How Butterworth low-pass filter can be applied on a digital signal (i.e. discrete representation of a signal)?

Question

I would like to know how to apply a low pass filter (Butterworth) to a digital signal. So, I have some values of the signal, let's say $S(t)$. Those values are equally spaced in time. I have read the Wikipedia's article, but I do not understand how to apply the transfer function $H$. I assume that I have to use a Z-transformation and apply it on $S$ but I am not certain to know how to do that. By the way, the part Digital implementation of this article is not really detailed.

Can someone help me to understand those things ? I would really appreciate any answers :)

P.S. I know there are functions in Python (e.g. scipy) that handle that issue, but I would like to know the mathematics behind it.

Thanks.

Answer 1

What you need to do is find a digital filter $H(z)$ that behaves similar to the $H(s)$ of your Butterworth filter. That is, you need to read about IIR implementation.

There are several methods described in signal processing books about this subject. I recommend you to check for example "Essentials of Digital Signal Processing" from Lathi and Green.

Two of the most common methods are the "Impulse invariance" and the "Bilinear transform".

• The impulse invariance requires you to find the impulse response $h(t)$ of your $H(s)$. After doing this, you sample it replacing $t$ by $nT$, to finally Z-transform your $h[n]$ into $H(z)$. Bare in mind that this method only works for band-limited filters (lowpass and bandpass) because of aliasing issues. You should determine an appropriate small enough T to avoid this.

• The bilinear transform, on the other hand, consists on applying the following mapping of your variable $s$ into $z$:

  $s=\frac{2}{T}(\frac{z-1}{z+1})$

  This method has a drawback as well, which is the frequency warping, you can read about it here https://en.wikipedia.org/wiki/Bilinear_transform#Frequency_warping, but is in general easy to counter by pre-warping your original $H(s)$.

Answer 2

In order to filter a discrete-time signal with a linear and time-invariant (LTI) filter, you need to implement a difference equation:

$$y[n]=-a_1y[n-1]-a_2y[n-2]-\ldots -a_Ny[n-N]+\\+b_0x[n]+b_1x[n-1]+\ldots+b_Nx[n-N]\tag{1}$$

where $x[n]$ is the input sequence, $y[n]$ is the output sequence, and $a_i$ and $b_i$ are the filter coefficients, which determine the properties of the filter (e.g., low pass or high pass, Butterworth or Chebyshev characteristic, etc.). The number of past values of $x[n]$ and $y[n]$ in $(1)$ is determined by the filter order $N$. The higher the order, the more degrees of freedom does the filter have, and the better it can approximate a desired response. In practice, very high orders are undesirable because of numerical problems.

The difference equation $(1)$ corresponds to a filter transfer function

$$H(z)=\frac{b_0+b_1z^{-1}+\ldots+b_Nz^{-N}}{1+a_1z^{-1}+\ldots+a_Nz^{-N}}\tag{2}$$

Some information on how to obtain the filter coefficients $a_i$ and $b_i$ is given in Axel Mancino's answer.