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11 ) ( ~ ) ( ~ ) ( ~ 2 1 3 n xb n xa n x + = ) ( ~ ) ( ~ ) ( ~ 2 1 3 k Xb k Xa k X + = 三、DFS的性质 1、线性： [ ] ) ( ~ ) ( ~ k X W m n x DFS mk N − = + [ ] nk N N n W m n x m n x DFS ∑ − = + = + 1 0 ) ( ~ ) ( ~ 2、序列移位： (1)时域移位 令i=n+m ik N m N m i mk N mk N ik N m N m i Wi x W W Wi x ∑ ∑ + − = − − + − = = = ∴ 1 1 ) ( ~ ) ( ~ 上式 ) ( ~ ) ( ~ 1 0 k X W Wi x W mk N ik N N i mk N − − = − = = ∑ [ ] ) ( ~ ) ( ~ n x W l k X IDFS l N n = + (2)频域移位
: ← 个S
12 3、周期卷积 （1）时域卷积 两个周期信号（序列）的卷积，只限在一个周期内卷积。 ) ( ~ ) ( ~ 1 1 k X n x → ) ( ~ ) ( ~ 2 2 k X n x → [ ] ) ( ~ ) ( ~ ) ( ~ ) ( ~ 2 1 2 1 k X k X n x n x DFS ⋅ = ⊗ 则， ∑ ∑ − = − = − = − = ⊗ 1 0 1 2 1 0 2 1 2 1 ) ( ~) ( ~ ) ( ~) ( ~ ) ( ~ ) ( ~ N m N m m n x m x m n x m x n x n x 而
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13 [ ] ) ( ~ ) ( ~ ) ( ~ ) ( ~ ) ), ( ( ) ( ~ ) ( ~ ) ( ~ ) ( ~ ) ( ~) ( ~ ) ( ~ ) ( ~ 2 1 1 0 2 1 2 1 2 1 ) ( 1 0 2 1 0 1 1 0 1 0 2 1 2 1 k X k X Wr x k X N W r x Wr x k X m n r W m n x W m x W m n x m x n x n x DFS rk N N r rk N rk N m N m r k m n N N n mk N N m N n nk N N m ⋅ = ⋅ = ⋅ = − = − ⋅ = − = ⊗ ∑ ∑ ∑ ∑ ∑ ∑ − = − − = − − = − = − = − = 为周期 均以 ，则原式 证： 令 如下 ) ( ~ ), ( ~ 2 1 n x n x 3 = N 1 2 0 1 . . .
〓
14 周期卷积 0 1 2 m ) ( ~1 m x ) ( ~ ) ( ~ 2 1 n x n x ⊗ .... .... n -3 -2 -1 0 1 2 3 4 5 1 2 ) ( ~2 m x − ... ... m 0 2 1
会sS囚囚 之 三<囚 三囚 之 z三 会S囚
15 (2)频域卷积定理 ) ( ~ ) ( ~ ) ( ~ 2 1 3 n x n x n x ⋅ = ) ( ~ ) ( ~ 1 ) ( ~ ) ( ~ 1 ) ( ~ 2 1 0 1 2 1 3 l k Xl X N k X k X N k X N l − = ⊗ = − ∑ = ) ( ~ ) ( ~ ) ( ~ 1 ) ( ~ 1 ) ( ~ ) ( ~ 1 ) ( ~ 1 ) ( ~ 2 1 ) ( 1 0 2 1 0 1 1 0 1 0 2 1 2 1 0 3 3 n x n x W l k X N W l X N W l k X l X N Wk X N n x l k n N N k nl N N l N k nk N N l N k nk N ⋅ = ⋅ − ⋅ ⋅ = ⋅ − ⋅ = = − − − = − − = − = − − = − = − ∑ ∑ ∑ ∑ ∑