Definition/

/explanation

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Example with cars crossing the bridge – average number during 5 minutes (example of convolution operation)

Sliding

average!

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Equivalent representation with the help of input

impulse response.

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Equivalent representation with the help of input

impulse response.

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Do not forget about the periodic properties in both spaces!

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Original part: Ideal filtering formula -IFF

If we consider the basic ideas of the filtering that were outlined in Chapter 5 then one can notice that the basic efforts of the author (and other authors) related to filtering is related to transformation input function to the desired output function. Therefore, one basic problem can be formulated

Is it possible to find a transformation that allows to transform any arbitrary function to another one having the same number of data points?

This remarkable transformation can be found based on some ideas related to the “fuzzy” calculus. Referring the

students to this manuscript we give only the final (the simplest) result:

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Example: How to transform directly the rectangle function to Dirichlet function?

Window Filtration

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Transformation of the part of the harmonic signal (red) to Dirichlet function (blue line)

Conclusion: if you know what you have in the beginning and want to have in the end (“pattern” function) then you can use the Ideal_Filt_Form! The stages of transformation (parameter M) is related to the purposes of this transformation.

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You can realize these transformations with the help of the IFF. Any “pattern” function can be calculated analytically and is transformed to “numbers”

I can suggest to realize of these transformation as an exercise.

Transform a triangle function y(T)=1-a* t/T , a=2,1.

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How to realize it with the help of the IFF?

See next page.

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