[人工智能]【线代&NumPy】第十一章 - 正交性2课后练习 | 离散傅里叶变换 | 逆离散傅里叶变换 | 简述并提供代码

💬 例1：Discrete Fourier Transform

import numpy as np

def DFT(x): # 离散傅里叶变换
    N = x.shape[0]
    n = np.arange(N)
    k = n.reshape((N, 1))
    M = np.exp(-2j * np.pi * k * n / N)
    return np.dot(M, x)

def IDFT(x): # 逆离散傅里叶变换
    N = x.shape[0]
    n = np.arange(N)
    k = n.reshape((N, 1))
    M = np.exp(2j * np.pi * k * n / N)/N
    return np.dot(M, x)

d = np.array([1, 0, 2, 1])
F = DFT(d)
print('离散傅里叶变换: ', F)
D = IDFT(F)
print('逆离散傅里叶变换结果: ', D)

?🚩 运行结果：

离散傅里叶变换: ?[ 4.+0.0000000e+00j -1.+1.0000000e+00j ?2.+1.2246468e-16j
?-1.-1.0000000e+00j]
逆离散傅里叶变换结果: ?[ 1.00000000e+00-1.91428435e-16j -1.11022302e-16-1.02440519e-16j ?
? 2.00000000e+00+7.73141507e-18j ?1.00000000e+00+1.63672859e-16j]

参考文献

Introduction to Linear Algebra, International 4 th Edition by Gilbert Strang, Wellesley Cambridge Press.

百度百科[EB/OL]. []. https://baike.baidu.com/

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