假设我有一些要拟合傅立叶级数的数据y.在此post上,Mermoz使用该系列的复杂格式发布了一个解决方案,并“用riemann和计算系数”.在另一个post上,通过FFT获得序列,并写下一个示例.

我尝试实现两种方法(下面的图像和代码-注意每次运行代码时,由于使用numpy.random.normal会生成不同的数据),但我想知道为什么我得到了不同的结果-黎曼方法似乎“错误地移位”,而FFT方法似乎“被压缩”.我也不确定我对系列“ tau”期间的定义.感谢您的关注.

我在Windows 7上将Spyder与Python 3.7.1结合使用

Example

import matplotlib.pyplot as plt
import numpy as np

# Assume x (independent variable) and y are the data.
# Arbitrary numerical values for question purposes:
start = 0
stop = 4
mean = 1
sigma = 2
N = 200
terms = 30 # number of terms for the Fourier series

x = np.linspace(start,stop,N,endpoint=True) 
y = np.random.normal(mean,sigma,len(x))

# Fourier series
tau = (max(x)-min(x)) # assume that signal length = 1 period (tau)

# From ref 1
def cn(n):
    c = y*np.exp(-1j*2*n*np.pi*x/tau)
    return c.sum()/c.size
def f(x,Nh):
    f = np.array([2*cn(i)*np.exp(1j*2*i*np.pi*x/tau) for i in range(1,Nh+1)])
    return f.sum()
y_Fourier_1 = np.array([f(t,terms).real for t in x])

# From ref 2
Y = np.fft.fft(y)
np.put(Y,range(terms+1,len(y)),0.0) # zero-ing coefficients above "terms"
y_Fourier_2 = np.fft.ifft(Y)

# Visualization
f,ax = plt.subplots()
ax.plot(x,y,color='lightblue',label = 'artificial data')
ax.plot(x,y_Fourier_1,label = ("'Riemann' series fit (%d terms)" % terms))
ax.plot(x,y_Fourier_2,label = ("'FFT' series fit (%d terms)" % terms))
ax.grid(True,color='dimgray',linestyle='--',linewidth=0.5)
ax.set_axisbelow(True)
ax.set_ylabel('y')
ax.set_xlabel('x')
ax.legend()