Butterworth lowpass filter

Question

In order to understand the behaviour of the butterworth (lowpass) filter I generated a "spectrum" consisting of 1's. I then went to time domain (ifft), applied the filter and transformed back (fft). I chose the window (maximum) frequency to be one. So i expected that the windowed spectrum should be equal to the actual spectrum of ones. Instead I got almost the spectrum, but at some point I got a huge line (I expect it to be spectral leakage?!). Also, the values are slightly moved up... Unfortunately I haven't understood how to solve this problem.

Here a sample source code from which I started:

from scipy import signal
import numpy as np
import matplotlib.pyplot as plt

wavelength = range(10000)
spectrum = [1]*len(wavelength)
filter_order = 3 #2
Wn = 1
inter = np.fft.ifft(spectrum) 
b,a = signal.butter(filter_order, Wn, btype='low', analog=False)
inter_w = signal.filtfilt(b, a, inter, method='gust') # filtered
spec_w = np.fft.fft(inter_w) # filtered spectrum

plt.plot(wavelength, spectrum, label='spec')
plt.plot(wavelength, spec_w, label='spec_w')
plt.legend()
plt.show()

And here a small plot. As stated I don't understand, why "spec_w" is (i) bigger than one and (ii) there is a peak that shouldn't be there.

Filter_order 3: Filter_order 2:

Answer 1

Everything you see are numerical artefacts. A "low pass" filter with Nyquist as cut-off frequency is simply a connection. However, you're asking for a recursive filter of order $2$ or $3$, so what you get is a filter with identical numerator and denominator polynomials (i.e., pole/zero cancellation). Check your filter coefficients, a and b should be identical.

So what you have is an unnecessarily complicated implementation of the identity operation, which results in numerical errors. The numerical errors are caused by the "filtering" operation (made worse by the double filtering implemented in filtfilt), and by the combination of FFT and IFFT.

Also note that you should plot the absolute value of spec_w, because due to numerical inaccuracies spec_w might have a small imaginary part and I don't know what your plot function does with a complex-valued vector.