**1. Introduction**

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278 Applications of Digital Signal Processing

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As the complexity of digital systems increases, the existing simulation-based quantization approaches soon become unaffordable due to the exceedingly long simulation times. Thus, it is necessary to develop optimized strategies aimed at significantly reducing the computation times required by the algorithms to find a valid solution (Clark et al., 2005; Hill, 2006). In this sense, interval-based computations are particularly well-suited to reduce the number of simulations required to quantize a digital system, since they are capable of evaluating a large number of numerical samples in a single interval-based simulation (Caffarena et al., 2009, 2010; López, 2004; López et al., 2007, 2008).

This chapter presents a review of the most common interval-based computation techniques, as well as some experiments that show their application to the analysis and design of digital Linear Time Invariant (LTI) systems. One of the main features of these computations is that they are capable of significantly reducing the number of simulations needed to characterize a digital system, at the expense of some additional complexity in the processing of each operation. On the other hand, one of the most important problems associated to these computations is interval oversizing (i.e., the computed bounds of the intervals are wider than required), so new descriptions and methods are continuously being proposed. In this sense, each description has its own features and drawbacks, making it suitable for a different type of processing.

The structure is as follows: Section 2 presents a general review of the main interval-based computation methods that have been proposed in the literature to perform fast evaluation of system descriptions. For each technique, the representation of the different types of computing elements is given, as well as the main advantages and disadvantages of each approach. Section 3 presents three groups of interval-based experiments: (i) a comparison of the results provided by two different interval-based approaches to show the main problem

Extended IA (EIA)

Parameterized IA

Centered Forms (CFs)

Directed Intervals (DIs)

Modal Intervals (MIs)

Mean Value Forms (MVFs)

Generalized IA (GIA)

Slopes Taylor Models (TMs)

Affine Arithmetic (AA)

In MIs (Gardenes, 1985; Gardenes & Trepat, 1980; SIGLA/X, 1999a, 1999b), each element is composed of one interval and a parameter called "modality" that indicates if the equation of the MIs holds for a single value of the interval or for all its values. These two descriptions are used to generate equations that bound the target function. If both descriptions exist and are equal, the result is exact. Among the publications on MIs, the underlying theoretical formulation and the justifications are given in (SIGLA/X, 1999a) and the applications, particularly for control systems, are given in (Armengol, et al., DX-2001; SIGLA/X, 1999b;

GIA (Hansen, 1975; Tupper, 1996) is based on limiting the regions of the represented domain using intervals with parameterizable endpoints, such as [1 – 2x, 3 + 4x] with x [0,1]. The authors define different types of parameterized intervals (constant, linear, quadratic, linear, multi-dimensional, functional and symbolic), but their analysis has focused on evaluating whether the target function is increasing or decreasing, concave or convex, in the region of interest using constant, linear and polynomial parameters. In the experiments, they have obtained the areas where the existence of the function is impossible, but they conclude that this type of analysis is too complex for parameterizations greater

In the different representations, CFs are based on representing a function as a Taylor Series expansion with one or more intervals that incorporate the uncertainties. Therefore, all these techniques are composed of one independent value (the central point of the function) and a

MVFs (Alefeld, 1984; Coconut\_Group, 2002; Moore, 1966; Neumaier, 1990; Schichl & Neumaier, 2002) are based on developing an expression of a first-order Taylor Series that

where *x* is the point or region where *f*(*x*) must be evaluated, *x*0 is the central point of the Taylor Series, and *Ix* is the interval that bounds the uncertainty range. The computation of the derivative is not complex when the function is polynomial, as it is usually the case in function approximation methods. Since the approximation error is quadratic, this method does not provide good results when the input intervals are large. However, if the input

*f* (*x*) = *f* (*x*0) + *f* ´(*x* )(*x* – *x*0) *fMVF* (*Ix*) = *f* (*x*0) + *f* ´( *Ix* ) (*Ix* – *x*0) (2)

set of summands that incorporate the intervals in the representation.

bounds the region of interest. The general expression is as follows:

intervals are small, it provides better results than traditional IA.

Fig. 1. Classification of interval-based computations methods.

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Aritmética

Interval Arithmetic (IA)

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than the linear case.

of interval-based computations; (ii) an analysis of the application of interval-based computations to measure and compare the sensitivity of the signals in the frequency domain; and (iii) an analysis of the application of interval-based techniques to the Monte-Carlo method. Finally, Section 4 concludes this work.
