DFT变换的性质
线性性质
[egin{aligned}
y[n]&=ax[n]+bw[n]xrightarrow{DFT}Y[k]=sum_{n=0}^{N-1}(ax[n]+bw[n])W_N^{kn}\
&=asum_{n=0}^{N-1}x[n]W_N^{kn}+bsum_{n=0}^{N-1}w[n]W_N^{kn} \
&=aX[k]+bW[k]
end{aligned}
]
时移性质
[egin{aligned}
x[n-n_0]&xrightarrow{DFT}sum_{n=0}^{N-1}x[<n-n_0>_N]e^{-jfrac{2pi k}{N}n} \
&xrightarrow{m=n-n_0}sum_{m=-n_0}^{N-n_0-1}x[<m>_N]e^{-jfrac{2pi 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
[egin{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
[egin{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
[egin{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:
[egin{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}=
egin{cases}
1, k -r = lN , \, l=0,1,...\
0, 其它
end{cases}
]
Parseval定理
[egin{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
]