Performance analysis of Gibbs sampling for Bayesian extracting sinusoids —This paper involves problems of estimating parameters of sinusoids from white noisy data by using Gibbs sampling (GS) in a Bayesian framework. Modifications of its algorithm is tested on data generated from synthetic signals and its performance is compared with conventional estimators such as Maximum Likelihood(ML) and Discrete Fourier Transform (DFT) under a variety of signal to noise ratio (SNR) and different length of data sampling (N), regarding to Cramér)Rao lower bound (CRLB). All simulation results show its effectiveness in frequency and amplitude estimation of sinusoids. —           . I. INTRODUCTION The sinusoidal frequency model embedded in noise is extensively important because of its wide applicability in many areas of science and engineering such as, modeling and manipulation of time)series from speech, audio to radar, seismology, nuclear magnetic resonance, communication problems and underwater acoustics[1]. We therefore address here a problem of estimating parameters of noisy sinusoids within a Bayesian inferential framework that provides a rigorous mathematical foundation for making inferences about them and a basis for quantifying uncertainties in their estimates. Under an assumption that a number of sinusoids is known a priori, several algorithms have already been applied to spectral analysis and parameter estimation problems, such as least) square fitting [2], maksimum likelihood(ML)[3], discrete Fourier transform(DFT) [4,5], and a periodogram [6]. After Jayness’ work [7], researchers in different fields of science have given much attention to the relationship between Bayesian inference and parameter estimation. Bretthorst and the others [8)16] have done excellent works in this area for the last sixteen years. In this paper, we consider analysis of Gibbs sampling[11, 12] for recovering sinusoids from noisy data and compare its performance with classical estimators, regarding to Cramér)Rao lower bound(CRLB), that is a limit on the best possible performance achievable by an unbiased estimator given a dataset[17]. For this purpose, a series of simulation studies with a variation in levels of noise and length of data sampling for a single sinusoid is set up. II. HARMONIC SIGNAL MODEL In many experiments, a discrete data set 1 2 { , ,..., }      = D denoted as an output of a physical system that we want to be modeled is sampled from an unknown function ()  at discrete times 1 {, ...., }     : () (;) , ( 1,..., ),           θ = = + = (1) where θ is a vector containing parameters that characterize behavior of physical system (;)  θ and that are usually unknown. The term   is assumed to be drawn from a known random process. The choice of the model function (,)  θ depends on the specific application, but we will consider here a superposition of  sinusoids: 1 (,) cos( ) sin( )              ω ω = = + ∑ θ . (2) where { } 2 ,        ∈ ℝ and ( ) 0,  ω π ∈ are amplitudes and angular frequencies, respectively. Hence, Equation (1) can be written in the matrix)vector form: = + D Ga e , (3) where D is ( 1)  × matrix of data points and e is ( 1)  × matrix of independent identically distributed Gaussian noise samples. G is ( 2)   × matrix whose each column is a basis function evaluated at each point of time series. The linear coefficient a is a (2 1)  × matrix whose components are arranged in order of coefficients of cosine and sine Received: June 10, 2019. Revised: September 24, 2021. Accepted: October 14, 2021. Published: November 23, 2021. Mehmet Cevri Department of Mathematics, Istanbul University, Istanbul, Turkey mcevri@gmail.com.tr Dursun Üstündag Department of Mathematics, Marmara University, Istanbul, Turkey dustundag@marmara.edu.tr INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES DOI: 10.46300/9101.2021.15.19 Volume 15, 2021 E-ISSN: 1998-0140 148