Filters
=======

In the signal processing jargon a filter is system that takes an input
signal and produces an output signal. In this chapter we restrict
ourselves to LTI filters [#filter]_.

In this section we first look at filters for discrete time signals:
the **digital filters**. This in contrast to **analog filters** that
work on continuous time signals.  Most filters nowadays are
implemented digitally and work on discrete time signals with a notable
exception of filters for audio signals and very high frequency radio
signals.


.. todo:: Rewrite of Filters chapter. Start with the basic types of
          filters.

	  Also include the block diagram description (bi-quad)


.. toctree::
   :hidden:

   digital_filters
   analog_filters
   exercises


..
  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 :download:`IPython
  notebook</PythonNotebooks/Filter Design.ipynb>`.
  This chapter will be written in these Sphinx lecture notes in due
  time.


.. toctree::

   filters_zdomain


   .. idealfilters
      transformingfilters
      analogfilters
      dicretizingfilters
      discreteIIRfilters


.. rubric:: Footnotes

.. [#filter] The notion of a filter is not uniquely defined in
             mathematics. Some applied branches of mathematics define
             a filter as an idempotent transformation on
             data. Idempotency is the mathematical notion that
             expresses that a filter after being applied once does not
             change the input anymore upon subsequent use. Think of a
             coffee filter for instance.