Abstract

A previously unstudied optimization problem concerning the summation of elements of numerical sequences \(Y\) and \(U\) of respective lengths \(N\) and \(q \leqslant N\) is considered. The task is to minimize the sum of differences between weighted convolutions of sequences of variable length (of at least \(q\)). In each difference, the minuend is a nonweighted autoconvolution of the sequence \(U\) extended to a variable length (by multiple repeats of its elements) and the subtrahend is a weighted convolution of this extended sequence and a subsequence of \(Y\). The variant of the problem with an optimized number of summed differences is analyzed. It is shown that the problem is equivalent to a problem of approximating the sequence \(Y\) by an element \(X\) of an exponential-size set of sequences. This set consists of all sequences of length \(N\) that include, as subsequences, a variable number \(M\) of admissible quasi-periodic (fluctuation) repeats of \(U\). Each quasi-periodic repeat is generated by admissible transformations of \(U\). These transformations are (i) a shift of \(U\) by a variable quantity that does not exceed \({{T}_{{\max}}} \leqslant N\) between neighboring repeats, and (ii) a variable extension mapping of \(U\) into a sequence of variable length defined in the form of repeats of elements of \(U\) with the multiplicity of these repeats being variable. The approximation criterion is the minimum of the sum of squared distances between the elements of the sequences. It is proved that the considered optimization problem, together with the approximation problem, is solvable in polynomial time. More specifically, it is shown that there exists an exact algorithm finding the solution of the problem in \(\mathcal{O}(T_{{\max}}^{3}N)\) time. If \({{T}_{{\max}}}\) is a fixed parameter of the problem, then the running time of the algorithm is linear. Examples of numerical simulation are used to illustrate the applicability of the algorithm for solving model application problems of noise-proof processing of ECG-like and PPG-like quasi-periodic signals (electrocardiogram-like and photoplethysmogram-like signals).

中文翻译：

---

最小化加权卷积之和的问题

摘要

考虑了先前未研究的关于各个长度\（N \）和\（q \ leqslant N \）的数字序列\（Y \）和\（U \）的元素求和的优化问题。任务是使可变长度（至少为（（q \））的序列的加权卷积之间的差之和最小。在每个差异中，被减数是扩展到可变长度（通过其元素的多次重复）的序列\（U \）的非加权自卷积，而次代数是此扩展序列和\（Y \ ）。分析了具有最佳总和差异的问题变体。可以看出，该问题等同于用指数大小的序列集的元素\（X \）近似序列\（Y \）的问题。该集合由所有长度为\（N \）的序列组成，这些序列包括作为子序列的可变数量\（M \）允许的\（U \）准周期（波动）重复。每个准周期重复都是通过\（U \）的可允许变换生成的。这些转换是（i）将\（U \）移位不超过\（{{T} _ {{\\ max}}} \ leqslant N \）的变量相邻重复序列，和（ii）的可变扩展映射之间\（U \）成的元件的重复的形式定义的可变长度的序列\（U \）与这些重复序列是可变的多重性。近似标准是序列元素之间的平方距离之和的最小值。证明了所考虑的优化问题以及近似问题在多项式时间内都是可解的。更具体地说，它表明存在一种精确的算法，可以在\（\ mathcal {O}（T _ {{{\ max}} ^ {3} N）\）时间内找到问题的解决方案。如果\（{{T} _ {{\ max}}} \）是问题的固定参数，则算法的运行时间是线性的。通过数值仿真的例子说明了该算法在解决类心电信号和类心电图准周期信号（心电图和光体积描记图信号）的抗噪处理模型应用问题时的适用性。