傅里叶或时域积分

Question

我正在努力理解信号数值积分的问题。基本上，我有一个信号，我想将其积分或执行，并将其反导作为时间的函数（用于获取磁场的拾波线圈的积分）。我尝试了两种不同的方法，原则上应该是一致的，但事实并非如此。我正在使用的代码如下。请注意，代码中的信号 y 之前已经使用巴特沃斯滤波进行了高通滤波（类似于此处所做的http://wiki.scipy.org/Cookbook/ButterworthBandpass）。信号和时间基准可以在此处下载（https://www.dropbox.com/s/fi5z38sae6j5410/Trial.npz?dl=0）

import scipy as sp
from scipy import integrate
from scipy import fftpack
data = np.load('trial.npz')
y = data['arr_1'] # this is the signal 
t = data['arr_0']
# integration using pfft
bI = sp.fftpack.diff(y-y.mean(),order=-1)
bI2= sp.integrate.cumtrapz(y-y.mean(),x=t)

现在，这两个信号（除了可以取出的最终不同的线性趋势之外）是不同的，或者更好的是动态地它们在相同的振荡时间下非常相似，但是两个信号之间存在大约 30 的系数，在这个意义上bI2 比 bI 低 30 倍（大约）。顺便说一句，我减去了两个信号的平均值，以确保它们是零均值信号，并在 IDL 中执行积分（均具有等效的 cumsumtrapz 和傅里叶域）给出与 bI2 兼容的值。任何线索都非常受欢迎

Answer 1

很难知道scipy.fftpack.diff()引擎盖下正在做什么。

为了尝试解决您的问题，我挖出了我不久前编写的一个旧频域积分函数。值得指出的是，在实践中，人们通常希望对某些参数有更多的控制，而不是scipy.fftpack.diff()给你提供的。例如，我的函数的f_lo和参数允许您对输入进行频带限制，以排除可能有噪声的非常低或非常高的频率。特别是嘈杂的低频会在积分过程中“爆炸”并淹没信号。您可能还想在时间序列的开始和结束处使用窗口来阻止频谱泄漏。f_hiintf()

我已经计算了bI2结果 ，并使用以下代码bI3集成了一次（为了简单起见，我假设了平均采样率）：intf()

import intf
from scipy import integrate
data = np.load(path)
y = data['arr_1']
t = data['arr_0']

bI2= sp.integrate.cumtrapz(y-y.mean(),x=t)
bI3 = intf.intf(y-y.mean(), fs=500458, f_lo=1, winlen=1e-2, times=1)

我绘制了 bI2 和 bI3：

尽管 bI2 中存在明显的分段线性趋势，但这两个时间序列具有相同的数量级和大致相同的形状。我知道这并不能解释 scipy 函数中发生的情况，但至少这表明频域方法不是问题。

intf下面完整粘贴了代码。

def intf(a, fs, f_lo=0.0, f_hi=1.0e12, times=1, winlen=1, unwin=False):
"""
Numerically integrate a time series in the frequency domain.

This function integrates a time series in the frequency domain using
'Omega Arithmetic', over a defined frequency band.

Parameters
----------
a : array_like
    Input time series.
fs : int
    Sampling rate (Hz) of the input time series.
f_lo : float, optional
    Lower frequency bound over which integration takes place.
    Defaults to 0 Hz.
f_hi : float, optional
    Upper frequency bound over which integration takes place.
    Defaults to the Nyquist frequency ( = fs / 2).
times : int, optional
    Number of times to integrate input time series a. Can be either 
    0, 1 or 2. If 0 is used, function effectively applies a 'brick wall' 
    frequency domain filter to a.
    Defaults to 1.
winlen : int, optional
    Number of seconds at the beginning and end of a file to apply half a 
    Hanning window to. Limited to half the record length.
    Defaults to 1 second.
unwin : Boolean, optional
    Whether or not to remove the window applied to the input time series
    from the output time series.

Returns
-------
out : complex ndarray
    The zero-, single- or double-integrated acceleration time series.

Versions
----------
1.1 First development version. 
    Uses rfft to avoid complex return values.
    Checks for even length time series; if not, end-pad with single zero.
1.2 Zero-means time series to avoid spurious errors when applying Hanning
    window.

"""

a = a - a.mean()                        # Convert time series to zero-mean
if np.mod(a.size,2) != 0:               # Check for even length time series
    odd = True
    a = np.append(a, 0)                 # If not, append zero to array
else:
    odd = False
f_hi = min(fs/2, f_hi)                  # Upper frequency limited to Nyquist
winlen = min(a.size/2, winlen)          # Limit window to half record length

ni = a.size                             # No. of points in data (int) 
nf = float(ni)                          # No. of points in data (float)
fs = float(fs)                          # Sampling rate (Hz)
df = fs/nf                              # Frequency increment in FFT
stf_i = int(f_lo/df)                    # Index of lower frequency bound
enf_i = int(f_hi/df)                    # Index of upper frequency bound

window = np.ones(ni)                    # Create window function
es = int(winlen*fs)                     # No. of samples to window from ends
edge_win = np.hanning(es)               # Hanning window edge 
window[:es/2] = edge_win[:es/2]
window[-es/2:] = edge_win[-es/2:]
a_w = a*window

FFTspec_a = np.fft.rfft(a_w)            # Calculate complex FFT of input
FFTfreq = np.fft.fftfreq(ni, d=1/fs)[:ni/2+1]

w = (2*np.pi*FFTfreq)                   # Omega
iw = (0+1j)*w                           # i*Omega

mask = np.zeros(ni/2+1)                 # Half-length mask for +ve freqs
mask[stf_i:enf_i] = 1.0                 # Mask = 1 for desired +ve freqs

if times == 2:                          # Double integration
    FFTspec = -FFTspec_a*w / (w+EPS)**3
elif times == 1:                        # Single integration
    FFTspec = FFTspec_a*iw / (iw+EPS)**2
elif times == 0:                        # No integration
    FFTspec = FFTspec_a
else:
    print 'Error'

FFTspec *= mask                         # Select frequencies to use

out_w = np.fft.irfft(FFTspec)           # Return to time domain

if unwin == True:
    out = out_w*window/(window+EPS)**2  # Remove window from time series
else:
    out = out_w

if odd == True:                         # Check for even length time series
    return out[:-1]                     # If not, remove last entry
else:        
    return out