.. _iirdes-butter:

Butterworth IIR Filter Design
=============================

:api:`LIQUID_IIRDES_BUTTER`
is a Butterworth filter.
This is an all-pole analog design that has a maximally flat magnitude
response in the pass-band.
The analog prototype interface is :api:`butter_azpkf()` which
computes the :math:`n` complex roots :math:`p_{a0},p_{a1},\ldots,p_{an-1}`
of the :math:`n^{th}`-order Butterworth polynomial,

.. math::
    p_{ak} =
        \omega_c \exp\left\{
            j \frac{\left(2k+n+1\right)\pi}{2n}
        \right\}
    :label: eqn-filter-iirdes-butter

for :math:`k=0,1,\ldots,n-1`.
Note that this results in a set of complex conjugate pairs such that
:math:`(-1)^n s_0 s_1 \cdots s_{n-1} = 1`.

An example Butterworth design can be seen in the code listing here:

.. code-block:: c

    #include <liquid/liquid.h>
    int main()
    {
        return 0;
    }

An example of a digital Butterworth filter response can be found in
the figure, below for an order 3:

.. qplot:: iirdes iirdes/iirdes.c
    :args: -t butter -n 7
    :width: 100%
    :name: fig-iirdes-butter-0
    :caption: Butterworth filter design, :math:`n=7`, :math:`f_c=0.2 F_s`

Notice how the performance changes as the order increases to 15:

.. qplot:: iirdes iirdes/iirdes.c
    :args: -t butter -n 15
    :width: 100%
    :name: fig-iirdes-butter-1
    :caption: Butterworth filter design, :math:`n=15`, :math:`f_c=0.1 F_s`

Notice how the poles change when the cutoff frequency drops to :math:`0.1 F_s`:

.. qplot:: iirdes iirdes/iirdes.c
    :args: -t butter -n 15 -f 0.1
    :width: 100%
    :name: fig-iirdes-butter-2
    :caption: Butterworth filter design, :math:`n=15`, :math:`f_c=0.1 F_s`