9.2 decimation-in-time FFT Algorithms X[-芝=∑n+∑ n=0 n even n odd Substitute variables n=2r for n even and n=2r+1 for odd N/2 X[F∑， 2r]+∑2r+H7 WW吹 N/2-point DFT 二 灯2r]W+W 灯2r+1]W2 N/21 G[]=∑gr]W g(r] r=0 hr] r=0 =-Gk猴H[=e及2-eā=mn N/2-1 H[∑h r=( G[k]and H[k]are the N/2-point DFTs ofeach subsequence: g[n]=x[2n]and h(n]=x2n1].015 9.2 decimation-in-time FFT Algorithms   2 (2 1) /2 1 /2 1 r 0 r 0 [ ] [ ] 2 2 1 N k N r r N N k X k x W x W r r − − = = + = +   + ◆Substitute variables n=2r for n even and n=2r+1 for odd ◆G[k] and H[k] are the N/2-pointDFTs of each subsequence: g[n]=x[2n] and h[n]=x[2n+1]   n even n odd 1 0 [ ] [ ] [ ] N N kn kn N kn n X k x nW x nW x nWN − = = = +       , k = + G k H k WN 1 1 r 0 r /2 / 0 / / 2 2 2 [ ] ] 2 1 [2 rk k rk N N N N N x W W x W r r − − = = = +   + 2 / 2 2 /2 2 2 N N N j j W W e e N   − − = = = , 0,1,..., -1 2 N n=   1 0 /2 /2 r [ ] N r N k G g k r W − = = N/2-pointDFT g[r] h[r] 2r N k k W WN , 0,1,.. - . ., 1 2 N k=   1 0 /2 /2 r [ ] N r N k H h k r W − = = N=2r