Finite Hankel SISO model reduction
==================================

.. admonition:: Tutorial goal

   Build a finite Hankel matrix from a stable IIR impulse response and construct lower-order rational models with the C++ backend.

.. 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
-------

This tutorial is the first executable bridge between the model-reduction theory page and
the package API.  It is deliberately finite-dimensional: a truncated Hankel matrix is built
from the impulse response, its singular values are inspected, and a lower-order Ho--Kalman
realization is recovered from the leading Hankel factors.

Exact AAK/Nehari theory is an infinite-dimensional optimal rational-approximation theory.
The implementation here is a practical finite-section reducer inspired by the same
Hankel-operator viewpoint, but it does not claim exact Nehari or AAK optimality.

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

Given an impulse response ``h[n]``, the finite Hankel matrix is

.. math::

   H_{ij}=h[i+j+1].

Its singular values measure input-output memory.  A reduced order ``r`` keeps the leading
singular directions and forms a balanced finite realization

.. math::

   H_0 \approx U_r \Sigma_r V_r^T,
   \qquad
   A_r = \Sigma_r^{-1/2}U_r^T H_1 V_r\Sigma_r^{-1/2}.

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

Look for Hankel singular-value decay, retained Hankel energy, impulse-response error, and whether the reduced denominator remains stable.

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

.. code-block:: bash

   python examples/finite_hankel_model_reduction.py

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

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