Nehari and AAK intuition from a finite SISO Hankel matrix
=========================================================

.. admonition:: Tutorial goal

   Show the finite-Hankel singular-value picture behind Nehari/AAK model-reduction theory within a finite-section scope.

.. 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 finite-Hankel reducer already available in the package is a practical baseline.  This
tutorial explains the finite-section Nehari/AAK perspective.  It calls ``finite_nehari_approximate_tail`` to
build a finite Hankel matrix from an anticausal tail and study its singular values.

The goal is to make the intuition precise.  A low-rank Hankel structure means that much of
the input-output memory is carried by a few modes.  In the finite matrix case, the next
singular value controls the best unconstrained rank-r spectral-norm error.  AAK/Nehari is
the rational/Hankel-structured analogue of that statement.

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

Given anticausal coefficients ``gamma_1, gamma_2, ...``, form the finite Hankel matrix

.. math::

   \Gamma =
   \begin{bmatrix}
     \gamma_1 & \gamma_2 & \gamma_3 & \cdots \\
     \gamma_2 & \gamma_3 & \gamma_4 & \cdots \\
     \gamma_3 & \gamma_4 & \gamma_5 & \cdots \\
     \vdots & \vdots & \vdots & \ddots
   \end{bmatrix}.

For a finite matrix, the Eckart--Young theorem gives

.. math::

   \min_{\operatorname{rank}(X)\le r} \|\Gamma-X\|_2 = \sigma_{r+1}(\Gamma).

The tutorial also Hankelizes the truncated SVD approximation by anti-diagonal averaging to
show the difference between unconstrained low-rank approximation and a Hankel-structured
approximation.

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

Check that the finite SVD error matches the next singular value, then compare it with the Hankelized approximation error.  This is a finite-section validation bridge and not an exact Nehari/AAK solver.

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

.. code-block:: bash

   python examples/nehari_aak_siso_toy.py

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

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