Decimation in Time Fast Fourier Transform Algorithm, Lecture notes of Digital Communication Systems

Download Decimation in Time Fast Fourier Transform Algorithm and more Lecture notes Digital Communication Systems in PDF only on Docsity!

Decimation in time FFT

• Split input into even and odd seq.

              

/ 2 1 0 / 2 / 2 1 0 / 2 / 2 1 0 ( 2 1 ) / 2 1 0 2 , , 1 0 [ 2 ] [ 2 1 ]

[ 2 ] [ 2 1 ]

[ ] [ ]

N r rk N k N N r rk N N r r k N N r rk N n even n odd N n kn N x r W W x r W x r W x r W X k x n W

  Assume 2

[ / 2 ] [ ] [ ], 0 , 1 ,..., / 2 1

[ ] [ ] [ ], 0 , 1 ,..., / 2 1

[ ] [ ], 0 , 1 ,..., 1

[ ] [ 2 ] [ 2 1 ]

/ 2 1 0 / 2 / 2 1 0 / 2 / 2 2 2 /( / 2 ) 

     N X k N G k W H k k N X k G k W H k k N or G k W H k k N X k x r W W x r W W e W k N k N k N N r rk N k N N r rk N N j N N

• N-point DFT is converted into two (N/2)-point DFT
  • May continue until 2-point DFT is reached

k N k N N N l lk N k N N l lk N N l lk N k N N l lk N

W W

H k h l W W h l W

G k g l W W g l W

        

/ 2 / 4 1 0 / 2 / 4 / 4 1 0 / 4 / 4 1 0 / 2 / 4 / 4 1 0 / 4

[ ] [ 2 ] [ 2 1 ]

[ ] [ 2 ] [ 2 1 ]

• Split N/2-point DFT into two (N/4)-point DFT