# Figure of Merits#

This page describes and defines Figure of Merits (FoM).

## Radar loop gain (RLG)#

### Signal-to-noise ratio (SNR)#

Let $$x(f, s, d)$$ be a (complex) point at a distance $$d$$ where the number of distances is $$N_d$$, in a sweep $$s$$ where the number of sweeps per frame is $$N_s$$, in a frame $$f$$ where the number of frames collected is $$N_f$$. Further, let $$x_S$$ be data collected with a known reflector in the range. This is used for the signal part of the SNR. Correspondingly, let $$x_N$$ 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: $$x(f, s, d) \rightarrow y(s', d)$$ where $$N_{s'} = N_f \cdot N_s$$.

Let the signal power $$S = \max_{d}(\text{mean}_{s'}(|y_S|^2))$$.

Let the noise power $$N = \text{mean}_{s', d}(|y_N|^2)$$. This is assuming $$E(y_N) = 0$$ and $$V(y_N)$$ is constant over $$(s', d)$$, where $$E$$ is the expectation value and $$V$$ is the variance.

Finally, let $$\text{SNR} = S/N$$. The SNR in decibel $$\text{SNR}_\text{dB} = 10 \cdot \log_{10}(S/N)$$.

Note

The expectation value of the SNR is not affected by the total number of sweeps $$N_{s'}$$ nor the number of distances $$N_d$$ measured.

### Radar loop gain (RLG)#

The radar loop gain is given by: $$\text{RLG}_\text{dB} = \text{SNR}_\text{dB} - \text{RCS}_\text{dB} + 40 \log_{10}(d)$$ where SNR is the signal-to-noise ratio, RLG is the radar loop gain, RCS is the radar cross-section, and $$d$$ is the distance to the target with the given RCS.

Note

Of the configuration parameters, RLG only depends on profile and HWAAS. Excluding the configuration, RLG may vary on other factors such as per unit variations, temperature, and the hardware integration.

### 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):

(10)#$\text{RLG/t}_\text{dB} = (\text{RLG}_\text{dB}\ |\ \text{HWAAS} = 1) - 10 \log_{10}(\tau_\text{sample})$

where $$(\text{RLG}_\text{dB}\ |\ \text{HWAAS} = 1)$$ is the base RLG, and $$\tau_\text{sample}$$ is the sample duration.

Note

RLG/t also depends on PRF. Higher PRF:s have higher RLG/t, which are generally more efficient.

## Radial resolution#

Radial resolution is described by the full width at half maximum (FWHM) envelope power.

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

Then, let the (average) envelope power

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

The FWHM of $$y$$ is what describes the radial resolution.

## Distance#

### Preliminaries#

Let $$d_{est}(f, d)$$ be the estimated distance to a target located at distance $$d$$, formed by processing frame $$f$$ in accordance with the steps outline in the distance detector documentation. $$f$$ is a single frame in a set of frames of size $$N_f$$.

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

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

$$\overline{e}(d)$$ 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 $$s$$ is added, $$\overline{e}(d,s)$$.

### Accuracy#

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

• Mean error($$\mu$$) and standard deviation($$\sigma$$).

• Mean absolute error($$\text{MAE}$$).

#### Mean and standard deviation#

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

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

#### Mean absolute error#

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

### 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, $$\sigma=\text{mean}_{s}(\text{std}_{d}(\overline{e}(d,s)))$$.

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:

(12)#$k = \left| \frac{\hat{x} - x}{x} \right|$

where $$x$$ is the actual temperature change and $$\hat{x}$$ 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.