Dual-comb spectroscopy allows for high-resolution spectra to be measured over broad bandwidths, but an essential requirement for coherent integration is the availability of a phase reference. Usually, this means that the combs’ phase and timing errors must be measured and either minimized by stabilization or removed by correction, limiting the technique’s applicability. We demonstrate that it is possible to extract the phase and timing signals of a multiheterodyne spectrum completely computationally, without any extra measurements or optical elements. These techniques are viable even when the relative linewidth exceeds the repetition rate difference and can tremendously simplify any dual-comb system. By reconceptualizing frequency combs in terms of the temporal structure of their phase noise, not their frequency stability, we can greatly expand the scope of multiheterodyne techniques.

https://advances.sciencemag.org/content/advances/2/11/e1601227.full.pdf

Computational multiheterodyne spectroscopy

Driessen (1994) and Christiansen (1994) independently showed that for a specific dynamic model, the proportional-integral phase-locked loop (PLL) has the same structure as the Kalman filter. In this paper, closed-form expressions of the corresponding Kalman gain values are derived both in acquisition and tracking modes of the PLL.

On phase-locked loops and Kalman filters

Characteristics of signal tracking Kalman filter (KF) approaches for a Global Navigation Satellite System (GNSS) receiver are presented. In particular the direct-state KF is designed by using a state vector for signal parameters not for signal parameter errors, so as to directly utilize the discriminator output as a measurement residual. This enables the reuse of the existing signal tracking loop structure by simply adjusting the filter coefficients based on a time-varying optimal Kalman gain. The characteristics of the direct-state KF together with the error-state KF are analyzed by comparing their noise bandwidth and open- and closed-loop transfer functions with those of a traditional method. These KF approaches are mathematically equivalent to the traditional method except for the use of time-varying gains. This fact provides an insight how to tune the designed KFs based on their expected performances.

Characteristics of Kalman Filters for GNSS Signal Tracking Loop

This paper presents a tuning method for an adaptive Kalman filter approach to efficiently track faded global navigation satellite system signals based on empirical knowledge of the signal-to-noise ratio with a special emphasis on equivalent noise bandwidths.

A Tuning Method Based on Signal-to-Noise Power Ratio for Adaptive PLL and its Relationship with Equivalent Noise Bandwidth

The loop latency compensated PLL filter comprising two additional feedback terms, a delayed phase compensation signal and a state compensation signal, that are provided as input of a PLL filter. Accordingly, the PLL filter input comprises two additional compensation input signals: the delayed phase compensation signal and the state compensation signal in addition to a phase estimated error output from a phase detector that is also coupled to the input of the PLL filter. Consequently, PLL filter thus is able to provide a latency compensated phase error control output that is fedback to control a phase mixer to generate a square waveform used to drive an A/D of the PLL in accordance with the principles of this invention. The loop latency compensated PLL of this invention thus minimizes the jitter of the PLL circuit, provides higher format efficiency, and also has reduced sensitivity to large bursty noises.

Loop latency compensated PLL filter

This paper demonstrates the equivalence between the timing loop used in the partial response maximum likelihood (PRML) disk-drive channel and a modified Kalman filter. The Kalman filter is shown to reach a steady-state gain that corresponds to the timing loop when in its steady-state tracking mode. The dependence of the steady-state gain on the noise and physical properties of the loop is investigated. The timing loop is then optimized using the properties of the Kalman filter.

G.S. Christiansen

Papers

Modeling of PRML timing loop as a Kalman filter