DLL Code Loop — Theory Validation¶

A theoretical-correctness check on track.Dll's non-coherent early-minus-late code discriminator (|E|-|L|)/(|E|+|L|).

Left — Code-detector S-curve. The open-loop discriminator (swept static code-phase error, bn → 0) follows the triangular-autocorrelation E-L reference (R(τ+s)-R(τ-s))/(R(τ+s)+R(τ-s)), R(τ)=max(0,1-|τ|): zero with a restoring (negative) slope at the lock, linear within the early-late span, and saturating beyond ±½ chip. 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.

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.