用于符号数学的 Python 库——sympy(二)：常用信号的Laplace变换

关于拉普拉斯变换的作用，可参考知乎
总的来说，拉普拉斯变换就是迫使函数满足绝对可积条件的傅里叶变换。

常用信号的Laplace变换

参考信号与系统/陈后金, 胡健, 薛健. ——2版.——北京：清华大学出版社；北京交通大学出版社, 2005.7(2017.3重印)第218-219页。

1. e − λ t u ( t ) e^{-\lambda t}u(t) e−λtu(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

lamb = symbols('\lambda')
F1 = laplace_transform(exp(-lamb*t)*Heaviside(t),t,s)

F1[0]

$\frac{1}{\lambda +s}$

F1[1]

$-\lambda$

2. e j ω 0 t u ( t ) e^{j\omega_0 t}u(t) ejω0​tu(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0 = symbols('\omega_0')
F1 = laplace_transform(exp(np.complex(0,1)*w0*t)*Heaviside(t),t,s)

F1[0]

$\frac{1}{-1.0i{\omega }_0+s}$

3. c o s ( ω 0 t ) u ( t ) cos(\omega_0 t)u(t) cos(ω0​t)u(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0 = symbols('\omega_0')
F1 = laplace_transform(cos(w0*t)*Heaviside(t),t,s)

F1[0]

$\frac{s}{{\omega }_0^2+s^2}$

4. s i n ( ω 0 T ) u ( t ) sin(\omega_0 T)u(t) sin(ω0​T)u(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0 = symbols('\omega_0')
F1 = laplace_transform(sin(w0*t)*Heaviside(t),t,s)

F1[0]

$\frac{{\omega }_0}{{\omega }_0^2+s^2}$

5. e − σ 0 t c o s ( ω t ) u ( t ) e^{-\sigma_0 t}cos(\omega t)u(t) e−σ0​tcos(ωt)u(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0,sigma0 = symbols('\omega_0 \sigma_0')
F1 = laplace_transform(exp(-sigma0*t)*cos(w0*t)*Heaviside(t),t,s)

F1[0]

$\frac{{\sigma }_0+s}{\left(-i{\omega }_0+{\sigma }_0+s\right)\left(i{\omega }_0+{\sigma }_0+s\right)}$

6. e − σ 0 t s i n ( ω t ) u ( t ) e^{-\sigma_0 t}sin(\omega t)u(t) e−σ0​tsin(ωt)u(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

w0,sigma0 = symbols('\omega_0 \sigma_0')
F1 = laplace_transform(exp(-sigma0*t)*sin(w0*t)*Heaviside(t),t,s)

F1[0]

$\frac{{\omega }_0}{\left(-i{\omega }_0+{\sigma }_0+s\right)\left(i{\omega }_0+{\sigma }_0+s\right)}$

7. δ ( t ) \delta(t) δ(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

F1 = laplace_transform(DiracDelta(t),t,s)

F1[0]

$1$

8. δ ( n ) ( t ) \delta^{(n)}(t) δ(n)(t)

n次求导无法计算，用多个数字代替，总结规律(不严谨) 结果看不懂

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

delta,n = symbols('\delta n')
F1 = laplace_transform(DiracDelta(t,2),t,s)

F1

${\mathcal{L}}_t\left[{\delta }^{\left(2\right)}\left(t\right)\right]\left(s\right)$

9. u ( t ) u(t) u(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

delta,n = symbols('\delta n')
F1 = laplace_transform(Heaviside(t),t,s)

F1[0]

$\frac{1}{s}$

10. t u ( t ) tu(t) tu(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

delta,n = symbols('\delta n')
F1 = laplace_transform(t*Heaviside(t),t,s)

F1[0]

$\frac{1}{s^2}$

11. t n u ( t ) t^n u(t) tnu(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

delta,n = symbols('\delta n')
F1 = laplace_transform(t**n*Heaviside(t),t,s)

F1[0]

$\frac{s^{-n}\Gamma \left(n+1\right)}{s}$

12. t e − λ t u ( t ) te^{-\lambda t}u(t) te−λtu(t)

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

lamb,n = symbols('\lambda n')
F1 = laplace_transform(t*exp(-lamb*t)*Heaviside(t),t,s)

F1[0]

$\frac{1}{{\left(\lambda +s\right)}^2}$

至于$\Gamma$函数
$$\Gamma (x)={\int }_0^{+\infty }{t}^{x-1}{e}^{-t}dt$$
当$x$为正整数时，
$$\Gamma (n)=(n-1)!$$

逆Laplace变换

逆Laplace变换与Laplace变换函数用法大致一致，举一个简单例子，对于$e^{-\lambda t}u(t)$的拉普拉斯变换，输出结果为
$\frac{1}{\lambda +s}$,那么对于$\frac{1}{\lambda +s}$的逆拉普拉斯变换，代码为

from sympy import *
from sympy.integrals import laplace_transform,inverse_laplace_transform

lamb,s,t = symbols('\lambda s t')
F1 = inverse_laplace_transform(1/(lamb+s),s,t)

F1

$e^{-t\mathrm{re}\left(\lambda \right)-it\mathrm{im}\left(\lambda \right)}\theta \left(t\right)$
结果的$\theta (t)$就是$u(t)$文章来源地址https://www.toymoban.com/news/detail-471419.html

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