Track Accuracy

This package does not contain a tracker. It runs no recursion, holds no state, and processes no detections; the design document lists a real-time DSP tracker as a non-goal and that line stands. What it computes is the steady-state result, which is algebra rather than recursion: under stationary noise and a constant revisit interval, the Kalman filter settles to gains and a covariance available in closed form. A designer needs that number long before any filter exists.

The chain is:

\[ \text{SNR, } B \;\rightarrow\; \sigma_R \qquad \text{SNR, } \theta_{3dB} \;\rightarrow\; \sigma_\theta \;\rightarrow\; \sigma_{cr} = R\,\sigma_\theta \]

\[ \sigma_w,\; T,\; A_{max} \;\rightarrow\; \Gamma \;\rightarrow\; (\alpha, \beta) \;\rightarrow\; \sigma_{pos},\; \sigma_{vel} \]

Every input already exists here, which is why the model belongs in this package rather than in a tracking library: the array sets the beamwidth, the radar equation sets the SNR, and the scheduler sets the revisit interval. Tracking libraries take the measurement covariance as given; none of them derive it from an aperture.

Measurement accuracy

\[ \sigma_R = \frac{\Delta R}{\sqrt{2\,\mathrm{SNR}}}, \qquad \Delta R = \frac{\alpha c}{2B} \]

\[ \sigma_\theta = \frac{\theta_{3dB}}{k_m \sqrt{2\,\mathrm{SNR}}}, \qquad \theta_\phi = \frac{\theta_B}{\cos\phi} \]

with \(k_m \approx 1.6\) the monopulse difference-pattern slope. The angle form is derived for SNR > 13 dB; below that the monopulse ratio is a biased estimate of the angle and the variance is optimistic. monopulse_snr_ok reports whether the case sits above the floor rather than silently extrapolating.

Thermal angle error combines in quadrature with the hardware pointing error from models/antenna/errors.py, so phase-shifter bits and calibration residue propagate all the way through to track accuracy.

The SNR convention

Two conventions appear in the literature and differ by a factor of two:

Source	Form	SNR means
POMR Eq. (18.33)	\(\sigma_R = \Delta R/\sqrt{\mathrm{SNR}}\)	\(2E/N_0\), matched-filter peak
Curry Eq. (8.6)	\(\sigma_R = \Delta R/\sqrt{2\,S/N}\)	\(E/N_0\)

They are algebraically identical. This package's range equation produces the Curry/Barton \(S/N\), so every function uses the \(\sqrt{2\,\mathrm{SNR}}\) form. POMR's angle relation already carries the factor of two, so range, angle, and Doppler end up on one convention.

Cross-range dominates

Angle accuracy is a fixed fraction of a beamwidth, so cross-range error grows linearly with range while range error does not. At X-band, 50 km, 20 dB SNR, 10 MHz bandwidth and a 64x64 array:

Quantity	Value
\(\sigma_R\)	1.04 m
\(\sigma_\theta\)	0.068°
\(\sigma_{cr}\)	59.5 m

Cross-range error is 57x the range error. Both terms carry \(1/\sqrt{\mathrm{SNR}}\), so more power or longer dwell improves each equally and leaves the ratio untouched. Only the aperture changes the ratio, by narrowing \(\theta_{3dB}\). Bandwidth, which buys range resolution cheaply, does nothing for cross-range at all.

Tracking index and steady-state gains

The random tracking index (Kalata 1984; POMR Eq. 19.47) is the single number that sets the steady-state filter:

\[ \Gamma = \frac{\sigma_v T^2}{\sigma_w} \]

the ratio of position uncertainty from target maneuverability to that from the sensor. The optimal gains follow in closed form and satisfy the Kalata relation \(\beta = 2(2-\alpha) - 4\sqrt{1-\alpha}\), with steady-state error

\[ \mathbf{P} = \sigma_w^2 \begin{bmatrix} \alpha & \beta/T \\ \beta/T & \dfrac{\beta(2\alpha-\beta)}{2(1-\alpha)T^2} \end{bmatrix} \]

so \(\sigma_{pos} = \sigma_w\sqrt{\alpha}\).

Rather than asking for a process-noise variance, the model takes a physical maneuver (\(A_{max}\), "the target pulls 4 g") and derives \(\sigma_v\) through POMR Eqs. (19.63)/(19.66).

Two correct covariance formulas

POMR Eq. (19.53) and Mahafza Eq. (11.94) disagree numerically because they answer different questions. Both are reproduced by the same fixed-gain covariance recursion, and both are emitted:

Metric	Formula	Meaning
track_pos_rms_*_m	POMR Eq. (19.53)	total error, maneuver included
track_vrr_*	Mahafza Eq. (11.94)	sensor-noise reduction, no maneuver

A third form circulating in the literature, \((2\alpha^2 + 2\beta + \alpha\beta)/(\alpha(4-2\alpha-\beta))\), is wrong: it exceeds unity, claiming that filtering amplifies noise. A canary test asserts this package does not implement it.

What the index tells a designer

The filter is applied per coordinate, and the two axes usually land in different regimes. For the 50 km case above against a 40 m/s² maneuver at 1 Hz revisit:

Axis	\(\sigma_w\)	\(\Gamma\)	\(\alpha\)	Track \(\sigma_{pos}\)
Range	1.04 m	26.0	0.996	1.04 m
Cross-range	59.5 m	0.60	0.660	48.3 m

On the range axis the index is high: the maneuver is enormous relative to a one-metre measurement, so the filter must trust each measurement almost completely and smoothing buys nothing. The precise range measurement is wasted on a maneuvering target. On the coarse cross-range axis filtering does help, improving 59.5 m to 48.3 m. Overall track quality is set by the aperture, not the waveform — which is a statement about the array, and therefore belongs in this package.

References

• Richards, Scheer & Holm, Principles of Modern Radar: Basic Principles, SciTech, 2010. Ch. 18 (measurements), Ch. 19 (tracking, W. D. Blair).
• Curry, Radar System Performance Modeling, 2nd ed., Artech House, 2005, ch. 8.
• Kalata, "The Tracking Index", IEEE Trans. AES 20(2), 174–182, 1984, doi:10.1109/TAES.1984.310438.
• Mahafza, Radar Systems Analysis and Design Using MATLAB, ch. 11.
• Barton & Ward, Handbook of Radar Measurement, Artech House, 1984.