*Article* **The Obtainable Uncertainty for the Frequency Evaluation of Tones with Different Spectral Analysis Techniques**

**Salvatore Dello Iacono, Giuseppe Di Leo, Consolatina Liguori \* and Vincenzo Paciello**

Department of Industrial Engineering, Università degli Studi di Salerno, Via Giovanni Paolo II, 132, 84084 Fisciano, Italy; sdelloiacono@unisa.it (S.D.I.); gdileo@unisa.it (G.D.L.); vpaciello@unisa.it (V.P.) **\*** Correspondence: tliguori@unisa.it

**Abstract:** Spectral analysis is successfully adopted in several fields. However, the requirements and the constraints of the different cases may be so varied that not only the tuning of the analysis parameters but also the choice of the most suitable technique can be a difficult task. For this reason, it is important that a designer of a measurement system for spectral analysis has knowledge about the behaviour of the different techniques with respect to the operating conditions. The case that will be considered is the realization of a numerical instrument for the real-time measurement of the spectral characteristics of a multi-tone signal (amplitude, frequency, and phase). For this purpose, different signal processing techniques can be used, that can be classified as parametric or nonparametric methods. The first class includes those methods that exploit the a priori knowledge about signal parameters, such as the spectral shape of the signal to be processed. Thus, a selfconfiguring procedure based on a parametric algorithm should include a preliminary evaluation of the number of components. The choice of the right method among several proposals in the literature is fundamental for any designer and, in particular, for the developers of spectral analysis software, for real-time applications and embedded devices where time and reliability constrains are arduous to fulfil. Different aspects should be considered: the desired level of accuracy, the available elaboration resources (memory depth and processing speed), and the signal parameters. The present paper details a comparison of some of the most effective methods available in the literature for the spectral analysis of signals (IFFT-2p, IFFT-3p, and IFFTc, all based on the use of an FFT algorithm, while improving the spectral resolution of the DFT with interpolation techniques and three parametric algorithms—MUSIC, ESPRIT, and IWPA). The methods considered for the comparison will be briefly described, and references to literature will be given for each one of them. Then, their behaviour will be analysed in terms of systematic contribution and uncertainty on the evaluated frequencies of the spectral tones of signals created from superimposed sinusoids and white Gaussian noise.

**Keywords:** digital signal processing; spectral resolution; frequency domain analysis; frequency– domain interpolation; frequency uncertainty
