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IEVref: | 351-45-20 | ID: | |

Language: | en | Status: Standard | |

Term: | unit impulse | ||

Synonym1: | Dirac impulse [Preferred] | ||

Synonym2: | |||

Synonym3: | |||

Symbol: | δ(t)
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Definition: | distribution, defined as the limit of a positive function, equal to zero outside a small interval containing the origin, the integral of which remains equal to one when this interval tends to zero $\delta (t)=\{\begin{array}{c}0\\ \infty \\ 0\end{array}\begin{array}{c}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}t<0\\ \text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}t=0\\ \text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}t>0\end{array}\text{\hspace{1em}}\text{with}{\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}\delta (t)}\text{\hspace{0.17em}}\mathrm{d}t=1$ SEE: Figure 4a) and IEC 60027-6. Note 1 to entry: A distribution assigns a number to any function The Dirac impulse does this according to $f\left({t}_{0}\right)={\displaystyle \underset{-\infty}{\overset{\infty}{\int}}\delta}\left(t-{t}_{0}\right)f\left(t\right)\text{d}t$. Note 2 to entry: Any shape with area 1 may be used for the definition of δ( $\frac{1}{\tau \cdot \sqrt{\pi}}\cdot {e}^{-\text{\hspace{0.17em}}\frac{{t}^{2}}{{\tau}^{2}}}$. Note 3 to entry: Any of the shapes mentioned in Note 2 to entry with Note 4 to entry: In control technology the Dirac function is mainly important for the definition of impulses and exclusively used as a function of time. Therefore the term Dirac impulse is used and the definition is adapted accordingly. | ||

Publication date: | 2013-11 | ||

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Internal notes: | 2017-06-02: Cleanup - Remove Attached Image 351-45-201_en.gif | ||

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$\delta (t)=\{\begin{array}{c}0\\ \infty \\ 0\end{array}\begin{array}{c}\text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}t<0\\ \text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}t=0\\ \text{\hspace{0.17em}}\text{for}\text{\hspace{0.17em}}t>0\end{array}\text{\hspace{1em}}\text{with}{\displaystyle \underset{-\infty}{\overset{+\infty}{\int}}\delta (t)}\text{\hspace{0.17em}}\mathrm{d}t=1$

SEE: Figure 4a) and IEC 60027-6.

Note 1 to entry: A distribution assigns a number to any function *f*(*t*), sufficiently smooth for *t* = *t*_{0} [see CEI 60050-103:2009, 103-03-05].

The Dirac impulse does this according to

$f\left({t}_{0}\right)={\displaystyle \underset{-\infty}{\overset{\infty}{\int}}\delta}\left(t-{t}_{0}\right)f\left(t\right)\text{d}t$.

Note 2 to entry: Any shape with area 1 may be used for the definition of δ(*t*), e.g. a rectangular pulse with width *τ* and height *τ*^{–1}, or a triangular pulse, as shown in Figure 4a), as well as a Gaussian function

$\frac{1}{\tau \cdot \sqrt{\pi}}\cdot {e}^{-\text{\hspace{0.17em}}\frac{{t}^{2}}{{\tau}^{2}}}$.

Note 3 to entry: Any of the shapes mentioned in Note 2 to entry with *τ* much smaller than the smallest time constant at work in the system under consideration may be used for a technical approximation of the Dirac impulse.

Note 4 to entry: In control technology the Dirac function is mainly important for the definition of impulses and exclusively used as a function of time. Therefore the term Dirac impulse is used and the definition is adapted accordingly.