From Wikipedia, the free encyclopedia
Triangular function The triangular function (also known as the triangle function , hat function , or tent function ) is defined either as:
tri
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max
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otherwise
{\displaystyle {\begin{aligned}\operatorname {tri} (t)=\land (t)\quad &{\overset {\underset {\mathrm {def} }{}}{=}}\ \max(1-|t|,0)\\&={\begin{cases}1-|t|,&|t|<1\\0,&{\mbox{otherwise}}\end{cases}}\end{aligned}}}
or, equivalently, as the convolution of two identical unit rectangular functions :
tri
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rect
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rect
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∞
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∞
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{\displaystyle {\begin{aligned}\operatorname {tri} (t)=\operatorname {rect} (t)*\operatorname {rect} (t)\quad &{\overset {\underset {\mathrm {def} }{}}{=}}\int _{-\infty }^{\infty }\mathrm {rect} (\tau )\cdot \mathrm {rect} (t-\tau )\ d\tau \\&=\int _{-\infty }^{\infty }\mathrm {rect} (\tau )\cdot \mathrm {rect} (\tau -t)\ d\tau .\end{aligned}}}
The triangular function can also be represented as the product of the rectangular and absolute value functions:
tri
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rect
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{\displaystyle \operatorname {tri} (t)=\operatorname {rect} (t)\left(\left|-t\right|+1\right)}
The function is useful in signal processing and communication systems engineering as a representation of an idealized signal, and as a prototype or kernel from which more realistic signals can be derived. It also has applications in pulse code modulation as a pulse shape for transmitting digital signals and as a matched filter for receiving the signals. It is also equivalent to the triangular window sometimes called the Bartlett window .
Scaling
For any parameter,
a
≠
0
{\displaystyle a\neq 0\,}
:
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{\displaystyle {\begin{aligned}\operatorname {tri} (t/a)&=\int _{-\infty }^{\infty }\mathrm {rect} (\tau )\cdot \mathrm {rect} (\tau -t/a)\ d\tau \\&={\begin{cases}1-|t/a|,&|t|<|a|\\0,&{\mbox{otherwise}}.\end{cases}}\end{aligned}}}
The transform is easily determined using the convolution property of Fourier transforms and the Fourier transform of the rectangular function :
F
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F
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⋅
F
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rect
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F
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2
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s
i
n
c
2
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f
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.
{\displaystyle {\begin{aligned}{\mathcal {F}}\{\operatorname {tri} (t)\}&={\mathcal {F}}\{\operatorname {rect} (t)*\operatorname {rect} (t)\}\\&={\mathcal {F}}\{\operatorname {rect} (t)\}\cdot {\mathcal {F}}\{\operatorname {rect} (t)\}\\&={\mathcal {F}}\{\operatorname {rect} (t)\}^{2}\\&=\mathrm {sinc} ^{2}(f).\end{aligned}}}
See also
