高斯的傅立叶变换不是高斯，但是那是错误的！ - Python [英] Fourier transform of a Gaussian is not a Gaussian, but thats wrong! - Python

问题描述

我正在尝试使用Numpy的fft函数，但是当我给该函数一个简单的高斯函数时，该高斯函数的fft不是高斯，而是接近但减半，因此每一半都位于x轴的两端

I am trying to utilize Numpy's fft function, however when I give the function a simple gausian function the fft of that gausian function is not a gausian, its close but its halved so that each half is at either end of the x axis.

我正在计算的高斯函数是 y = exp(-x ^ 2)

The Gaussian function I'm calculating is y = exp(-x^2)

这是我的代码:

from cmath import *
from numpy import multiply
from numpy.fft import fft
from pylab import plot, show

""" Basically the standard range() function but with float support """
def frange (min_value, max_value, step):
    value = float(min_value)
    array = []
    while value < float(max_value):
        array.append(value)
        value += float(step)
    return array


N = 256.0 # number of steps
y = []
x = frange(-5, 5, 10/N)

# fill array y with values of the Gaussian function   
cache = -multiply(x, x)
for i in cache: y.append(exp(i))

Y = fft(y)

# plot the fft of the gausian function
plot(x, abs(Y))
show()

结果不太正确，因为高斯函数的FFT应该是高斯函数本身...

The result is not quite right, cause the FFT of a Gaussian function should be a Gaussian function itself...

推荐答案

np.fft.fft以所谓的标准顺序"返回结果:(

np.fft.fft returns a result in so-called "standard order": (from the docs)

如果A = fft(a, n)，则A[0] 包含零频项( 信号的均值)，通常 对于真实输入而言，完全是真实的.然后 A[1:n/2]包含 正频率项，以及 A[n/2+1:]包含 负频率项，按 负频率递减.

If A = fft(a, n), then A[0] contains the zero-frequency term (the mean of the signal), which is always purely real for real inputs. Then A[1:n/2] contains the positive-frequency terms, and A[n/2+1:] contains the negative-frequency terms, in order of decreasingly negative frequency.

函数np.fft.fftshift将结果重新排列为大多数人期望的顺序(这对于绘制很有用):

The function np.fft.fftshift rearranges the result into the order most humans expect (and which is good for plotting):

例程np.fft.fftshift(A) 转变变换及其 归零频率 中间的组件...

The routine np.fft.fftshift(A) shifts transforms and their frequencies to put the zero-frequency components in the middle...

因此使用np.fft.fftshift:

import matplotlib.pyplot as plt
import numpy as np

N = 128
x = np.arange(-5, 5, 10./(2 * N))
y = np.exp(-x * x)
y_fft = np.fft.fftshift(np.abs(np.fft.fft(y))) / np.sqrt(len(y))
plt.plot(x,y)
plt.plot(x,y_fft)
plt.show()

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