附录 1 傅里叶变换简表
原书第 217 页
附录1 傅里叶变换简表
| f(t) | F( \omega ) | |||
|---|---|---|---|---|
| 1 | $ \cos \omega_{0}t $ | $ \pi [\delta(\omega + \omega_{0}) + \delta(\omega - \omega_{0})] $ | ||
| 2 | $ \sin \omega_{0}t $ | j $ \pi [\delta(\omega + \omega_{0}) - \delta(\omega - \omega_{0})] $ | ||
| 3 | $ \frac{\sin \omega_{0}t}{\pi t} $ | $ \begin{cases}1,&\mid\omega\mid\leq\omega_{0},\\0,&\mid\omega\mid>\omega_{0}\end{cases} $ | ||
| 4 | u(t) | $ \frac{1}{j\omega} + \pi\delta(\omega) $ | ||
| 5 | u(t-c) | $ \frac{1}{j\omega} e^{-j\omega c} + \pi\delta(\omega) $ | ||
| 6 | u(t) \cdot t | $ -\frac{1}{\omega^{2}} + \pi j\delta'(\omega) $ | ||
| 7 | u(t) \cdot t^{n} | \frac{n!}{(j\omega)^{n+1}} + \pi j^{n}\delta^{(n)}(\omega) | ||
| 8 | u(t) \sin at | \frac{a}{a^{2} - \omega^{2}} + \frac{\pi}{2j}[\delta(\omega-a) - \delta(\omega+a)] | ||
| 9 | u(t) \cos at | \frac{j\omega}{a^{2} - \omega^{2}} + \frac{\pi}{2}[\delta(\omega-a) + \delta(\omega+a)] | ||
| 10 | u(t) e^{-\beta t} (\beta>0) | \frac{1}{\beta + j\omega} | ||
| 11 | u(t) e^{j\omega t} | \frac{1}{j(\omega-a)} + \pi\delta(\omega-a) | ||
| 12 | u(t-c) e^{j\omega t} | \frac{1}{j(\omega-a)} e^{-j(\omega-a)c} + \pi\delta(\omega-a) | ||
| 13 | u(t) e^{j\omega t} t^{n} | \frac{n!}{[j(\omega-a)]^{n+1}} + \pi j^{n}\delta^{(n)}(\omega-a) | ||
| 14 | e^{a | t | } (Re a<0) | \frac{-2a}{\omega^{2} + a^{2}} |
原书第 218 页
| f(t) | F( \omega ) | |||||
|---|---|---|---|---|---|---|
| 15 | \delta (t) | 1 | ||||
| 16 | \delta (t-c) | $ e^{-j\omega c} $ | ||||
| 17 | \delta '(t) | j \omega | ||||
| 18 | \delta (n)(t) | (j \omega )n | ||||
| 19 | \delta (n)(t-c) | (j \omega )n $ e^{-j\omega c} $ | ||||
| 20 | 1 | 2 \pi \delta ( \omega ) | ||||
| 21 | t | 2 \pi j \delta ' ( \omega ) | ||||
| 22 | t^n | 2 \pi j^n \delta (n)( \omega ) | ||||
| 23 | $ e^{j\alpha t} $ | 2 \pi \delta ( \omega -a) | ||||
| 24 | t^n e^{j\alpha t} | 2 \pi j^n \delta (n)( \omega -a) | ||||
| 25 | \frac{1}{a^2+t^2} (Re a<0) | -\frac{\pi}{a}e^a | \omega | |||
| 26 | \frac{t}{(a^2+t^2)^2} (Re a<0) | \frac{j\omega \pi}{2a}e^a | \omega | |||
| 27 | \frac{e^{j\alpha t}}{a^2+t^2} (Re a<0, b为实数) | -\frac{\pi}{a}e^a | \omega -b | |||
| 28 | \frac{\cos bt}{a^2+t^2} (Re a<0, b为实数) | -\frac{\pi}{2a}[ e^a | \omega -b | + e^a | \omega +b | ] |
| 29 | \frac{\sin bt}{a^2+t^2} (Re a<0, b为实数) | -\frac{\pi}{2aj}[ e^a | \omega -b | - e^a | \omega +b | ] |
| 30 | \frac{\sinh at}{\sinh \pi t} (- \pi <a<\pi) | \frac{\sin a}{\cosh \omega + \cos a} | ||||
| 31 | \frac{\sinh at}{\cosh \pi t} (- \pi <a<\pi) | \frac{\sin \omega}{2} \sin \frac{\omega}{2} - 2j \frac{\cosh \omega + \cos a}{ } |
原书第 219 页
续表
| f(t) | F( \omega ) | |||||||
|---|---|---|---|---|---|---|---|---|
| 32 | $ \frac{\cosh at}{\cosh \pi t} $ (- \pi <a<\pi) | $ 2 \frac{\cos \frac{a}{2}\cosh \frac{\omega}{2}}{\cosh \omega+\cos a} $ | ||||||
| 33 | $ \frac{1}{\cosh at} $ | $ \frac{\pi}{a}\frac{1}{\cosh \frac{\pi \omega}{2a}} $ | ||||||
| 34 | $ \sin at^{2} $ (a>0) | $ \sqrt{\frac{\pi}{a}}\cos\left(\frac{\omega^{2}}{4a}+\frac{\pi}{4}\right) $ | ||||||
| 35 | $ \cos at^{2} $ (a>0) | $ \sqrt{\frac{\pi}{a}}\cos\left(\frac{\omega^{2}}{4a}-\frac{\pi}{4}\right) $ | ||||||
| 36 | $ \frac{1}{t}\sin at $ (a>0) | $ \{\begin{array}{l}\pi, \quad | \omega | \leq a, \\ 0, \quad | \omega | > a\end{array}. $ | ||
| 37 | $ \frac{1}{t^{2}}\sin^{2}at $ (a>0) | $ \{\begin{array}{l}\pi(a-\frac{ | \omega | }{2}), \quad | \omega | \leq 2a, \\ 0, \quad | \omega | > 2a\end{array}. $ |
| 38 | $ \frac{\sin at}{\sqrt{ | t | }} $ | $ j\sqrt{\frac{\pi}{2}}\left(\frac{1}{\sqrt{ | \omega+a | }}-\frac{1}{\sqrt{ | \omega-a | }}\right) $ |
| 39 | $ \frac{\cos at}{\sqrt{ | t | }} $ | $ \sqrt{\frac{\pi}{2}}\left(\frac{1}{\sqrt{ | \omega+a | }}+\frac{1}{\sqrt{ | \omega-a | }}\right) $ |
| 40 | $ \frac{1}{\sqrt{ | t | }} $ | $ \sqrt{\frac{2\pi}{\omega}} $ | ||||
| 41 | sgn t | $ \frac{2}{j\omega} $ | ||||||
| 42 | $ e^{-at^{2}} $ (Re a>0) | $ \sqrt{\frac{\pi}{a}}e^{\frac{\omega^{2}}{4a}} $ | ||||||
| 43 | t | $ -\frac{2}{\omega^{2}} $ | ||||||
| 44 | $ \frac{1}{ | t | } $ | $ \frac{\sqrt{2\pi}}{ | \omega | } $ |