巴特沃思滤波器-输出x（-1）？

Question

I am trying to apply a Butterworth filter as in this great reply How to implement band-pass Butterworth filter with Scipy.signal.butter . However, when I use the functions from there, the result seems to be the wrong way round (x(-1)):

FIGURE: applied Butterworth filter

What is wrong? (I assume this is wrong?)

from scipy.signal import butter, lfilter

def butter_bandpass(lowcut, highcut, fs, order=5):
    nyq = 0.5 * fs
    low = lowcut / nyq
    high = highcut / nyq
    b, a = butter(order, [low, high], btype='band')
    return b, a


def butter_bandpass_filter(data, lowcut, highcut, fs, order=5):
    b, a = butter_bandpass(lowcut, highcut, fs, order=order)
    y = lfilter(b, a, data)
    return y

x=np.array(range(100))
y1=np.array([math.sin(2*3.14*xx/3) for xx in x])
y2=np.array([2*math.sin(2*3.14*xx/30) for xx in x])
y0=x*0.05
y=y1+y2+y0

lowcut=1./10000000.
highcut=1./20.

plt.figure(3)
plt.clf()

plt.plot(x, y, label='Noisy signal',lw=2.5,color='blue')
plt.plot(x,y0,ls='--',color='blue')
plt.plot(x,y1,ls='--',color='blue')
plt.plot(x,y2,ls='--',color='blue')

ysm = butter_bandpass_filter(y, lowcut, highcut, fs, order=6)

plt.plot(x, ysm, label='Filtered signal',lw=2.5,color='red')
plt.grid(True)
plt.axis('tight')
plt.legend(loc='upper left')

plt.show()

Answer 1

Nothing is wrong. What you are seeing is the normal phase shift created by an IIR filter.

If that phase shift is unacceptable, one option is to change this line:

y = lfilter(b, a, data)

to

y = filtfilt(b, a, data)

and add filtfilt to the names imported from scipy.signal . filtfilt applies the same filter twice, once forward and once backward, so the phase shifts "cancel". If you use filtfilt , you can lower the order of the filter because you are applying it twice.

Another option is to use a FIR filter instead of an IIR filter. Questions about the behavior of filters are probably best asked over at dsp.stackexchange.com.