?DFT变换的性质

线性性质

\[ \begin{aligned} y[n]&=ax[n]+bw[n]\xrightarrow{DFT}Y[k]=\sum_{n=0}^{N-1}(ax[n]+bw[n])W_N^{kn}\&=a\sum_{n=0}^{N-1}x[n]W_N^{kn}+b\sum_{n=0}^{N-1}w[n]W_N^{kn} \&=aX[k]+bW[k] \end{aligned} \]

时移性质

\[ \begin{aligned} x[n-n_0]&\xrightarrow{DFT}\sum_{n=0}^{N-1}x[<n-n_0>_N]e^{-j\frac{2\pi k}{N}n} \&\xrightarrow{m=n-n_0}\sum_{m=-n_0}^{N-n_0-1}x[<m>_N]e^{-j\frac{2\pi k}{N}(m+n_0)} \&=W_{N}^{kn_0}\sum_{m=0}^{N-1}x[m]W_{N}^{km} \&=W_{N}^{kn_0}X[k] \end{aligned} \]

频移性质

\[ \begin{aligned} W_N^{-k_0n}x[n]\xrightarrow{DFT}\sum_{n=0}^{N-1}x[n]W_N^{(k-k_0)n}=X[<k-k_0>_N] \end{aligned} \]

时域反转

\[ \begin{aligned} x[<-n>_N]&\xrightarrow{DFT}\sum_{n=0}^{N-1}x[<-n>_N]W_{N}^{kn} \&\xrightarrow{m=-n}\sum_{m=-(N-1)}^{0}x[<m>_N]W_{N}^{-km} \&=\sum_{m=0}^{N-1}x[m]W_{N}^{-km} \&=X[<-k>_N] \end{aligned} \]

时域共轭

\[ \begin{aligned} x^{*}[n]&\xrightarrow{DFT}\sum_{n=0}^{N-1}x^{*}[n]W_N^{kn} \&=(\sum_{n=0}^{N-1}x[n]W_N^{-kn})^{*} \&=X^{*}[<-k>_N] \end{aligned} \]

由上面两个可以推得
\[ \color{red}x^{*}[<-n>_N]\xrightarrow{DFT}X^{*}[k] \]

对称性质

\[ x_{cs}[n]=\frac{1}{2}(x[n]+x^{*}[<-n>_N])\xrightarrow{DFT}\frac{1}{2}(X[k]+X^{*}[k])=X_{re}[k] \]
\[ x_{ca}[n]=\frac{1}{2}(x[n]-x^{*}[<-n>_N])\xrightarrow{DFT}\frac{1}{2}(X[k]-X^{*}[k])=jX_{im}[k] \]
\[ x_{re}[n]=\frac{1}{2}(x[n]+x^{*}[n])\xrightarrow{DFT}\frac{1}{2}(X[k]+X^{*}[<-k>_N])=X_{cs}[k] \]
\[ jx_{im}[n]=\frac{1}{2}(x[n]-x^{*}[n])\xrightarrow{DFT}\frac{1}{2}(X[k]-X^{*}[<-k>_N])=X_{ca}[k] \]

卷积性质

??假设\(x[n],w[n]\)都是长度为\(N\)的有限长序列，它们的DFT分别为\(X[k],W[k]\),假设它们的有值区间为\(0 \leq n \leq N-1?\),那么它们进行圆周卷积的DFT为:
\[ \begin{aligned} x[n]\otimes w[n]&=\sum_{m=0}^{N-1}x[m]w[<n-m>_N] \&\xrightarrow{DFT}\sum_{n=0}^{N-1}\sum_{m=0}^{N-1}x[m]w[<n-m>_N]W_N^{kn} \&=\sum_{m=0}^{N-1}x[m]\sum_{n=0}^{N-1}\frac{1}{N}\sum_{r=0}^{N-1}W[r]W_N^{r(n-m)}W_N^{kn} \&=\sum_{m=0}^{N-1}x[m]\sum_{r=0}^{N-1}W[r]W_N^{km}(\frac{1}{N}\sum_{n=0}^{N-1}W_N^{k-r}) \&=\sum_{m=0}^{N-1}x[m]W_N^{km}W[k] \&=X[k]W[k] \end{aligned} \]

上式中用到了
\[ \frac{1}{N}\sum_{n=0}^{N-1}W_N^{k-r}= \begin{cases} 1, k -r = lN , \, l=0,1,...\0, 其它 \end{cases} \]

Parseval定理

\[ \begin{aligned} \sum_{n=0}^{N-1}x[n]y^{*}[n]&=\sum_{n=0}^{N-1}x[n](\frac{1}{N}\sum_{k=0}^{N-1}Y[k]W_N^{-kn})^{*}\&=\frac{1}{N}\sum_{k=0}^{N-1}Y^{*}[k]\sum_{n=0}^{N-1}x[n]W_N^{kn}\&=\frac{1}{N}\sum_{k=0}^{N-1}X[k]Y^{*}[k] \end{aligned} \]

特别的，当\(x[n]=y[n]?\)时
\[ \sum_{n=0}^{N-1}\vert x[n]\vert^2=\frac{1}{N}\sum_{k=0}^{N-1}\vert X[k]\vert^2 \]

13 DFT变换的性质

原文：https://www.cnblogs.com/LastKnight/p/10958058.html