Distortion properties of the interval spectrum of IPFM generated heartbeats for heart rate variability analysis

The integral pulse frequency modulation (IPFM) model converts a continuous-time signal into a modulated series of event times, often represented as a pulse train. The IPFM process is important to the field of heart rate variability (HRV) as a simple model of the sinus modulation of heart rate. In this paper, we discuss the distortion properties associated with employing the interval spectrum for the recovery of the input signal from an IPFM process's output pulse train. The results state, in particular for HRV, how precisely the interval spectrum can be used to infer the modulation signal responsible for a series of heartbeats. We have developed a detailed analytical approximation of the interval spectrum of an IPFM process with multiple sinusoids as the input signal. Employing this result, we describe the structure and the distortion of the interval spectrum. The distortion properties of the interval spectrum are investigated systematically for a pair of frequency components. The effects of linear and nonlinear distortion of the fundamentals, the overall contribution of harmonic components to the total power, the relative contribution of "folded back" power due to aliasing and the total distortion of the input spectrum are investigated. We also provide detailed comparisons between the interval spectrum and the spectrum of counts (SOC). The spectral distortion is significant enough that caution should be taken when interpreting the interval spectrum, especially for high frequencies or large modulation amplitudes. Nevertheless, the distortion levels are not significantly larger than those of the SOC. Therefore, the spectrum of intervals may be considered a viable technique that suffers more distortion than the SOC.