Abstract—This paper presents an algorithm to estimate the parameters of two closely spaced sinusoids, providing a frequency resolution that is more than 800 times greater than that obtained by using the Discrete Fourier Transform (DFT). The strategy uses a highly optimized grid search approach to accurately estimate frequency, amplitude and phase of both sinusoids, keeping at the same time the computational effort at reasonable levels. The proposed method has three main characteristics: 1) a high frequency resolution; 2) frequency, amplitude and phase are all estimated at once using one single package; 3) it does not rely on any statistical assumption or constraint. Potential applications to this strategy include the difficult task of resolving coincident partials of instruments in musical signals. Keywords—Closely spaced sinusoids, high-resolution parameter estimation, optimized grid search. I. INTRODUCTION HE problem of estimating the parameters of sinusoidal signals has been intensively studied for many decades due to its importance in many practical situations. A few examples of applications are transient disturbance in power systems; channel prediction in communications; estimation of direction of arrival in radar and sonar; audio, speech and image processing; condition monitoring of engineering structures and systems; and nuclear magnetic resonance. As a result of such a scenario, there is a huge amount of literature treating aspects ranging from the proposition of estimation methods to the development of performance bounds, analysis of accuracy and computational effort. This paper deals with just one kind of sinusoidal estimation problem: the simultaneous estimation of amplitude, frequency and phase of two sinusoids with extremely close frequencies and using only one set of observations (just one snapshot). Although this particular problem also has significance in a variety of applications, the background motivation here is the complete identification of all sinusoids present in small pieces of musical signals, aiming applications such as the single channel source separation problem [1]. Signals produced by many musical instruments exhibit strong local periodicities modeled as a sum of harmonically Jayme Garcia Arnal Barbedo (corresponding author) and Amauri Lopes are with Department of Communication of the School of Electrical and Computer Engineering of the State University of Campinas (Unicamp), C.P. 6101, CEP: 13.083-970, Campinas - SP – Brazil (phone: +55-19-3521-3703; e-mails: {jgab,amauri}@decom.fee.unicamp.br). related sinusoids. Since most songs are played by simultaneous harmonically related instruments, there may exist sinusoidal components almost coincident in frequency for a certain period of time. Such situation can occur in two main cases: 1) when two instruments play the same note, the fundamental frequency component and corresponding partials probably have very closely spaced frequencies; 2) if the instruments play harmonically related notes, then the nearly coincident frequencies can occur at some harmonic components. Exact spectral coincidence is highly unlikely due to different characteristics of instruments and musicians, and, if it occurs, it will last just for a few milliseconds. Therefore, any algorithm aiming to separate the signal of each instrument must deal with sinusoids with very close frequencies. It is worth noting that estimating all three sinusoidal parameters is paramount for many musical signal processing tools, because the human perception of a song is usually closely linked to the way different sources and their respective partials interact, and such an interaction strongly depends on those parameters. Even for this particular sinusoidal estimation problem there are several estimation methods which can be classified in a number of ways depending on their theoretical supports. One possible way to classify such methods into categories is presented next, together with some of related work: 1- correlation-based techniques [2-7]; 2- methods derived from the maximum likelihood function [8-15]; 3- methods based on rational models [4-6,16,17]; methods based on subspace properties [18-28]; algorithms that use spectral properties or filtering [5,13,14,17,29-36]; least squares-based methods [37,38] and so on. Many of those propositions also use iterative or adaptive procedures [7,8,30]. Finally, there are some papers that test and compare a variety of methods [39- 42]. All those propositions have good performance for certain conditions in which their underlying assumptions hold, but they may fail when facing some specific conditions such as, for example, extremely closely spaced frequencies and/or limited number of observations. In this context, this work presents the development of a new strategy to estimate the parameters of two sinusoids under the conditions aforementioned. The basic idea is to define a fine grid of points in the 6-dimensional parameter space (six parameters) and to compose a two-sinusoid mathematical model for each point. The choice of the best point is carried out by comparing the waveform corresponding to each model Estimating Frequency, Amplitude and Phase of Two Sinusoids with Very Close Frequencies Jayme G. A. Barbedo, Amauri Lopes T World Academy of Science, Engineering and Technology International Journal of Electronics and Communication Engineering Vol:3, No:11, 2009 2026 International Scholarly and Scientific Research & Innovation 3(11) 2009 scholar.waset.org/1307-6892/4390 International Science Index, Electronics and Communication Engineering Vol:3, No:11, 2009 waset.org/Publication/4390