Signal processing - Why a signal is not completely filtered out at my cutoff frequency?

Question

I need to filter a signal. I want to keep frequencies between 0 and 51Hz. The following is the code that I am using, part of it is taken from these two questions (Python: Designing a time-series filter after Fourier analysis, Creating lowpass filter in SciPy - understanding methods and units):

def butter_lowpass(cutoff, fs, order=5):
    nyq = 0.5 * fs
    normal_cutoff = cutoff / nyq
    b, a = butter(order, normal_cutoff, btype='low', analog=False)
    return b, a

def butter_lowpass_filter(data, cutoff, fs, order=5):
    b, a = butter_lowpass(cutoff, fs, order=order)
    y = lfilter(b, a, data)
    return y

    # y is the original signal
    # getting the unbiased signal
    y = list(np.array(y)-sts.mean(y))
    # saving the original signal
    y_before = y

    # x is the time vector

    # for the spectrum of the original signal
    yps_before = np.abs(np.fft.fft(y_before))**2
    yfreqs_before = np.fft.fftfreq(6000, d = 0.001)
    y_idx_before = np.argsort(yfreqs_before)

    #Filtering
    order = 8
    fs = 1000.0       
    cutoff = 50.0  
    y = butter_lowpass_filter(y, cutoff, fs, order)

    # for the spectrum of the filtered signal
    yps = np.abs(np.fft.fft(y))**2
    yfreqs = np.fft.fftfreq(6000, d = 0.001)
    y_idx = np.argsort(yfreqs)


    fig = plt.figure(figsize=(14,10))
    fig.suptitle(file_name, fontsize=20)
    plt.plot(yfreqs_before[y_idx_before], yps_before[y_idx_before], 'k-', label='original spectrum',linewidth=0.5)
    plt.plot(yfreqs[y_idx], yps[y_idx], 'r-', linewidth=2, label='filtered spectrum')
    plt.xlabel('Frequency [Hz]')
    plt.yscale('log')
    plt.grid()
    plt.legend()
    plt.show()

The result of this code is a filtered signal, however, that is the spectrum comparison:

As you can see the spectrum looks good after 100Hz, however, between 50Hz and about 100Hz there is still a component. Thus I tried to use a filter with higher order (20), but as output, I get a really strange spectrum:

So, I know that a filter can not and will never be perfect, but to me, this seems a little too much. In my experience, I was always able to obtain a pretty good filtered signal at my cutoff frequency. Any advice?

Matthieu Brucher Matthieu Brucher · Accepted Answer · 2019-01-25T16:42:06

Cut off frequency is usually where the drop for the transfer function is -6dB.

Increasing the order will make the filter steeper, but it will also add artifacts depending on the filter type. Usually, you have bigger phase issues (numerical issues, phase change is directly proportional to the order, ripples...).

Here, I don't know, it seems to follow the original curve quite well until the cutoff.

That being said, 20-order filters are also very steep and there will be numerical issues due to small numbers in the coefficients. The usual trick is to cascade 2-order filters and make the global filter follow the same curve (you can have a look at Linkwitz-Riley filters). Be aware as well as these filters are LTI, so you can't modify parameters on the fly (that would not be LTI).

Signal processing - Why a signal is not completely filtered out at my cutoff frequency?

1 Answers