DFT变换的性质
DFT变换的性质
- 线性性质
- 时移性质
- 频移性质
- 时域反转
- 时域共轭
- 对称性质
- 卷积性质
- Parseval定理
线性性质
y [ n ] = a x [ n ] + b w [ n ] → D F T Y [ k ] = ∑ n = 0 N − 1 ( a x [ n ] + b w [ n ] ) W N k n = a ∑ n = 0 N − 1 x [ n ] W N k n + b ∑ n = 0 N − 1 w [ n ] W N k n = a X [ k ] + b W [ k ] \begin{aligned} y[n]&=ax[n]+bw[n]\xrightarrow{DFT}Y[k]=\sum_{n=0}^{N-1}(ax[n]+bw[n])W_N^{kn}\\ &=a\sum_{n=0}^{N-1}x[n]W_N^{kn}+b\sum_{n=0}^{N-1}w[n]W_N^{kn} \\ &=aX[k]+bW[k] \end{aligned} y[n]=ax[n]+bw[n]DFTY[k]=n=0∑N−1(ax[n]+bw[n])WNkn=an=0∑N−1x[n]WNkn+bn=0∑N−1w[n]WNkn=aX[k]+bW[k]
时移性质
x [ n − n 0 ] → D F T ∑ n = 0 N − 1 x [ < n − n 0 > N ] e − j 2 π k N n → m = n − n 0 ∑ m = − n 0 N − n 0 − 1 x [ < m > N ] e − j 2 π k N ( m + n 0 ) = W N k n 0 ∑ m = 0 N − 1 x [ m ] W N k m = W N k n 0 X [ k ] \begin{aligned} x[n-n_0]&\xrightarrow{DFT}\sum_{n=0}^{N-1}x[<n-n_0>_N]e^{-j\frac{2\pi k}{N}n} \\ &\xrightarrow{m=n-n_0}\sum_{m=-n_0}^{N-n_0-1}x[<m>_N]e^{-j\frac{2\pi k}{N}(m+n_0)} \\ &=W_{N}^{kn_0}\sum_{m=0}^{N-1}x[m]W_{N}^{km} \\ &=W_{N}^{kn_0}X[k] \end{aligned} x[n−n0]DFTn=0∑N−1x[<n−n0>N]e−jN2πknm=n−n0m=−n0∑N−n0−1x[<m>N]e−jN2πk(m+n0)=WNkn0m=0∑N−1x[m]WNkm=WNkn0X[k]
频移性质
W N − k 0 n x [ n ] → D F T ∑ n = 0 N − 1 x [ n ] W N ( k − k 0 ) n = X [ < k − k 0 > N ] \begin{aligned} W_N^{-k_0n}x[n]\xrightarrow{DFT}\sum_{n=0}^{N-1}x[n]W_N^{(k-k_0)n}=X[<k-k_0>_N] \end{aligned} WN−k0nx[n]DFTn=0∑N−1x[n]WN(k−k0)n=X[<k−k0>N]
时域反转
x [ < − n > N ] → D F T ∑ n = 0 N − 1 x [ < − n > N ] W N k n → m = − n ∑ m = − ( N − 1 ) 0 x [ < m > N ] W N − k m = ∑ m = 0 N − 1 x [ m ] W N − k m = X [ < − k > N ] \begin{aligned} x[<-n>_N]&\xrightarrow{DFT}\sum_{n=0}^{N-1}x[<-n>_N]W_{N}^{kn} \\ &\xrightarrow{m=-n}\sum_{m=-(N-1)}^{0}x[<m>_N]W_{N}^{-km} \\ &=\sum_{m=0}^{N-1}x[m]W_{N}^{-km} \\ &=X[<-k>_N] \end{aligned} x[<−n>N]DFTn=0∑N−1x[<−n>N]WNknm=−nm=−(N−1)∑0x[<m>N]WN−km=m=0∑N−1x[m]WN−km=X[<−k>N]
时域共轭
x ∗ [ n ] → D F T ∑ n = 0 N − 1 x ∗ [ n ] W N k n = ( ∑ n = 0 N − 1 x [ n ] W N − k n ) ∗ = X ∗ [ < − k > N ] \begin{aligned} x^{*}[n]&\xrightarrow{DFT}\sum_{n=0}^{N-1}x^{*}[n]W_N^{kn} \\ &=(\sum_{n=0}^{N-1}x[n]W_N^{-kn})^{*} \\ &=X^{*}[<-k>_N] \end{aligned} x∗[n]DFTn=0∑N−1x∗[n]WNkn=(n=0∑N−1x[n]WN−kn)∗=X∗[<−k>N]
由上面两个可以推得
x ∗ [ < − n > N ] → D F T X ∗ [ k ] \color{red}x^{*}[<-n>_N]\xrightarrow{DFT}X^{*}[k] x∗[<−n>N]DFTX∗[k]
对称性质
x c s [ n ] = 1 2 ( x [ n ] + x ∗ [ < − n > N ] ) → D F T 1 2 ( X [ k ] + X ∗ [ k ] ) = X r e [ k ] \color{red}x_{cs}[n]=\frac{1}{2}(x[n]+x^{*}[<-n>_N])\xrightarrow{DFT}\frac{1}{2}(X[k]+X^{*}[k])=X_{re}[k] xcs[n]=21(x[n]+x∗[<−n>N])DFT21(X[k]+X∗[k])=Xre[k]
x c a [ n ] = 1 2 ( x [ n ] − x ∗ [ < − n > N ] ) → D F T 1 2 ( X [ k ] − X ∗ [ k ] ) = j X i m [ k ] \color{red}x_{ca}[n]=\frac{1}{2}(x[n]-x^{*}[<-n>_N])\xrightarrow{DFT}\frac{1}{2}(X[k]-X^{*}[k])=jX_{im}[k] xca[n]=21(x[n]−x∗[<−n>N])DFT21(X[k]−X∗[k])=jXim[k]
x r e [ n ] = 1 2 ( x [ n ] + x ∗ [ n ] ) → D F T 1 2 ( X [ k ] + X ∗ [ < − k > N ] ) = X c s [ k ] \color{red}x_{re}[n]=\frac{1}{2}(x[n]+x^{*}[n])\xrightarrow{DFT}\frac{1}{2}(X[k]+X^{*}[<-k>_N])=X_{cs}[k] xre[n]=21(x[n]+x∗[n])DFT21(X[k]+X∗[<−k>N])=Xcs[k]
j x i m [ n ] = 1 2 ( x [ n ] − x ∗ [ n ] ) → D F T 1 2 ( X [ k ] − X ∗ [ < − k > N ] ) = X c a [ k ] \color{red}jx_{im}[n]=\frac{1}{2}(x[n]-x^{*}[n])\xrightarrow{DFT}\frac{1}{2}(X[k]-X^{*}[<-k>_N])=X_{ca}[k] jxim[n]=21(x[n]−x∗[n])DFT21(X[k]−X∗[<−k>N])=Xca[k]
卷积性质
假设 x [ n ] , w [ n ] x[n],w[n] x[n],w[n]都是长度为 N N N的有限长序列，它们的DFT分别为 X [ k ] , W [ k ] X[k],W[k] X[k],W[k],假设它们的有值区间为 0 ≤ n ≤ N − 1 0 \leq n \leq N-1 0≤n≤N−1,那么它们进行圆周卷积的DFT为:
x [ n ] N ◯ w [ n ] = ∑ m = 0 N − 1 x [ m ] w [ < n − m > N ] → D F T ∑ n = 0 N − 1 ∑ m = 0 N − 1 x [ m ] w [ < n − m > N ] W N k n = ∑ m = 0 N − 1 x [ m ] ∑ n = 0 N − 1 1 N ∑ r = 0 N − 1 W [ r ] W N r ( n − m ) W N k n = ∑ m = 0 N − 1 x [ m ] ∑ r = 0 N − 1 W [ r ] W N k m ( 1 N ∑ n = 0 N − 1 W N k − r ) = ∑ m = 0 N − 1 x [ m ] W N k m W [ k ] = X [ k ] W [ k ] \begin{aligned} x[n]\text{\textcircled{N}}w[n]&=\sum_{m=0}^{N-1}x[m]w[<n-m>_N] \\ &\xrightarrow{DFT}\sum_{n=0}^{N-1}\sum_{m=0}^{N-1}x[m]w[<n-m>_N]W_N^{kn} \\ &=\sum_{m=0}^{N-1}x[m]\sum_{n=0}^{N-1}\frac{1}{N}\sum_{r=0}^{N-1}W[r]W_N^{r(n-m)}W_N^{kn} \\ &=\sum_{m=0}^{N-1}x[m]\sum_{r=0}^{N-1}W[r]W_N^{km}(\frac{1}{N}\sum_{n=0}^{N-1}W_N^{k-r}) \\ &=\sum_{m=0}^{N-1}x[m]W_N^{km}W[k] \\ &=X[k]W[k] \end{aligned} x[n]N◯w[n]=m=0∑N−1x[m]w[<n−m>N]DFTn=0∑N−1m=0∑N−1x[m]w[<n−m>N]WNkn=m=0∑N−1x[m]n=0∑N−1N1r=0∑N−1W[r]WNr(n−m)WNkn=m=0∑N−1x[m]r=0∑N−1W[r]WNkm(N1n=0∑N−1WNk−r)=m=0∑N−1x[m]WNkmW[k]=X[k]W[k]
上式中用到了
1 N ∑ n = 0 N − 1 W N k − r = { 1 , k − r = l N , l = 0 , 1 , . . . 0 , 其 它 \frac{1}{N}\sum_{n=0}^{N-1}W_N^{k-r}= \begin{cases} 1, k -r = lN , \, l=0,1,...\\ 0, 其它 \end{cases} N1n=0∑N−1WNk−r={1,k−r=lN,l=0,1,...0,其它
Parseval定理
∑ n = 0 N − 1 x [ n ] y ∗ [ n ] = ∑ n = 0 N − 1 x [ n ] ( 1 N ∑ k = 0 N − 1 Y [ k ] W N − k n ) ∗ = 1 N ∑ k = 0 N − 1 Y ∗ [ k ] ∑ n = 0 N − 1 x [ n ] W N k n = 1 N ∑ k = 0 N − 1 X [ k ] Y ∗ [ k ] \begin{aligned} \sum_{n=0}^{N-1}x[n]y^{*}[n]&=\sum_{n=0}^{N-1}x[n](\frac{1}{N}\sum_{k=0}^{N-1}Y[k]W_N^{-kn})^{*}\\ &=\frac{1}{N}\sum_{k=0}^{N-1}Y^{*}[k]\sum_{n=0}^{N-1}x[n]W_N^{kn}\\ &=\frac{1}{N}\sum_{k=0}^{N-1}X[k]Y^{*}[k] \end{aligned} n=0∑N−1x[n]y∗[n]=n=0∑N−1x[n](N1k=0∑N−1Y[k]WN−kn)∗=N1k=0∑N−1Y∗[k]n=0∑N−1x[n]WNkn=N1k=0∑N−1X[k]Y∗[k]
特别的，当 x [ n ] = y [ n ] x[n]=y[n] x[n]=y[n]时
∑ n = 0 N − 1 ∣ x [ n ] ∣ 2 = 1 N ∑ k = 0 N − 1 ∣ X [ k ] ∣ 2 \sum_{n=0}^{N-1}\vert x[n]\vert^2=\frac{1}{N}\sum_{k=0}^{N-1}\vert X[k]\vert^2 n=0∑N−1∣x[n]∣2=N1k=0∑N−1∣X[k]∣2