Polynomial-Time Solvability of One Optimization Problem Induced by Processing and Analyzing Quasiperiodic ECG and PPG Signals

Abstract

This paper is devoted to an unexplored discrete optimization problem, which can be interpreted as a problem of least mean squares approximation of some observed discrete-time signal (a numerical time series) by an unobservable quasiperiodic (almost periodic) pulse signal generated by a pulse with a given pattern (reference) shape. Quasiperiodicity is understood, first, in the sense of admissible fluctuations of the interval between repetitions of the reference pulse, and second, in the sense of admissible nonlinear time expansions of its reference shape. Such problems are common in biomedical applications related to monitoring and analyzing electrocardiogram (ECG), photoplethysmogram (PPG), and several other signals. In the optimization model, the number of generated (admissible or approximating) quasiperiodic pulse sequences grows exponentially with the duration of the discrete-time signal (i.e., with the number of points in the time series). The size of the admissible solutions set also grows exponentially. However, despite that exponential growth, we have constructively proved the optimization problem polynomial-time solvability. Namely, we propose an algorithm that finds an optimal solution to the problem in $\mathcal {O} (T_{\max }^{3}N)$ time; where N is the duration of the observed signal (the number of points in the time series), $T_{\max } \le N$ is a positive integer number which bounds the fluctuations of the repetition period. If $T_{\max }$ is a part of the input, then the algorithm’s running time is $\mathcal {O} (N^{4})$, i.e., the algorithm is polynomial. If $T_{\max }$ is a fixed parameter (that is typical for applications), then the running-time of the algorithm is $\mathcal {O} (N)$, i.e., the algorithm is linear in time. Numerical simulation examples demonstrate the robustness of the algorithm in the presence of additive noise.

A. Kel’manov—Deceased 1 December 2019.