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

In this article, we use the interval mathematics and targeted rounding by specific functions to establish a framework for interval quantization. The function approximation F-1d, that maps x to an interval [x(1), x(2))] such that x(1) is the largest floating point number less than or equal to x and x(2) 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 (N-j) represent the specific quantization levels (n(j)), 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 N-j, then the quantized signal will be X-q [n] = N-j. Moreover, for the interval quantization error (E[n] = X-q[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.