标签:frac 三角函数 变换 傅里叶 beta leftrightarrow alpha omega sin
傅里叶变换
$$e^{-at}u(t) \leftrightarrow \frac{1}{a+jw}$$ $$te^{-at}u(t) \leftrightarrow \frac{1}{(a+jw)^2}$$ $$|t| \leftrightarrow -\frac{2}{w^2}$$ $$\delta(t) \leftrightarrow 1$$ $$1 \leftrightarrow 2\pi\delta(\omega)$$ $$e^{-at}u(t) \leftrightarrow \pi\delta(\omega)+\frac{1}{j\omega}$$ $$cos\omega_0t \leftrightarrow \pi[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]$$
$$cos\omega_0t·u(t) \leftrightarrow \frac{\pi}{2}[\delta(\omega-\omega_0)+\delta(\omega+\omega_0)]+\frac{j\omega}{\omega_0^2-\omega^2}$$ $$sin\omega_0t \leftrightarrow j\pi[\delta(\omega+\omega_0)-\delta(\omega-\omega_0)]$$
$$sin\omega_0t·u(t) \leftrightarrow \frac{\pi}{2j}[\delta(\omega-\omega_0)-\delta(\omega+\omega_0)]+\frac{\omega_0}{\omega_0^2-\omega^2}$$ $$e^{-at}·sin\omega_0t·u(t) \leftrightarrow \frac{\omega_0}{(a+jw)^2+\omega_0^2}$$
$$e^{j\omega_0t} \leftrightarrow 2\pi\delta(\omega-\omega_0)$$ $$\frac{W}{2\pi}Sa\frac{Wt}{2} \leftrightarrow rect\frac{\omega}{W}$$ $$rect\frac{t}{\tau} \leftrightarrow \tau Sa\frac{\omega \tau}{2}$$
$$\begin{cases} \displaystyle{1-\frac{|t|}{\tau}}& |t|<\tau\\ 0& |t|>\tau \end{cases} \leftrightarrow \tau(Sa\frac{\omega \tau}{2})^2$$ $$e^{-\frac{t^2}{2\sigma^2}} \leftrightarrow \sqrt{2\pi}\sigma e^{-\frac{\sigma^2\omega^2}{2}}$$ $$\delta_T(t) \leftrightarrow \omega_0\delta_{\omega_0}(\omega), \omega_0=\frac{2\pi}{T}$$
$$t·u(t) \leftrightarrow -\frac{1}{\omega^2}+j\pi\delta'(\omega)$$
三角函数公式
积化和差
$$sin\alpha cos\beta=\frac{1}{2}[sin(\alpha+\beta)+sin(\alpha-\beta)]$$
$$cos\alpha sin\beta=\frac{1}{2}[sin(\alpha+\beta)-sin(\alpha-\beta)]$$
$$cos\alpha cos\beta=\frac{1}{2}[cos(\alpha+\beta)+cos(\alpha-\beta)]$$
$$sin\alpha sin\beta=-\frac{1}{2}[cos(\alpha+\beta)-sin(\alpha-\beta)]$$
和差化积(推导是上式角度和、差的一半)
$$sin\alpha+sin\beta=2sin\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2}$$
$$sin\alpha-sin\beta=2cos\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2}$$
$$cos\alpha+cos\beta=2cos\frac{\alpha+\beta}{2}cos\frac{\alpha-\beta}{2}$$
$$cos\alpha-cos\beta=-2sin\frac{\alpha+\beta}{2}sin\frac{\alpha-\beta}{2}$$
标签:frac,
三角函数,
变换,
傅里叶,
beta,
leftrightarrow,
alpha,
omega,
sin
From: https://www.cnblogs.com/asandstar/p/17472185.html