Triangular function

Encyclopedia : T : TR : TRI : Triangular function

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

[\operatorname(t) = \and (t) = \begin1 - |t|; & |t| < 1 \\0 & \mbox \end ]

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

[\operatorname(t) = \operatorname(t) * \operatorname(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.

The unitary Fourier transforms of the triangular function are:

[\frac}\int_^\infty \textrm(t)e^ \, dt]	[= \sqrt \left( \frac(\frac)}} \right)^2]
	[=\frac}\cdot \mathrm^2\left(\frac\right)], in terms of the normalized sinc function

[\int_^\infty \mathrm(t)\cdot e^ \, dt \ = \ \mathrm^2(f)]

These results follow from the Fourier transform of the rectangular function, and the convolution property of the Fourier transform.

Triangular function

See also