离散时间傅里叶变换与离散傅里叶变换

离散时间傅里叶变换

单一变元形式

单一变元离散时间傅里叶变换的定义

$$a[n]\stackrel{\mathscr F}{\leftrightarrow}F(\omega): \begin{cases} F(\omega)&=&\sum\limits_{n=-\infty}^\infty a[n]e^{-\mathrm i\omega n} \\ a[n]&=&\dfrac1{2\pi}\int_0^{2\pi}F(w)e^{i\omega n}\,\mathrm d\omega \end{cases}$$

单一变元离散时间傅里叶变换的性质

1. 线性
   若
   $$a[n]\stackrel{\mathscr F}{\leftrightarrow}f_a(\omega) \\ b[n]\stackrel{\mathscr F}{\leftrightarrow}f_b(\omega)$$
   且 $c$，$d$ 为常数，则
   $$c\cdot a[n]+d\cdot b[n]\stackrel{\mathscr F}{\leftrightarrow}c\cdot f_a(\omega)+d\cdot f_d(\omega)$$
2. 时移性
   若
   $$a[n]\stackrel{\mathscr F}{\leftrightarrow}F(\omega)$$
   则
   $$a[n+n_0]\stackrel{\mathscr F}{\leftrightarrow}e^{\mathrm i\omega n_0} F(\omega)$$
3. 频移性
   若
   $$a[n]\stackrel{\mathscr F}{\leftrightarrow}F(\omega)$$
   则
   $$e^{-\mathrm i\omega_0n} a[n]\stackrel{\mathscr F}{\leftrightarrow}F(\omega+\omega_0)$$

多变元形式

多变元离散时间傅里叶变换的定义

$$a[n_1,n_2,\cdots,n_p]\stackrel{\mathscr F}{\leftrightarrow}F(\omega_1,\omega_2,\cdots,\omega_p): \begin{cases} F(\omega_1,\omega_2,\cdots,\omega_p)&=&\sum\limits_{n_1=-\infty}^\infty\sum\limits_{n_2=-\infty}^\infty\cdots\sum\limits_{n_p=-\infty}^\infty a[n_1,n_2,\cdots,n_p]e^{-\mathrm i(\omega_1n_1+\omega_2n_2+\cdots+\omega_pn_p)} \\ \\ a[n_1,n_2,\cdots,n_p]&=&\dfrac1{(2\pi)^p}\int_0^{2\pi}\int_0^{2\pi}\cdots\int_0^{2\pi}F(\omega_1,\omega_2,\cdots,\omega_p)e^{i(\omega_1n_1+\omega_2n_2+\cdots+\omega_pn_p)}\\&&\mathrm d\omega_1\mathrm d\omega_2\cdots\mathrm d\omega_p \end{cases}$$

多变元离散时间傅里叶变换的性质

1. 线性
   若
   $$a[n_1,n_2,\cdots,n_p]\stackrel{\mathscr F}{\leftrightarrow}f_a(\omega_1,\omega_2,\cdots,\omega_p) \\ b[n_1,n_2,\cdots,n_p]\stackrel{\mathscr F}{\leftrightarrow}f_b(\omega_1,\omega_2,\cdots,\omega_p)$$
   且 $c$，$d$ 为常数，
   则
   $$c\cdot a[n_1,n_2,\cdots,n_p]+d\cdot b[n_1,n_2,\cdots,n_p] \stackrel{\mathscr F}{\leftrightarrow}c\cdot f_a(\omega_1,\omega_2,\cdots,\omega_p)+d\cdot f_b(\omega_1,\omega_2,\cdots,\omega_p)$$
2. 时移性
   若
   $$a[n_1,n_2,\cdots,n_p]\stackrel{\mathscr F}{\leftrightarrow}F(\omega_1,\omega_2,\cdots,\omega_p)$$
   则
   $$a[n_1+n_{10},n_2+n_{20},\cdots,n_p+n_{p0}]\stackrel{\mathscr F}{\leftrightarrow} e^{\mathrm i(\omega_1n_{10}+\omega_2n_{20}+\cdots+\omega_pn_{p0})} F(\omega_1,\omega_2,\cdots,\omega_p)$$
3. 频移性
   若
   $$a[n_1,n_2,\cdots,n_p]\stackrel{\mathscr F}{\leftrightarrow}F(\omega_1,\omega_2,\cdots,\omega_p)$$
   则
   $$e^{-\mathrm i\omega_{10}n_1+\omega_{20}n_2+\cdots+\omega_{p0}n_p} a[n_1,n_2,\cdots,n_p]\stackrel{\mathscr F}{\leftrightarrow}F(\omega_1+\omega_{10},\omega_2+\omega_{20},\cdots,\omega_p+\omega_{p0})$$

常用积分公式

$$\begin{cases} \int_0^{2\pi}\dfrac1{A-\cos x}\,\mathrm dx=\dfrac{2\pi}{\sqrt{A^2-1}},&(A>1) \\ \int_0^{2\pi}\dfrac{\cos mx}{A-\cos x}\,\mathrm dx=\dfrac{2\pi}{\sqrt{A^2-1}}\Big( A-\sqrt{A^2-1} \Big),&(A>1) \\ \int_0^{2\pi}\dfrac{\sin mx}{A-\cos x}\,\mathrm dx=0,&(A>1) \\ \int_0^{2\pi}\dfrac{1-\cos mx}{1-\cos x}\,\mathrm dx=2\pi\begin{vmatrix}m\end{vmatrix} & \end{cases}$$ $$\int_0^{2\pi}\dfrac{P(x)}{(A-\cos x)^n}\,\mathrm dx$$ $$\dfrac{\partial}{\partial A}\int_0^{2\pi}\dfrac{P(x)}{(A-\cos x)^n}\,\mathrm dx$$

离散时间傅里叶变换的作用

离散傅里叶变换

单一变元形式

单一变元离散傅里叶变换的定义

单一变元离散傅里叶变换的性质

多变元形式

多边缘离散傅里叶变换的定义

多边缘离散傅里叶变换的性质

常用求和公式

$$\begin{cases} \dfrac1N\sum\limits_{k=0}^{N-1}\dfrac{\cos\dfrac{2\pi km}N}{A-\cos\dfrac{2\pi k}N}=\dfrac1{\sqrt{A^2-1}}\cdot\dfrac{(A-\sqrt{A^2-1})^{\begin{vmatrix}m\end{vmatrix}}+(A-\sqrt{A^2-1}^{N-\begin{vmatrix}m\end{vmatrix}})}{1-(A-\sqrt{A^2-1})^N},&(A>1) \\ \dfrac1N\sum\limits_{k=0}^{N-1}\dfrac{1-\cos\dfrac{2\pi km}N}{1-\cos \dfrac{2\pi k}N}=\dfrac{\begin{vmatrix}m\end{vmatrix}(N-\begin{vmatrix}m\end{vmatrix})}{N} \\ \dfrac1N\sum\limits_{k=0}^{N-1}\dfrac{\sin\dfrac{2\pi km}N}{A-\cos\dfrac{2\pi k}N}=0,&(A>1) \end{cases}$$

离散傅里叶变换的作用