DLL Code Loop — Theory Validation¶

A theoretical-correctness check on track.Dll's prompt-normalized early-minus-late power code discriminator 0.5·(|E|²-|L|²)/|P|² (clamped to ±1).

Left — Code-detector S-curve. The open-loop discriminator (swept static code-phase error, bn → 0) follows the reference built from the triangular code autocorrelation R(τ)=max(0,1-|τ|), clip(0.5·(R(τ+s)²-R(τ-s)²)/R(τ)², -1, 1): zero with a restoring (negative) slope at the lock, and saturating at ±1 by the half-chip point. The fractional-boundary integrate-and-dump overlap-weights the lone sample straddling each chip transition, so the curve is smooth and antisymmetric to round-off at any sps — no integer-sample code-phase staircase, giving the loop true sub-chip resolution. (The only visible gap between measured and reference is at the clamp knee, where the continuous triangular model departs most from the discrete, fractional-boundary-weighted correlation.)

Right — Code-error variance vs SNR. At the lock the early-late discriminator variance follows a clean 1/SNR law (the per-epoch code-error noise) — the measurements lie on the line across four decades of SNR.

This completes the three tracking loops' theory validation: the Costas carrier loop, the Gardner timing loop, and the DLL code loop.

Source: src/doppler/examples/dll_theory_demo.py; tests in src/doppler/track/tests/test_theory_dll.py.