A minimization problem for the sum of weighted convolutions' difference and a novel approach to the inverse problem of ECG- and PPG-signals recovering

In this paper, we consider an unstudied problem of approximation of an observed pulse train by by a quasiperiodic signal generated by a pulse with a given pattern (reference) shape. The quasiperiodicity allows variation of time intervals between repetitions of the pattern pulse, as well as nonlinear expansions of the pattern in time. Such inverse problems are common in electrocardiogram (ECG) and photoplethysmogram (PPG) features extraction. The following two variants of the problem are considered. In the first variant, the number of the pulse repetitions is unknown, while in the second one, that number is given. The polynomial-time solvability of the both variants of the problem is constructively proved.