3.离散傅里叶级数(DFS) (2)频域 (n)=x(m)2(n)<03>X(k)=X1(k)eX1(k) 时域相乘<>频域周期卷积
（2）频域 ( ) ~ ( ) ~ ( ) ~ ( ) ~ ( ) ~ ( ) ~ 1 2 1 1 x n x n x n X k X k X k DFS = ⎯→ = 时域相乘⎯→频域周期卷积 3.离散傅里叶级数(DFS)
3.离散傅里叶级数(DFS) DFS: (m)<>X(k) 实际情况:x()—>x(7),Vn x(n)=x (n), n=0, 1 , N-1 那么, x(n).0≤n≤N-1 X(k,0≤k≤N-1
( ) ~ ( ) ~ DFS： x n ⎯→ X k 实际情况： xa (t) ⎯→ xa (nT),n x(n)= xa (nT), n = 0,1,, N −1 △ 那么， x(n),0 n N −1 X (k),0 k N −1 3.离散傅里叶级数(DFS)
4.离散傅里叶变换 、DFT的定义 x(n)周期延拓 x(n+)=x(n),0≤n≤N-1,v=0,±,±2, x(n)0≤n≤N-1 10,n<0,n≥N =x(n)R(m)x(m)主值序列 由DFS变换[3-17式] X(k)=∑x(m)W=∑x(m)WMk∈I 显然 X(k=X(k+N) 仅有N个独立值
一、DFT的定义 ( + ) = ( ),0 −1, = 0,1,2, ~ 令 x n lN x n n N l ( ) ( ) ~ 0 , 0, ( ),0 1 ~ ( ) x n R n n n N x n n N x n = N − = ~ x(n)主值序列 由DFS变换[3-17式] X k x n W x n W k I N n kn N N n kn = N = − = − = ( ) ( ) ~ ( ) ~ 1 0 1 0 显然 ( ) ~ ( ) ~ X k = X k + N 仅有N个独立值 x(n) 的周期延拓 4.离散傅里叶变换
4.离散傅里叶变换 △ 令X(k)=X(k)R(n) 则有X(k)=∑x(n)0≤k≤N-1 即x(m),0≤n≤N-1—)Y(k),0≤k≤N 问题:X(k)0≤k≤N-1-→x(m)0≤n≤N-1 x(n)=x(nR(n) N-1 N ∑X(W如R、(n) ∑X(kW X(k)—>x(m) N 0<k<N-10<n<N-1 0<k<N-1
令 ( ) ( ) 0 1 1 0 = − − = X k x n W k N N n kn 则有 N ( ) ( ) ~ X (k) X k RN n △ = 即 x(n),0 n N −1 ⎯→X(k),0 k N −1 问题: X(k),0 k N −1 ⎯→x(n),0 n N −1 ? ( ) ( ) ~ x(n) = x n RN n ( ) ( ) 1 ~ 1 0 X k W R n N N N k kn N = − = − 0 1 0 1 ( ) ( ) − − ⎯→ k N n N X k x n 0 1 ( ) 1 1 0 − = − = − k N X k W N N k kn N 4.离散傅里叶变换
4.离散傅里叶变换 归纳起来: X(k)=∑xOnW0≤k≤N DFT(n) x(n)=1∑X(k)00≤k≤N-1 k=0 IDFTIX(k)I
归纳起来： ( ) ( ) ( ) 0 1 1 0 DFT x n X k x n W k N N n kn N △ = = − − = ( ) ( ) 0 1 1 ( ) 1 0 IDFT X k X k W k N N x n N k kn N △ = = − − = − 4.离散傅里叶变换