Triangular function

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Triangular function
Triangular function

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

\operatorname{tri}(t) = \and (t) = \begin{cases} 1 - |t|; & |t| < 1 \\ 0 & \mbox{otherwise} \end{cases}

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

\operatorname{tri}(t) = \operatorname{rect}(t) * \operatorname{rect}(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{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \textrm{tri}(t)e^{-i \omega t} \, dt = \sqrt{2\pi} \left( \frac{\textrm{sinc}(\frac{\omega}{2\pi})}{\sqrt{2\pi}} \right)^2
=\frac{1}{\sqrt{2\pi}}\cdot \mathrm{sinc}^2\left(\frac{\omega}{2\pi}\right),   in terms of the normalized sinc function
\int_{-\infty}^\infty \mathrm{tri}(t)\cdot e^{-i 2\pi f t} \, dt \ = \ \mathrm{sinc}^2(f)


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

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