Image Processing Reference
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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
the accuracy of the statistical estimators in the general case.
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 step can be avoided.
4. Conclusions and future work
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
of noise terms of the affine forms under a reasonable limit.
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 IA-
based 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 interval-
based simulations can reduce the number of samples of the simulations, but the edges of the
distributions are softened by this type of processing.
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
stability, and fast and accurate determination of finite wordlength effects.
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.
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