Reconstructing the Spectrum of a Polyharmonic Signal under Slow Fluctuations in the Sampling Period

Introduction. Polyharmonic signals with a line spectrum are often encountered in practical problems. Among the examples are signals from sensors monitoring rotating elements of mechanical systems, heart rate signals, or signals of radio systems with pulse-to-pulse repetition-period staggering. Due to instable frequencies of the signal harmonics or due to fluctuations in the sampling period, the line spectrum is disrupted. These distortions can be considered as a consequence of changes in the local time scale of the processed signal. This interpretation makes it possible to use scale-invariant transforms to reconstruct the signal spectrum. Methods for reconstructing the spectrum of a signal, the sampling moments of which are unknown a priori, are based on a preliminary reconstruction of the signal and subsequent estimation of its spectrum. Existing algorithms for reconstructing a signal sampled on an uneven time grid with unknown nodes are characterized by a high computational complexity due to their iterative nature and reliance on optimization search methods.

Aim. To synthesize a non-iteration algorithm for reconstructing the spectrum of a polyharmonic discrete signal under the assumption of slow changes in the sampling period.

Materials and methods. To solve the problem, the digital Lamperti transform is implemented. The quality assessment of the proposed algorithm is realized via computer simulation using a test signal known from the literature on the digital spectral analysis.

Results. The conducted computer simulation of the proposed algorithm has proven its feasibility. The line structure of the test signal spectrum, which was distorted by slow changes in the sampling period with an amplitude of 20 % of the mean value of the sampling period, was completely restored. Errors of the frequency and power estimates of individual signal harmonics exhibit values comparable with those derived from the spectrum estimate when the signal is evenly spaced. The error in estimating the sampling period comprised 5 % of its mean value.

Conclusion. A new iteration free algorithm for reconstructing the line spectrum of a discrete polyharmonic signal is proposed. The algorithm uses the scale invariant Lamperti transform. The synthesized algorithm can be used in a simple iterative procedure to estimate changes in the sampling period.

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