5.3. Filters¶

In the course of 2016-2017 we have discussed:

The Laplace Transform as a generalization of the Fourier domain. The Laplace transform takes a signal \(x(t)\) and represents it as function \(X(s)\) in the complex \(s\)-domain where \(X(j\omega)\) is the Fourier transform of \(x(t)\).
The characterization of analog filters is most often done in the \(s\)-domain. In this chapter we will look at several filter prototypes characterized in the \(s\)-domain.
Starting from a prototype for a low-pass filter we can transform the filter to act as a high-pass filter, or as a band-pass or a band-reject (notch) filter. Other types of filters are the shelf-filters.
Then we take an analog filter and transform it to a digital filter. This is done by mapping the \(s\)-domain onto the \(z\)-domain where the imaginary axis in the \(s\)-plane (where the Fourier transform is) onto the unit circle in the \(z\)-domain (where the DT Fourier transform is).
Furthermore we show examples throughout this chapter where we will use the Scipy Signal processing package. We also show the principle of using a digital filter on a long stream of data in which the data is to be processed in chunks (say 1024 samples) at a time.

A lot of the above is available now in a IPython notebook. This chapter will be written in these Sphinx lecture notes in due time.