Capon/MVDR spectral estimation
==============================

.. admonition:: Tutorial goal

   Use an inverse-covariance Capon spectrum to resolve nearby tones.

.. 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
-------

Capon/MVDR spectra are useful as a high-resolution diagnostic.  They are not a replacement
for every periodogram or AR model, but they are visually helpful when nearby narrowband
components are hard to separate.

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

For steering vector ``a(ω)`` and loaded covariance matrix ``R``, the Capon spectrum is

.. math::

   \hat S_{\mathrm{Capon}}(\omega)=\frac{1}{a(\omega)^H R^{-1} a(\omega)}.

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

The main figure compares periodogram, AR, and Capon curves.  The covariance-eigenvalue plot helps diagnose whether the covariance estimate is well conditioned.

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

.. code-block:: bash

   python examples/capon_spectrum_demo.py

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

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