Information granulation 179 Kybernetes, Vol. 30 No. 2, 2001, pp. 179-192. # MCB University Press, 0368-492X Information granulation and signal quantization Witold Pedrycz Department of Electrical & Computer Engineering, University of Alberta, Edmonton, Canada, and Adam Gacek Institute of Medical Technology and Equipment (ITAM), Zabrze, Poland Keywords Cybernetics, Fuzzy sets, Codes, Information, Signal processing Abstract Shows that signal quantization can be conveniently captured and quantified in the language of information granules. Optimal codebooks exploited in any signal quantization (discretization) lend themselves to the underlying fundamental issues of information granulation. The paper elaborates on and contrasts between various forms of information granulation such as set theory, shadowed sets, and fuzzy sets. It is revealed that a set-based codebook can be easily enhanced by the use of the shadowed sets. This also raises awareness about the performance of the quantization process and helps increase its quality by defining additional elements of the codebook and specifying their range of applicability. We show how different information granules contribute to the performance of signal quantification. The role of clustering techniques giving rise to information granules is also analyzed. Some pertinent theoretical results are derived. It is shown that fuzzy sets defined in terms of piecewise linear membership functions with 1 / 2 overlap between any two adjacent terms of the codebook give rise to the effect of lossless quantization. The study addresses both scalar and multivariable quantization. Numerical studies are included to help illustrate the quantization mechanisms carried out in the setting of granular computing. 1. Introduction Signal quantization is one of the fundamental activities of signal processing (Madisetti and Williams, 1998; Oppenheim and Schafer, 1989) that implies an efficient transmission and storage of signals. The literature in this area is abundant with a significant number of classic and commonly cited results, cf. Gray (1984), Gersho and Gray (1992), Linde et al. (1978), Lloyd (1982). The underlying overall scheme of signal quantization can be portrayed as in Figure 1. The essence of the quantization is in the formation of the codebook. Instead of transmitting an original highly dimensional datum, an index of the element of the codebook (prototype) that matches the datum to the highest extent is transmitted. At the receiver end, the tag is decoded, namely the corresponding element of the codebook is generated. The quantization process iseitherlosslessorlossy(GershoandGray,1992).Inthefirstcase,thedecoded datumisexactlythesameastheonebeingoriginallytransmitted.Inthesecond case, the result of transmission (x Ã) differs from the original datum (x) and a distance function||.||between these two serves as a suitable measure of losses introducedbythequantizationprocess. The current issue and full text archive of this journal is available at http://www.emerald-library.com/ft Support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.