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

Alexander Kel’manov, Sergey Khamidullin, Liudmila Mikhailova, Pavel Ruzankin

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

2 Citations (Scopus)

## 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 (formula presented) time; where N is the duration of the observed signal (the number of points in the time series), (formula presented) is a positive integer number which bounds the fluctuations of the repetition period. If (formula presented) is a part of the input, then the algorithm’s running time is (formula presented), i.e., the algorithm is polynomial. If (formula presented) is a fixed parameter (that is typical for applications), then the running-time of the algorithm is (formula presented), i.e., the algorithm is linear in time. Numerical simulation examples demonstrate the robustness of the algorithm in the presence of additive noise.

Original language English Optimization and Applications - 10th International Conference, OPTIMA 2019, Revised Selected Papers Milojica Jaćimović, Michael Khachay, Vlasta Malkova, Mikhail Posypkin Springer Gabler 88-101 14 9783030386023 https://doi.org/10.1007/978-3-030-38603-0_7 Published - 1 Jan 2020 10th International Conference on Optimization and Applications, OPTIMA 2019 - Petrovac, MontenegroDuration: 30 Sep 2019 → 4 Oct 2019

### Publication series

Name Communications in Computer and Information Science 1145 CCIS 1865-0929 1865-0937

### Conference

Conference 10th International Conference on Optimization and Applications, OPTIMA 2019 Montenegro Petrovac 30.09.2019 → 04.10.2019

## Keywords

• Discrete optimization problem
• ECG
• Linear-time algorithm and Pulse train signal processing
• Polynomial-time solvability
• PPG
• Quasiperiodic