How sid works

sid follows two complementary paths: a frequency-domain path for non-parametric estimation, and a time-domain path for parametric state-space identification. Both paths share a single mathematical specification and cross-language reference test vectors.

Frequency-domain path

The core spectral estimators use the Blackman-Tukey method: compute biased cross-covariances between input and output, apply a Hann lag window, then transform via FFT. The transfer function is the cross-spectrum / input auto-spectrum ratio; asymptotic variance formulas (Ljung, 1999) provide per-frequency uncertainty. When multiple trajectories are provided, covariances are ensemble-averaged before forming the ratio, reducing variance by a factor of L without sacrificing frequency resolution.

See freq_bt / sidFreqBT for the entry point, and the Specification for the full mathematical derivation.

State-space path

The COSMIC algorithm (Carvalho et al., 2022) identifies discrete-time LTV models x(k+1) = A(k) x(k) + B(k) u(k) by solving a block-tridiagonal regularized least-squares problem in O(N) time. Multiple trajectories — including variable-length sequences — are pooled into the data matrices. When only outputs are observed, Output-COSMIC alternates between state estimation (RTS smoother) and dynamics identification, converging to a joint optimum. Bayesian uncertainty quantification propagates through to frozen transfer functions G(ω, k) for direct comparison with non-parametric frequency estimates.

See ltv_disc / sidLTVdisc for the entry point, and the COSMIC notes for derivations.