The Fast Fourier Transform (Fft)

background image

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

Z

bNM

]

M

1

1b

;

M

H

bL

}NM

1

]

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

=

u

N

1

)

1

=

N

1

1b

;

M

H

bL

}NM

1

]

M

1

3

1b

M

H

bL

}NM

1

]

M

M HYHQ

1

3

1b

M

H

bL

}NM

1

]

M

M RGG

background image

1

1

b

;

C

H

bL

}NC

1

]

C

1

1

b

;

C

H

bL

}NC

1

]

C

1

1

b

;

C

H

bL

}NC

1

]

C

1

H

bL

}N

1

1

b

;

C

H

bL

}NC

1

]

C


„

1

)

1

=

N


H

bL

}N

1

„

1

)

1

=

N


t

)

1

=

u

N


H

bL

}N

1

t

)

1

=

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 ]

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

H

bL

}NC

1

]

C

1

3

1

b

C

H

bL

}NC

1

]

C

C U

1

3

1

b

C

H

bL

}NC

1

]

C

C U



%

#

1

1

b

;

U

H

bL

}NU

1

]

U


&

$


H

bL

}N

1


%

#

1

1

b

;

U

H

bL

}NU

1

]

U


&

$


„

1

)

1

=

N


H

bL

}N

1

„

1

)

1

=

N

D Q G D OV R

1

t

)

1

=

u

N

1

1

b

;

C

H

bL

}NC

1

]

C

background image

1

3

1

b

C

H

bL

}NC

1

]

C

C U

1

3

1

b

C

H

bL

}NC

1

]

C

C U

1

1

b

;

U

H

bL

}NU

1

]

U

1

H

bL

}N

1

1

b

;

U

H

bL

}NU

1

]

U


„

1

)

1

=

N


H

bL

}N

1

„

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


H

bL

}N

1

„

1

)

1

=

N


H

bL

}N

1

„„

1

)

1

=

N

H

bL

}N

1

„

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

H

bL

}N

1

)

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 H

bL

}N

1

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

background image

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 H

bL

}N

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 AN W K H Q

H

bL

}N

1

H

bL }

H

bL

}AN

1

H

bL

}A

N

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

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

background image

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 \


Wyszukiwarka

Podobne podstrony:
Properties Of The Classical Fourier Transform, Some Examp
Efficient VLSI architectures for the biorthogonal wavelet transform by filter bank and lifting sc
6 i 7 Fourier Transform Properties
Dane Rudhyar THE ASTROLOGY OF TRANSFORMATION
Practical Analysis Techniques of Polymer Fillers by Fourier Transform Infrared Spectroscopy (FTIR)
5 Fourier transform
Efficient VLSI architectures for the biorthogonal wavelet transform by filter bank and lifting sc
iec cetral office on the fast track process 2007
Matlab Fourier Transformation
The Nazi Party in the Fast East, 1931 45
The Fast and the Furious
AES New Transducer Technology A R T Evolution Of The Air Motion Transformer Principle
Let s go to the fast food restaurant 2
The Mission of The One Star Transforming Planet Earth to Planet Star by Alloya N Huckfield (2003)
The Fast and the Furious 3 Tokyo Drift
fourier transformata
The Fast and The Furious 4
Alchemy The Art of Transformation by Paracelsus
The Fast And The Furious 2

więcej podobnych podstron