Метод уточнения значения частотного сдвига спектра, полученного в условиях малой разрешающей способности анализатора
Authors: Аглиуллин А.Ф. , Аглиуллин Т.А. , Валеев Б.И. , Куликова Е.В. , Мальцев А.В. , Морозов О.Г. , Сахабутдинов А.Ж. 
Published: February 2025

В статье предложен новый метод определения частотного сдвига спектра, полученного в условиях малой разрешающей способности анализатора, позволяющий за счет математической обработки сигнала, описывающего спектр, определять его частотный сдвиг с разрешающей способностью на два порядка большей, чем приборная. Метод основан на представлении сигнала, описывающего сдвинутый по частоте спектр в виде непрерывной всюду дифференцируемой функции, разложении её в ряд Тейлора, аппроксимации его производных конечными разностями заданного порядка и определении частотного сдвига методом наименьших квадратов. Дано математическое обоснование метода, приведены результаты математического моделирования. Обсуждены перспективы применения предложенного метода.

Introduction. Determining the frequency shift of the spectrum (FSS) caused by various physical influences is a pressing challenge in many scientific and technological fields. However, existing devices, such as optoelectronic spectrum analyzers, often lack the necessary resolution to accurately measure FSS. To address this issue, it is necessary to develop mathematical methods for estimating FSS while improving measurement accuracy. While increasing the resolution of optical-electronic spectrum analyzers can directly address this challenge, doing so requires costly component upgrades that may be prohibitively expensive. This problem is particularly relevant for interrogation systems of the central wavelength of fiber Bragg gratings (FBGs). The aim of this paper is to develop and evaluate a signal processing method for analyzing the spectrum of a Gaussian fiber Bragg grating (FBG) under low instrumental resolution conditions. The method should enable accurate determination of the central wavelength position of the FBG, with a possible spectrum shift of ±0.5 nm (±60 GHz) on either side of the unperturbed lattice position. Methods. To address this challenge, we assume that external physical impacts on the Gaussian FBG only affect its amplitude and FSS, without distorting the Gaussian shape. This approach accounts for fluctuations in the radiation source's power level and dynamic losses. We also assume that the FSS and amplitude changes occur simultaneously. Our study proposes a mathematical algorithm for determining the coordinates of refined shift vector D along the frequency and amplitude axes. Although our main goal is to find the FSS along the frequency axis, we approach the problem in a general case. Conclusion. although the mathematical apparatus used may seem simple, the proposed algorithm has demonstrated the ability to surpass other methods by one and a half orders of magnitude in estimating the position of the central FBG wavelength, and by almost three orders of magnitude in estimating the FSS value obtained through low-resolution spectrum analysis. This algorithm ensures that the FSS of FBG can be determined with an error of three orders of magnitude greater than its resolution. Additionally, the method has the advantage of being independent of the intensity of fluctuations of the probing radiation spectrum. The paper also discusses the potential applications of the developed method.