**5. Conclusions**

This paper examined the promise of various time-frequency representation methods at conducting real-time high-rate state estimation. In particular, five methods were compared: the short-term Fourier transform (STFT), wavelet transform (WT), Wigner–Ville distribution (WVD), synchrosqueezing transform (SST), and multi-SST (MSST). The performance of the methods was assessed using high-rate experimental data produced on the DROPBEAR (Dynamic Reproduction of Projectiles in Ballistic Environments for Advanced Research) testbed. Such data included acceleration measurements of a beam with a cart located at fixed positions ("fixed cart configuration") sampled at 1 kHz, and with a cart moving between two locations ("moving cart configuration"), sampled at 25 kHz.

Results from the fixed cart configuration show that both the STFT and WT methods could be performed significantly faster than the three other methods, with the WT outperforming other methods in terms of convergence speed. Under the moving cart experiment, both the STFT performed similarly in terms of frequency estimation precision, but with the STFT being computationally faster to implement. The WVD failed at identifying the fundamental frequency, while the SST and MSST had unacceptable computation times, attributable to the longer window lengths that were necessary for implementing the methods. The SST and MSST can achieve good energy concentration and estimation in fixed cart configuration, but not for the moving-cart configuration.

Overall, it appears that the WT would be a better candidate for real-time applicability to high-rate state-estimation given its relatively faster convergence, but this may come at the cost of lower precision on the estimation depending on circumstances. The performance of the WT is yet to be assessed for strongly time-varying systems to characterize high-rate mechanisms undergoing sudden and high amplitude changes in their dynamics. It should also be noted that this paper limited the investigation to only five methods over a very specific experimental dataset and that, while results point towards the WT method as a possible path to high-rate applications, different conclusions could be drawn in a different environment, in particular for systems dominated by higher frequencies. In general, it is envisioned that applications to the high-rate problem would come in the form of advanced yet computationally fast algorithms inspired by the STFT or WT methods and that their implementations in field-programmable gate arrays (FPGAs) would significantly improve their performance in terms of computation time.

**Author Contributions:** The authors made the following contributions: Conceptualization: J.Y., S.L., and A.S.; methodology: J.Y. and S.L.; software: J.Y.; resources: J.D.; writing—original draft preparation: J.Y. and P.S.; writing—review and editing: S.L. and A.S.; visualization: J.Y.; supervision: A.S. and S.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work presented in this paper is partially funded by the National Science Foundation under award number CCSS-1937460. Their support is gratefully acknowledged. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the sponsors.

**Conflicts of Interest:** The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
