使用sympy实现傅里叶变换

from sympy import fourier_transform, inverse_fourier_transform
from sympy.abc import t, omega
from sympy import exp, plot, Piecewise

# 定义原始信号（示例使用矩形函数）
f = Piecewise((0, t < -1), (0, t > 1), (1, True))

# 计算傅里叶变换
F = fourier_transform(f, t, omega)

print("原始信号:")
plot(f, (t, -2, 2), title='Original Signal')

print("\n傅里叶变换结果:")
print(F)

原始信号:

傅里叶变换结果:
Piecewise((sin(2piomega)/(pi*omega), ((omega > -oo) | (omega > 0)) & ((omega > -oo) | (omega < oo)) & ((omega > 0) | (omega < 0)) & ((omega < 0) | (omega < oo))), (2, omega >= 0))

原函数：
$\{\begin{array}{ll}0& \text{for}t>1\vee t<-1\\ 1& \text{otherwise}\end{array}$

变换后函数：
$\{\begin{array}{ll}\frac{\sin \left(2\pi \omega \right)}{\pi \omega }& \text{for}\left(\omega >-\infty \vee \omega >0\right)\wedge \left(\omega >-\infty \vee \omega <\infty \right)\wedge \left(\omega >0\vee \omega <0\right)\wedge \left(\omega <0\vee \omega <\infty \right)\\ 2& \text{for}\omega \ge 0\end{array}$