DFS Digital Signal Processing--DFT/FFT Algorithms )=DST=是We亨 k=-00,十00 n=0 0可=之k字 n=-o0,+o0 队=e帝 N-1 X()=DFS[(n)】=∑n)wg k=-00,+00 0=ne人芝tewg, n=0 n=-o0,+o0 上游充通大￥
Digital Signal Processing—— DFT/FFT Algorithms DFS 1 2 0 ( ) [ ( )] ( ) N j nk N n X k DFS x n x n e 1 2 0 1 ( ) [ ( )] ( ) N j kn N k x n IDFS X k X k e N 2 j N W e N 1 0 ( ) [ ( )] ( ) N nk N n X k DFS x n x n W 1 0 1 ( ) [ ( )] ( ) N nk N k x n IDFS X k X k W N k , k , n , n
DFS-Example Digital Signal Processing--DFT/FFT Algorithms x(n)=R(n),(n)=>x(n+4r) r三00 (n) N=4 )-立0wg-公。e停-1+e月 n=0 X(0)=2 x0=1+e5-1-j 3 x(2)=1+ex=0 (3=1+e=1+j 上浒充通大粤
Digital Signal Processing—— DFT/FFT Algorithms DFS-Example 2 ( ) ( ), ( ) ( 4 ) r x n R n x n x n r N 4 3 1 2 4 2 4 0 0 ( ) ( ) 1 j nk j k kn n n X k x n W e e X (0) 2 2 (1) 1 1 j X e j (2) 1 0 j X e 3 2 (3) 1 1 j X e j
Properties of DFS Digital Signal Processing--DFT/FFT Algorithms Linear N2 元(n)→元(K),元，(n)产2(k) 元(n)=a成(n)+bx2(n)→产3(k)=aX1(m)+br2(n) Time shift (n))(k)◆ (n+m)←→Wm(化) (n)→京（飞）◆(k+)→WW(n) ∑0n+m）p='∑Ua- N-1+m n=0 n'=m W-1 =W∑n)W=Ww文( n=0 上游文通大
Digital Signal Processing—— DFT/FFT Algorithms Properties of DFS Linear Time shift 1 2 1 1 2 2 ( ) ( ), ( ) ( ) N N x n X k x n X k 3 1 2 3 1 2 ( ) ( ) ( ) ( ) ( ) ( ) N x n ax n bx n X k aX n bX n x n X k ( ) ( ) ( ) ( ) km N x n m W X k x n X k ( ) ( ) ( ) ( ) nl X k l W x n N 1 1 ( ) 0 ( ) ( ) N N m kn k n m N N n n m x n m W x n W 1 0 ( ) ( ) N km kn km N N N n W x n W W X k
Properties of DFS Digital Signal Processing--DFT/FFT Algorithms circular convolution N-I 元m)=∑(m)元，(n-m)元，(K)=产()2() m=0 文，(k)=∑元，(n)W (ms.(n-mlo n-0 m=0 )-(m艺n-mw m=0 京，()=∑(m)W,()=产，(k)X,(k) 上泽充通大
Digital Signal Processing—— DFT/FFT Algorithms Properties of DFS circular convolution 1 3 1 2 3 1 2 0 ( ) ( ) ( ) ( ) ( ) ( ) N N m x n x m x n m X k X k X k 1 3 3 0 ( ) ( ) N kn N n X k x n W 1 1 1 2 0 0 ( ) ( ) N N kn N n m x m x n m W 1 1 3 1 2 0 0 ( ) ( ) ( ) N N kn N m n X k x m x n m W 1 3 1 2 1 2 0 ( ) ( ) ( ) ( ) ( ) N km N m X k x m W X k X k X k
Properties of DFS Digital Signal Processing--DFT/FFT Algorithms 元，0m）=(m元，(m）今元，()=文，()*产2(k) N =艺，k- =艺，成-0 上游充通大
Digital Signal Processing—— DFT/FFT Algorithms Properties of DFS 3 1 2 3 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) N x n x n x n X k X k X k N 1 1 2 0 1 ( ) ( ) N l X l X k l N 1 2 1 0 1 ( ) ( ) N l X l X k l N