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Triangular function

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Triangular function

The triangular function (also known as the triangle function, hat function, or tent function) is defined either as:

Contents

tri ⁡ ( t ) = ∧ ( t ) = d e f   max ( 1 − | t | , 0 ) = { 1 − | t | , | t | < 1 0 , otherwise

or, equivalently, as the convolution of two identical unit rectangular functions:

tri ⁡ ( t ) = rect ⁡ ( t ) ∗ rect ⁡ ( t ) = d e f ∫ − ∞ ∞ r e c t ( τ ) ⋅ r e c t ( t − τ )   d τ = ∫ − ∞ ∞ r e c t ( τ ) ⋅ r e c t ( τ − t )   d τ .

The triangular function can also be represented as the product of the rectangular and absolute value functions:

tri ⁡ ( t ) = rect ⁡ ( t / 2 ) ( 1 − | t | )

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  :

tri ⁡ ( t / a ) = ∫ − ∞ ∞ r e c t ( τ ) ⋅ r e c t ( τ − t / a )   d τ = { 1 − | t / a | , | t | < | a | 0 , otherwise .

Fourier transform

The transform is easily determined using the convolution property of Fourier transforms and the Fourier transform of the rectangular function:

F { tri ⁡ ( t ) } = F { rect ⁡ ( t ) ∗ rect ⁡ ( t ) } = F { rect ⁡ ( t ) } ⋅ F { rect ⁡ ( t ) } = F { rect ⁡ ( t ) } 2 = s i n c 2 ( f ) ,

where sinc ⁡ ( x ) = sin ⁡ ( π x ) / ( π x ) is the normalized sinc function.

Alternative definition

Note that in some cases the triangle function may be defined to have a base of length 1 instead of length 2:

tri ⁡ ( t ) = ∧ ( t ) = d e f   max ( 1 − | 2 t | , 0 ) = { 1 − | 2 t | , | 2 t | < 1 0 , otherwise

References

Triangular function Wikipedia
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