Low-pass Chebyshev type-I filter with Scipy - numpy

Question

I am reading a paper, trying to reproduce the results of the paper. In this paper, they use a low-pass Chebyshev type-I filter on the raw data. And they give those parameters.
Sampling frequency = 32Hz, Fcut=0.25Hz, Apass = 0.001dB, Astop = -100dB, Fstop = 2Hz, Order of the filter = 5. I found some materials help me understand these parameters
But when I take a look at the scipy.signal.cheby1. The parameters required by this function are different.
cheby1(N, rp, Wn, btype='low', analog=False, output='ba')
Here N:The order of the filter; btype: type of filter, in my case, it is 'lowpass'; analog=False, because the data is sampled, so it is digital; output: specifies the type of output. But I am not sure about rp, Wn.
In the documentation, it says:
rp : float
The maximum ripple allowed below unity gain in the passband. Specified in decibels, as a positive number.
Wn : array_like
A scalar or length-2 sequence giving the critical frequencies. For Type I filters, this is the point in the transition band at which the gain first drops below -rp. For digital filters, Wn is normalized from 0 to 1, where 1 is the Nyquist frequency, pi radians/sample. (Wn is thus in half-cycles / sample.) For analog filters, Wn is an angular frequency (e.g. rad/s).
According to this question:
How To apply a filter to a signal in python
I know how I can use the filter. But I don't know how to create a filter which has the same parameters as mentioned above. I don't know how to convert these parameters and provide them to the function in Scipy.

Answer 1

Take a look at the wikipedia page on the Type I Chebyshev filter. Note that your plot illustrates the characteristics of a general filter. A lowpass Type I Chebyshev filter, however, has no ripple in the stop band.
You have three available parameters for the design of a Type I Chebyshev filter: the filter order, the ripple factor, and the cutoff frequency. These are the first three parameters of scipy.signal.cheby1:
The first argument of cheby1 is the order of the filter.
The second argument, rp, corresponds to δ in the wikipedia page, and is apparently what you called Apass.
The third argument is wn, the cutoff frequency expressed as a fraction of the Nyquist frequency. In your case, you could write something like
fs = 32 # Sample rate (Hz)
fcut = 0.25 # Desired filter cutoff frequency (Hz)
# Cutoff frequency relative to the Nyquist
wn = fcut / (0.5*fs)
Once those three parameters are chosen, all the other characteristics
(e.g. transition band width, Astop, Fstop, etc) are determined. So it appears that the specification that you give, "Sampling frequency = 32Hz, Fcut=0.25Hz, Apass = 0.001dB, Astop = -100dB, Fstop = 2Hz, Order of the filter = 5", are not compatible with a Type I Chebyshev filter. In particular, I get a gain of approximately -78 dB at 2 Hz.
(If you increase the order to 6, then the gain at 2 Hz is approximately -103.)
Here's a complete script, followed by the plot that it generates. The plot shows just the pass band, but you can change the arguments of the xlim and ylim functions to see more.
import numpy as np
from scipy.signal import cheby1, freqz
import matplotlib.pyplot as plt
# Sampling parameters
fs = 32 # Hz
# Desired filter parameters
order = 5
Apass = 0.001 # dB
fcut = 0.25 # Hz
# Normalized frequency argument for cheby1
wn = fcut / (0.5*fs)
b, a = cheby1(order, Apass, wn)
w, h = freqz(b, a, worN=8000)
plt.figure(1)
plt.plot(0.5*fs*w/np.pi, 20*np.log10(np.abs(h)))
plt.axvline(fcut, color='r', alpha=0.2)
plt.plot([0, fcut], [-Apass, -Apass], color='r', alpha=0.2)
plt.xlim(0, 0.3)
plt.xlabel('Frequency (Hz)')
plt.ylim(-5*Apass, Apass)
plt.ylabel('Gain (dB)')
plt.grid()
plt.title("Chebyshev Type I Lowpass Filter")
plt.tight_layout()
plt.show()