IEEE SIGNAL PROCESSING LETTERS, VOL. 14, NO. 2, FEBRUARY 2007 81 Optimum Rate-Distortion Dictionary Selection for Compression of Atomic Decompositions of Electric Disturbance Signals Michel P. Tcheou, Student Member, IEEE, Lisandro Lovisolo, Eduardo A. B. da Silva, Senior Member, IEEE, Marco A. M. Rodrigues, Member, IEEE, and Paulo S. R. Diniz, Fellow, IEEE Abstract—In this letter, we address rate-distortion-optimum compression of signals from electric power system disturbances, using atomic decompositions. Usually, such optimization is ob- tained assuming a single dictionary and consists of finding the best compromise between the quantization of the coefficients in the atomic decomposition and its number of terms. Here, several parameterized dictionaries are used instead. This allows the selection of the dictionary leading to the best rate-distortion (R-D) compromise. Distinct dictionaries correspond to different quantizers for the parameters of the atoms. Side information must be transmitted in order to indicate the dictionary employed. The R-D performance in this case depends on a complex interplay between the quantizers of the parameters of the atoms and the co- efficient quantizers. Using a training stage, we select a reduced set of parameter and coefficient quantizers that give near-optimum R-D performance. Simulation results show that the proposed scheme indeed achieves near-optimum R-D performance with low computational complexity in the coding stage. Index Terms—Atomic decompositions, dictionaries, electric dis- turbance signals, rate-distortion optimization. I. INTRODUCTION A TOMIC decompositions represent signals using linear combinations of functions (atoms) drawn from a dictio- nary. They are important tools for the compression of several signal sources [1]–[4]. When based on a redundant dictionary, atomic decompositions can provide good adaptive signal ap- proximations. The approximation is adaptive since the atoms are selected from the dictionary according to the signal being decomposed. The use of highly redundant dictionaries enables efficient decompositions of a wide range of signals. Several methods have been used to obtain these representations [5], such as the matching pursuit (MP) algorithm [6]. A signal can be approximated by an atomic decomposition as (1) Manuscript received May 12, 2006; revised June 10, 2006. The associate ed- itor coordinating the review of this manuscript and approving it for publication was Dr. Xiang-Gen Xia. M. P. Tcheou and M. A. M. Rodrigues are the Centro de Pesquisas de Energia Elétrica-CEPEL, Rio de Janeiro, RJ, 21941-590, Brazil (e-mail: pompeu@cepel.br; mamr@cepel.br). L. Lovisolo is with the Universidade Estadual do Rio de Janeiro, Rio de Janeiro, RJ, 20550-900, Brazil (e-mail: lisandro@uerj.br). E. A. B. da Silva and P. S. R. Diniz are with the Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21945-970, Brazil (e-mail: eduardo@lps.ufrj.br; diniz@lps.ufrj.br). Digital Object Identifier 10.1109/LSP.2006.882117 The atoms are selected from a redundant dictionary , being indexed by the mapping . If is the dictionary car- dinality, this mapping is defined as . In compression applications, one encodes the number of terms , the coefficients , and atom indexes . The optimum R-D tradeoff is achieved by finding a compromise between the number of expansion elements and the quantization of each co- efficient [7]. Instead of using a single dictionary, consider a set of redundant dictionaries described as , where is the number of dictionaries included in . This framework is illustrated in Fig. 1. In this case, the dictionary used must be indicated to the decoder as side information. The optimum R-D performance corresponds to the tradeoff among the bits spent on side information, atom indexes, and coefficients leading to the minimum distortion. The solution to this tradeoff usually involves high computational demands, which increase with the number of distinct dictionaries in . In this letter, we address atomic decompositions of electric power system disturbance signals using parameterized dictio- naries [4]. The atoms consist of pieces of damped sinusoids that can be defined by a set of five parameters (frequency, damping factor, phase, start, and ending times). For coding, the parame- ters must be quantized. Referring to Fig. 1, each quantizer for the parameter space defines one dictionary . One possible way to obtain the optimum R-D tradeoff would be to find the optimum coefficient quantizer and number of terms for each dictionary (parameter quantizers) and choosing the one with the best performance. However, some combinations of coefficient quan- tizers and dictionaries lead to poor performance, irrespective of the signal being compressed. We avoid such combinations by employing a reduced set of dictionaries belonging to the convex hull of operational R-D characteristics of signals from a training set. The results show that the proposed strategy allows to achieve near-optimal R-D performance with comparatively low compu- tational complexity. II. COMPRESSION OF ELECTRIC DISTURBANCE SIGNALS USING PARAMETERIZED DICTIONARIES Large electric power systems consist of complex intercon- nected grids where several players perform distinct functions such as generation, transmission, and distribution of electrical power. For each player, it is crucial that any disturbance be monitored, not only to meet regulatory requirements but also to identify its causes. With the wide availability of dedicated monitoring devices known as digital fault recorders (DFRs), an increasing number of events can be acquired. Therefore, specific compression methods for reducing the file sizes of the DFR data are needed. Typical DFR data are formed by voltage and current 1070-9908/$25.00 © 2007 IEEE