Finite Nehari tail to rational model
====================================

.. admonition:: Tutorial goal

   Connect the finite Nehari tail approximation to a low-order recursive/rational SISO model.

.. note::

   New to the terminology? See the :doc:`lattice DSP concept map <../../algorithms/concept_map>` and the :doc:`causality/data-use guide <../../theory/causality_and_data_use>` for how online, offline, block, and MIMO examples should be read.

Context
-------

The previous Nehari/AAK toy tutorial stops at the finite Hankel matrix.  This tutorial takes
one more conservative step: after calling ``finite_nehari_approximate_tail``, it fits a
short linear recurrence to the Hankelized tail and realizes that recurrence as a stable
rational/IIR impulse response.

This is the practical bridge from Hankel singular values to recursive filters.  It still is
not a full infinite-dimensional AAK or Nehari solver, but it shows how a low-rank Hankel
tail can become a small rational model with poles inside the unit circle.

Key idea and equations
----------------------

A finite-rank Hankel tail is generated by a sum of exponentials,

.. math::

   \gamma_n \approx \sum_{i=1}^r c_i p_i^n, \qquad |p_i|<1.

Equivalently, the tail satisfies a linear recurrence

.. math::

   \gamma_n + a_1\gamma_{n-1}+\cdots+a_r\gamma_{n-r}\approx 0.

The fitted recurrence gives a scalar denominator whose roots are the poles ``p_i``.

How to read the result
----------------------

Look for singular-value decay, agreement between the Hankelized tail and the rational realization, and fitted poles inside the unit circle.

Run command
-----------

.. code-block:: bash

   python examples/finite_nehari_rational_bridge.py

Source code
-----------

.. literalinclude:: ../../../examples/finite_nehari_rational_bridge.py
   :language: python
   :linenos: