Costas Loop — Theory Validation¶

A theoretical-correctness check on track.Costas's decision-directed BPSK phase discriminator, e = sign(Re P)·Im(P)/|P|.

Left — Phase-detector S-curve. Driving the loop open-loop (bandwidth → 0) with a noiseless prompt exp(jφ) at a swept static phase error and reading the discriminator traces the analytical characteristic e(φ) = sign(cos φ)·sin φ to ~5e-8: zero with unit slope (Kd = 1) at the φ = 0 lock, the 180° BPSK data ambiguity at ±180°, and the unstable nulls at ±90° where the hard decision flips.

Right — Phase-error variance vs SNR. At φ = 0 the discriminator variance follows the BPSK squaring-loss law σ_e² = 1/(2ρ)·(1 + 1/(2ρ)) in the high-SNR regime (ratio ≈ 0.98 for SNR ≥ 10 dB). Because doppler's discriminator is normalised by |P| it is bounded to [-1,1] and so falls below the (divergent) law at low SNR — shown for honesty rather than hidden.

Source: src/doppler/examples/costas_theory_demo.py; tests in src/doppler/track/tests/test_theory_costas.py.