=================
Block Error Rate
=================

.. inheritance-diagram:: hermespy.modem.evaluators.BlockErrorEvaluator hermespy.modem.evaluators.BlockErrorArtifact hermespy.modem.evaluators.BlockErrorEvaluation
   :parts: 1
   :top-classes: hermespy.core.monte_carlo.Evaluator, hermespy.core.monte_carlo.Evaluation, hermespy.core.monte_carlo.Artifact
    
Considering two linked modems denoted by :math:`(\alpha)` and :math:`(\beta)`,
with modem :math:`(\alpha)` transmitting a bit stream

.. math::

   \mathbf{b}_{\mathrm{Tx}}^{(\alpha)} = \left[ b_{\mathrm{Tx}}^{(\alpha,1)}, b_{\mathrm{Tx}}^{(\alpha,2)}, \ldots, b_{\mathrm{Tx}}^{(\alpha,B)} \right]^{\mathsf{T}} \in \lbrace 0, 1 \rbrace^{B}
   
and modem :math:`(\beta)` receiving a bit stream

.. math::

   \mathbf{b}_{\mathrm{Rx}}^{(\beta)} = \left[ b_{\mathrm{Rx}}^{(\beta,1)}, b_{\mathrm{Rx}}^{(\beta,2)}, \ldots, b_{\mathrm{Rx}}^{(\beta,B)} \right]^{\mathsf{T}} \in \lbrace 0, 1 \rbrace^{B}

which can be partitioned into :math:`L` bit block segments of equal length

.. math::

   \mathbf{b}_{\mathrm{Tx}}^{(\alpha)} &= \left[ b_{\mathrm{B,Tx}}^{(\alpha,1)\mathsf{T}}, b_{\mathrm{B,Tx}}^{(\alpha,2)\mathsf{T}}, \ldots, b_{\mathrm{B,Tx}}^{(\alpha,L)\mathsf{T}} \right] \in \lbrace 0, 1 \rbrace^{B} \\
   \mathbf{b}_{\mathrm{Rx}}^{(\beta)} &= \left[ b_{\mathrm{B,Rx}}^{(\beta,1)\mathsf{T}}, b_{\mathrm{B,Rx}}^{(\beta,2)\mathsf{T}}, \ldots, b_{\mathrm{B,Rx}}^{(\beta,L)\mathsf{T}} \right] \in \lbrace 0, 1 \rbrace^{B}

Hermes defines the block error rate (BLER) as the exepcted number of block errors between the streams, i.e.,

.. math::

   \mathrm{BLER}^{(\alpha,\beta)} = \mathbb{E} \lbrace \| b_{\mathrm{B,Tx}}^{(\alpha,l)} - b_{\mathrm{B,Rx}}^{(\alpha,l)} \|_2^2 > 0 \rbrace \ \text{.}

In practice, the number of bits :math:`B` may differ between transmitter and receiver.
In this case, the shorter bit stream is padded with zeros.

The following minimal examples outlines how to configure this evaluator
within the context of a simulation campaign:

.. literalinclude:: ../scripts/examples/modem_evaluators_bler.py
   :language: python
   :linenos:
   :lines: 10-31

.. autoclass:: hermespy.modem.evaluators.BlockErrorEvaluator

.. autoclass:: hermespy.modem.evaluators.BlockErrorArtifact

.. autoclass:: hermespy.modem.evaluators.BlockErrorEvaluation

.. footbibliography::