Tent function

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) dtau &= int_{-infty}^infty mathrm{rect}(tau) cdot mathrm{rect}(tau-t) dtau . 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.

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) dtau &= begin{cases} 1 - |t/a|, & |t| < |a| 0, & mbox{otherwise} . end{cases} end{align}

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}

See also

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