.. _handbook-a121-fom: Figure of Merits ================ This page describes and defines Figure of Merits (FoM). .. _handbook-a121-fom-rlg: Radar loop gain (RLG) --------------------- Signal-to-noise ratio (SNR) ^^^^^^^^^^^^^^^^^^^^^^^^^^^ Let :math:`x(f, s, d)` be a (complex) point at a 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`. Further, let :math:`x_S` be data collected with a known reflector in the range. This is used for the signal part of the SNR. Correspondingly, let :math:`x_N` be the data collected with the transmitter (TX) disabled. This is used for the noise part of the SNR. The frame/sweep distinction is not necessary for the SNR definition. The frames can simply be concatenated: :math:`x(f, s, d) \rightarrow y(s', d)` where :math:`N_{s'} = N_f \cdot N_s`. Let the signal power :math:`S = \max_{d}(\text{mean}_{s'}(|y_S|^2))`. Let the noise power :math:`N = \text{mean}_{s', d}(|y_N|^2)`. This is assuming :math:`E(y_N) = 0` and :math:`V(y_N)` is constant over :math:`(s', d)`, where :math:`E` is the expectation value and :math:`V` is the variance. Finally, let :math:`\text{SNR} = S/N`. The SNR in decibel :math:`\text{SNR}_\text{dB} = 10 \cdot \log_{10}(S/N)`. .. note:: The expectation value of the SNR is **not** affected by the total number of sweeps :math:`N_{s'}` nor the number of distances :math:`N_d` measured. Radar loop gain (RLG) ^^^^^^^^^^^^^^^^^^^^^ The radar loop gain is given by: :math:`\text{RLG}_\text{dB} = \text{SNR}_\text{dB} - \text{RCS}_\text{dB} + 40 \log_{10}(d)` where SNR is the signal-to-noise ratio, RLG is the radar loop gain, RCS is the radar cross-section, and :math:`d` is the distance to the target with the given RCS. .. note:: Of the configuration parameters, RLG only depends on :ref:`profile ` and HWAAS. Excluding the configuration, RLG may vary on other factors such as per unit variations, temperature, and the hardware integration. .. _handbook-a121-fom-base-rlg: Base RLG ^^^^^^^^ *Base* RLG is the RLG (equivalent) for HWAAS = 1. SNR and therefore also RLG scales linearly with HWAAS, meaning a 3dB increase for every doubling of HWAAS. Radar loop gain per time (RLG/t) ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Measurement time is not accounted for in the RLG metrics. To account for it, we define the *radar loop gain per time (RLG/t)*: .. math:: :label: \text{RLG/t}_\text{dB} = (\text{RLG}_\text{dB}\ |\ \text{HWAAS} = 1) - 10 \log_{10}(\tau_\text{sample}) where :math:`(\text{RLG}_\text{dB}\ |\ \text{HWAAS} = 1)` is the :ref:`base RLG`, and :math:`\tau_\text{sample}` is the :ref:`sample duration`. .. note:: RLG/t also depends on :class:`PRF `. Higher PRF:s have higher RLG/t, which are generally more efficient. .. _handbook-a121-fom-radial-resolution: Radial resolution ----------------- Radial resolution is described by the `full width at half maximum (FWHM) `_ 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:`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.