Discrete Fourier Transform
1. 离散傅里叶变换的表示
离散傅里叶变换就是有限长序列的离散频域表示,即有限长序列的离散傅里叶级数。
- x ( n ) x(n) x(n) 看作是周期为 N N N 的周期序列 x ~ ( n ) \tilde{x}(n) x~(n) 的一个周期
- x ~ ( n ) \tilde{x}(n) x~(n) 看作是 以 N N N 为周期 x ( n ) {x}(n) x(n) 的周期延拓
x ~ ( n ) = ∑ r = − ∞ ∞ x ( n + r N ) = x ( ( n ) ) N \tilde{x}(n) = \sum_{r=-\infty}^{\infty}x(n + rN) = x((n))_N x~(n)=r=−∞∑∞x(n+rN)=x((n))N
x ~ ( n ) \tilde{x}(n) x~(n) 的第一个周期称 [ 0 , N − 1 ] [ 0,N-1 ] [0,N−1]为主值区间, x ( n ) {x}(n) x(n) 是 x ~ ( n ) \tilde{x}(n) x~(n) 的主值大小序列。
- 矩形序列
R ( n ) = { 1 , 0 ≤ n ≤ N − 1 0 , e l s e R(n) = \begin{cases} 1, & 0 \leq n\leq N-1 \\ 0, & else \end{cases} R(n)={ 1,0,0≤n≤N−1else
则有 x ( n ) = x ~ ( n ) ⋅ R ( n ) x(n) = \tilde{x}(n) \cdot R(n) x(n)=x~(n)⋅R(n)
离散傅里叶变换对
D F S : X ( k ) = D F S [ x ( n ) ] = ∑ 0 N − 1 x ( n ) e − j 2 π N k n = ∑ n = 0 N − 1 x ( n ) W N n k = X ~ ( k ) R N ( k ) , k = 0 , 1.. , N − 1 DFS: \ {X}(k)=DFS[{x}(n)] = \sum_0^{N-1}{x}(n)e^{-j\frac{2 \pi}{N}kn}=\sum_{n=0}^{N-1}x(n)W_N^{nk}=\tilde{X}(k)R_N(k), \ \ \ k=0,1..,N-1 DFS: X(k)=DFS[x(n)]=0∑N−1x(n)e−jN2πkn=n=0∑N−1x(n)WNnk=X~(k)RN(k), k=0,1..,N−1
I D F S : x ( n ) = I D F S [ X ( k ) ] = 1 N ∑ n = 0 N − 1 X ( k ) e j 2 π N k n = 1 N ∑ n = 0 N − 1 X ( k ) W N − n k = x ~ ( n ) R N ( n ) , n = . . N − 1 IDFS: \ {x}(n)=IDFS[{X}(k)] =\frac{1}{N} \sum_{n=0}^{N-1}{X}(k)e^{j\frac{2 \pi}{N}kn}=\frac{1}{N}\sum_{n=0}^{N-1}X(k)W_N^{-nk}=\tilde{x}(n)R_N(n),n=..N-1 IDFS: x(n)=IDFS[X(k)]=N1n=0∑N−1X(k)ejN2πkn=N1n=0∑N−1X(k)WN−nk=x~(n)RN(n),n=..N−1
2.离散傅里叶变换的性质
- 线性
- D F T [ a ⋅ x 1 ( n ) + b ⋅ x 2 ( n ) ] = a ⋅ X 1 ( k ) + b ⋅ X 2 ( k ) DFT[a \cdot {x}_1(n) + b \cdot {x}_2(n)] = a \cdot {X}_1(k) + b \cdot {X}_2(k) DFT[a⋅x1(n)+b⋅x2(n)]=a⋅X1(k)+b⋅X2(k)
- 序列的圆周(循环)移位
- x m ( n ) = x ( ( n + m ) ) N R N ( n ) , m > 0 左移 , m < 0 右移 x_m(n)=x((n+m))_NR_{N}(n), \ \ \ \ m>0 左移, \ m < 0 右移 xm(n)=x((n+m))NRN(n), m>0左移, m<0右移
- X m ( k ) = D F T [ x m ( n ) ] = D F T [ x ( ( n + m ) ) N R N ( n ) ] = W N − m k X ( K ) X_m(k) = DFT[x_m(n)] = DFT[x((n+m))_NR_N(n)] = W_N^{-mk}X(K) Xm(k)=DFT[xm(n)]=DFT[x((n+m))NRN(n)]=WN−mkX(K)
e g . eg. eg. x ( n ) = { 1 , 2 , 3 , 4 ; n = 0 , 1 , 2 , 3 } 求 x 2 ( n ) = ? x(n) = \{1,2,3,4;n=0,1,2,3\}\ \ \ \ 求x_2(n)=? x(n)={ 1,2,3,4;n=0,1,2,3} 求x2(n)=?
x 2 ( n ) = x ( ( n + 2 ) ) 4 R 4 ( n ) = { 3 , 4 , 1 , 2 ; n = 0 , 1 , 2 , 3 } x_2(n)=x((n+2))_4R_4(n)=\{3,4,1,2;n=0,1,2,3\} x2(n)=x((n+2))4R4(n)={ 3,4,1,2;n=0,1,2,3}
- 对偶性(对称定理)
- D F T [ X ( n ) ] = N x ( N − k ) DFT[X(n)]=Nx(N-k) DFT[X(n)]=Nx(N−k)
- 反转(反褶)定理
- KaTeX parse error: Can't use function '$' in math mode at position 7: x(N-n)$̲\Leftrightarrow…
- 序列的总和
- 时间序列 x ( n ) x(n) x(n) 中各取样值的总和等于其离散傅里叶变换 X ( k ) X(k) X(k) 在 k = 0 k=0 k=0 时的值。
- S U M [ x ( n ) ] = ∑ n = 0 N − 1 x ( n ) = X ( k ) ∣ k = 0 = X ( 0 ) SUM[x(n)]=\sum_{n=0}^{N-1}x(n)=X(k)|_{k=0} = X(0) SUM[x(n)]=n=0∑N−1x(n)=X(k)∣k=0=X(0)
- 若 x ( n ) x(n) x(n) 的离散傅里叶变换为 X ( k ) X(k) X(k),则 x ( n ) x(n) x(n) 的初始值 x ( 0 ) x(0) x(0) 为频谱序列各取样值 X ( k ) X(k) X(k) 的总和除以 N N N
- ∵ x ( n ) = I D F S [ X ( k ) ] = 1 N ∑ n = 0 N − 1 X ( k ) e j 2 π N k n ∴ x ( 0 ) = 1 N ∑ k = 0 N − 1 X ( k ) \because \ {x}(n)=IDFS[{X}(k)] =\frac{1}{N} \sum_{n=0}^{N-1}{X}(k)e^{j\frac{2 \pi}{N}kn} \ \ \ \ \therefore x(0) = \frac{1}{N}\sum_{k=0}^{N-1}X(k) ∵ x(n)=IDFS[X(k)]=N1n=0∑N−1X(k)ejN2πkn ∴x(0)=N1k=0∑N−1X(k)
- 共轭对称性
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- 有限长序列的奇偶对称及其DFT
- 若序列是奇对称的, 即 x ( n ) = − x ( − n ) = − x ( N − n ) X ( k ) = − X ( N − n ) x(n) = -x(-n) = -x(N-n) \ \ \ \ \ X(k) = -X(N-n) x(n)=−x(−n)=−x(N−n) X(k)=−X(N−n)
- 若序列是偶对称的,即 x ( n ) = x ( N − n ) X ( k ) = X ( N − k ) x(n)=x(N-n) \ \ \ \ \ X(k) = X(N-k) x(n)=x(N−n) X(k)=X(N−k)
- 判断方法
- 共轭(反)对称分量
- x e p ( n ) = x ~ e ( n ) R N ( n ) = 1 2 [ x ( ( n ) ) N + x ∗ ( ( N − n ) ) N ] R N ( n ) {x}_{ep}(n) = \tilde{x}_e(n)R_N(n) = \frac{1}{2}[x((n))_N + x^*((N-n))_N]R_N(n) xep(n)=x~e(n)RN(n)=21[x((n))N+x∗((N−n))N]RN(n) x o p ( n ) = x ~ o ( n ) R N ( n ) = 1 2 [ x ( ( n ) ) N − x ∗ ( ( N − n ) ) N ] R N ( n ) {x}_{op}(n) = \tilde{x}_o(n)R_N(n) = \frac{1}{2}[x((n))_N - x^*((N-n))_N]R_N(n) xop(n)=x~o(n)RN(n)=21[x((n))N−x∗((N−n))N]RN(n)
x ( n ) = x e p ( n ) + x o p ( n ) x(n)=x_{ep}(n)+x_{op}(n) x(n)=xep(n)+xop(n)
- 对称性质
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共轭复数的DFT D F T [ x ∗ ( n ) ] = X ∗ ( ( − k ) ) N R N ( k ) = X ∗ ( ( N − k ) ) N R N ( k ) DFT[x^*(n)] = X^*((-k))_NR_N(k)=X^*((N-k))_NR_N(k) DFT[x∗(n)]=X∗((−k))NRN(k)=X∗((N−k))NRN(k)
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序列实部的DFT D F T { R e [ x ( n ) ] } = X e p ( k ) = 1 2 [ X ( ( k ) ) N + X ∗ ( ( N − k ) ) N ] R N ( k ) DFT\{Re[x(n)]\} = X_{ep}(k) = \frac{1}{2}[X((k))_N + X^*((N-k))_N]R_N(k) DFT{ Re[x(n)]}=Xep(k)=21[X((k))N+X∗((N−k))N]RN(k)复序列的实部的DFT等于序列的DFT 的圆周共轭对称分量
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序列虚部的DFT D F T { j I m [ x ( n ) ] } = X o p ( k ) = 1 2 [ X ( ( k ) ) N − X ∗ ( ( N − k ) ) N ] R N ( k ) DFT\{jIm[x(n)]\} = X_{op}(k) = \frac{1}{2}[X((k))_N - X^*((N-k))_N]R_N(k) DFT{ jIm[x(n)]}=Xop(k)=21[X((k))N−X∗((N−k))N]RN(k)复序列的虚部乘以j的DFT等于序列的 DFT的圆周共轭反对称分量
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圆周共轭对称序列DFT X e p ( k ) = X e p ∗ ( ( N − k ) ) N R N ( k ) X_{ep}(k)=X_{ep}^*((N-k))_NR_N(k) Xep(k)=Xep∗((N−k))NRN(k) X e p ( k ) X_{ep}(k) Xep(k)实部偶对称,虚部奇对称
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圆周共轭反对称序列DFT X o p ( k ) = − X o p ∗ ( ( N − k ) ) N R N ( k ) X_{op}(k)=-X_{op}^*((N-k))_NR_N(k) Xop(k)=−Xop∗((N−k))NRN(k) X o p ( k ) X_{op}(k) Xop(k)实部奇对称,虚部偶对称
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