在傅立叶或时域集成

尼古拉·范安内洛(Nicola Vianello)

我正在努力理解信号的数值积分问题。基本上,我有一个要积分或要执行的信号,它是反导随时间的函数(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,这些方法原则上应该保持一致,但不一致。我正在使用的代码如下。请注意,代码中的信号y之前已使用Butterworth滤波进行了高通滤波(类似于此处http://wiki.scipy.org/Cookbook/ButterworthBandpass所做的操作)。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(除了最终可能出现的线性趋势不同)不同,或者更好的是动态变化,它们在相同的振荡时间下非常相似,但是在这两个信号之间大约有30的因数bI2比bI低30倍(大约)。顺便说一句,我已经减去了两个信号的均值,以确保它们是零均值信号,并在IDL中进行积分(均具有等效的cumsumtrapz和在傅立叶域中),从而得出与bI2兼容的值。任何线索都受到欢迎

模糊鸭

很难知道scipy.fftpack.diff()在发动机罩下正在做什么

为了尝试解决您的问题,我挖掘了我之前写的一个旧的频域积分函数。值得指出的是,实际上,人们通常希望对某些参数的控制要多于scipy.fftpack.diff()您提供的控制。例如,函数f_lof_hi参数intf()允许您对输入进行频带限制,以排除可能会产生噪声的非常低或非常高的频率。特别是在集成过程中,嘈杂的低频可能会“爆炸”,并使信号不堪重负。您可能还希望在时间序列的开始和结束时使用一个窗口来停止频谱泄漏。

我已经计算出bI2了一个结果,bI3intf()使用以下代码进行了一次集成(为简单起见,我假设使用平均采样率):

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: 在此处输入图片说明

尽管bI2中存在明显的分段线性趋势,但两个时间序列的数量级相同,形状大致相同。我知道这并不能解释scipy函数中发生的事情,但是至少这表明频域方法不是问题。

的代码intf完整粘贴在下面。

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

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