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:

$\begin{align} \operatorname{tri}(t) = \and (t) \quad &\overset{\underset{\mathrm{def}}{}}{=} \ \max(1 - |t|, 0) \\ &= \begin{cases} 1 - |t|, & |t| < 1 \\ 0, & \mbox{otherwise} \end{cases} \end{align}$

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

$\begin{align} \operatorname{tri}(t) = \operatorname{rect}(t) * \operatorname{rect}(t) \quad &\overset{\underset{\mathrm{def}}{}}{=} \int_{-\infty}^\infty \mathrm{rect}(\tau) \cdot \mathrm{rect}(t-\tau)\ d\tau\\ &= \int_{-\infty}^\infty \mathrm{rect}(\tau) \cdot \mathrm{rect}(\tau-t)\ d\tau . \end{align}$

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.

[edit] Scaling

For any parameter, $a \ne 0\,$ :

$\begin{align} \operatorname{tri}(t/a) &= \int_{-\infty}^\infty \mathrm{rect}(\tau) \cdot \mathrm{rect}(\tau - t/a)\ d\tau \\ &= \begin{cases} 1 - |t/a|, & |t| < |a| \\ 0, & \mbox{otherwise} . \end{cases} \end{align}$

[edit] Fourier transform

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

$\begin{align} \mathcal{F}\{\operatorname{tri}(t)\} &= \mathcal{F}\{\operatorname{rect}(t) * \operatorname{rect}(t)\}\\ &= \mathcal{F}\{\operatorname{rect}(t)\}\cdot \mathcal{F}\{\operatorname{rect}(t)\}\\ &= \mathcal{F}\{\operatorname{rect}(t)\}^2\\ &= \mathrm{sinc}^2(f) . \end{align}$