*********************
A121 Figures of Merit
*********************

The :doc:`sparse_iq` applies to A121 sensors.
The definitions below apply **only** to the A121 sensor.

.. _rdac-a121-fom-radial-resolution:

Radial resolution
=================

Radial resolution is described by the `full width at half maximum (FWHM) <https://en.wikipedia.org/wiki/Full_width_at_half_maximum>`_ envelope power.

Let
:math:`x(f, s, d)`
be a (complex) point
at a radial distance :math:`d` where the number of distances is :math:`N_d`,
in a sweep :math:`s` where the number of sweeps per frame is :math:`N_s`,
in a frame :math:`f` where the number of frames collected is :math:`N_f`.

Then, let the (average) envelope power

.. math::
    :label:

    y(d) = |\text{mean}_{f,s}(x)|^2

The FWHM of :math:`y` is what describes the radial resolution.


Distance
========

Preliminaries
-------------

Let :math:`d_{est}(f, d)` be the estimated distance to a target located at distance :math:`d`,
formed by processing frame :math:`f` in accordance with the steps outline in the
:doc:`/detectors/a121/distance_detector` documentation.
:math:`f` is a single frame in a set of frames of size :math:`N_f`.

Next, let :math:`e(f, d)=d_{est}(f, d) - d` be the estimation error of a single frame/measurement.

Lastly, form the mean error by averaging over the frames, :math:`\overline{e}(d)=\text{mean}_{f}(e(f,d))`.

:math:`\overline{e}(d)` describes the average error for a single sensor.
The metrics calculated in the following sections are based on data from a set of sensors.
To indicate what sensor the mean error is associated with, the argument :math:`s` is added,
:math:`\overline{e}(d,s)`.

Accuracy
--------

The distance estimation accuracy is characterized through the following two sets of metrics:

- Mean error(:math:`\mu`) and standard deviation(:math:`\sigma`).
- Mean absolute error(:math:`\text{MAE}`).

Mean and standard deviation
^^^^^^^^^^^^^^^^^^^^^^^^^^^

The mean error for a set of sensors is given by :math:`\mu=\text{mean}_{d,s}(\overline{e}(d,s))`.

The standard deviation for a set of sensors is given by :math:`\sigma=\text{std}_{d,s}(\overline{e}(d,s))`.

Mean absolute error
^^^^^^^^^^^^^^^^^^^

The mean absolute error for a set of sensor is given by :math:`\text{MAE}=mean_{d,s}(|\overline{e}(d,s)|)`.

Linearity
---------

Linearity refers to the variation in the distance estimate error as a function of the distance to the target.

The distance linearity is characterized through the mean of the standard deviation of the estimation error
over a number of distances, :math:`\sigma=\text{mean}_{s}(\text{std}_{d}(\overline{e}(d,s)))`.

The distance linearity is evaluated over two sets of distances:

- Micro: A number of distances within a few wavelengths.
- Macro: A number of distances over many wavelengths.

Temperature sensing
===================

The accuracy of the built-in temperature sensor is described by the *relative deviation*:

.. math::
    :label:

    k = \left| \frac{\hat{x} - x}{x} \right|

where :math:`x` is the actual temperature change and :math:`\hat{x}` is the measured temperature change.

The evaluated temperature span is typically the range from -40°C to 105°C.

.. note::

    The built-in temperature sensor is not designed for *absolute* measurements and should therefore not be used for that.
    For this reason, the absolute accuracy is not described as a FoM.