The input and output (Pattern) functions can be obtained initially by different methods including the algebraic transformations.
Smoothing of Data
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Example:
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The properties of the L-transform:
Linearity property
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Idea in general case is clear. It is necessary to decompose the transfer function on to simple fractions. Let us demonstrate this idea for the denominator presented by
the polynomial of the second order
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Illustration the concept of stability
In order to demonstrate the general case we need to add some additional properties of the L-transform
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The general idea – decomposition of the transfer function on a set of a simple fractions
Stability and instability solutions!?
Other roots? – degenerated, repeated roots? For this aim it is necessary to recall other properties of Laplace transform
Differentiation of the L-image
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Integration of original and L-image
Couple of examples
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Retardation theorem
Example – combination of the step-functions
Find the plot of this function! |
L-image of this function |
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Periodic rectangle signal
For periodic triangle signal we obtain
Translation theorem
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Multiplication/convolution theorem
Duhamel integral
The generalized multiplication theorem (Efros theorem) |
The prove |
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