Exact computation of lock-in amplifier outputs for arbitrary frequency modulations using Gauss-Chebyshev quadrature.

Authors

Dennis Lönard, Stefan Johansson, Alena Erlenbach, Jonas Gutsche, Artur Widera

Published in

The Review of scientific instruments. Volume 97. Issue 9. Sep 01, 2026.

Abstract

The technique of lock-in amplification is widely used across many fields to separate a signal from a noisy background. The mathematical descriptions of frequency-modulated signals are, however, often approximated in a regime of small modulation deviation, potentially leading to inaccurate calculations of optimal modulation parameters. Here, we present an approximation-free numerical approach based on Gauss-Chebyshev quadrature that is able to calculate the lock-in amplifier output for any modulation type, modulation function, input signal, and modulation deviation. Our numerical technique is exact, straightforward, fast to compute, and general for any lock-in-amplified experiment. We apply our findings as an example to nitrogen-vacancy center-based magnetometry and use them to determine the optimal modulation parameters for this use case. Our analysis is accompanied by an open-source Python package containing a reference implementation of our numerical approach.

PMID:
42714283
Bibliographic data and abstract were imported from PubMed on 09 Sep 2026.

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