A Framework for Interval Quantization and Application to Interval Based Algorithms in Digital Signal Processing

• Published in: Fundamenta informaticae Vol. 112; no. 4; pp. 337 - 363
• Main Authors: Santana, Fabiana T., Santana, Fágner L., Guerreiro, Ana Maria G., Dória Neto, Adrião D., Santiago, Regivan H. N.
• Format: Journal Article
• Language: English
• Published: London, England SAGE Publications 01.01.2011
• Subjects: Approximation; Errors; Floating point arithmetic; Intervals; Mathematical analysis; Quantization; Sampling; Interval mathematics; interval representation; interval sampling; interval quantization; interval coding
• ISSN: 0169-2968; 1875-8681
• DOI: 10.3233/FI-2011-593

In this article, we use the interval mathematics and targeted rounding by specific functions to establish a framework for interval quantization. The function approximation FId, that maps x to an interval [x1, x2] such that x1 is the largest floating point number less than or equal to x and x2 is the smallest floating point number greater than or equal to x, is used to establish the sampling interval and the levels of interval quantization. We show that the interval quantization levels (Nj) represent the specific quantization levels (nj), that are comparable, according to Kulisch- Miranker order and are disjoint two by two. If an interval signal X[n] intercepts a quantization interval level Nj, then the quantized signal will be Xq[n] = Nj. Moreover, for the interval quantization error (E[n] = Xq[n] − X[n]) an estimate is shown due to the quantization step and the number of levels. It is also presented the definition of interval coding, in which the number of required bits depends on the amount of quantization levels. Finally, in an example can be seen that the the interval quantization level represent the classical quantization levels and the interval error represents the classical quantization error.