IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 50, NO. 6, DECEMBER 2003 1109 Neural Network Technique for the Joint Time–Frequency Analysis of Distorted Signal Manuele Bertoluzzo, Giuseppe S. Buja, Fellow, IEEE, Simone Castellan, and Pietro Fiorentin Abstract—Nonstationary distorted signals need to be analyzed in both the time and frequency domains to determine their charac- teristics. In this paper, a technique based on a neural network (NN) is presented which has the merit of providing such an analysis in real time. After arranging a suitable NN, the algorithm utilized to carry out the analysis is illustrated. Then, expressions assessing the dynamic behavior and the steady-state accuracy of the technique are derived. From the expressions, the influence of the NN param- eters on the technique performance is readily recognized. As an example, the technique is applied to the analysis of the time evolu- tion of the current harmonics absorbed by a diode rectifier and the results are compared with those obtained by the short-time Fourier transform. Index Terms—Distorted signal, harmonic analysis, neural net- work (NN) application. I. INTRODUCTION I N PRINCIPLE, a signal can be represented in an infinite number of ways. Time is the natural variable in the signal occurrence and the signal representation in the time domain is, hence, widely used. However, some interesting characteristics of a signal can not be easily focused in the time domain and the map of the signal from the time into another domain makes them explicit. For example, the periodicity characteristics are high- lighted in the frequency domain where the signal is represented as a composition of harmonically related sinusoidal functions. Traditionally, signals have been analyzed either as a function of the time or of the frequency, not of both. For instance, the fre- quency analysis is commonly carried out regardless of the time, in spite of the fact that the majority of the signals are nonsta- tionary and would require a joint time–frequency analysis. In general, a signal can be written as the linear combination of elementary functions that make a base of the domain to which the signal belongs. Each coefficient in the linear combination is the signal projection on the corresponding elementary func- tion and describes how much of the elementary function is in- cluded in the signal. When the signal has to be characterized in the time and in the frequency simultaneously, the elementary functions should be localized in both time and frequency do- main. This is not the case of the Fourier series since it uses sinu- Manuscript received October 8, 2002; revised March 13, 2003. Abstract pub- lished on the Internet September 17, 2003. M. Bertoluzzo, G. S. Buja, and P. Fiorentin are with the Department of Electrical Engineering, University of Padova, 35131 Padova, Italy (e-mail: bertoluzzo@die.unipd.it; giuseppe.buja@unipd.it; pietro.fiorentin@unipd.it). S. Castellan is with the Department of Electrotechnics, Electronics and Computer Science, University of Trieste, 34127 Trieste, Italy (e-mail: simone. castellan@deei.units.it). Digital Object Identifier 10.1109/TIE.2003.819577 soidal functions, which are fully concentrated in the frequency domain and do not have any location in the time domain, with the well-known outcome that the Fourier series is not able to give any information on the time evolution of a signal. Being the representations in time and frequency domains re- lated via the Fourier transform, the time and frequency behav- iors of a signal are not independent. In particular, its time dura- tion and frequency bandwidth can not be made arbitrarily small simultaneously. For this reason the time–frequency analysis of a signal is commonly carried out by time-located elementary functions such as the windowed harmonically related complex sinusoidal functions , used by the windowed Fourier transform, or the scaled function , used by the wavelets transform. In many industrial appliances the behavior of a signal is a se- quence of transients and steady states. During steady states the signal is considered to be periodic and is analyzed by the Fourier series using sinusoidal functions as a base. As stated before, they alone are not suitable for representing the signal during the transients. To overcome this deadlock, the most useful so- lution is to describe the signal by a sum of harmonically re- lated sinusoidal functions with magnitudes depending on the time and to calculate the magnitudes by the short-time Fourier transform (STFT) [1]. Recently, the convenience of utilizing a neural network (NN) [2] to calculate the harmonic magnitudes as well as their time evolution has been recognized by many au- thors [3]–[5], especially in the field of power electronics. The existing literature, however, limits itself to the implementation of suitable NN structures and to the illustration of the resultant time–frequency analysis without supporting it with a mathemat- ical approach helping the designers in arranging an appropriate NN and in selecting its parameters. It is the purpose of this paper to give such support by developing an analytical tool useful in understanding the processing at the basis of the NN time–fre- quency analysis of a signal. The paper is organized as follows. Section II defines the char- acteristics of the signals considered in the paper and briefly re- calls STFT. Section III presents topology and operation of an NN built for the joint time–frequency analysis of a signal. Sec- tion IV formulates the equations by which the NN carries out the harmonic analysis of a signal as a function of the time. Sec- tion V establishes the performance of the NN technique and the influence of the NN parameters. Section VI reports the applica- tion of the NN technique to the analysis of the current absorbed by a diode rectifier with time-variant load and compares the re- sults with those obtained by STFT. Section VII concludes the paper. 0278-0046/03$17.00 © 2003 IEEE