**5. Acknowledgment**

This work has been partially supported by the Ministerio de Ciencia e Innovación of Spain under project TEC2009-14219-C03-02, and the E.T.S.I. Telecomunicación of the Universidad Politécnica de Madrid under the FastCFD project.

### **6. References**

294 Applications of Digital Signal Processing

the PDFs, although these effects can be reduced by selecting smaller intervals or by preprocessing the probability function. In particular, normal distributions are better defined (due to the Central Limit Theorem) and, if the widths of the intervals are significantly smaller than the variance of the distribution, the differences with respect to the theoretical PDFs are smaller than with numerical simulations using the same number of samples. In the second part, the evolution of the mean and the variance of the mean and variance estimators has been studied for a normal PDF using the Monte-Carlo method for different interval widths. These estimators behave similarly than their numerical counterparts (slightly better in most cases), but the mean of the variance increases when the interval widths are greater than 1/8 of the variance of the distribution. Moreover, the increased complexity associated to the interval-based computations does not seem to compensate the small improvement of

In summary, interval-based simulations are preferred when the PDFs are being evaluated, but these improvements are not significant when only the statistical parameters are computed. If the distributions contain edges (for example in the uniform or histogram-based distributions), a pre-processing or post-processing stage can be included to cancel the smoothing performed by the interval sets. Otherwise (such in normally distributed signals),

This chapter has presented a detailed review of the interval-based simulation techniques and their application to the analysis and design of DSP systems. First, the main extensions of the traditional IA have been explained, and AA has been selected as the most suitable arithmetic for the simulation of linear systems. MAA has also been introduced for the analysis of nonlinear systems, but in this case it is particularly important to keep the number

Second, three groups of experiments have been performed. In the first group, a simple IIR filter has been simulated using IA and AA to detail the causes of the oversizing of the IAbased simulations, and to determine why AA is particularly well suited to solve this problem. In the second group, different deterministic traces have been simulated using intervals of different widths in some or all the samples. This experiment has revealed the most sensitive frequencies to the small variations of the signals. In the third group, the effect of including intervals in the computation of the statistical parameters using the Monte-Carlo method has been studied. Thanks to these experiments, it has been shown that intervalbased simulations can reduce the number of samples of the simulations, but the edges of the

Finally, it is important to remark that interval-based simulations can significantly reduce the computation times in the analysis of DSP systems. Due to their features, they are particularly well suited to perform rapid system modeling, verification of the system

This work has been partially supported by the Ministerio de Ciencia e Innovación of Spain under project TEC2009-14219-C03-02, and the E.T.S.I. Telecomunicación of the Universidad

stability, and fast and accurate determination of finite wordlength effects.

the accuracy of the statistical estimators in the general case.

of noise terms of the affine forms under a reasonable limit.

distributions are softened by this type of processing.

Politécnica de Madrid under the FastCFD project.

this step can be avoided.

**5. Acknowledgment** 

**4. Conclusions and future work** 


http://caneos.mcmaster.ca/solvers/GLOB:GLOBSOL/


**Part 4** 

**DSP Algorithms and Discrete Transforms** 

Hoefkens, J. (2001), *Verified Methods for Differential Algebraic Equations*.

