Short-time DFT computation by a modified radix-4 decimation-in-frequency algorithm Dan-El A. Montoya a , J.A. Rosendo Macías b,n , A. Gómez-Expósito b a School of Electrical Engineering, Central University of Venezuela, Caracas, Los Chaguaramos,1041 Venezuela b Department of Electrical Engineering, University of Seville, Camino de los Descubrimientos s/n, 41012 Seville, Spain article info Article history: Received 22 January 2013 Accepted 14 June 2013 Available online 27 June 2013 Keywords: Short-time DFT Moving-window DFT Time-dependent DFT Running DFT Fixed-time-origin STDFT Decimation-in-frequency abstract This work presents a radix-4 decimation-in-frequency algorithm for the efficient compu- tation of the short-time, discrete Fourier transform, which makes use of radix-4 butterflies with time-varying coefficients arising from a fixed time origin. The proposed scheme is successfully compared with existing competing algorithms in terms of computational cost. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Many applications in electrical engineering require the spectrum of a time-varying or non-stationary signal to be computed. Since the seminal paper by Cooley and Tukey [1], introducing the most popular family of discrete Fourier transform (DFT) algorithms, a formidable research effort has been devoted to develop other signal processing algo- rithms, including two algorithms to compute the spectrum of time-varying signals [2,3]. Moving-window DFT, known also in the literature as short-time, time-dependent, sliding or running DFT, pro- vides time-varying or nonstationary frequency spectrum of signals [4,5]. This transform, known by the acronym STDFT, finds applications in radar, sonar, speech or data communication signal processing [6,7], digital protection of power systems [8], or real-time control of electronic power converters [9]. There are two ways of defining the STDFT. The first one, called moving-time-origin STDFT, can be interpreted as the DFT of a running signal as viewed through a time window. In this case the window is held fixed while the signal slides over the window. The second one, called fixed-time- origin STDFT, shifts the window along the signal while keeping the time origin for Fourier analysis fixed at the initial time origin of the signal [4]. In both cases, the DFT is computed on a windowed signal which takes the last N available samples of the signal at instant n. In the naive and usual approach, where the DFT of the last N samples is computed at each time instant, N log 2 N complex adds and ðN=2Þlog 2 N complex multiplications are required [4]. Most of the research on this topic is devoted to recursive algorithms, in which the spectrum at any instant is related to previously computed spectra [10–13]. However, from the practical implementation point of view, recursive algorithms present long-term anomalous behavior [8,14,15] which may preclude its use in certain applications. On the other hand, there are two efficient non- recursive algorithms to compute STDFTs [2,3]. The first one, based on a moving-time-origin decimation-in-time Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/sigpro Signal Processing 0165-1684/$ -see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.06.019 n Corresponding author. Tel.: +34 954481276; fax: +34 954487284. E-mail address: rosendo@us.es (J.A. Rosendo Macías). Signal Processing 94 (2014) 81–89