Multi-Sine Fitting Algorithm enhancement for sinusoidal signal characterization Domenico Luca Carní ⁎, Giuseppe Fedele Department of Electronics, Computer and System Science, University of Calabria, Italy abstract article info Article history: Received 26 October 2010 Accepted 16 March 2011 Available online 30 March 2011 Keywords: Multi-sine fitting ADC testing Non-uniform sampling In the Multi-Sine Fitting Algorithm (MSFA) based on four parameters sine fitting a problem is the evaluation of the initial condition of the fundamental harmonic frequency, to guarantee the convergence since it is not assured for each initial condition. To overcome this problem, a method devoted to the improved evaluation of initial condition for the MSFA is presented. In particular, the method is based on the algebraic derivative approach in the frequency domain. Numerical and experimental tests confirm the potentiality of the method and the advantages in term of higher accuracy, in comparison with others methods. © 2011 Elsevier B.V. All rights reserved. 1. Introduction The sinusoidal parameters estimation has an important role in signal processing for electronic device characterization. Indeed, information about specific characteristics of the device under test can be obtained according to the deformation affecting the output signal corresponding to the sinusoidal input. In the IEEE standard 1241 [1] a sine fitting algorithm is recom- mended to be used in the test of the Analog to Digital Converters (ADCs). In this standard two different versions of the sine fitting algorithm are proposed. The first one is the three parameters sine fitting algorithm, henceforth 3SFA, based on the assumption that the sinusoidal signal frequency is known. It is able also to evaluate amplitude, phase of the fundamental harmonic and the dc offset. The second one is the four parameters sine fitting algorithm, namely 4SFA, based on the assump- tion that the sinusoidal signal frequency is unknown. The problem is that the acquisition system should be characterized by both resolution and linearity higher than the characteristics of the signal under test. Moreover, in the experimental tests, the output signal can be constituted by multi-harmonic components. Applied to a multi-harmonic signal, both 3SFA and 4SFA perform the estimation with low accuracy because they fit the sine wave into the set of non- sinusoidal samples [2]. As a consequence, no admissible errors can arise in frequency, amplitude, and phase estimation. Many techniques are proposed in literature to acquire the signal with high resolution and linearity. In [3] the problem is shifted into the simpler one of digitizing, through the low resolution and high frequency ADC, the signal obtained by the comparison between the signal under test and a reference one. The acquired signal is characterized by a non-uniform sampling that can not be easily analyzed in the frequency domain. In fact, spurious frequencies arising in the frequency domain analysis of a non-uniform sampled signal. In [2] two different versions of the Multi-Sine Fitting Algorithm (MSFA) are proposed to estimate the parameters of multi-harmonic signal such as amplitude, phase of each harmonic and the dc offset. The first version is based on the modified version of 3SFA and the Least-Square Method is used to minimize the residual between the multi-harmonics signal and the sampled one. The second version of the MSFA uses the modified version of 4SFA. The Gauss–Newton gradient-search procedure is used to estimate the frequency of the first harmonic by starting from an initial coarse value. A problem is the evaluation of the initial condition of the fundamental harmonic frequency, and both amplitude and phase of each harmonic to guarantee the convergence. The convergence is not assured for each value of the initial conditions. Indeed, it is highly dependent on the initial frequency and the number of samples used [4]. If the algorithm converges, the number of iterations is highly dependent on the initial conditions. Moreover, it can converge to local minimum instead of the global one. In [2] the initial value of the frequency is estimated using the interpolated Discrete Fourier Transform (DFT) [5]. Successively, to estimate the initial condition of amplitude and phase of each harmonic and the dc offset value, the modified 3SFA is used. It can be noted that the frequency estimation based on DFT can give wrong estimation in the case of non-uniform sampling [3]. In the paper, to overcome the problem of the initial condition estimation, the approach based on the use of the algebraic derivative method in the frequency domain is used [26], [7]. It does not require the assumption of uniform sampling and permits to increase the convergence speed of the MSFA based on 4SFA. The paper is organized as follows. The aspects justifying the change to be introduced in the MSFA are presented. The fundamental aspects of the algorithm for the frequency estimation are summarized. Computer Standards & Interfaces 34 (2012) 535–540 ⁎ Corresponding author. E-mail addresses: dlcarni@deis.unical.it (D.L. Carní), fedele@si.deis.unical.it (G. Fedele). 0920-5489/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.csi.2011.03.003 Contents lists available at ScienceDirect Computer Standards & Interfaces journal homepage: www.elsevier.com/locate/csi