Extended Kalman Smoothing of Free Spin Precession Signals for Accurate Magnetic Field Determination

Technical Summary: Extended Kalman Smoothing of Free Spin Precession Signals for Accurate Magnetic Field Determination

Problem Statement
Accurate determination of static magnetic flux density relies on measuring the free spin precession frequency ($f_L$) of polarized nuclei, such as $^3$He, via the relation $f_L = \frac{\gamma}{2\pi}B_0$. While the gyromagnetic ratio $\gamma$ for $^3$He is known with extreme precision (relative uncertainty $<10^{-9}$), the overall accuracy of magnetic field measurements is currently limited by the precision of frequency estimation from Nuclear Magnetic Resonance (NMR) signals. Traditional methods, primarily nonlinear least-squares fitting (specifically the sine-cos-fit or SCF), face significant challenges when applied to Free Precession Decay (FPD) signals. These signals exhibit exponential amplitude decay and time-dependent frequency shifts due to magnetic field drifts. Conventional approaches often require dividing data into fixed-length blocks to assume constant amplitude and frequency, creating a trade-off: short blocks adapt to drifts but suffer from high variance due to noise, while long blocks reduce noise but fail to track rapid frequency changes or amplitude decay, leading to model mismatch and bias.

Methodology
The authors propose a novel application of the Extended Kalman Smoother (EKS) to estimate time-varying frequency and amplitude from FPD signals. Unlike the standard Kalman Filter which operates in real-time, the EKS utilizes all available measurements (past and future) to refine state estimates, making it ideal for post-processing sensor data.

The methodology involves the following key components:

1. State-Space Modeling: The system is modeled in the frequency domain rather than the time domain. The signal is segmented into non-overlapping blocks. For each block, the state vector $x_k$ includes the amplitude $A_k$, the frequency offset $\delta f_k$, and phase $\phi_k$. The evolution of these states is modeled as integrated random walks (for amplitude and frequency) and numerical integration (for phase).
2. Measurement Equation: Instead of fitting the time-domain signal directly, the EKS compares the model prediction to the Discrete Fourier Transform (DFT) coefficients of the signal blocks. This transforms the additive Gaussian noise of the time-domain signal into independent Gaussian noise components in the frequency domain, significantly reducing noise variance.
3. Parameter Optimization: The static parameters of the model (process noise covariance $Q$, measurement noise covariance $R$, and initial values) are optimized automatically using an Expectation-Maximization (EM) algorithm. This allows the method to adapt to signal characteristics without manual tuning of block lengths or noise parameters.
4. Reference Method: The EKS is benchmarked against the standard Sine-Cos-Fit (SCF) method, which uses variable projection to separate linear and nonlinear parameters. In the SCF approach, the optimal block length is determined retrospectively in simulations to minimize Mean Squared Error (MSE), representing a "best-case" scenario for the traditional method.

Key Contributions

• Algorithmic Innovation: The paper introduces a frequency-domain EKS framework specifically adapted for tracking harmonic signals with slowly varying frequency and decaying amplitude, utilizing Fourier coefficients as the measurement model.
• Automated Adaptation: The method eliminates the need for manual selection of block lengths or noise parameters, which are critical and often difficult tuning steps in least-squares fitting. The EM algorithm handles the optimization of process and measurement noise covariances.
• Comprehensive Benchmarking: The study provides a rigorous comparison between EKS and SCF using both extensive simulations (varying Signal-to-Noise Ratio and frequency drift diffusion constants) and real experimental data from a $^3$He magnetometer.

Results

• Simulation Studies: Across a wide range of Signal-to-Noise Ratios (SNR) and frequency drift rates, the EKS systematically reduced estimation errors compared to the SCF. In regimes with moderate to high SNR and evolving frequencies, the EKS improved frequency estimates by a factor of approximately 2 (indicated by a $\rho_{RMSE,f} \approx -1$). The SCF only outperformed EKS in regimes of very low SNR and negligible frequency drift, where it approaches the Cramér–Rao lower bound.
• Amplitude Estimation: The EKS significantly outperformed the SCF in estimating signal amplitude, as the SCF's assumption of constant amplitude within fixed blocks leads to degradation when the signal decays exponentially.
• Experimental Validation: Applied to experimental data from a $^3$He spin precession measurement (duration ~4 hours), the EKS demonstrated superior tracking of frequency fluctuations. While SCF estimates with large block lengths failed to resolve rapid frequency variations (e.g., an upward swerve at ~300s) and exhibited unreliable uncertainty intervals, the EKS maintained consistent tracking. Notably, the EKS achieved uncertainty intervals comparable to the SCF using block lengths nearly two orders of magnitude smaller, demonstrating higher temporal resolution without sacrificing precision.

Significance
The paper claims that the EKS-based analysis substantially improves the accuracy of frequency estimation in NMR-based magnetometry, which is the foundational step for absolute magnetic field determination. By effectively filtering noise and adapting to signal decay and frequency drifts automatically, the method enhances the stability and accuracy of high-precision measurements. The authors emphasize that this versatile approach is particularly valuable for tracking magnetic field fluctuations over time, a critical requirement for modern magnetometry applications. The work provides a robust, easy-to-use implementation that does not rely on system-dependent parameters, making it a promising candidate for replacing current state-of-the-art least-squares methods in ultra-high-precision NMR measurements.