Spectral information support for unmanned ground vehicles

Abstract

The paper investigates the application of Parseval’s theorem to digital signal processing and its use for relating the mean-square value of a signal to the integral of its power spectral density. Particular attention is given to the relationship between the square of the root-mean-square amplitude and the area under the one-sided power spectral density curve, which represents the total mean-square power of the signal over the analyzed frequency range. A controlled multi-harmonic signal is used to examine how sampling conditions, additive noise, and spectral estimation affect the preservation and interpretation of signal power. Several computational cases are considered to analyze the influence of sampling at the Nyquist boundary, increased sampling frequency, stochastic noise, and spectral smoothing by Welch’s method. The results demonstrate that Parseval’s theorem provides an exact power-consistency relation for the discrete signal represented by the sampled data, but this consistency alone does not guarantee faithful representation of the original continuous physical signal. Adequate sampling frequency, measurement bandwidth, and spectral estimation methodology are therefore essential for physically meaningful spectral analysis. The study also demonstrates that the integrated power spectral density can be used as a quantitative criterion for verifying the consistency of time-domain and frequency-domain calculations. The relationship between Parseval’s theorem and the Wiener–Khinchin theorem is clarified by distinguishing the conservation of mean-square quantities from the statistical description of stationary random processes. The developed signal-processing approach is further considered as an information-support framework for vibration monitoring of unmanned ground vehicles, where measured vibration signals can be transformed into frequency-domain representations for subsequent monitoring, diagnostics, and condition assessment. Parseval’s theorem is proposed as an internal consistency check for this transition from measured time-domain vibration data to their spectral representation.