numpy的fft结果的幅度要乘以采样周期吗? [英] amplitude of numpy's fft results is to be multiplied by sampling period?

问题描述

我尝试通过一个示例来验证我对Numpy FFT的理解:exp(-pi*t^2)的傅立叶变换应为exp(-pi*f^2)，如果在直接变换上未应用任何缩放.

I try to validate my understanding of Numpy's FFT with an example: the Fourier transform of exp(-pi*t^2) should be exp(-pi*f^2) when no scaling is applied on the direct transform.

但是，我发现要获得此结果，我需要将FFT的结果乘以因子dt，这是函数上两个采样点之间的时间间隔.我不明白为什么.有人可以帮忙吗?

However, I find that to obtain this result I need to multiply the result of FFT by a factor dt, which is the time interval between two sample points on my function. I don't understand why. Can anybody help ?

这是示例代码:

# create data
N = 4097
T = 100.0
t = linspace(-T/2,T/2,N)
f = exp(-pi*t**2)

# perform FT and multiply by dt
dt = t[1]-t[0]
ft = fft(f)  * dt      
freq = fftfreq( N, dt )
freq = freq[:N/2+1]

# plot results
plot(freq,abs(ft[:N/2+1]),'o')
plot(freq,exp(-pi*freq**2),'r')
legend(('numpy fft * dt', 'exact solution'),loc='upper right')
xlabel('f')
ylabel('amplitude')
xlim(0,1.4)

推荐答案

请注意，您不是在计算连续时间傅立叶变换，计算机是处理离散数据的，Numpy也是如此，如果您查看 numpy.fft.fft 文档说:

Be careful, you are not computing the continuous time Fourier transform, computers work with discrete data, so does Numpy, if you take a look to numpy.fft.fft documentation it says:

numpy.fft.fft(a，n = None，axis = -1)[源代码]

numpy.fft.fft(a, n=None, axis=-1)[source]

计算一维离散傅立叶变换.

Compute the one-dimensional discrete Fourier Transform.

此函数使用高效的快速傅立叶变换(FFT)算法计算一维n点离散傅立叶变换(DFT)

This function computes the one-dimensional n-point discrete Fourier Transform (DFT) with the efficient Fast Fourier Transform (FFT) algorithm

这意味着您正在计算方程式定义的DFT:

That means that your are computing the DFT which is defined by equation:

连续时间傅立叶变换的定义为:

the continuous time Fourier transform is defined by:

如果您进行数学运算以查找它们之间的关系:

And if you do the maths to look for the relationship between them:

如您所见，有一个常数因子1/N恰好是您的标度值dt(x[n] - x[n-1]，其中n在[0，T]区间中等于1/N).

As you can see there is a constant factor 1/N which is exactly your scale value dt (x[n] - x[n-1] where n is in [0,T] interval is equivalent to 1/N).

仅对代码进行注释，导入所有内容from numpy import *而不是使用以下方法不是一个好习惯:

Just a comment on your code, it is not a good practice to import everything from numpy import * instead use:

import numpy as np
import matplotlib.pyplot as plt

# create data
N = 4097
T = 100.0
t = np.linspace(-T/2,T/2,N)
f = np.exp(-np.pi*t**2)

# perform FT and multiply by dt
dt = t[1]-t[0]
ft = np.fft.fft(f) * dt      
freq = np.fft.fftfreq(N, dt)
freq = freq[:N/2+1]

# plot results
plt.plot(freq, np.abs(ft[:N/2+1]),'o')
plt.plot(freq, np.exp(-np.pi * freq**2),'r')
plt.legend(('numpy fft * dt', 'exact solution'), loc='upper right')
plt.xlabel('f')
plt.ylabel('amplitude')
plt.xlim([0, 1.4])
plt.show()

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