

C U 

1 

1

b

;

U

HbL}NU1 ]U

1 HbL

}N

1

b

;

U

HbL}NU1 ]U

1 )1 =

N

HbL}N1

1 )1 =

N

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

1 )1 =

w

N

1 )1 =

N

HbL}N1

1 )1 =

N

HbL

}N 1

1 )1 =

N

 HbL}N1

1 )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 U G H U G LJ LW V LQ W K H E LQ D U \ H [S D Q V LR Q

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 1 G LP H Q V LR Q D O V X E YH F W R U R I = W R

ZK LF K

1 )1 LV D S S OLH G 7K X V = LQ F OX G H V ]M V X F K W K D W M LV G LYLV LE OH E \ 

] LQ F OX G H V ]M V X F K W K D W ZK H Q M LV G LYLG H G E \ W K H U H P D LQ G H U LV D Q G V R

R Q

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

V W H S

)1 = N )1 = N  HbL}N1 )1 = N

D W R W D O R I 1 P X OW LS OLF D W LR Q V ZLW K W K H ID F W R U HbL}N1 D U H U H TX LU H G ,W V K R X OG

E H Q R W H G K R ZH YH U W K D W R Q W K H U LJ K W K D Q G V LG H W K H VXEVFULSW N V K R X OG E H

F R P S X W H G ? P R G X OR 1 W K D W LV LI N y 1 W K H Q ZH V K R X OG V X E W U D F W 1 

W R R E W D LQ D Q LQ W H J H U LQ W K H U D Q J H W R 1 b 7K H OD W W H U LV MX V W D Q

R E V H U YD W LR Q LW G R H V Q R W D oH F W W K H Q X 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

Trang 4

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}N1 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 AN W K H Q

HbL}N1 HbL }HbL}AN1 HbL}A1 N

,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 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

S U R F H V V G H V F 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 U H

W K D Q D K X Q G 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 \

 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 Q R Q H F D Q

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

Trang 5

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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