f_p = 40. flim = (1., sfreq / 2.) # limits for plotting ############################################################################### # Take for example an ideal low-pass filter, which would give a value of 1 in # the pass-band (up to frequency :math:`f_p`) and a value of 0 in the stop-band # (down to frequency :math:`f_s`) such that :math:`f_p=f_s=40` Hz here # (shown to a lower limit of -60 dB for simplicity): nyq = sfreq / 2. # the Nyquist frequency is half our sample rate freq = [0, f_p, f_p, nyq] gain = [1, 1, 0, 0] third_height = np.array(plt.rcParams['figure.figsize']) * [1, 1. / 3.] ax = plt.subplots(1, figsize=third_height)[1] plot_ideal_filter(freq, gain, ax, title='Ideal %s Hz lowpass' % f_p, flim=flim) ############################################################################### # This filter hypothetically achieves zero ripple in the frequency domain, # perfect attenuation, and perfect steepness. However, due to the discontunity # in the frequency response, the filter would require infinite ringing in the # time domain (i.e., infinite order) to be realized. Another way to think of # this is that a rectangular window in frequency is actually sinc_ function # in time, which requires an infinite number of samples, and thus infinite # time, to represent. So although this filter has ideal frequency suppression, # it has poor time-domain characteristics. # # Let's try to naïvely make a brick-wall filter of length 0.1 sec, and look # at the filter itself in the time domain and the frequency domain: n = int(round(0.1 * sfreq)) + 1
f_p = 40. flim = (1., sfreq / 2.) # limits for plotting ############################################################################### # Take for example an ideal low-pass filter, which would give a value of 1 in # the pass-band (up to frequency :math:`f_p`) and a value of 0 in the stop-band # (down to frequency :math:`f_s`) such that :math:`f_p=f_s=40` Hz here # (shown to a lower limit of -60 dB for simplicity): nyq = sfreq / 2. # the Nyquist frequency is half our sample rate freq = [0, f_p, f_p, nyq] gain = [1, 1, 0, 0] third_height = np.array(plt.rcParams['figure.figsize']) * [1, 1. / 3.] ax = plt.subplots(1, figsize=third_height)[1] plot_ideal_filter(freq, gain, ax, title='Ideal %s Hz lowpass' % f_p, flim=flim) ############################################################################### # This filter hypothetically achieves zero ripple in the frequency domain, # perfect attenuation, and perfect steepness. However, due to the discontunity # in the frequency response, the filter would require infinite ringing in the # time domain (i.e., infinite order) to be realized. Another way to think of # this is that a rectangular window in frequency is actually sinc_ function # in time, which requires an infinite number of samples, and thus infinite # time, to represent. So although this filter has ideal frequency suppression, # it has poor time-domain characteristics. # # Let's try to naïvely make a brick-wall filter of length 0.1 sec, and look # at the filter itself in the time domain and the frequency domain: n = int(round(0.1 * sfreq)) + 1