Loading... $$ \begin{array}{ll} {e^{j \omega_0 t}} & {e^{j \omega_0 n}} \\ \hline \begin{array}{l} \text { Distinct signals for } \\ \text { distinct } \omega_0 \end{array} & \begin{array}{l} \text { Identical signals for } \\ \text { values of } \omega_0 \text { separated } \\ \text { by multiples of } 2 \pi \end{array} \\ \hline \text { Periodic for any } \omega_0 & \begin{array}{l} \text { Only if } \omega_0=2 \pi \mathrm{m} / \mathrm{N} \text { for } \\ \text { some integers } \mathrm{N}>0 \text { and } \mathrm{m} \end{array} \\ \hline \begin{array}{l} \text { Fundamental } \\ \text { frequency } \omega_0 \end{array} & \omega_0 / \mathrm{m} \\ \hline \begin{array}{l} \text { Fundamental } \\ \text { period } 2 \pi / \omega_0 \end{array} & \mathrm{~N}=\mathrm{m}\left(2 \pi / \omega_0\right) \end{array}\begin{align*} x(t)&=\sum^{\infty}_{k=-\infty}a_ke^{jk\omega_0t}\\ x(t)&=a_0+2\sum^{\infty}_{k=1}A_kcos(k\omega_0t+\theta_k)\\ x(t)&=a_0+2\sum^{\infty}_{k=1}[B_kcosk\omega_0t-C_ksink\omega_0t] \end{align*} \begin{gathered} \int xsinnxdx=\frac{-xcosnx}{n}+\frac{sinnx}{n^2} \ \ \ \ \int xcosnxdx=\frac{xsinnx}{n}+\frac{cosnx}{n^2}\\ \int x^2sinnxdx=-\frac{x^2cosnx}{n}+\frac{2xsinnx}{n^2}+\frac{2cosnx}{n^3}\ \ \ \ \int x^2cosnxdx=\frac{x^2sinnx}{n}+\frac{2xcosnx}{n^2}-\frac{2sinnx}{n^3}\\ \int e^{-j t w} t d t = -\frac{e^{-j t w}(1+j t w)}{j^2 w^2} ,\int e^{-j t w} t^2 d t=-\frac{e^{-j t w}\left(2+2 j t w+j^2 t^2 w^2\right)}{j^3 w^3} \end{gathered} $$ 若$x(t)$是实信号$a_k=a_{-k}^*$,$\displaystyle a_n =\frac{1}{T}\int^T_0 x(t)e^{-jn\omega_0 t},a_k=\frac{1}{N}\sum_{<N>}x[n]z^n,z=e^{-jk\omega_0n}$ 狄利克雷条件1.可积且绝对可积2.在任何有限区间内,$x(t)$只有有限个起伏变化。也就是说,在任何单个周期内,$x(t)$最大值和最小值的数目有限。3.在$x(t)$的任何有限区间内,只有有限个不连续点,而且在这些不连续点上,函数是有限值。 <img src="https://heaticy-1310163554.cos.ap-shanghai.myqcloud.com/markdown/image-20230403202230289.png" alt="image-20230403202230289" style="zoom: 33%;" style=""><img src="https://heaticy-1310163554.cos.ap-shanghai.myqcloud.com/markdown/image-20230403202317157.png" alt="image-20230403202317157" style="zoom: 33%;" style=""> | $f(t)$ | $F(\omega)$ | | -------------------- | ----------------------------------------------------------------------------------------------------------------------- | | $e^{-at}u(t)$ | $\displaystyle\frac{1}{a+j\omega}$ | | $te^{-at}u(t)$ | $\displaystyle\frac{1}{(a+j\omega)^2}$ | | $ | t | | $\delta (t)$ | $1$ | | $1$ | $2\pi \delta(\omega)$ | | $u(t)$ | $\pi\delta (\omega)+\frac{1}{jw}$ | | $\cos\omega_0 tu(t)$ | $\displaystyle\frac{\pi}{2}[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]+\frac{j\omega}{w^2_0-\omega^2}F(\omega)$ | | $\sin\omega_0 tu(t)$ | $\displaystyle\frac{\pi}{j2}[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]+\frac{\omega_0}{w^2_0-\omega^2}F(\omega)$ | | $\cos \omega_0 t$ | $\pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$ | | $\sin\omega_0 t$ | $j\pi [\delta(\omega+\omega_0)-\delta(\omega-\omega_0)]$ | <img src="https://heaticy-1310163554.cos.ap-shanghai.myqcloud.com/markdown/image-20230403194425494.png" alt="image-20230403194425494" style="zoom: 67%;" style=""> $\mathcal{F}[f'(t)]=i\omega \mathcal{F}[f(t)]$ $\mathcal{F}[f^{(k)}(t)]=(i\omega)^{k}\mathcal{F}[f(t)]$ $\mathcal F^{-1}[i\omega F(\omega)]=(\mathcal F^{-1}(F(\omega))', \ \ \ \mathcal F^{-1}[(i\omega)^k F(\omega)]=(F^{-1}(F(\omega))^{(k)}$ $$ F'(\omega)=-i\mathcal{F}[tf(t)],\ \ \mathcal{F}[tf(t)]=iF'(\omega)\ \ ,F^{(k)}=(-i)^k\mathcal{F}[t^kf(t)],\ \ \mathcal{F}[t^kf(t)]=i^kF^{(k)}(\omega) $$ $$ \mathcal{F}[\int^{t}_{-\infty}f(\tau)d\tau]=\frac1{i\omega}\mathcal F(f(t))+\pi \mathcal F(f(t))|_{\omega=0}\delta(\omega)\ \ \ \ \ \ \ \mathcal{F}^{-1}[\frac{1}{i\omega }F(\omega)]=\int^{t}_{-\infty}f(\tau)d\tau $$ $$ \mathcal{F}[f(t)*g(t)]=\mathcal{F}(f(t))\mathcal{F}(g(t))\ \ \ \ \ \ \ \ \ \ \ \ \mathcal{F}^{-1}[F(\omega)*G(\omega)]=2\pi f(t)g(t) $$ $$ x(t) \stackrel{\mathcal{F}}{\longleftrightarrow} X(j \omega) \longleftrightarrow x^*(t) \stackrel{\mathcal{F}}{\longleftrightarrow} X^*(-j \omega) $$ $$ F(-\omega)=F^*(\omega) \Longleftrightarrow \begin{aligned} & \mathcal{R} e\{F(\omega)\}=\mathcal{R} e\{F(- \omega)\} \\ & \mathcal{J} m\{F(\omega)\}=-\mathcal{J} m\{F(-\omega)\} \end{aligned} $$ $$ \begin{aligned} & x(t)=x_e(t)+x_o(t) \\ & \mathcal{F}\{x(t)\}=\mathcal{F}\left\{x_e(t)\right\}+\mathcal{F}\left\{x_o(t)\right\} \\ \end{aligned} \Longleftrightarrow \begin{aligned}\mathcal{O} d\{x(t)\}\stackrel{\mathcal{F}}{\longleftrightarrow} j \mathcal{J} m\{X(j \omega)\}\\\mathcal{E} v\{x(t)\} \stackrel{\mathcal{F}}{\longleftrightarrow} \mathcal{R} e\{X(j \omega)\}\end{aligned} $$ <img src="https://heaticy-1310163554.cos.ap-shanghai.myqcloud.com/markdown/image-20230403223556066.png" alt="image-20230403223556066" style="zoom: 50%;" style=""> 最后修改:2023 年 08 月 30 日 © 允许规范转载 打赏 赞赏作者 支付宝微信 赞 如果觉得我的文章对你有用,请随意赞赏