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
\[\sum_{n=0}^{N-1}\vert x[n]\vert^2=\frac{1}{N}\sum_{k=0}^{N-1}\vert X[k]\vert^2
\]
