Spectra of data sampled at frequency-modulated rates in application to cardiovascular signals: Part 1 analytical derivation of the spectra

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Abstract
Beat-to-beat cardiovascular signals, e.g. a series of systolic pressure values, can be considered as time series which are pulse amplitude modulated (PAM) and pulse frequency modulated (PFM). The latter process, due to variations in heart rate, causes the series to become non-uniformly spaced in time. If PAM is to be quantified by spectral analysis, the influence of PFM must be known. An analytical expression is therefore derived for the spectrum of sinusoids which are sampled according to the output event series of a linear integral pulse frequency modulator (IPFM). We conclude that two spectral components arise at the difference and sum of the PFM and PAM frequencies, fp±fx, with amplitudes proportional to the PFM modulation depth. These components appear as a DC component and as a first harmonic if both modulating frequencies are equal. In addition, a cluster of spectral components appears around the mean pulse frequency fo (i.e. mean heart rate), at frequencies fo-nfp±fx, which may leak into the signal band. From these theoretical considerations, we conclude that the amplitude spectrum of a sinusoidally varying systolic blood pressure series can contain up to 20–30% spurious components, owing to the heart rate modulation process.

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