kshana 0.8.0

Open hybrid quantum/classical PNT performance simulator
Documentation 
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// SPDX-License-Identifier: Apache-2.0
//! Two-state Kalman clock estimator.
//!
//! State `x = [phase error (s), fractional-frequency error (1/s)]`. The clock is
//! the standard two-state model (Brown & Hwang; Zucca & Tavella, Metrologia 2005):
//!
//! ```text
//!   d(phase)/dt = freq + white-FM driving   (PSD q_wf)
//!   d(freq)/dt  = random-walk-FM driving     (PSD q_rw)
//! ```
//!
//! Over a step `dt` the transition is `F = [[1, dt], [0, 1]]` and the exact
//! (van Loan) process-noise covariance is
//!
//! ```text
//!   Q = q_wf * [[dt, 0], [0, 0]] + q_rw * [[dt^3/3, dt^2/2], [dt^2/2, dt]].
//! ```
//!
//! These `Q` terms are exactly the per-step noise the simulator injects (phase
//! white-FM increment variance `q_wf*dt`; frequency random-walk increment
//! variance `q_rw*dt`), so the filter is consistent with the truth model.
//!
//! Coasting from a known state (`P = 0`, no measurements) the phase-error variance
//! grows to `P[0,0](T) = q_wf*T + q_rw*T^3/3` — exactly the analytic holdover error
//! growth (the `q_rw*T^3/3` term is the random-walk-FM relation of NIST SP 1065).
//! The filter therefore reproduces the analytic limit while also yielding an
//! online uncertainty (1-sigma) bound usable for integrity.
//!
//! The estimator is fully deterministic: no random sampling, so a given sequence
//! of `predict`/`update` calls is bit-for-bit reproducible.

/// Two-state (phase, frequency) Kalman clock estimator.
#[derive(Clone, Debug)]
pub struct KalmanClock {
    x: [f64; 2],      // [phase error (s), frequency error (1/s)]
    p: [[f64; 2]; 2], // state covariance
    q_wf: f64,        // white-FM PSD (s^2/s)
    q_rw: f64,        // random-walk-FM PSD ((1/s)^2/s)
    r: f64,           // phase-measurement noise variance (s^2)
}

impl KalmanClock {
    /// New filter for white-FM PSD `q_wf`, random-walk-FM PSD `q_rw`, and phase
    /// measurement-noise variance `r` (s^2). Starts from a perfectly known state
    /// (zero covariance); seed an initial uncertainty with [`with_initial_cov`].
    ///
    /// [`with_initial_cov`]: Self::with_initial_cov
    pub fn new(q_wf: f64, q_rw: f64, r: f64) -> Self {
        Self {
            x: [0.0, 0.0],
            p: [[0.0, 0.0], [0.0, 0.0]],
            q_wf,
            q_rw,
            r,
        }
    }

    /// Builder: set the initial phase- and frequency-error variances (diagonal P).
    pub fn with_initial_cov(mut self, phase_var: f64, freq_var: f64) -> Self {
        self.p = [[phase_var, 0.0], [0.0, freq_var]];
        self
    }

    /// Time update over `dt`: propagate state and covariance, adding process noise.
    pub fn predict(&mut self, dt: f64) {
        if dt <= 0.0 {
            return;
        }
        // x = F x, with F = [[1, dt], [0, 1]].
        self.x[0] += dt * self.x[1];

        // P = F P F^T.
        let p = self.p;
        // F P:
        let fp = [
            [p[0][0] + dt * p[1][0], p[0][1] + dt * p[1][1]],
            [p[1][0], p[1][1]],
        ];
        // (F P) F^T, with F^T = [[1, 0], [dt, 1]]:
        let mut np = [
            [fp[0][0] + dt * fp[0][1], fp[0][1]],
            [fp[1][0] + dt * fp[1][1], fp[1][1]],
        ];

        // + Q (exact van Loan discretisation).
        let (dt2, dt3) = (dt * dt, dt * dt * dt);
        np[0][0] += self.q_wf * dt + self.q_rw * dt3 / 3.0;
        np[0][1] += self.q_rw * dt2 / 2.0;
        np[1][0] += self.q_rw * dt2 / 2.0;
        np[1][1] += self.q_rw * dt;
        self.p = np;
    }

    /// Measurement update from a phase observation `z` (s). Scalar update with
    /// observation matrix `H = [1, 0]` and the filter's measurement-noise
    /// variance `r`.
    pub fn update(&mut self, z: f64) {
        self.update_with_r(z, self.r);
    }

    /// Measurement update from a phase observation `z` (s) using an explicit
    /// measurement-noise variance `r` (s^2) for this update only — e.g. a noisier
    /// re-anchor (optical time-transfer) versus GNSS disciplining.
    pub fn update_with_r(&mut self, z: f64, r: f64) {
        let s = self.p[0][0] + r;
        if s <= 0.0 {
            return;
        }
        let k = [self.p[0][0] / s, self.p[1][0] / s]; // Kalman gain
        let innov = z - self.x[0];
        self.x[0] += k[0] * innov;
        self.x[1] += k[1] * innov;

        // P = (I - K H) P, H = [1, 0].
        let p = self.p;
        self.p = [
            [(1.0 - k[0]) * p[0][0], (1.0 - k[0]) * p[0][1]],
            [p[1][0] - k[1] * p[0][0], p[1][1] - k[1] * p[0][1]],
        ];
    }

    /// Estimated phase error (s).
    pub fn phase_est(&self) -> f64 {
        self.x[0]
    }
    /// Estimated fractional-frequency error (1/s).
    pub fn freq_est(&self) -> f64 {
        self.x[1]
    }
    /// Phase-error variance (s^2) — the filter's online uncertainty.
    pub fn phase_var(&self) -> f64 {
        self.p[0][0]
    }
    /// Phase-error 1-sigma uncertainty (s).
    pub fn phase_sigma(&self) -> f64 {
        self.p[0][0].max(0.0).sqrt()
    }
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn coasting_covariance_matches_analytic_holdover() {
        // From a known state (P=0), coasting N steps with no measurement must grow
        // the phase-error variance to exactly q_wf*T + q_rw*T^3/3, the frequency
        // variance to q_rw*T, and the cross term to q_rw*T^2/2.
        let (q_wf, q_rw, r) = (1e-24, 1e-30, 1e-20);
        let dt = 1.0;
        let n = 100usize;
        let t = n as f64 * dt;
        let mut kf = KalmanClock::new(q_wf, q_rw, r);
        for _ in 0..n {
            kf.predict(dt);
        }
        let expected_phase_var = q_wf * t + q_rw * t.powi(3) / 3.0;
        let rel = (kf.phase_var() - expected_phase_var).abs() / expected_phase_var;
        assert!(
            rel < 1e-9,
            "phase_var={} expected={expected_phase_var}",
            kf.phase_var()
        );
    }

    #[test]
    fn pure_random_walk_fm_coast_is_q_rw_t_cubed_over_three() {
        // With white FM off, the coast variance is exactly q_rw*T^3/3 — the
        // random-walk-FM holdover relation (sigma_x^2(T) = q_rw*T^3/3).
        let q_rw = 2e-31;
        let dt = 0.5;
        let n = 200usize;
        let t = n as f64 * dt;
        let mut kf = KalmanClock::new(0.0, q_rw, 1e-20);
        for _ in 0..n {
            kf.predict(dt);
        }
        let expected = q_rw * t.powi(3) / 3.0;
        let rel = (kf.phase_var() - expected).abs() / expected;
        assert!(
            rel < 1e-9,
            "phase_var={} expected={expected}",
            kf.phase_var()
        );
    }

    #[test]
    fn measurement_pulls_estimate_and_shrinks_covariance() {
        // After a coast, a precise measurement (small r) corrects the estimate
        // towards the observation and reduces the phase-error variance.
        let mut kf = KalmanClock::new(1e-24, 1e-30, 1e-26);
        for _ in 0..50 {
            kf.predict(1.0);
        }
        let var_before = kf.phase_var();
        kf.update(3e-12);
        assert!(kf.phase_var() < var_before, "covariance did not shrink");
        // With r << P, the estimate should sit close to the measurement.
        assert!(
            (kf.phase_est() - 3e-12).abs() < 3e-13,
            "phase_est={}",
            kf.phase_est()
        );
    }

    #[test]
    fn perfect_repeated_measurements_drive_variance_down() {
        // Repeated zero-noise-limit measurements at a stationary truth converge the
        // phase variance towards the measurement floor.
        let mut kf = KalmanClock::new(1e-26, 1e-32, 1e-24).with_initial_cov(1e-18, 1e-24);
        for _ in 0..200 {
            kf.predict(1.0);
            kf.update(0.0);
        }
        assert!(kf.phase_var() < 1e-22, "phase_var={}", kf.phase_var());
        assert!(kf.phase_est().abs() < 1e-9);
    }

    #[test]
    fn predict_update_sequence_is_deterministic() {
        let run = || {
            let mut kf = KalmanClock::new(1e-24, 1e-30, 1e-22);
            for i in 0..100 {
                kf.predict(1.0);
                if i % 5 == 0 {
                    kf.update(1e-13 * i as f64);
                }
            }
            (kf.phase_est(), kf.phase_var())
        };
        assert_eq!(run(), run());
    }
}