RLS-style lattice-ladder identification

Tutorial goal

Compare RLS-style adaptation with NLMS on a small stable identification problem.

Note

New to the terminology? See the lattice DSP concept map and the causality/data-use guide for how online, offline, block, and MIMO examples should be read.

Context

RLS updates can converge faster than NLMS when the input is correlated, at the cost of more state and computation. This example keeps the denominator stable and focuses on the adaptive numerator/tap behavior.

Key idea and equations

RLS maintains an inverse covariance estimate P and uses a gain vector that depends on the current regressor and forgetting factor.

How to read the result

Compare the final errors and convergence behavior for the RLS and NLMS paths.

Run command

python examples/rls_lattice_identification.py

Source code

"""Fixed-denominator RLS adaptation of stable lattice/IIR numerator taps."""

from __future__ import annotations

import numpy as np

from lattice_dsp import LatticeIIR, LatticeLadderNLMS, LatticeLadderRLS


def main() -> None:
    rng = np.random.default_rng(0)
    n = 5000
    x = rng.normal(size=n)
    reflection = [0.55, -0.25]
    target_taps = [0.25, -0.15, 0.65]
    desired = np.asarray(LatticeIIR(reflection, target_taps).process(x), dtype=float)

    nlms = LatticeLadderNLMS(reflection, [0.0, 0.0, 0.0], mu=0.12)
    rls = LatticeLadderRLS(reflection, [0.0, 0.0, 0.0], forgetting_factor=0.995)

    _, e_nlms = nlms.process_adapt(x, desired)
    _, e_rls = rls.process_adapt(x, desired)

    print("target taps:", np.round(target_taps, 4))
    print("NLMS taps:  ", np.round(nlms.taps, 4))
    print("RLS taps:   ", np.round(rls.taps, 4))
    print("tail MSE NLMS:", float(np.mean(np.asarray(e_nlms)[-1000:] ** 2)))
    print("tail MSE RLS: ", float(np.mean(np.asarray(e_rls)[-1000:] ** 2)))


if __name__ == "__main__":
    main()