**7. References**

362 Applications of Digital Signal Processing

The difficulty in designing and debugging of such systems necessitates practical simulation tools and simplified metrics such as the entropy measures described here. Both of these entropy metrics are potentially useful during system development as an *analysis of alternatives* or *design space exploration* tool. If several concurrent options are available for a design, either of these metric could provide criteria to establish the least complex design. In that sense, it serves as a simple utility function to measure unwanted or creeping complexity

A close connection exists between the entropy metrics and usage modelling (Whittaker, 1994) for program verification. Most non-exhaustive testing requires a mix of tests taken during nominal conditions along with tests sampled according to potentially rare conditions. This consideration takes into account the number of test vectors and the path coverage for testing. Any characterization at this stage will provide useful inputs to generating a stochastic measure of reliability. This could incorporate stochastic usage models and a log-likelihood metric is often used to compare between two state space probabilities. Between an entropic measure and a usage model, we can cover the temporal

As a diagnostic tool, the context switching metric can also detect potential complexities during execution. Since the FFT can easily compute in real-time for typical sample sizes *N*, parallel execution of the entropy algorithm with the context switching data can reveal deviations from expected operation. For example, if an execution profile shows a high regularity of frequent context switches during some interval and then transitions to a more irregular sequence of switches with the same overall density, the expected entropy measure will definitely increase. In that sense, the entropy metric measures an intrinsic property of the signal, and that strictly speaking, density fluctuations such as expected increases in the rate of context switches will not influence the measure. In other words, density alone does

By the same token, the multi-scale metric has obvious benefit for detecting long term complexity changes or short-term bursts buried in a nominally sampled signal. The idea of using frequency domain entropy for diagnostics of complex machinery is further explored

We described a complexity metric for concurrent software controlled systems or concurrent realizations of behaviour. The novel approach of creating a complexity metric for context switching involves the analysis of the switching frequency spectrum. We take a Fourier transform of the temporally distributed context switching events (*c(t)*) and treat that as a probability density function in frequency space (i.e. a normalized power spectrum). Then

The context switching metric can be used during system development as an analysis of alternative utility function. If several design options or algorithms are available, the context-

the entropy (*S*) of *p(f)* will generate a simple complexity measure.

and path dimensions of a program's execution and its programmatic complexity.

much like a duty-cycle utilization measures processor contention.

**4.1 Applications to design** 

**4.2 Applications to test and verification** 

**4.3 Applications to diagnostics** 

not affect the complexity.

in (Shen, 2000).

**5. Conclusion** 


**0**

**18**

**A Description of Experimental Design on the**

The Fourier series representation of a function is a classic representation which is widely used to approximate real functions (Stein & Shakarchi, 2003). In digital signal processing (Oppenheim & Schafer, 1975), the sampling theorem states that any real valued function *f* can be reconstructed from a sequence of values of *f* that are discretely sampled with a frequency at least twice as high as the maximum frequency of the spectrum of *f* . This theorem can also be applied to functions over finite domains (Stankovic & Astola, 2007; Takimoto & Maruoka, 1997). Then, the range of frequencies of *f* can be expressed in more detail by using a bounded set instead of the maximum frequency. A function whose range of frequencies is confined to a bounded set *I* is referred to as "bandlimited to *I*". Ukita et al. obtained a sampling theorem for bandlimited functions over Boolean (Ukita et al., 2003) and *GF*(*q*)*<sup>n</sup>* domains (Ukita et al., 2010a), where *q* is a prime power and *GF*(*q*) is Galois field of order *q*. The sampling theorem can be applied in various fields as well as in digital signal processing, and one of the fields is

In most areas of scientific research, experimentation is a major tool for acquiring new knowledge or a better understanding of the target phenomenon. Experiments usually aim to study how changes in various factors affect the response variable of interest (Cochran & Cox, 1992; Toutenburg & Shalabh, 2009). Since the model used most often at present in experimental design is expressed through the effect of each factor, it is easy to understand how each factor affects the response variable. However, since the model contains redundant parameters and is not expressed in terms of an orthonormal system, a considerable amount of

In this chapter, we propose that the model of experimental design be expressed as an orthonormal system, and show that the model contains no redundant parameters. Then, the model is expressed by using Fourier coefficients instead of the effect of each factor. As there is an abundance of software for calculating the Fourier transform, such a system allows for a straightforward implementation of the procedures for estimating the Fourier coefficients by using Fourier transform. In addition, the effect of each factor can be easily obtained from the Fourier coefficients (Ukita & Matsushima, 2011). Therefore, it is possible to implement easily the estimation procedures as well as to understand how each factor affects the response variable in a model based on an orthonormal system. Moreover, the analysis of variance can also be performed in a model based on an orthonormal system (Ukita et al., 2010b). Hence,

time is often necessary to implement the procedure for estimating the effects.

**1. Introduction**

the experimental design.

**Basis of an Orthonormal System**

Yoshifumi Ukita1 and Toshiyasu Matsushima2

<sup>1</sup>*Yokohama College of Commerce*

<sup>2</sup>*Waseda University*

*Japan*

