Reflection coefficients as a stability coordinate system
========================================================

.. admonition:: Tutorial goal

   Show why bounded scalar reflection/PARCOR coefficients are a practical stability coordinate system.

.. 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
-------

Scalar IIR stability is hard to read from direct denominator coefficients.  In
lattice coordinates, the all-pole stability condition is exposed stage by stage.
This tutorial compares reflection-parameterized denominators with a few direct
denominator examples near the unit-circle boundary.

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

For a scalar all-pole denominator built by the Schur step-up recursion,

.. math::

   |k_i| < 1 \quad \text{for every stage}

keeps the poles inside the unit disk.  Direct denominator coefficients do not
provide an equivalent per-coefficient test.

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

Compare the maximum reflection magnitude with the maximum pole radius.  The direct denominator rows show why pole or reflection diagnostics are still needed outside lattice coordinates.

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

.. code-block:: bash

   python examples/reflection_coefficients_stability_demo.py

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

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