用Python模拟傅立叶分析

import numpy as np
import matplotlib.pyplot as plt

# 定义一个周期为T的函数
T = 2 * np.pi
def f(t):
    return np.sin(t) + 0.5 * np.sin(2 * t) + 0.2 * np.sin(3 * t)

# 生成时间序列
t = np.linspace(0, T, 1000)

# 计算傅立叶级数
n = 100  # 傅立叶级数的阶数
a0 = 1 / T * np.sum(f(t))
an = np.zeros(n)
bn = np.zeros(n)
for i in range(1, n+1):
    an[i-1] = 2 / T * np.sum(f(t) * np.cos(i * t))
    bn[i-1] = 2 / T * np.sum(f(t) * np.sin(i * t))

# 根据傅立叶级数重构原函数
f_reconstructed = np.zeros_like(t)
f_reconstructed += a0 / 2
for i in range(1, n+1):
    f_reconstructed += an[i-1] * np.cos(i * t) + bn[i-1] * np.sin(i * t)

# 绘制原函数和重构函数的图像
plt.plot(t, f(t), label='Original Function')
plt.plot(t, f_reconstructed, label='Reconstructed Function')
plt.legend()
plt.show()